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Motivation </div> </a> <ul id="toc-Motivation-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Definition" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Definition"> <div class="vector-toc-text"> 1.2 Definition </div> </a> <ul id="toc-Definition-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Simple_examples" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Simple_examples"> <div class="vector-toc-text"> 1.3 Simple examples </div> </a> <ul id="toc-Simple_examples-sublist" class="vector-toc-list"> <li id="toc-The_real_numbers" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#The_real_numbers"> <div class="vector-toc-text"> 1.3.1 The real numbers </div> </a> <ul id="toc-The_real_numbers-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Metrics_on_Euclidean_spaces" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Metrics_on_Euclidean_spaces"> <div class="vector-toc-text"> 1.3.2 Metrics on Euclidean spaces </div> </a> <ul id="toc-Metrics_on_Euclidean_spaces-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Subspaces" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Subspaces"> <div class="vector-toc-text"> 1.3.3 Subspaces </div> </a> <ul id="toc-Subspaces-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> </ul> </li> <li id="toc-History" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#History"> <div class="vector-toc-text"> 2 History </div> </a> <ul id="toc-History-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Basic_notions" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Basic_notions"> <div class="vector-toc-text"> 3 Basic notions </div> </a> <button aria-controls="toc-Basic_notions-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Basic notions subsection </button> <ul id="toc-Basic_notions-sublist" class="vector-toc-list"> <li id="toc-The_topology_of_a_metric_space" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#The_topology_of_a_metric_space"> <div class="vector-toc-text"> 3.1 The topology of a metric space </div> </a> <ul id="toc-The_topology_of_a_metric_space-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Convergence" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Convergence"> <div class="vector-toc-text"> 3.2 Convergence </div> </a> <ul id="toc-Convergence-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Completeness" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Completeness"> <div class="vector-toc-text"> 3.3 Completeness </div> </a> <ul id="toc-Completeness-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Bounded_and_totally_bounded_spaces" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Bounded_and_totally_bounded_spaces"> <div class="vector-toc-text"> 3.4 Bounded and totally bounded spaces </div> </a> <ul id="toc-Bounded_and_totally_bounded_spaces-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Compactness" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Compactness"> <div class="vector-toc-text"> 3.5 Compactness </div> </a> <ul id="toc-Compactness-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Functions_between_metric_spaces" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Functions_between_metric_spaces"> <div class="vector-toc-text"> 4 Functions between metric spaces </div> </a> <button aria-controls="toc-Functions_between_metric_spaces-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Functions between metric spaces subsection </button> <ul id="toc-Functions_between_metric_spaces-sublist" class="vector-toc-list"> <li id="toc-Isometries" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Isometries"> <div class="vector-toc-text"> 4.1 Isometries </div> </a> <ul id="toc-Isometries-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Continuous_maps" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Continuous_maps"> <div class="vector-toc-text"> 4.2 Continuous maps </div> </a> <ul id="toc-Continuous_maps-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Uniformly_continuous_maps" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Uniformly_continuous_maps"> <div class="vector-toc-text"> 4.3 Uniformly continuous maps </div> </a> <ul id="toc-Uniformly_continuous_maps-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Lipschitz_maps_and_contractions" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Lipschitz_maps_and_contractions"> <div class="vector-toc-text"> 4.4 Lipschitz maps and contractions </div> </a> <ul id="toc-Lipschitz_maps_and_contractions-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Quasi-isometries" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Quasi-isometries"> <div class="vector-toc-text"> 4.5 Quasi-isometries </div> </a> <ul id="toc-Quasi-isometries-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Notions_of_metric_space_equivalence" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Notions_of_metric_space_equivalence"> <div class="vector-toc-text"> 4.6 Notions of metric space equivalence </div> </a> <ul id="toc-Notions_of_metric_space_equivalence-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Metric_spaces_with_additional_structure" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Metric_spaces_with_additional_structure"> <div class="vector-toc-text"> 5 Metric spaces with additional structure </div> </a> <button aria-controls="toc-Metric_spaces_with_additional_structure-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Metric spaces with additional structure subsection </button> <ul id="toc-Metric_spaces_with_additional_structure-sublist" class="vector-toc-list"> <li id="toc-Normed_vector_spaces" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Normed_vector_spaces"> <div class="vector-toc-text"> 5.1 Normed vector spaces </div> </a> <ul id="toc-Normed_vector_spaces-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Length_spaces" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Length_spaces"> <div class="vector-toc-text"> 5.2 Length spaces </div> </a> <ul id="toc-Length_spaces-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Riemannian_manifolds" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Riemannian_manifolds"> <div class="vector-toc-text"> 5.3 Riemannian manifolds </div> </a> <ul id="toc-Riemannian_manifolds-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Metric_measure_spaces" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Metric_measure_spaces"> <div class="vector-toc-text"> 5.4 Metric measure spaces </div> </a> <ul id="toc-Metric_measure_spaces-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Further_examples_and_applications" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Further_examples_and_applications"> <div class="vector-toc-text"> 6 Further examples and applications </div> </a> <button aria-controls="toc-Further_examples_and_applications-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Further examples and applications subsection </button> <ul id="toc-Further_examples_and_applications-sublist" class="vector-toc-list"> <li id="toc-Graphs_and_finite_metric_spaces" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Graphs_and_finite_metric_spaces"> <div class="vector-toc-text"> 6.1 Graphs and finite metric spaces </div> </a> <ul id="toc-Graphs_and_finite_metric_spaces-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Distances_between_mathematical_objects" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Distances_between_mathematical_objects"> <div class="vector-toc-text"> 6.2 Distances between mathematical objects </div> </a> <ul id="toc-Distances_between_mathematical_objects-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Hausdorff_and_Gromov–Hausdorff_distance" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Hausdorff_and_Gromov–Hausdorff_distance"> <div class="vector-toc-text"> 6.3 Hausdorff and Gromov–Hausdorff distance </div> </a> <ul id="toc-Hausdorff_and_Gromov–Hausdorff_distance-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Miscellaneous_examples" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Miscellaneous_examples"> <div class="vector-toc-text"> 6.4 Miscellaneous examples </div> </a> <ul id="toc-Miscellaneous_examples-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Constructions" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Constructions"> <div class="vector-toc-text"> 7 Constructions </div> </a> <button aria-controls="toc-Constructions-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Constructions subsection </button> <ul id="toc-Constructions-sublist" class="vector-toc-list"> <li id="toc-Product_metric_spaces" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Product_metric_spaces"> <div class="vector-toc-text"> 7.1 Product metric spaces </div> </a> <ul id="toc-Product_metric_spaces-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Quotient_metric_spaces" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Quotient_metric_spaces"> <div class="vector-toc-text"> 7.2 Quotient metric spaces </div> </a> <ul id="toc-Quotient_metric_spaces-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Generalizations_of_metric_spaces" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Generalizations_of_metric_spaces"> <div class="vector-toc-text"> 8 Generalizations of metric spaces </div> </a> <button aria-controls="toc-Generalizations_of_metric_spaces-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Generalizations of metric spaces subsection </button> <ul id="toc-Generalizations_of_metric_spaces-sublist" class="vector-toc-list"> <li id="toc-Extended_metrics" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Extended_metrics"> <div class="vector-toc-text"> 8.1 Extended metrics </div> </a> <ul id="toc-Extended_metrics-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Metrics_valued_in_structures_other_than_the_real_numbers" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Metrics_valued_in_structures_other_than_the_real_numbers"> <div class="vector-toc-text"> 8.2 Metrics valued in structures other than the real numbers </div> </a> <ul id="toc-Metrics_valued_in_structures_other_than_the_real_numbers-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Pseudometrics" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Pseudometrics"> <div class="vector-toc-text"> 8.3 Pseudometrics </div> </a> <ul id="toc-Pseudometrics-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Quasimetrics" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Quasimetrics"> <div class="vector-toc-text"> 8.4 Quasimetrics </div> </a> <ul id="toc-Quasimetrics-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Metametrics_or_partial_metrics" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Metametrics_or_partial_metrics"> <div class="vector-toc-text"> 8.5 Metametrics or partial metrics </div> </a> <ul id="toc-Metametrics_or_partial_metrics-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Semimetrics" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Semimetrics"> <div class="vector-toc-text"> 8.6 Semimetrics </div> </a> <ul id="toc-Semimetrics-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Premetrics" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Premetrics"> <div class="vector-toc-text"> 8.7 Premetrics </div> </a> <ul id="toc-Premetrics-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Pseudoquasimetrics" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Pseudoquasimetrics"> <div class="vector-toc-text"> 8.8 Pseudoquasimetrics </div> </a> <ul id="toc-Pseudoquasimetrics-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Metrics_on_multisets" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Metrics_on_multisets"> <div class="vector-toc-text"> 8.9 Metrics on multisets </div> </a> <ul id="toc-Metrics_on_multisets-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> 9 See also </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Notes" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Notes"> <div class="vector-toc-text"> 10 Notes </div> </a> <ul id="toc-Notes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Citations" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Citations"> <div class="vector-toc-text"> 11 Citations </div> </a> <ul id="toc-Citations-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> 12 References </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> 13 External links </div> </a> <ul id="toc-External_links-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" > Toggle the table of contents </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading">Metric space</h1> <div 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Available in 61 languages" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-61" aria-hidden="true" > 61 languages </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D9%81%D8%B6%D8%A7%D8%A1_%D9%85%D8%AA%D8%B1%D9%8A" title="فضاء متري – Arabic" lang="ar" hreflang="ar" data-title="فضاء متري" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target">العربية</a></li><li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://ast.wikipedia.org/wiki/Espaciu_m%C3%A9tricu" title="Espaciu métricu – Asturian" lang="ast" hreflang="ast" data-title="Espaciu métricu" data-language-autonym="Asturianu" data-language-local-name="Asturian" class="interlanguage-link-target">Asturianu</a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%9C%D0%B5%D1%82%D1%80%D0%B8%D1%87%D0%BD%D0%BE_%D0%BF%D1%80%D0%BE%D1%81%D1%82%D1%80%D0%B0%D0%BD%D1%81%D1%82%D0%B2%D0%BE" title="Метрично пространство – Bulgarian" lang="bg" hreflang="bg" data-title="Метрично пространство" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target">Български</a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Espai_m%C3%A8tric" title="Espai mètric – Catalan" lang="ca" hreflang="ca" data-title="Espai mètric" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target">Català</a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%9C%D0%B5%D1%82%D1%80%D0%B8%D0%BA%C4%83%D0%BB%D0%BB%C4%83_%D1%83%C3%A7%D0%BB%C4%83%D1%85" title="Метрикăллă уçлăх – Chuvash" lang="cv" hreflang="cv" data-title="Метрикăллă уçлăх" data-language-autonym="Чӑвашла" data-language-local-name="Chuvash" class="interlanguage-link-target">Чӑвашла</a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Metrick%C3%BD_prostor" title="Metrický prostor – Czech" lang="cs" hreflang="cs" data-title="Metrický prostor" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target">Čeština</a></li><li class="interlanguage-link interwiki-cy mw-list-item"><a href="https://cy.wikipedia.org/wiki/Gofod_metrig" title="Gofod metrig – Welsh" lang="cy" hreflang="cy" data-title="Gofod metrig" data-language-autonym="Cymraeg" data-language-local-name="Welsh" class="interlanguage-link-target">Cymraeg</a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Metrisk_rum" title="Metrisk rum – Danish" lang="da" hreflang="da" data-title="Metrisk rum" data-language-autonym="Dansk" data-language-local-name="Danish" class="interlanguage-link-target">Dansk</a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Metrischer_Raum" title="Metrischer Raum – German" lang="de" hreflang="de" data-title="Metrischer Raum" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target">Deutsch</a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Meetriline_ruum" title="Meetriline ruum – Estonian" lang="et" hreflang="et" data-title="Meetriline ruum" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target">Eesti</a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%9C%CE%B5%CF%84%CF%81%CE%B9%CE%BA%CF%8C%CF%82_%CF%87%CF%8E%CF%81%CE%BF%CF%82" title="Μετρικός χώρος – Greek" lang="el" hreflang="el" data-title="Μετρικός χώρος" data-language-autonym="Ελληνικά" data-language-local-name="Greek" class="interlanguage-link-target">Ελληνικά</a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Espacio_m%C3%A9trico" title="Espacio métrico – Spanish" lang="es" hreflang="es" data-title="Espacio métrico" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target">Español</a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Metrika_spaco" title="Metrika spaco – Esperanto" lang="eo" hreflang="eo" data-title="Metrika spaco" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target">Esperanto</a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Espazio_metriko" title="Espazio metriko – Basque" lang="eu" hreflang="eu" data-title="Espazio metriko" data-language-autonym="Euskara" data-language-local-name="Basque" class="interlanguage-link-target">Euskara</a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D9%81%D8%B6%D8%A7%DB%8C_%D9%85%D8%AA%D8%B1%DB%8C" title="فضای متری – Persian" lang="fa" hreflang="fa" data-title="فضای متری" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target">فارسی</a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Espace_m%C3%A9trique" title="Espace métrique – French" lang="fr" hreflang="fr" data-title="Espace métrique" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target">Français</a></li><li class="interlanguage-link interwiki-ga mw-list-item"><a href="https://ga.wikipedia.org/wiki/Sp%C3%A1s_m%C3%A9adrach" title="Spás méadrach – Irish" lang="ga" hreflang="ga" data-title="Spás méadrach" data-language-autonym="Gaeilge" data-language-local-name="Irish" class="interlanguage-link-target">Gaeilge</a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Espazo_m%C3%A9trico" title="Espazo métrico – Galician" lang="gl" hreflang="gl" data-title="Espazo métrico" data-language-autonym="Galego" data-language-local-name="Galician" class="interlanguage-link-target">Galego</a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EA%B1%B0%EB%A6%AC_%EA%B3%B5%EA%B0%84" title="거리 공간 – Korean" lang="ko" hreflang="ko" data-title="거리 공간" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target">한국어</a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D5%84%D5%A5%D5%BF%D6%80%D5%AB%D5%AF%D5%A1%D5%AF%D5%A1%D5%B6_%D5%BF%D5%A1%D6%80%D5%A1%D5%AE%D5%B8%D6%82%D5%A9%D5%B5%D5%B8%D6%82%D5%B6" title="Մետրիկական տարածություն – Armenian" lang="hy" hreflang="hy" data-title="Մետրիկական տարածություն" data-language-autonym="Հայերեն" data-language-local-name="Armenian" class="interlanguage-link-target">Հայերեն</a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Metri%C4%8Dki_prostor" title="Metrički prostor – Croatian" lang="hr" hreflang="hr" data-title="Metrički prostor" data-language-autonym="Hrvatski" data-language-local-name="Croatian" class="interlanguage-link-target">Hrvatski</a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Ruang_metrik" title="Ruang metrik – Indonesian" lang="id" hreflang="id" data-title="Ruang metrik" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target">Bahasa Indonesia</a></li><li class="interlanguage-link interwiki-is mw-list-item"><a href="https://is.wikipedia.org/wiki/Fir%C3%B0r%C3%BAm" title="Firðrúm – Icelandic" lang="is" hreflang="is" data-title="Firðrúm" data-language-autonym="Íslenska" data-language-local-name="Icelandic" class="interlanguage-link-target">Íslenska</a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Spazio_metrico" title="Spazio metrico – Italian" lang="it" hreflang="it" data-title="Spazio metrico" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target">Italiano</a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%9E%D7%A8%D7%97%D7%91_%D7%9E%D7%98%D7%A8%D7%99" title="מרחב מטרי – Hebrew" lang="he" hreflang="he" data-title="מרחב מטרי" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target">עברית</a></li><li class="interlanguage-link interwiki-kn mw-list-item"><a href="https://kn.wikipedia.org/wiki/%E0%B2%AE%E0%B3%86%E0%B2%9F%E0%B3%8D%E0%B2%B0%E0%B2%BF%E0%B2%95%E0%B3%8D_%E0%B2%86%E0%B2%95%E0%B2%BE%E0%B2%B6" title="ಮೆಟ್ರಿಕ್ ಆಕಾಶ – Kannada" lang="kn" hreflang="kn" data-title="ಮೆಟ್ರಿಕ್ ಆಕಾಶ" data-language-autonym="ಕನ್ನಡ" data-language-local-name="Kannada" class="interlanguage-link-target">ಕನ್ನಡ</a></li><li class="interlanguage-link interwiki-ka mw-list-item"><a href="https://ka.wikipedia.org/wiki/%E1%83%9B%E1%83%94%E1%83%A2%E1%83%A0%E1%83%98%E1%83%99%E1%83%A3%E1%83%9A%E1%83%98_%E1%83%A1%E1%83%98%E1%83%95%E1%83%A0%E1%83%AA%E1%83%94" title="მეტრიკული სივრცე – Georgian" lang="ka" hreflang="ka" data-title="მეტრიკული სივრცე" data-language-autonym="ქართული" data-language-local-name="Georgian" class="interlanguage-link-target">ქართული</a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%9C%D0%B5%D1%82%D1%80%D0%B8%D0%BA%D0%B0%D0%BB%D1%8B%D2%9B_%D0%BA%D0%B5%D2%A3%D1%96%D1%81%D1%82%D1%96%D0%BA" title="Метрикалық кеңістік – Kazakh" lang="kk" hreflang="kk" data-title="Метрикалық кеңістік" data-language-autonym="Қазақша" data-language-local-name="Kazakh" class="interlanguage-link-target">Қазақша</a></li><li class="interlanguage-link interwiki-lb mw-list-item"><a href="https://lb.wikipedia.org/wiki/Metresche_Raum" title="Metresche Raum – Luxembourgish" lang="lb" hreflang="lb" data-title="Metresche Raum" data-language-autonym="Lëtzebuergesch" data-language-local-name="Luxembourgish" class="interlanguage-link-target">Lëtzebuergesch</a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Metrin%C4%97_erdv%C4%97" title="Metrinė erdvė – Lithuanian" lang="lt" hreflang="lt" data-title="Metrinė erdvė" data-language-autonym="Lietuvių" data-language-local-name="Lithuanian" class="interlanguage-link-target">Lietuvių</a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Metrikus_t%C3%A9r" title="Metrikus tér – Hungarian" lang="hu" hreflang="hu" data-title="Metrikus tér" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target">Magyar</a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%9C%D0%B5%D1%82%D1%80%D0%B8%D1%87%D0%BA%D0%B8_%D0%BF%D1%80%D0%BE%D1%81%D1%82%D0%BE%D1%80" title="Метрички простор – Macedonian" lang="mk" hreflang="mk" data-title="Метрички простор" data-language-autonym="Македонски" data-language-local-name="Macedonian" class="interlanguage-link-target">Македонски</a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Ruang_metrik" title="Ruang metrik – Malay" lang="ms" hreflang="ms" data-title="Ruang metrik" data-language-autonym="Bahasa Melayu" data-language-local-name="Malay" class="interlanguage-link-target">Bahasa Melayu</a></li><li class="interlanguage-link interwiki-my mw-list-item"><a href="https://my.wikipedia.org/wiki/%E1%80%A1%E1%80%90%E1%80%AD%E1%80%AF%E1%80%84%E1%80%BA%E1%80%B8%E1%80%86" title="အတိုင်းဆ – Burmese" lang="my" hreflang="my" data-title="အတိုင်းဆ" data-language-autonym="မြန်မာဘာသာ" data-language-local-name="Burmese" class="interlanguage-link-target">မြန်မာဘာသာ</a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Metrische_ruimte" title="Metrische ruimte – Dutch" lang="nl" hreflang="nl" data-title="Metrische ruimte" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target">Nederlands</a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E8%B7%9D%E9%9B%A2%E7%A9%BA%E9%96%93" title="距離空間 – Japanese" lang="ja" hreflang="ja" data-title="距離空間" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target">日本語</a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Metrisk_rom" title="Metrisk rom – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Metrisk rom" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegian Bokmål" class="interlanguage-link-target">Norsk bokmål</a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Metrisk_rom" title="Metrisk rom – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Metrisk rom" data-language-autonym="Norsk nynorsk" data-language-local-name="Norwegian Nynorsk" class="interlanguage-link-target">Norsk nynorsk</a></li><li class="interlanguage-link interwiki-pms mw-list-item"><a href="https://pms.wikipedia.org/wiki/Spassi_m%C3%A9trich" title="Spassi métrich – Piedmontese" lang="pms" hreflang="pms" data-title="Spassi métrich" data-language-autonym="Piemontèis" data-language-local-name="Piedmontese" class="interlanguage-link-target">Piemontèis</a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Przestrze%C5%84_metryczna" title="Przestrzeń metryczna – Polish" lang="pl" hreflang="pl" data-title="Przestrzeń metryczna" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target">Polski</a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Espa%C3%A7o_m%C3%A9trico" title="Espaço métrico – Portuguese" lang="pt" hreflang="pt" data-title="Espaço métrico" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target">Português</a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Spa%C8%9Biu_metric" title="Spațiu metric – Romanian" lang="ro" hreflang="ro" data-title="Spațiu metric" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target">Română</a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%9C%D0%B5%D1%82%D1%80%D0%B8%D1%87%D0%B5%D1%81%D0%BA%D0%BE%D0%B5_%D0%BF%D1%80%D0%BE%D1%81%D1%82%D1%80%D0%B0%D0%BD%D1%81%D1%82%D0%B2%D0%BE" title="Метрическое пространство – Russian" lang="ru" hreflang="ru" data-title="Метрическое пространство" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target">Русский</a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Metric_space" title="Metric space – Simple English" lang="en-simple" hreflang="en-simple" data-title="Metric space" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target">Simple English</a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Metrick%C3%BD_priestor" title="Metrický priestor – Slovak" lang="sk" hreflang="sk" data-title="Metrický priestor" data-language-autonym="Slovenčina" data-language-local-name="Slovak" class="interlanguage-link-target">Slovenčina</a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Metri%C4%8Dni_prostor" title="Metrični prostor – Slovenian" lang="sl" hreflang="sl" data-title="Metrični prostor" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target">Slovenščina</a></li><li class="interlanguage-link interwiki-ckb mw-list-item"><a href="https://ckb.wikipedia.org/wiki/%D8%A8%DB%86%D8%B4%D8%A7%DB%8C%DB%8C%DB%8C_%D9%85%DB%95%D8%AA%D8%B1%DB%8C" title="بۆشاییی مەتری – Central Kurdish" lang="ckb" hreflang="ckb" data-title="بۆشاییی مەتری" data-language-autonym="کوردی" data-language-local-name="Central Kurdish" class="interlanguage-link-target">کوردی</a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%9C%D0%B5%D1%82%D1%80%D0%B8%D1%87%D0%BA%D0%B8_%D0%BF%D1%80%D0%BE%D1%81%D1%82%D0%BE%D1%80" title="Метрички простор – Serbian" lang="sr" hreflang="sr" data-title="Метрички простор" data-language-autonym="Српски / srpski" data-language-local-name="Serbian" class="interlanguage-link-target">Српски / srpski</a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Metrinen_avaruus" title="Metrinen avaruus – Finnish" lang="fi" hreflang="fi" data-title="Metrinen avaruus" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target">Suomi</a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Metriskt_rum" title="Metriskt rum – Swedish" lang="sv" hreflang="sv" data-title="Metriskt rum" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target">Svenska</a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%AE%E0%AF%86%E0%AE%9F%E0%AF%8D%E0%AE%B0%E0%AE%BF%E0%AE%95%E0%AF%8D_%E0%AE%B5%E0%AF%86%E0%AE%B3%E0%AE%BF" title="மெட்ரிக் வெளி – Tamil" lang="ta" hreflang="ta" data-title="மெட்ரிக் வெளி" data-language-autonym="தமிழ்" data-language-local-name="Tamil" class="interlanguage-link-target">தமிழ்</a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Metrik_uzay" title="Metrik uzay – Turkish" lang="tr" hreflang="tr" data-title="Metrik uzay" data-language-autonym="Türkçe" data-language-local-name="Turkish" class="interlanguage-link-target">Türkçe</a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%9C%D0%B5%D1%82%D1%80%D0%B8%D1%87%D0%BD%D0%B8%D0%B9_%D0%BF%D1%80%D0%BE%D1%81%D1%82%D1%96%D1%80" title="Метричний простір – Ukrainian" lang="uk" hreflang="uk" data-title="Метричний простір" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target">Українська</a></li><li class="interlanguage-link interwiki-ur mw-list-item"><a href="https://ur.wikipedia.org/wiki/%D8%A8%D8%AD%D8%B1_%D9%81%D8%B6%D8%A7" title="بحر فضا – Urdu" lang="ur" hreflang="ur" data-title="بحر فضا" data-language-autonym="اردو" data-language-local-name="Urdu" class="interlanguage-link-target">اردو</a></li><li class="interlanguage-link interwiki-vec mw-list-item"><a href="https://vec.wikipedia.org/wiki/Spassio_m%C3%A8trico" title="Spassio mètrico – Venetian" lang="vec" hreflang="vec" data-title="Spassio mètrico" data-language-autonym="Vèneto" data-language-local-name="Venetian" class="interlanguage-link-target">Vèneto</a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/Kh%C3%B4ng_gian_m%C3%AAtric" title="Không gian mêtric – Vietnamese" lang="vi" hreflang="vi" data-title="Không gian mêtric" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamese" class="interlanguage-link-target">Tiếng Việt</a></li><li class="interlanguage-link interwiki-fiu-vro mw-list-item"><a href="https://fiu-vro.wikipedia.org/wiki/Meetriline_ruum" title="Meetriline ruum – Võro" lang="vro" hreflang="vro" data-title="Meetriline ruum" data-language-autonym="Võro" data-language-local-name="Võro" class="interlanguage-link-target">Võro</a></li><li class="interlanguage-link interwiki-zh-classical mw-list-item"><a href="https://zh-classical.wikipedia.org/wiki/%E5%BA%A6%E9%87%8F%E7%A9%BA%E9%96%93" title="度量空間 – Literary Chinese" lang="lzh" hreflang="lzh" data-title="度量空間" data-language-autonym="文言" data-language-local-name="Literary Chinese" class="interlanguage-link-target">文言</a></li><li class="interlanguage-link interwiki-wuu mw-list-item"><a href="https://wuu.wikipedia.org/wiki/%E5%BA%A6%E9%87%8F%E7%A9%BA%E9%97%B4" title="度量空间 – Wu" lang="wuu" hreflang="wuu" data-title="度量空间" data-language-autonym="吴语" data-language-local-name="Wu" class="interlanguage-link-target">吴语</a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E5%BA%A6%E9%87%8F%E7%A9%BA%E9%96%93" title="度量空間 – Cantonese" lang="yue" hreflang="yue" data-title="度量空間" data-language-autonym="粵語" data-language-local-name="Cantonese" class="interlanguage-link-target">粵語</a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E5%BA%A6%E9%87%8F%E7%A9%BA%E9%97%B4" title="度量空间 – Chinese" lang="zh" hreflang="zh" data-title="度量空间" data-language-autonym="中文" data-language-local-name="Chinese" class="interlanguage-link-target">中文</a></li> </ul> <div class="after-portlet after-portlet-lang"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q180953#sitelinks-wikipedia" title="Edit interlanguage links" class="wbc-editpage">Edit links</a></div> </div> </div> </div> </header> <div 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data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">From Wikipedia, the free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Mathematical space with a notion of distance</div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Manhattan_distance.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/08/Manhattan_distance.svg/200px-Manhattan_distance.svg.png" decoding="async" width="200" height="200" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/08/Manhattan_distance.svg/300px-Manhattan_distance.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/08/Manhattan_distance.svg/400px-Manhattan_distance.svg.png 2x" data-file-width="283" data-file-height="283" /></a><figcaption>The <a href="/wiki/Two-dimensional_Euclidean_space" class="mw-redirect" title="Two-dimensional Euclidean space">plane</a> (a set of points) can be equipped with different metrics. In the <a href="/wiki/Taxicab_geometry" title="Taxicab geometry">taxicab metric</a> the red, yellow and blue paths have the same <a href="/wiki/Arc_length" title="Arc length">length</a> (12), and are all shortest paths. In the <a href="/wiki/Euclidean_metric" class="mw-redirect" title="Euclidean metric">Euclidean metric</a>, the green path has length <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 6{\sqrt {2}}\approx 8.49}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>6</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mo>≈</mo> <mn>8.49</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 6{\sqrt {2}}\approx 8.49}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6d4363249f648e42df783c885be8eb92338f2af2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.493ex; height:3.009ex;" alt="{\displaystyle 6{\sqrt {2}}\approx 8.49}">, and is the unique shortest path, whereas the red, yellow, and blue paths still have length 12.</figcaption></figure> In <a href="/wiki/Mathematics" title="Mathematics">mathematics</a>, a metric space is a <a href="/wiki/Set_(mathematics)" title="Set (mathematics)">set</a> together with a notion of <a href="/wiki/Distance" title="Distance">distance</a> between its <a href="/wiki/Element_(mathematics)" title="Element (mathematics)">elements</a>, usually called <a href="/wiki/Point_(geometry)" title="Point (geometry)">points</a>. The distance is measured by a <a href="/wiki/Function_(mathematics)" title="Function (mathematics)">function</a> called a metric or distance function.<a href="#cite_note-FOOTNOTEČech196942-1">[1]</a> Metric spaces are the most general setting for studying many of the concepts of <a href="/wiki/Mathematical_analysis" title="Mathematical analysis">mathematical analysis</a> and <a href="/wiki/Geometry" title="Geometry">geometry</a>. The most familiar example of a metric space is <a href="/wiki/3-dimensional_Euclidean_space" class="mw-redirect" title="3-dimensional Euclidean space">3-dimensional Euclidean space</a> with its usual notion of distance. Other well-known examples are a <a href="/wiki/Sphere" title="Sphere">sphere</a> equipped with the <a href="/wiki/Angular_distance" title="Angular distance">angular distance</a> and the <a href="/wiki/Hyperbolic_plane" class="mw-redirect" title="Hyperbolic plane">hyperbolic plane</a>. A metric may correspond to a <a href="/wiki/Conceptual_metaphor" title="Conceptual metaphor">metaphorical</a>, rather than physical, notion of distance: for example, the set of 100-character Unicode strings can be equipped with the <a href="/wiki/Hamming_distance" title="Hamming distance">Hamming distance</a>, which measures the number of characters that need to be changed to get from one string to another. Since they are very general, metric spaces are a tool used in many different branches of mathematics. Many types of mathematical objects have a natural notion of distance and therefore admit the structure of a metric space, including <a href="/wiki/Riemannian_manifold" title="Riemannian manifold">Riemannian manifolds</a>, <a href="/wiki/Normed_vector_space" title="Normed vector space">normed vector spaces</a>, and <a href="/wiki/Graph_(discrete_mathematics)" title="Graph (discrete mathematics)">graphs</a>. In <a href="/wiki/Abstract_algebra" title="Abstract algebra">abstract algebra</a>, the <a href="/wiki/P-adic_numbers" class="mw-redirect" title="P-adic numbers">p-adic numbers</a> arise as elements of the <a href="/wiki/Completion_(metric_space)" class="mw-redirect" title="Completion (metric space)">completion</a> of a metric structure on the <a href="/wiki/Rational_numbers" class="mw-redirect" title="Rational numbers">rational numbers</a>. Metric spaces are also studied in their own right in metric geometry<a href="#cite_note-FOOTNOTEBuragoBuragoIvanov2001-2">[2]</a> and analysis on metric spaces.<a href="#cite_note-FOOTNOTEHeinonen2001-3">[3]</a> Many of the basic notions of <a href="/wiki/Mathematical_analysis" title="Mathematical analysis">mathematical analysis</a>, including <a href="/wiki/Ball_(mathematics)" title="Ball (mathematics)">balls</a>, <a href="/wiki/Complete_metric_space" title="Complete metric space">completeness</a>, as well as <a href="/wiki/Uniform_continuity" title="Uniform continuity">uniform</a>, <a href="/wiki/Lipschitz_continuity" title="Lipschitz continuity">Lipschitz</a>, and <a href="/wiki/H%C3%B6lder_continuity" class="mw-redirect" title="Hölder continuity">Hölder continuity</a>, can be defined in the setting of metric spaces. Other notions, such as <a href="/wiki/Continuous_function" title="Continuous function">continuity</a>, <a href="/wiki/Compactness" class="mw-redirect" title="Compactness">compactness</a>, and <a href="/wiki/Open_set" title="Open set">open</a> and <a href="/wiki/Closed_set" title="Closed set">closed sets</a>, can be defined for metric spaces, but also in the even more general setting of <a href="/wiki/Topological_space" title="Topological space">topological spaces</a>. <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Definition_and_illustration">Definition and illustration</h2>[<a href="/w/index.php?title=Metric_space&action=edit&section=1" title="Edit section: Definition and illustration">edit</a>]</div> <div class="mw-heading mw-heading3"><h3 id="Motivation">Motivation</h3>[<a href="/w/index.php?title=Metric_space&action=edit&section=2" title="Edit section: Motivation">edit</a>]</div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Great-circle_distance_vs_straight_line_distance.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/5f/Great-circle_distance_vs_straight_line_distance.svg/220px-Great-circle_distance_vs_straight_line_distance.svg.png" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/5f/Great-circle_distance_vs_straight_line_distance.svg/330px-Great-circle_distance_vs_straight_line_distance.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/5f/Great-circle_distance_vs_straight_line_distance.svg/440px-Great-circle_distance_vs_straight_line_distance.svg.png 2x" data-file-width="298" data-file-height="298" /></a><figcaption>A diagram illustrating the great-circle distance (in cyan) and the straight-line distance (in red) between two points P and Q on a sphere.</figcaption></figure> To see the utility of different notions of distance, consider the <a href="/wiki/Surface_of_the_Earth" class="mw-redirect" title="Surface of the Earth">surface of the Earth</a> as a set of points. We can measure the distance between two such points by the length of the <a href="/wiki/Great-circle_distance" title="Great-circle distance">shortest path along the surface</a>, "<a href="/wiki/As_the_crow_flies" title="As the crow flies">as the crow flies</a>"; this is particularly useful for shipping and aviation. We can also measure the straight-line distance between two points through the Earth's interior; this notion is, for example, natural in <a href="/wiki/Seismology" title="Seismology">seismology</a>, since it roughly corresponds to the length of time it takes for seismic waves to travel between those two points. The notion of distance encoded by the metric space axioms has relatively few requirements. This generality gives metric spaces a lot of flexibility. At the same time, the notion is strong enough to encode many intuitive facts about what distance means. This means that general results about metric spaces can be applied in many different contexts. Like many fundamental mathematical concepts, the metric on a metric space can be interpreted in many different ways. A particular metric may not be best thought of as measuring physical distance, but, instead, as the cost of changing from one state to another (as with <a href="/wiki/Wasserstein_metric" title="Wasserstein metric">Wasserstein metrics</a> on spaces of <a href="/wiki/Measure_(mathematics)" title="Measure (mathematics)">measures</a>) or the degree of difference between two objects (for example, the <a href="/wiki/Hamming_distance" title="Hamming distance">Hamming distance</a> between two strings of characters, or the <a href="/wiki/Gromov%E2%80%93Hausdorff_convergence" title="Gromov–Hausdorff convergence">Gromov–Hausdorff distance</a> between metric spaces themselves). <div class="mw-heading mw-heading3"><h3 id="Definition">Definition</h3>[<a href="/w/index.php?title=Metric_space&action=edit&section=3" title="Edit section: Definition">edit</a>]</div> Formally, a metric space is an <a href="/wiki/Ordered_pair" title="Ordered pair">ordered pair</a> (M, d) where M is a set and d is a metric on M, i.e., a <a href="/wiki/Function_(mathematics)" title="Function (mathematics)">function</a><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d\,\colon M\times M\to \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mspace width="thinmathspace" /> <mo>:</mo> <mi>M</mi> <mo>×</mo> <mi>M</mi> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d\,\colon M\times M\to \mathbb {R} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fa7a2e6b602e9a9325d834170bf5b841bf3bf9cf" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:15.654ex; height:2.176ex;" alt="{\displaystyle d\,\colon M\times M\to \mathbb {R} }">satisfying the following axioms for all points <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x,y,z\in M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> <mo>∈</mo> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x,y,z\in M}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ba283a127121ad64c98d3f69ced0ac4a86ec6414" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.924ex; height:2.509ex;" alt="{\displaystyle x,y,z\in M}">:<a href="#cite_note-FOOTNOTEBuragoBuragoIvanov20011-4">[4]</a><a href="#cite_note-FOOTNOTEGromov2007xv-5">[5]</a> <ol><li>The distance from a point to itself is zero: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d(x,x)=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d(x,x)=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f8b0b392db0a4abf6c47dcd5b35f90592401e756" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.979ex; height:2.843ex;" alt="{\displaystyle d(x,x)=0}"></li> <li>(Positivity) The distance between two distinct points is always positive: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{If }}x\neq y{\text{, then }}d(x,y)>0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtext>If </mtext> </mrow> <mi>x</mi> <mo>≠</mo> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>, then </mtext> </mrow> <mi>d</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{If }}x\neq y{\text{, then }}d(x,y)>0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/14c273c8a68c3cbc06a20d236f26fb83043fd941" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.851ex; height:2.843ex;" alt="{\displaystyle {\text{If }}x\neq y{\text{, then }}d(x,y)>0}"></li> <li>(<a href="/wiki/Symmetric_function" title="Symmetric function">Symmetry</a>) The distance from x to y is always the same as the distance from y to x: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d(x,y)=d(y,x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>d</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo>,</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d(x,y)=d(y,x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c7fea33d0e60116abd16287351eb6bf142a61fdd" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.187ex; height:2.843ex;" alt="{\displaystyle d(x,y)=d(y,x)}"></li> <li>The <a href="/wiki/Triangle_inequality" title="Triangle inequality">triangle inequality</a> holds: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d(x,z)\leq d(x,y)+d(y,z)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>≤</mo> <mi>d</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>d</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d(x,z)\leq d(x,y)+d(y,z)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/43ae751284c2944886e1effbfe4e0c1293f98419" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.263ex; height:2.843ex;" alt="{\displaystyle d(x,z)\leq d(x,y)+d(y,z)}">This is a natural property of both physical and metaphorical notions of distance: you can arrive at z from x by taking a detour through y, but this will not make your journey any shorter than the direct path.</li></ol> If the metric d is unambiguous, one often refers by <a href="/wiki/Abuse_of_notation" title="Abuse of notation">abuse of notation</a> to "the metric space M". By taking all axioms except the second, one can show that distance is always non-negative:<math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0=d(x,x)\leq d(x,y)+d(y,x)=2d(x,y)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo>=</mo> <mi>d</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>≤</mo> <mi>d</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>d</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo>,</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>2</mn> <mi>d</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0=d(x,x)\leq d(x,y)+d(y,x)=2d(x,y)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dbb5c81eebd08b8155f33c70761a4f619b436e6d" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:40.812ex; height:2.843ex;" alt="{\displaystyle 0=d(x,x)\leq d(x,y)+d(y,x)=2d(x,y)}">Therefore the second axiom can be weakened to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle {\text{If }}x\neq y{\text{, then }}d(x,y)\neq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtext>If </mtext> </mrow> <mi>x</mi> <mo>≠</mo> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>, then </mtext> </mrow> <mi>d</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>≠</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle {\text{If }}x\neq y{\text{, then }}d(x,y)\neq 0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a711db1f66ee741f7620e40bafcf4c905c6d7f21" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.851ex; height:2.843ex;" alt="{\textstyle {\text{If }}x\neq y{\text{, then }}d(x,y)\neq 0}"> and combined with the first to make <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle d(x,y)=0\iff x=y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>d</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> <mspace width="thickmathspace" /> <mo stretchy="false">⟺</mo> <mspace width="thickmathspace" /> <mi>x</mi> <mo>=</mo> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle d(x,y)=0\iff x=y}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8f45cfc9536a8e2fc8a1a5a7128fc148c4fbb125" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.286ex; height:2.843ex;" alt="{\textstyle d(x,y)=0\iff x=y}">.<a href="#cite_note-6">[6]</a> <div class="mw-heading mw-heading3"><h3 id="Simple_examples">Simple examples</h3>[<a href="/w/index.php?title=Metric_space&action=edit&section=4" title="Edit section: Simple examples">edit</a>]</div> <div class="mw-heading mw-heading4"><h4 id="The_real_numbers">The real numbers</h4>[<a href="/w/index.php?title=Metric_space&action=edit&section=5" title="Edit section: The real numbers">edit</a>]</div> The <a href="/wiki/Real_number" title="Real number">real numbers</a> with the distance function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d(x,y)=|y-x|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>y</mi> <mo>−</mo> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d(x,y)=|y-x|}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d4ccdaa16fa9887b70273ee18e3b1b2f20d61d3f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.262ex; height:2.843ex;" alt="{\displaystyle d(x,y)=|y-x|}"> given by the <a href="/wiki/Absolute_difference" title="Absolute difference">absolute difference</a> form a metric space. Many properties of metric spaces and functions between them are generalizations of concepts in <a href="/wiki/Real_analysis" title="Real analysis">real analysis</a> and coincide with those concepts when applied to the real line. <div class="mw-heading mw-heading4"><h4 id="Metrics_on_Euclidean_spaces">Metrics on Euclidean spaces</h4>[<a href="/w/index.php?title=Metric_space&action=edit&section=6" title="Edit section: Metrics on Euclidean spaces">edit</a>]</div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Minkowski_distance_examples.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/eb/Minkowski_distance_examples.svg/220px-Minkowski_distance_examples.svg.png" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/eb/Minkowski_distance_examples.svg/330px-Minkowski_distance_examples.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/eb/Minkowski_distance_examples.svg/440px-Minkowski_distance_examples.svg.png 2x" data-file-width="512" data-file-height="512" /></a><figcaption>Comparison of Chebyshev, Euclidean and taxicab distances for the hypotenuse of a 3-4-5 triangle on a chessboard</figcaption></figure> The Euclidean plane <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e150115ab9f63023215109595b76686a1ff890fd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.732ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{2}}"> can be equipped with many different metrics. The <a href="/wiki/Euclidean_distance" title="Euclidean distance">Euclidean distance</a> familiar from school mathematics can be defined by <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d_{2}((x_{1},y_{1}),(x_{2},y_{2}))={\sqrt {(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>,</mo> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mo stretchy="false">(</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d_{2}((x_{1},y_{1}),(x_{2},y_{2}))={\sqrt {(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ea0ba1812589f13bde6abe0610dd32061edb0f1d" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:49.42ex; height:4.843ex;" alt="{\displaystyle d_{2}((x_{1},y_{1}),(x_{2},y_{2}))={\sqrt {(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}}}.}"> The <a href="/wiki/Taxicab_geometry" title="Taxicab geometry">taxicab or Manhattan distance</a> is defined by <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d_{1}((x_{1},y_{1}),(x_{2},y_{2}))=|x_{2}-x_{1}|+|y_{2}-y_{1}|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>,</mo> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d_{1}((x_{1},y_{1}),(x_{2},y_{2}))=|x_{2}-x_{1}|+|y_{2}-y_{1}|}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e7b3acf7666b59a8c0c458360ec598ea129f1777" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:43.309ex; height:2.843ex;" alt="{\displaystyle d_{1}((x_{1},y_{1}),(x_{2},y_{2}))=|x_{2}-x_{1}|+|y_{2}-y_{1}|}"> and can be thought of as the distance you need to travel along horizontal and vertical lines to get from one point to the other, as illustrated at the top of the article. The maximum, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{\infty }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L^{\infty }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b9ab400cc4dfd865180cd84c72dc894ca457671f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.458ex; height:2.343ex;" alt="{\displaystyle L^{\infty }}">, or <a href="/wiki/Chebyshev_distance" title="Chebyshev distance">Chebyshev distance</a> is defined by <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d_{\infty }((x_{1},y_{1}),(x_{2},y_{2}))=\max\{|x_{2}-x_{1}|,|y_{2}-y_{1}|\}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>,</mo> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>=</mo> <mo movablelimits="true" form="prefix">max</mo> <mo fence="false" stretchy="false">{</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo fence="false" stretchy="false">}</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d_{\infty }((x_{1},y_{1}),(x_{2},y_{2}))=\max\{|x_{2}-x_{1}|,|y_{2}-y_{1}|\}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7af57ea864a570149bc3a0482a3a7111812c7905" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:49.622ex; height:2.843ex;" alt="{\displaystyle d_{\infty }((x_{1},y_{1}),(x_{2},y_{2}))=\max\{|x_{2}-x_{1}|,|y_{2}-y_{1}|\}.}"> This distance does not have an easy explanation in terms of paths in the plane, but it still satisfies the metric space axioms. It can be thought of similarly to the number of moves a <a href="/wiki/King_(chess)" title="King (chess)">king</a> would have to make on a <a href="/wiki/Chess" title="Chess">chess</a> <a href="/wiki/Board_game" title="Board game">board</a> to travel from one point to another on the given space. In fact, these three distances, while they have distinct properties, are similar in some ways. Informally, points that are close in one are close in the others, too. This observation can be quantified with the formula <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d_{\infty }(p,q)\leq d_{2}(p,q)\leq d_{1}(p,q)\leq 2d_{\infty }(p,q),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>p</mi> <mo>,</mo> <mi>q</mi> <mo stretchy="false">)</mo> <mo>≤</mo> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>p</mi> <mo>,</mo> <mi>q</mi> <mo stretchy="false">)</mo> <mo>≤</mo> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>p</mi> <mo>,</mo> <mi>q</mi> <mo stretchy="false">)</mo> <mo>≤</mo> <mn>2</mn> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>p</mi> <mo>,</mo> <mi>q</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d_{\infty }(p,q)\leq d_{2}(p,q)\leq d_{1}(p,q)\leq 2d_{\infty }(p,q),}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5ef8a8fb50768d565fbf6cec4727b4fda84fc56" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:42.128ex; height:2.843ex;" alt="{\displaystyle d_{\infty }(p,q)\leq d_{2}(p,q)\leq d_{1}(p,q)\leq 2d_{\infty }(p,q),}"> which holds for every pair of points <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p,q\in \mathbb {R} ^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>,</mo> <mi>q</mi> <mo>∈</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p,q\in \mathbb {R} ^{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1ee72fcef6aa20ae237e8d7242fef4eb83de566e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:8.935ex; height:3.009ex;" alt="{\displaystyle p,q\in \mathbb {R} ^{2}}">. A radically different distance can be defined by setting <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d(p,q)={\begin{cases}0,&{\text{if }}p=q,\\1,&{\text{otherwise.}}\end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo stretchy="false">(</mo> <mi>p</mi> <mo>,</mo> <mi>q</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mn>0</mn> <mo>,</mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>if </mtext> </mrow> <mi>p</mi> <mo>=</mo> <mi>q</mi> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> <mo>,</mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>otherwise.</mtext> </mrow> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d(p,q)={\begin{cases}0,&{\text{if }}p=q,\\1,&{\text{otherwise.}}\end{cases}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/765a68868cb7a61922dced5212dc20949d561cb1" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:26.247ex; height:6.176ex;" alt="{\displaystyle d(p,q)={\begin{cases}0,&{\text{if }}p=q,\\1,&{\text{otherwise.}}\end{cases}}}"> Using <a href="/wiki/Iverson_bracket" title="Iverson bracket">Iverson brackets</a>, <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d(p,q)=[p\neq q]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo stretchy="false">(</mo> <mi>p</mi> <mo>,</mo> <mi>q</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">[</mo> <mi>p</mi> <mo>≠</mo> <mi>q</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d(p,q)=[p\neq q]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e025bbb02a33d44a0c2373007bd8e878240f2dc4" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.028ex; height:2.843ex;" alt="{\displaystyle d(p,q)=[p\neq q]}"> In this discrete metric, all distinct points are 1 unit apart: none of them are close to each other, and none of them are very far away from each other either. Intuitively, the discrete metric no longer remembers that the set is a plane, but treats it just as an undifferentiated set of points. All of these metrics make sense on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c510b63578322050121fe966f2e5770bea43308d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.897ex; height:2.343ex;" alt="{\displaystyle \mathbb {R} ^{n}}"> as well as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e150115ab9f63023215109595b76686a1ff890fd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.732ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{2}}">. <div class="mw-heading mw-heading4"><h4 id="Subspaces">Subspaces</h4>[<a href="/w/index.php?title=Metric_space&action=edit&section=7" title="Edit section: Subspaces">edit</a>]</div> Given a metric space (M, d) and a <a href="/wiki/Subset" title="Subset">subset</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\subseteq M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>⊆</mo> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\subseteq M}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3817146332479574af6e80d38b1247eada04ae83" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.284ex; height:2.343ex;" alt="{\displaystyle A\subseteq M}">, we can consider A to be a metric space by measuring distances the same way we would in M. Formally, the induced metric on A is a function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d_{A}:A\times A\to \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mo>:</mo> <mi>A</mi> <mo>×</mo> <mi>A</mi> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d_{A}:A\times A\to \mathbb {R} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d1d52f04717ef0fb3463da72303831852a125990" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:16.23ex; height:2.509ex;" alt="{\displaystyle d_{A}:A\times A\to \mathbb {R} }"> defined by <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d_{A}(x,y)=d(x,y).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>d</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d_{A}(x,y)=d(x,y).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/382fe6588674cd4853889d6b327c321d3fd01039" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.292ex; height:2.843ex;" alt="{\displaystyle d_{A}(x,y)=d(x,y).}"> For example, if we take the two-dimensional sphere S2 as a subset of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{3}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f936ddf584f8f3dd2a0ed08917001b7a404c10b5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.732ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{3}}">, the Euclidean metric on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{3}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f936ddf584f8f3dd2a0ed08917001b7a404c10b5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.732ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{3}}"> induces the straight-line metric on S2 described above. Two more useful examples are the open interval (0, 1) and the closed interval [0, 1] thought of as subspaces of the real line. <div class="mw-heading mw-heading2"><h2 id="History">History</h2>[<a href="/w/index.php?title=Metric_space&action=edit&section=8" title="Edit section: History">edit</a>]</div> <a href="/wiki/Arthur_Cayley" title="Arthur Cayley">Arthur Cayley</a>, in his article "On Distance", extended metric concepts beyond Euclidean geometry into domains bounded by a conic in a projective space. His <a href="/wiki/Distance" title="Distance">distance</a> was given by logarithm of a <a href="/wiki/Cross_ratio" class="mw-redirect" title="Cross ratio">cross ratio</a>. Any projectivity leaving the conic stable also leaves the cross ratio constant, so isometries are implicit. This method provides models for <a href="/wiki/Elliptic_geometry" title="Elliptic geometry">elliptic geometry</a> and <a href="/wiki/Hyperbolic_geometry" title="Hyperbolic geometry">hyperbolic geometry</a>, and <a href="/wiki/Felix_Klein" title="Felix Klein">Felix Klein</a>, in several publications, established the field of <a href="/wiki/Non-euclidean_geometry" class="mw-redirect" title="Non-euclidean geometry">non-euclidean geometry</a> through the use of the <a href="/wiki/Cayley-Klein_metric" class="mw-redirect" title="Cayley-Klein metric">Cayley-Klein metric</a>. The idea of an abstract space with metric properties was addressed in 1906 by <a href="/wiki/Ren%C3%A9_Maurice_Fr%C3%A9chet" title="René Maurice Fréchet">René Maurice Fréchet</a><a href="#cite_note-7">[7]</a> and the term metric space was coined by <a href="/wiki/Felix_Hausdorff" title="Felix Hausdorff">Felix Hausdorff</a> in 1914.<a href="#cite_note-8">[8]</a><a href="#cite_note-9">[9]</a><a href="#cite_note-10">[10]</a> Fréchet's work laid the foundation for understanding <a href="/wiki/Convergence_of_random_variables" title="Convergence of random variables">convergence</a>, <a href="/wiki/Continuity_equation" title="Continuity equation">continuity</a>, and other key concepts in non-geometric spaces. This allowed mathematicians to study functions and sequences in a broader and more flexible way. This was important for the growing field of functional analysis. Mathematicians like Hausdorff and <a href="/wiki/Stefan_Banach" title="Stefan Banach">Stefan Banach</a> further refined and expanded the framework of metric spaces. Hausdorff introduced <a href="/wiki/Topological_space" title="Topological space">topological spaces</a> as a generalization of metric spaces. Banach's work in <a href="/wiki/Functional_analysis" title="Functional analysis">functional analysis</a> heavily relied on the metric structure. Over time, metric spaces became a central part of <a href="/wiki/History_of_mathematics" title="History of mathematics">modern mathematics</a>. They have influenced various fields including <a href="/wiki/Topology" title="Topology">topology</a>, <a href="/wiki/Geometry" title="Geometry">geometry</a>, and <a href="/wiki/Applied_mathematics" title="Applied mathematics">applied mathematics</a>. Metric spaces continue to play a crucial role in the study of abstract mathematical concepts. <div class="mw-heading mw-heading2"><h2 id="Basic_notions">Basic notions</h2>[<a href="/w/index.php?title=Metric_space&action=edit&section=9" title="Edit section: Basic notions">edit</a>]</div> A distance function is enough to define notions of closeness and convergence that were first developed in <a href="/wiki/Real_analysis" title="Real analysis">real analysis</a>. Properties that depend on the structure of a metric space are referred to as metric properties. Every metric space is also a <a href="/wiki/Topological_space" title="Topological space">topological space</a>, and some metric properties can also be rephrased without reference to distance in the language of topology; that is, they are really <a href="/wiki/Topological_property" title="Topological property">topological properties</a>. <div class="mw-heading mw-heading3"><h3 id="The_topology_of_a_metric_space">The topology of a metric space</h3>[<a href="/w/index.php?title=Metric_space&action=edit&section=10" title="Edit section: The topology of a metric space">edit</a>]</div> For any point x in a metric space M and any real number r > 0, the <a href="/wiki/Ball_(mathematics)" title="Ball (mathematics)">open ball</a> of radius r around x is defined to be the set of points that are strictly less than distance r from x: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B_{r}(x)=\{y\in M:d(x,y)<r\}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mi>y</mi> <mo>∈</mo> <mi>M</mi> <mo>:</mo> <mi>d</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo><</mo> <mi>r</mi> <mo fence="false" stretchy="false">}</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B_{r}(x)=\{y\in M:d(x,y)<r\}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1c23ac48e61a11d09e030c3fbbee290b9eaea2c8" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:31.014ex; height:2.843ex;" alt="{\displaystyle B_{r}(x)=\{y\in M:d(x,y)<r\}.}"> This is a natural way to define a set of points that are relatively close to x. Therefore, a set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N\subseteq M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> <mo>⊆</mo> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N\subseteq M}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cfe9ab1adee54a74e4a9595b31e0f3302b9e4c2e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.604ex; height:2.343ex;" alt="{\displaystyle N\subseteq M}"> is a <a href="/wiki/Neighborhood_(mathematics)" class="mw-redirect" title="Neighborhood (mathematics)">neighborhood</a> of x (informally, it contains all points "close enough" to x) if it contains an open ball of radius r around x for some r > 0. An open set is a set which is a neighborhood of all its points. It follows that the open balls form a <a href="/wiki/Base_(topology)" title="Base (topology)">base</a> for a topology on M. In other words, the open sets of M are exactly the unions of open balls. As in any topology, <a href="/wiki/Closed_set" title="Closed set">closed sets</a> are the complements of open sets. Sets may be both open and closed as well as neither open nor closed. This topology does not carry all the information about the metric space. For example, the distances d1, d2, and d∞ defined above all induce the same topology on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e150115ab9f63023215109595b76686a1ff890fd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.732ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{2}}">, although they behave differently in many respects. Similarly, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }"> with the Euclidean metric and its subspace the interval (0, 1) with the induced metric are <a href="/wiki/Homeomorphism" title="Homeomorphism">homeomorphic</a> but have very different metric properties. Conversely, not every topological space can be given a metric. Topological spaces which are compatible with a metric are called <a href="/wiki/Metrizable_space" title="Metrizable space">metrizable</a> and are particularly well-behaved in many ways: in particular, they are <a href="/wiki/Paracompact_space" title="Paracompact space">paracompact</a><a href="#cite_note-11">[11]</a> <a href="/wiki/Hausdorff_space" title="Hausdorff space">Hausdorff spaces</a> (hence <a href="/wiki/Normal_space" title="Normal space">normal</a>) and <a href="/wiki/First-countable_space" title="First-countable space">first-countable</a>.<a href="#cite_note-12">[a]</a> The <a href="/wiki/Nagata%E2%80%93Smirnov_metrization_theorem" title="Nagata–Smirnov metrization theorem">Nagata–Smirnov metrization theorem</a> gives a characterization of metrizability in terms of other topological properties, without reference to metrics. <div class="mw-heading mw-heading3"><h3 id="Convergence">Convergence</h3>[<a href="/w/index.php?title=Metric_space&action=edit&section=11" title="Edit section: Convergence">edit</a>]</div> <a href="/wiki/Limit_of_a_sequence" title="Limit of a sequence">Convergence of sequences</a> in Euclidean space is defined as follows: <dl><dd>A sequence (xn) converges to a point x if for every ε > 0 there is an integer N such that for all n > N, d(xn, x) < ε.</dd></dl> Convergence of sequences in a topological space is defined as follows: <dl><dd>A sequence (xn) converges to a point x if for every open set U containing x there is an integer N such that for all n > N, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{n}\in U}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>∈</mo> <mi>U</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{n}\in U}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9552d871d5e5c3d4f0cdee55ea24cf6253ed5e85" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.171ex; height:2.509ex;" alt="{\displaystyle x_{n}\in U}">.</dd></dl> In metric spaces, both of these definitions make sense and they are equivalent. This is a general pattern for <a href="/wiki/Topological_property" title="Topological property">topological properties</a> of metric spaces: while they can be defined in a purely topological way, there is often a way that uses the metric which is easier to state or more familiar from real analysis. <div class="mw-heading mw-heading3"><h3 id="Completeness">Completeness</h3>[<a href="/w/index.php?title=Metric_space&action=edit&section=12" title="Edit section: Completeness">edit</a>]</div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Complete_metric_space" title="Complete metric space">Complete metric space</a></div> Informally, a metric space is complete if it has no "missing points": every sequence that looks like it should converge to something actually converges. To make this precise: a sequence (xn) in a metric space M is <a href="/wiki/Cauchy_sequence" title="Cauchy sequence">Cauchy</a> if for every ε > 0 there is an integer N such that for all m, n > N, d(xm, xn) < ε. By the triangle inequality, any convergent sequence is Cauchy: if xm and xn are both less than ε away from the limit, then they are less than 2ε away from each other. If the converse is true—every Cauchy sequence in M converges—then M is complete. Euclidean spaces are complete, as is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e150115ab9f63023215109595b76686a1ff890fd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.732ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{2}}"> with the other metrics described above. Two examples of spaces which are not complete are (0, 1) and the rationals, each with the metric induced from <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }">. One can think of (0, 1) as "missing" its endpoints 0 and 1. The rationals are missing all the irrationals, since any irrational has a sequence of rationals converging to it in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }"> (for example, its successive decimal approximations). These examples show that completeness is not a topological property, since <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }"> is complete but the homeomorphic space (0, 1) is not. This notion of "missing points" can be made precise. In fact, every metric space has a unique <a href="/wiki/Completion_(metric_space)" class="mw-redirect" title="Completion (metric space)">completion</a>, which is a complete space that contains the given space as a <a href="/wiki/Dense_set" title="Dense set">dense</a> subset. For example, [0, 1] is the completion of (0, 1), and the real numbers are the completion of the rationals. Since complete spaces are generally easier to work with, completions are important throughout mathematics. For example, in abstract algebra, the <a href="/wiki/P-adic_numbers" class="mw-redirect" title="P-adic numbers">p-adic numbers</a> are defined as the completion of the rationals under a different metric. Completion is particularly common as a tool in <a href="/wiki/Functional_analysis" title="Functional analysis">functional analysis</a>. Often one has a set of nice functions and a way of measuring distances between them. Taking the completion of this metric space gives a new set of functions which may be less nice, but nevertheless useful because they behave similarly to the original nice functions in important ways. For example, <a href="/wiki/Weak_solution" title="Weak solution">weak solutions</a> to <a href="/wiki/Differential_equation" title="Differential equation">differential equations</a> typically live in a completion (a <a href="/wiki/Sobolev_space" title="Sobolev space">Sobolev space</a>) rather than the original space of nice functions for which the differential equation actually makes sense. <div class="mw-heading mw-heading3"><h3 id="Bounded_and_totally_bounded_spaces">Bounded and totally bounded spaces</h3>[<a href="/w/index.php?title=Metric_space&action=edit&section=13" title="Edit section: Bounded and totally bounded spaces">edit</a>]</div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Diameter_of_a_Set.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fc/Diameter_of_a_Set.svg/220px-Diameter_of_a_Set.svg.png" decoding="async" width="220" height="262" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fc/Diameter_of_a_Set.svg/330px-Diameter_of_a_Set.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fc/Diameter_of_a_Set.svg/440px-Diameter_of_a_Set.svg.png 2x" data-file-width="344" data-file-height="409" /></a><figcaption>Diameter of a set.</figcaption></figure> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Bounded_set" title="Bounded set">Bounded set</a></div> A metric space M is bounded if there is an r such that no pair of points in M is more than distance r apart.<a href="#cite_note-13">[b]</a> The least such r is called the <style data-mw-deduplicate="TemplateStyles:r1238216509">.mw-parser-output .vanchor>:target~.vanchor-text{background-color:#b1d2ff}@media screen{html.skin-theme-clientpref-night .mw-parser-output .vanchor>:target~.vanchor-text{background-color:#0f4dc9}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .vanchor>:target~.vanchor-text{background-color:#0f4dc9}}</style><a href="/wiki/Diameter" title="Diameter">diameter</a> of M. The space M is called precompact or <a href="/wiki/Totally_bounded" class="mw-redirect" title="Totally bounded">totally bounded</a> if for every r > 0 there is a finite <a href="/wiki/Cover_(topology)" title="Cover (topology)">cover</a> of M by open balls of radius r. Every totally bounded space is bounded. To see this, start with a finite cover by r-balls for some arbitrary r. Since the subset of M consisting of the centers of these balls is finite, it has finite diameter, say D. By the triangle inequality, the diameter of the whole space is at most D + 2r. The converse does not hold: an example of a metric space that is bounded but not totally bounded is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e150115ab9f63023215109595b76686a1ff890fd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.732ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{2}}"> (or any other infinite set) with the discrete metric. <div class="mw-heading mw-heading3"><h3 id="Compactness">Compactness</h3>[<a href="/w/index.php?title=Metric_space&action=edit&section=14" title="Edit section: Compactness">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Compact_space" title="Compact space">Compact space</a></div> Compactness is a topological property which generalizes the properties of a closed and bounded subset of Euclidean space. There are several equivalent definitions of compactness in metric spaces: <ol><li>A metric space M is compact if every open cover has a finite subcover (the usual topological definition).</li> <li>A metric space M is compact if every sequence has a convergent subsequence. (For general topological spaces this is called <a href="/wiki/Sequentially_compact_space" title="Sequentially compact space">sequential compactness</a> and is not equivalent to compactness.)</li> <li>A metric space M is compact if it is complete and totally bounded. (This definition is written in terms of metric properties and does not make sense for a general topological space, but it is nevertheless topologically invariant since it is equivalent to compactness.)</li></ol> One example of a compact space is the closed interval [0, 1]. Compactness is important for similar reasons to completeness: it makes it easy to find limits. Another important tool is <a href="/wiki/Lebesgue%27s_number_lemma" title="Lebesgue's number lemma">Lebesgue's number lemma</a>, which shows that for any open cover of a compact space, every point is relatively deep inside one of the sets of the cover. <div class="mw-heading mw-heading2"><h2 id="Functions_between_metric_spaces">Functions between metric spaces</h2>[<a href="/w/index.php?title=Metric_space&action=edit&section=15" title="Edit section: Functions between metric spaces">edit</a>]</div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Functions_between_metric_spaces.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/9c/Functions_between_metric_spaces.svg/280px-Functions_between_metric_spaces.svg.png" decoding="async" width="280" height="210" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/9c/Functions_between_metric_spaces.svg/420px-Functions_between_metric_spaces.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/9c/Functions_between_metric_spaces.svg/560px-Functions_between_metric_spaces.svg.png 2x" data-file-width="540" data-file-height="405" /></a><figcaption><a href="/wiki/Euler_diagram" title="Euler diagram">Euler diagram</a> of types of functions between metric spaces.</figcaption></figure> Unlike in the case of topological spaces or algebraic structures such as <a href="/wiki/Group_(mathematics)" title="Group (mathematics)">groups</a> or <a href="/wiki/Ring_(mathematics)" title="Ring (mathematics)">rings</a>, there is no single "right" type of <a href="/wiki/Morphism" title="Morphism">structure-preserving function</a> between metric spaces. Instead, one works with different types of functions depending on one's goals. Throughout this section, suppose that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (M_{1},d_{1})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (M_{1},d_{1})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/94d3c0aca0772489db301d6c287dc52c5545bc75" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.415ex; height:2.843ex;" alt="{\displaystyle (M_{1},d_{1})}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (M_{2},d_{2})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (M_{2},d_{2})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b05a4c68a67ff10c60270286f08ee4917de29200" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.415ex; height:2.843ex;" alt="{\displaystyle (M_{2},d_{2})}"> are two metric spaces. The words "function" and "map" are used interchangeably. <div class="mw-heading mw-heading3"><h3 id="Isometries">Isometries</h3>[<a href="/w/index.php?title=Metric_space&action=edit&section=16" title="Edit section: Isometries">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Isometry" title="Isometry">Isometry</a></div> One interpretation of a "structure-preserving" map is one that fully preserves the distance function: <dl><dd>A function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:M_{1}\to M_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>:</mo> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">→</mo> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f:M_{1}\to M_{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/46f53db9ee82416bb7380eee882afeb7533bf639" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.446ex; height:2.509ex;" alt="{\displaystyle f:M_{1}\to M_{2}}"> is distance-preserving<a href="#cite_note-FOOTNOTEBuragoBuragoIvanov20012-14">[12]</a> if for every pair of points x and y in M1, <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d_{2}(f(x),f(y))=d_{1}(x,y).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d_{2}(f(x),f(y))=d_{1}(x,y).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bab26e06458a11228b6022d106fe307a480a0d66" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.104ex; height:2.843ex;" alt="{\displaystyle d_{2}(f(x),f(y))=d_{1}(x,y).}"></dd></dl> It follows from the metric space axioms that a distance-preserving function is injective. A bijective distance-preserving function is called an isometry.<a href="#cite_note-15">[13]</a> One perhaps non-obvious example of an isometry between spaces described in this article is the map <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:(\mathbb {R} ^{2},d_{1})\to (\mathbb {R} ^{2},d_{\infty })}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>:</mo> <mo stretchy="false">(</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>,</mo> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mo stretchy="false">(</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>,</mo> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f:(\mathbb {R} ^{2},d_{1})\to (\mathbb {R} ^{2},d_{\infty })}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d98ff99747477211f9aa50fc39961e47c4b9613a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.328ex; height:3.176ex;" alt="{\displaystyle f:(\mathbb {R} ^{2},d_{1})\to (\mathbb {R} ^{2},d_{\infty })}"> defined by <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x,y)=(x+y,x-y).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>y</mi> <mo>,</mo> <mi>x</mi> <mo>−</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x,y)=(x+y,x-y).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b5a6fc13924420432747020d500f1f2aefbad8dd" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.847ex; height:2.843ex;" alt="{\displaystyle f(x,y)=(x+y,x-y).}"> If there is an isometry between the spaces M1 and M2, they are said to be isometric. Metric spaces that are isometric are <a href="/wiki/Isomorphism" title="Isomorphism">essentially identical</a>. <div class="mw-heading mw-heading3"><h3 id="Continuous_maps">Continuous maps</h3>[<a href="/w/index.php?title=Metric_space&action=edit&section=17" title="Edit section: Continuous maps">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Continuous_function_(topology)" class="mw-redirect" title="Continuous function (topology)">Continuous function (topology)</a></div> On the other end of the spectrum, one can forget entirely about the metric structure and study <a href="/wiki/Continuous_function_(topology)" class="mw-redirect" title="Continuous function (topology)">continuous maps</a>, which only preserve topological structure. There are several equivalent definitions of continuity for metric spaces. The most important are: <ul><li>Topological definition. A function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\,\colon M_{1}\to M_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mspace width="thinmathspace" /> <mo>:</mo> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">→</mo> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f\,\colon M_{1}\to M_{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6aa1f72d391488d26c0f48c7c0ace1aa256daa7e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.93ex; height:2.509ex;" alt="{\displaystyle f\,\colon M_{1}\to M_{2}}"> is continuous if for every open set U in M2, the <a href="/wiki/Preimage" class="mw-redirect" title="Preimage">preimage</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f^{-1}(U)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>U</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f^{-1}(U)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8467d2b1831e3e4ef040a899742a9d6db350f90" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.245ex; height:3.176ex;" alt="{\displaystyle f^{-1}(U)}"> is open.</li> <li><a href="/wiki/Sequential_continuity" class="mw-redirect" title="Sequential continuity">Sequential continuity</a>. A function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\,\colon M_{1}\to M_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mspace width="thinmathspace" /> <mo>:</mo> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">→</mo> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f\,\colon M_{1}\to M_{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6aa1f72d391488d26c0f48c7c0ace1aa256daa7e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.93ex; height:2.509ex;" alt="{\displaystyle f\,\colon M_{1}\to M_{2}}"> is continuous if whenever a sequence (xn) converges to a point x in M1, the sequence <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x_{1}),f(x_{2}),\ldots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>,</mo> <mi>f</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>,</mo> <mo>…</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x_{1}),f(x_{2}),\ldots }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6fbef3751f2171351fb9383c14c9993bec6eef30" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.735ex; height:2.843ex;" alt="{\displaystyle f(x_{1}),f(x_{2}),\ldots }"> converges to the point f(x) in M2.</li></ul> <dl><dd>(These first two definitions are not equivalent for all topological spaces.)</dd></dl> <ul><li>ε–δ definition. A function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\,\colon M_{1}\to M_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mspace width="thinmathspace" /> <mo>:</mo> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">→</mo> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f\,\colon M_{1}\to M_{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6aa1f72d391488d26c0f48c7c0ace1aa256daa7e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.93ex; height:2.509ex;" alt="{\displaystyle f\,\colon M_{1}\to M_{2}}"> is continuous if for every point x in M1 and every ε > 0 there exists δ > 0 such that for all y in M1 we have <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d_{1}(x,y)<\delta \implies d_{2}(f(x),f(y))<\varepsilon .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo><</mo> <mi>δ</mi> <mspace width="thickmathspace" /> <mo stretchy="false">⟹</mo> <mspace width="thickmathspace" /> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo><</mo> <mi>ε</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d_{1}(x,y)<\delta \implies d_{2}(f(x),f(y))<\varepsilon .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/687b9947d7d93aee4b5878bf8ab3e46555f210ac" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:36.721ex; height:2.843ex;" alt="{\displaystyle d_{1}(x,y)<\delta \implies d_{2}(f(x),f(y))<\varepsilon .}"></li></ul> A <a href="/wiki/Homeomorphism" title="Homeomorphism">homeomorphism</a> is a continuous bijection whose inverse is also continuous; if there is a homeomorphism between M1 and M2, they are said to be homeomorphic. Homeomorphic spaces are the same from the point of view of topology, but may have very different metric properties. For example, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }"> is unbounded and complete, while (0, 1) is bounded but not complete. <div class="mw-heading mw-heading3"><h3 id="Uniformly_continuous_maps">Uniformly continuous maps</h3>[<a href="/w/index.php?title=Metric_space&action=edit&section=18" title="Edit section: Uniformly continuous maps">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Uniform_continuity" title="Uniform continuity">Uniform continuity</a></div> A function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\,\colon M_{1}\to M_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mspace width="thinmathspace" /> <mo>:</mo> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">→</mo> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f\,\colon M_{1}\to M_{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6aa1f72d391488d26c0f48c7c0ace1aa256daa7e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.93ex; height:2.509ex;" alt="{\displaystyle f\,\colon M_{1}\to M_{2}}"> is <a href="/wiki/Uniform_continuity" title="Uniform continuity">uniformly continuous</a> if for every real number ε > 0 there exists δ > 0 such that for all points x and y in M1 such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d(x,y)<\delta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo><</mo> <mi>δ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d(x,y)<\delta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/95e7bd68b6630a4968db4b0e2152b76a75056a23" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.691ex; height:2.843ex;" alt="{\displaystyle d(x,y)<\delta }">, we have <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d_{2}(f(x),f(y))<\varepsilon .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo><</mo> <mi>ε</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d_{2}(f(x),f(y))<\varepsilon .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8140d93a391f5eb8705fd6ad6322faa406771ae7" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.596ex; height:2.843ex;" alt="{\displaystyle d_{2}(f(x),f(y))<\varepsilon .}"> The only difference between this definition and the ε–δ definition of continuity is the order of quantifiers: the choice of δ must depend only on ε and not on the point x. However, this subtle change makes a big difference. For example, uniformly continuous maps take Cauchy sequences in M1 to Cauchy sequences in M2. In other words, uniform continuity preserves some metric properties which are not purely topological. On the other hand, the <a href="/wiki/Heine%E2%80%93Cantor_theorem" title="Heine–Cantor theorem">Heine–Cantor theorem</a> states that if M1 is compact, then every continuous map is uniformly continuous. In other words, uniform continuity cannot distinguish any non-topological features of compact metric spaces. <div class="mw-heading mw-heading3"><h3 id="Lipschitz_maps_and_contractions">Lipschitz maps and contractions</h3>[<a href="/w/index.php?title=Metric_space&action=edit&section=19" title="Edit section: Lipschitz maps and contractions">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Lipschitz_continuity" title="Lipschitz continuity">Lipschitz continuity</a></div> A <a href="/wiki/Lipschitz_continuity" title="Lipschitz continuity">Lipschitz map</a> is one that stretches distances by at most a bounded factor. Formally, given a real number K > 0, the map <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\,\colon M_{1}\to M_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mspace width="thinmathspace" /> <mo>:</mo> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">→</mo> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f\,\colon M_{1}\to M_{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6aa1f72d391488d26c0f48c7c0ace1aa256daa7e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.93ex; height:2.509ex;" alt="{\displaystyle f\,\colon M_{1}\to M_{2}}"> is K-Lipschitz if <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d_{2}(f(x),f(y))\leq Kd_{1}(x,y)\quad {\text{for all}}\quad x,y\in M_{1}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>≤</mo> <mi>K</mi> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>for all</mtext> </mrow> <mspace width="1em" /> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>∈</mo> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d_{2}(f(x),f(y))\leq Kd_{1}(x,y)\quad {\text{for all}}\quad x,y\in M_{1}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a8dd7e0a4bcf513c8571bcb3fae31fcc35f7a8cd" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:47.306ex; height:2.843ex;" alt="{\displaystyle d_{2}(f(x),f(y))\leq Kd_{1}(x,y)\quad {\text{for all}}\quad x,y\in M_{1}.}"> Lipschitz maps are particularly important in metric geometry, since they provide more flexibility than distance-preserving maps, but still make essential use of the metric.<a href="#cite_note-FOOTNOTEGromov2007xvii-16">[14]</a> For example, a curve in a metric space is <a href="/wiki/Arc_length" title="Arc length">rectifiable</a> (has finite length) if and only if it has a Lipschitz reparametrization. A 1-Lipschitz map is sometimes called a nonexpanding or <a href="/wiki/Metric_map" title="Metric map">metric map</a>. Metric maps are commonly taken to be the morphisms of the <a href="/wiki/Category_of_metric_spaces" title="Category of metric spaces">category of metric spaces</a>. A K-Lipschitz map for K < 1 is called a <a href="/wiki/Contraction_mapping" title="Contraction mapping">contraction</a>. The <a href="/wiki/Banach_fixed-point_theorem" title="Banach fixed-point theorem">Banach fixed-point theorem</a> states that if M is a complete metric space, then every contraction <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:M\to M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>:</mo> <mi>M</mi> <mo stretchy="false">→</mo> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f:M\to M}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9887c379b4c7403189b8bd4c2eae331f47d33717" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.714ex; height:2.509ex;" alt="{\displaystyle f:M\to M}"> admits a unique <a href="/wiki/Fixed_point_(mathematics)" title="Fixed point (mathematics)">fixed point</a>. If the metric space M is compact, the result holds for a slightly weaker condition on f: a map <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:M\to M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>:</mo> <mi>M</mi> <mo stretchy="false">→</mo> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f:M\to M}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9887c379b4c7403189b8bd4c2eae331f47d33717" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.714ex; height:2.509ex;" alt="{\displaystyle f:M\to M}"> admits a unique fixed point if <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d(f(x),f(y))<d(x,y)\quad {\mbox{for all}}\quad x\neq y\in M_{1}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo><</mo> <mi>d</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>for all</mtext> </mstyle> </mrow> <mspace width="1em" /> <mi>x</mi> <mo>≠</mo> <mi>y</mi> <mo>∈</mo> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d(f(x),f(y))<d(x,y)\quad {\mbox{for all}}\quad x\neq y\in M_{1}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fc66e3706c76037136fec484554f245e2ed76ea2" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:45.21ex; height:2.843ex;" alt="{\displaystyle d(f(x),f(y))<d(x,y)\quad {\mbox{for all}}\quad x\neq y\in M_{1}.}"> <div class="mw-heading mw-heading3"><h3 id="Quasi-isometries">Quasi-isometries</h3>[<a href="/w/index.php?title=Metric_space&action=edit&section=20" title="Edit section: Quasi-isometries">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Quasi-isometry" title="Quasi-isometry">Quasi-isometry</a></div> A <a href="/wiki/Quasi-isometry" title="Quasi-isometry">quasi-isometry</a> is a map that preserves the "large-scale structure" of a metric space. Quasi-isometries need not be continuous. For example, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e150115ab9f63023215109595b76686a1ff890fd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.732ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{2}}"> and its subspace <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} ^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} ^{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a807ab4cb3de13a66771b5a303aca31e0391e6aa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.605ex; height:2.676ex;" alt="{\displaystyle \mathbb {Z} ^{2}}"> are quasi-isometric, even though one is connected and the other is discrete. The equivalence relation of quasi-isometry is important in <a href="/wiki/Geometric_group_theory" title="Geometric group theory">geometric group theory</a>: the <a href="/wiki/%C5%A0varc%E2%80%93Milnor_lemma" title="Švarc–Milnor lemma">Švarc–Milnor lemma</a> states that all spaces on which a group <a href="/wiki/Geometric_group_action" title="Geometric group action">acts geometrically</a> are quasi-isometric.<a href="#cite_note-FOOTNOTEMargalitThomas2017-17">[15]</a> Formally, the map <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\,\colon M_{1}\to M_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mspace width="thinmathspace" /> <mo>:</mo> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">→</mo> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f\,\colon M_{1}\to M_{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6aa1f72d391488d26c0f48c7c0ace1aa256daa7e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.93ex; height:2.509ex;" alt="{\displaystyle f\,\colon M_{1}\to M_{2}}"> is a quasi-isometric embedding if there exist constants A ≥ 1 and B ≥ 0 such that <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{A}}d_{2}(f(x),f(y))-B\leq d_{1}(x,y)\leq Ad_{2}(f(x),f(y))+B\quad {\text{ for all }}\quad x,y\in M_{1}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>A</mi> </mfrac> </mrow> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>−</mo> <mi>B</mi> <mo>≤</mo> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>≤</mo> <mi>A</mi> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>+</mo> <mi>B</mi> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mtext> for all </mtext> </mrow> <mspace width="1em" /> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>∈</mo> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{A}}d_{2}(f(x),f(y))-B\leq d_{1}(x,y)\leq Ad_{2}(f(x),f(y))+B\quad {\text{ for all }}\quad x,y\in M_{1}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86d80138a711860401a8e82fedc649d7d086de93" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:76.798ex; height:5.343ex;" alt="{\displaystyle {\frac {1}{A}}d_{2}(f(x),f(y))-B\leq d_{1}(x,y)\leq Ad_{2}(f(x),f(y))+B\quad {\text{ for all }}\quad x,y\in M_{1}.}"> It is a quasi-isometry if in addition it is quasi-surjective, i.e. there is a constant C ≥ 0 such that every point in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M_{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d5d4dffae5ee0db4cc433e252ee9ed7530e5cf0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.308ex; height:2.509ex;" alt="{\displaystyle M_{2}}"> is at distance at most C from some point in the image <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(M_{1})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(M_{1})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1c16e62abf9983afb653dbdfc69e9da5319b27f9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.396ex; height:2.843ex;" alt="{\displaystyle f(M_{1})}">. <div class="mw-heading mw-heading3"><h3 id="Notions_of_metric_space_equivalence">Notions of metric space equivalence</h3>[<a href="/w/index.php?title=Metric_space&action=edit&section=21" title="Edit section: Notions of metric space equivalence">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Equivalence_of_metrics" title="Equivalence of metrics">Equivalence of metrics</a></div> Given two metric spaces <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (M_{1},d_{1})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (M_{1},d_{1})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/94d3c0aca0772489db301d6c287dc52c5545bc75" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.415ex; height:2.843ex;" alt="{\displaystyle (M_{1},d_{1})}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (M_{2},d_{2})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (M_{2},d_{2})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b05a4c68a67ff10c60270286f08ee4917de29200" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.415ex; height:2.843ex;" alt="{\displaystyle (M_{2},d_{2})}">: <ul><li>They are called homeomorphic (topologically isomorphic) if there is a <a href="/wiki/Homeomorphism" title="Homeomorphism">homeomorphism</a> between them (i.e., a continuous <a href="/wiki/Bijection" title="Bijection">bijection</a> with a continuous inverse). If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{1}=M_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M_{1}=M_{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8513d91a8cca09687f54732350d1b5a2128d8a2d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.715ex; height:2.509ex;" alt="{\displaystyle M_{1}=M_{2}}"> and the identity map is a homeomorphism, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d_{1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4cccb5a6a2f1acab4ca255e0be86c224ed82282a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.263ex; height:2.509ex;" alt="{\displaystyle d_{1}}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d_{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9276f8f68c5c23329de74ad76e69f6801358fb1f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.263ex; height:2.509ex;" alt="{\displaystyle d_{2}}"> are said to be topologically equivalent.</li> <li>They are called uniformic (uniformly isomorphic) if there is a <a href="/wiki/Uniform_isomorphism" title="Uniform isomorphism">uniform isomorphism</a> between them (i.e., a uniformly continuous bijection with a uniformly continuous inverse).</li> <li>They are called bilipschitz homeomorphic if there is a bilipschitz bijection between them (i.e., a Lipschitz bijection with a Lipschitz inverse).</li> <li>They are called isometric if there is a (bijective) <a href="/wiki/Isometry" title="Isometry">isometry</a> between them. In this case, the two metric spaces are essentially identical.</li> <li>They are called quasi-isometric if there is a <a href="/wiki/Quasi-isometry" title="Quasi-isometry">quasi-isometry</a> between them.</li></ul> <div class="mw-heading mw-heading2"><h2 id="Metric_spaces_with_additional_structure">Metric spaces with additional structure</h2>[<a href="/w/index.php?title=Metric_space&action=edit&section=22" title="Edit section: Metric spaces with additional structure">edit</a>]</div> <div class="mw-heading mw-heading3"><h3 id="Normed_vector_spaces">Normed vector spaces</h3>[<a href="/w/index.php?title=Metric_space&action=edit&section=23" title="Edit section: Normed vector spaces">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Normed_vector_space" title="Normed vector space">Normed vector space</a></div> A <a href="/wiki/Normed_vector_space" title="Normed vector space">normed vector space</a> is a vector space equipped with a <a href="/wiki/Norm_(mathematics)" title="Norm (mathematics)">norm</a>, which is a function that measures the length of vectors. The norm of a vector v is typically denoted by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lVert v\rVert }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">‖</mo> <mi>v</mi> <mo fence="false" stretchy="false">‖</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lVert v\rVert }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e373cfcd31951e395710bc6b431120c3c3931745" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.453ex; height:2.843ex;" alt="{\displaystyle \lVert v\rVert }">. Any normed vector space can be equipped with a metric in which the distance between two vectors x and y is given by <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d(x,y):=\lVert x-y\rVert .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>:=</mo> <mo fence="false" stretchy="false">‖</mo> <mi>x</mi> <mo>−</mo> <mi>y</mi> <mo fence="false" stretchy="false">‖</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d(x,y):=\lVert x-y\rVert .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fa80d85ddbb90dda9fde16b8d5e4058857c1d2a2" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.587ex; height:2.843ex;" alt="{\displaystyle d(x,y):=\lVert x-y\rVert .}"> The metric d is said to be induced by the norm <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lVert {\cdot }\rVert }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>⋅</mo> </mrow> <mo fence="false" stretchy="false">‖</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lVert {\cdot }\rVert }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7d4e4508f912075db0aca27e0e828fbcadd9e067" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.972ex; height:2.843ex;" alt="{\displaystyle \lVert {\cdot }\rVert }">. Conversely,<a href="#cite_note-FOOTNOTENariciBeckenstein201147–66-18">[16]</a> if a metric d on a <a href="/wiki/Vector_space" title="Vector space">vector space</a> X is <ul><li>translation invariant: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d(x,y)=d(x+a,y+a)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>d</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>a</mi> <mo>,</mo> <mi>y</mi> <mo>+</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d(x,y)=d(x+a,y+a)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b177c9d38083f4e7e19df135504670d7213d3887" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.327ex; height:2.843ex;" alt="{\displaystyle d(x,y)=d(x+a,y+a)}"> for every x, y, and a in X; and</li> <li><a href="/wiki/Absolute_homogeneity" class="mw-redirect" title="Absolute homogeneity"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238216509">absolutely homogeneous</a>: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d(\alpha x,\alpha y)=|\alpha |d(x,y)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo stretchy="false">(</mo> <mi>α</mi> <mi>x</mi> <mo>,</mo> <mi>α</mi> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>d</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d(\alpha x,\alpha y)=|\alpha |d(x,y)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1ae5f628515d47184a8d50a5bbb1caf78f454d29" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.944ex; height:2.843ex;" alt="{\displaystyle d(\alpha x,\alpha y)=|\alpha |d(x,y)}"> for every x and y in X and real number α;</li></ul> then it is the metric induced by the norm <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lVert x\rVert :=d(x,0).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">‖</mo> <mi>x</mi> <mo fence="false" stretchy="false">‖</mo> <mo>:=</mo> <mi>d</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lVert x\rVert :=d(x,0).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dc798453419e3222a827999903ce8bc240459937" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.598ex; height:2.843ex;" alt="{\displaystyle \lVert x\rVert :=d(x,0).}"> A similar relationship holds between <a href="/wiki/Seminorm" title="Seminorm">seminorms</a> and <a href="/wiki/Pseudometric_space" title="Pseudometric space">pseudometrics</a>. Among examples of metrics induced by a norm are the metrics d1, d2, and d∞ on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e150115ab9f63023215109595b76686a1ff890fd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.732ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{2}}">, which are induced by the <a href="/wiki/Manhattan_norm" class="mw-redirect" title="Manhattan norm">Manhattan norm</a>, the <a href="/wiki/Euclidean_norm" class="mw-redirect" title="Euclidean norm">Euclidean norm</a>, and the <a href="/wiki/Maximum_norm" class="mw-redirect" title="Maximum norm">maximum norm</a>, respectively. More generally, the <a href="/wiki/Kuratowski_embedding" title="Kuratowski embedding">Kuratowski embedding</a> allows one to see any metric space as a subspace of a normed vector space. Infinite-dimensional normed vector spaces, particularly spaces of functions, are studied in <a href="/wiki/Functional_analysis" title="Functional analysis">functional analysis</a>. Completeness is particularly important in this context: a complete normed vector space is known as a <a href="/wiki/Banach_space" title="Banach space">Banach space</a>. An unusual property of normed vector spaces is that <a href="/wiki/Linear_transformation" class="mw-redirect" title="Linear transformation">linear transformations</a> between them are continuous if and only if they are Lipschitz. Such transformations are known as <a href="/wiki/Bounded_operator" title="Bounded operator">bounded operators</a>. <div class="mw-heading mw-heading3"><h3 id="Length_spaces">Length spaces</h3>[<a href="/w/index.php?title=Metric_space&action=edit&section=24" title="Edit section: Length spaces">edit</a>]</div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Approximate_arc_length.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/11/Approximate_arc_length.svg/220px-Approximate_arc_length.svg.png" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/11/Approximate_arc_length.svg/330px-Approximate_arc_length.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/11/Approximate_arc_length.svg/440px-Approximate_arc_length.svg.png 2x" data-file-width="360" data-file-height="360" /></a><figcaption>One possible approximation for the arc length of a curve. The approximation is never longer than the arc length, justifying the definition of arc length as a <a href="/wiki/Supremum" class="mw-redirect" title="Supremum">supremum</a>.</figcaption></figure> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Intrinsic_metric" title="Intrinsic metric">Intrinsic metric</a></div> A <a href="/wiki/Curve" title="Curve">curve</a> in a metric space (M, d) is a continuous function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma :[0,T]\to M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>γ</mi> <mo>:</mo> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mi>T</mi> <mo stretchy="false">]</mo> <mo stretchy="false">→</mo> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma :[0,T]\to M}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f51598fa2d3e48c963d4e56c0af969afaa51f549" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.382ex; height:2.843ex;" alt="{\displaystyle \gamma :[0,T]\to M}">. The <a href="/wiki/Arc_length" title="Arc length">length</a> of γ is measured by <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L(\gamma )=\sup _{0=x_{0}<x_{1}<\cdots <x_{n}=T}\left\{\sum _{k=1}^{n}d(\gamma (x_{k-1}),\gamma (x_{k}))\right\}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> <mo stretchy="false">(</mo> <mi>γ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munder> <mo movablelimits="true" form="prefix">sup</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> <mo>=</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo><</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo><</mo> <mo>⋯</mo> <mo><</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mi>T</mi> </mrow> </munder> <mrow> <mo>{</mo> <mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mi>d</mi> <mo stretchy="false">(</mo> <mi>γ</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>,</mo> <mi>γ</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mrow> <mo>}</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L(\gamma )=\sup _{0=x_{0}<x_{1}<\cdots <x_{n}=T}\left\{\sum _{k=1}^{n}d(\gamma (x_{k-1}),\gamma (x_{k}))\right\}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/57fa5c5287e046af0e8484e05fa038b82aae6d36" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:49.548ex; height:7.509ex;" alt="{\displaystyle L(\gamma )=\sup _{0=x_{0}<x_{1}<\cdots <x_{n}=T}\left\{\sum _{k=1}^{n}d(\gamma (x_{k-1}),\gamma (x_{k}))\right\}.}"> In general, this supremum may be infinite; a curve of finite length is called rectifiable.<a href="#cite_note-FOOTNOTEBuragoBuragoIvanov2001Definition_2.3.1-19">[17]</a> Suppose that the length of the curve γ is equal to the distance between its endpoints—that is, it is the shortest possible path between its endpoints. After reparametrization by arc length, γ becomes a <a href="/wiki/Geodesic" title="Geodesic">geodesic</a>: a curve which is a distance-preserving function.<a href="#cite_note-FOOTNOTEMargalitThomas2017-17">[15]</a> A geodesic is a shortest possible path between any two of its points.<a href="#cite_note-22">[c]</a> A geodesic metric space is a metric space which admits a geodesic between any two of its points. The spaces <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\mathbb {R} ^{2},d_{1})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>,</mo> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\mathbb {R} ^{2},d_{1})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2362b4a4d7e9b64ee7e2262dee7de30d2403e299" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.839ex; height:3.176ex;" alt="{\displaystyle (\mathbb {R} ^{2},d_{1})}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\mathbb {R} ^{2},d_{2})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>,</mo> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\mathbb {R} ^{2},d_{2})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/14ffad81c43f03ddc159b7d4a7b620dc195aafce" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.839ex; height:3.176ex;" alt="{\displaystyle (\mathbb {R} ^{2},d_{2})}"> are both geodesic metric spaces. In <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\mathbb {R} ^{2},d_{2})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>,</mo> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\mathbb {R} ^{2},d_{2})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/14ffad81c43f03ddc159b7d4a7b620dc195aafce" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.839ex; height:3.176ex;" alt="{\displaystyle (\mathbb {R} ^{2},d_{2})}">, geodesics are unique, but in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\mathbb {R} ^{2},d_{1})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>,</mo> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\mathbb {R} ^{2},d_{1})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2362b4a4d7e9b64ee7e2262dee7de30d2403e299" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.839ex; height:3.176ex;" alt="{\displaystyle (\mathbb {R} ^{2},d_{1})}">, there are often infinitely many geodesics between two points, as shown in the figure at the top of the article. The space M is a <a href="/wiki/Length_space" class="mw-redirect" title="Length space">length space</a> (or the metric d is intrinsic) if the distance between any two points x and y is the infimum of lengths of paths between them. Unlike in a geodesic metric space, the infimum does not have to be attained. An example of a length space which is not geodesic is the Euclidean plane minus the origin: the points (1, 0) and (-1, 0) can be joined by paths of length arbitrarily close to 2, but not by a path of length 2. An example of a metric space which is not a length space is given by the straight-line metric on the sphere: the straight line between two points through the center of the Earth is shorter than any path along the surface. Given any metric space (M, d), one can define a new, intrinsic distance function dintrinsic on M by setting the distance between points x and y to be the infimum of the d-lengths of paths between them. For instance, if d is the straight-line distance on the sphere, then dintrinsic is the great-circle distance. However, in some cases dintrinsic may have infinite values. For example, if M is the <a href="/wiki/Koch_snowflake" title="Koch snowflake">Koch snowflake</a> with the subspace metric d induced from <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e150115ab9f63023215109595b76686a1ff890fd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.732ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{2}}">, then the resulting intrinsic distance is infinite for any pair of distinct points. <div class="mw-heading mw-heading3"><h3 id="Riemannian_manifolds">Riemannian manifolds</h3>[<a href="/w/index.php?title=Metric_space&action=edit&section=25" title="Edit section: Riemannian manifolds">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Riemannian_manifold" title="Riemannian manifold">Riemannian manifold</a></div> A <a href="/wiki/Riemannian_manifold" title="Riemannian manifold">Riemannian manifold</a> is a space equipped with a Riemannian <a href="/wiki/Metric_tensor" title="Metric tensor">metric tensor</a>, which determines lengths of <a href="/wiki/Tangent_space" title="Tangent space">tangent vectors</a> at every point. This can be thought of defining a notion of distance infinitesimally. In particular, a differentiable path <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma :[0,T]\to M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>γ</mi> <mo>:</mo> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mi>T</mi> <mo stretchy="false">]</mo> <mo stretchy="false">→</mo> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma :[0,T]\to M}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f51598fa2d3e48c963d4e56c0af969afaa51f549" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.382ex; height:2.843ex;" alt="{\displaystyle \gamma :[0,T]\to M}"> in a Riemannian manifold M has length defined as the integral of the length of the tangent vector to the path: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L(\gamma )=\int _{0}^{T}|{\dot {\gamma }}(t)|dt.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> <mo stretchy="false">(</mo> <mi>γ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>γ</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>d</mi> <mi>t</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L(\gamma )=\int _{0}^{T}|{\dot {\gamma }}(t)|dt.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/78f26517d50ba283d6685065612192c5f2f10d11" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:19.901ex; height:6.176ex;" alt="{\displaystyle L(\gamma )=\int _{0}^{T}|{\dot {\gamma }}(t)|dt.}"> On a connected Riemannian manifold, one then defines the distance between two points as the infimum of lengths of smooth paths between them. This construction generalizes to other kinds of infinitesimal metrics on manifolds, such as <a href="/wiki/Sub-Riemannian_manifold" title="Sub-Riemannian manifold">sub-Riemannian</a> and <a href="/wiki/Finsler_manifold" title="Finsler manifold">Finsler metrics</a>. The Riemannian metric is uniquely determined by the distance function; this means that in principle, all information about a Riemannian manifold can be recovered from its distance function. One direction in metric geometry is finding purely metric (<a href="/wiki/Synthetic_geometry" title="Synthetic geometry">"synthetic"</a>) formulations of properties of Riemannian manifolds. For example, a Riemannian manifold is a <a href="/wiki/CAT(k)_space" title="CAT(k) space">CAT(k) space</a> (a synthetic condition which depends purely on the metric) if and only if its <a href="/wiki/Sectional_curvature" title="Sectional curvature">sectional curvature</a> is bounded above by k.<a href="#cite_note-FOOTNOTEBuragoBuragoIvanov2001127-23">[20]</a> Thus CAT(k) spaces generalize upper curvature bounds to general metric spaces. <div class="mw-heading mw-heading3"><h3 id="Metric_measure_spaces">Metric measure spaces</h3>[<a href="/w/index.php?title=Metric_space&action=edit&section=26" title="Edit section: Metric measure spaces">edit</a>]</div> Real analysis makes use of both the metric on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c510b63578322050121fe966f2e5770bea43308d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.897ex; height:2.343ex;" alt="{\displaystyle \mathbb {R} ^{n}}"> and the <a href="/wiki/Lebesgue_measure" title="Lebesgue measure">Lebesgue measure</a>. Therefore, generalizations of many ideas from analysis naturally reside in <a href="/w/index.php?title=Metric_measure_space&action=edit&redlink=1" class="new" title="Metric measure space (page does not exist)">metric measure spaces</a>: spaces that have both a <a href="/wiki/Measure_(mathematics)" title="Measure (mathematics)">measure</a> and a metric which are compatible with each other. Formally, a metric measure space is a metric space equipped with a <a href="/wiki/Borel_regular_measure" title="Borel regular measure">Borel regular measure</a> such that every ball has positive measure.<a href="#cite_note-FOOTNOTEHeinonen2007191-24">[21]</a> For example Euclidean spaces of dimension n, and more generally n-dimensional Riemannian manifolds, naturally have the structure of a metric measure space, equipped with the <a href="/wiki/Lebesgue_measure" title="Lebesgue measure">Lebesgue measure</a>. Certain <a href="/wiki/Fractal" title="Fractal">fractal</a> metric spaces such as the <a href="/wiki/Sierpi%C5%84ski_gasket" class="mw-redirect" title="Sierpiński gasket">Sierpiński gasket</a> can be equipped with the α-dimensional <a href="/wiki/Hausdorff_measure" title="Hausdorff measure">Hausdorff measure</a> where α is the <a href="/wiki/Hausdorff_dimension" title="Hausdorff dimension">Hausdorff dimension</a>. In general, however, a metric space may not have an "obvious" choice of measure. One application of metric measure spaces is generalizing the notion of <a href="/wiki/Ricci_curvature" title="Ricci curvature">Ricci curvature</a> beyond Riemannian manifolds. Just as CAT(k) and <a href="/wiki/Alexandrov_space" title="Alexandrov space">Alexandrov spaces</a> generalize sectional curvature bounds, <a href="/w/index.php?title=RCD_space&action=edit&redlink=1" class="new" title="RCD space (page does not exist)">RCD spaces</a> are a class of metric measure spaces which generalize lower bounds on Ricci curvature.<a href="#cite_note-25">[22]</a> <div class="mw-heading mw-heading2"><h2 id="Further_examples_and_applications">Further examples and applications</h2>[<a href="/w/index.php?title=Metric_space&action=edit&section=27" title="Edit section: Further examples and applications">edit</a>]</div> <div class="mw-heading mw-heading3"><h3 id="Graphs_and_finite_metric_spaces">Graphs and finite metric spaces</h3>[<a href="/w/index.php?title=Metric_space&action=edit&section=28" title="Edit section: Graphs and finite metric spaces">edit</a>]</div> A <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238216509">metric space is discrete if its induced topology is the <a href="/wiki/Discrete_topology" class="mw-redirect" title="Discrete topology">discrete topology</a>. Although many concepts, such as completeness and compactness, are not interesting for such spaces, they are nevertheless an object of study in several branches of mathematics. In particular, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238216509">finite metric spaces (those having a <a href="/wiki/Finite_set" title="Finite set">finite</a> number of points) are studied in <a href="/wiki/Combinatorics" title="Combinatorics">combinatorics</a> and <a href="/wiki/Theoretical_computer_science" title="Theoretical computer science">theoretical computer science</a>.<a href="#cite_note-26">[23]</a> Embeddings in other metric spaces are particularly well-studied. For example, not every finite metric space can be <a href="/wiki/Isometry" title="Isometry">isometrically embedded</a> in a Euclidean space or in <a href="/wiki/Hilbert_space" title="Hilbert space">Hilbert space</a>. On the other hand, in the worst case the required distortion (bilipschitz constant) is only logarithmic in the number of points.<a href="#cite_note-27">[24]</a><a href="#cite_note-28">[25]</a> For any <a href="/wiki/Graph_(discrete_mathematics)" title="Graph (discrete mathematics)">undirected connected graph</a> G, the set V of vertices of G can be turned into a metric space by defining the <a href="/wiki/Distance_(graph_theory)" title="Distance (graph theory)">distance</a> between vertices x and y to be the length of the shortest edge path connecting them. This is also called shortest-path distance or geodesic distance. In <a href="/wiki/Geometric_group_theory" title="Geometric group theory">geometric group theory</a> this construction is applied to the <a href="/wiki/Cayley_graph" title="Cayley graph">Cayley graph</a> of a (typically infinite) <a href="/wiki/Finitely-generated_group" class="mw-redirect" title="Finitely-generated group">finitely-generated group</a>, yielding the <a href="/wiki/Word_metric" title="Word metric">word metric</a>. Up to a bilipschitz homeomorphism, the word metric depends only on the group and not on the chosen finite generating set.<a href="#cite_note-FOOTNOTEMargalitThomas2017-17">[15]</a> <div class="mw-heading mw-heading3"><h3 id="Distances_between_mathematical_objects">Distances between mathematical objects</h3>[<a href="/w/index.php?title=Metric_space&action=edit&section=29" title="Edit section: Distances between mathematical objects">edit</a>]</div> In modern mathematics, one often studies spaces whose points are themselves mathematical objects. A distance function on such a space generally aims to measure the dissimilarity between two objects. Here are some examples: <ul><li>Functions to a metric space. If X is any set and M is a metric space, then the set of all <a href="/wiki/Bounded_function" title="Bounded function">bounded functions</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\colon X\to M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>:</mo> <mi>X</mi> <mo stretchy="false">→</mo> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f\colon X\to M}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01174240f796f3f567c3a311ff2fddc1e05edd9f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.349ex; height:2.509ex;" alt="{\displaystyle f\colon X\to M}"> (i.e. those functions whose image is a <a href="/wiki/Bounded_subset" class="mw-redirect" title="Bounded subset">bounded subset</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}">) can be turned into a metric space by defining the distance between two bounded functions f and g to be <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d(f,g)=\sup _{x\in X}d(f(x),g(x)).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo stretchy="false">(</mo> <mi>f</mi> <mo>,</mo> <mi>g</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munder> <mo movablelimits="true" form="prefix">sup</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo>∈</mo> <mi>X</mi> </mrow> </munder> <mi>d</mi> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d(f,g)=\sup _{x\in X}d(f(x),g(x)).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d88e98e11dc5c2deab1273d66dc5596b19d21aea" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:26.819ex; height:4.509ex;" alt="{\displaystyle d(f,g)=\sup _{x\in X}d(f(x),g(x)).}"> This metric is called the <a href="/wiki/Uniform_metric" class="mw-redirect" title="Uniform metric">uniform metric</a> or supremum metric.<a href="#cite_note-FOOTNOTEÓ_Searcóid2006107-29">[26]</a> If M is complete, then this <a href="/wiki/Function_space" title="Function space">function space</a> is complete as well; moreover, if X is also a topological space, then the subspace consisting of all bounded <a href="/wiki/Continuous_function_(topology)" class="mw-redirect" title="Continuous function (topology)">continuous</a> functions from X to M is also complete. When X is a subspace of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c510b63578322050121fe966f2e5770bea43308d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.897ex; height:2.343ex;" alt="{\displaystyle \mathbb {R} ^{n}}">, this function space is known as a <a href="/wiki/Classical_Wiener_space" title="Classical Wiener space">classical Wiener space</a>.</li> <li><a href="/wiki/String_metric" title="String metric">String metrics</a> and <a href="/wiki/Edit_distance" title="Edit distance">edit distances</a>. There are many ways of measuring distances between <a href="/wiki/String_(computer_science)" title="String (computer science)">strings of characters</a>, which may represent <a href="/wiki/Sentence_(linguistics)" title="Sentence (linguistics)">sentences</a> in <a href="/wiki/Computational_linguistics" title="Computational linguistics">computational linguistics</a> or <a href="/wiki/Code_word_(communication)" title="Code word (communication)">code words</a> in <a href="/wiki/Coding_theory" title="Coding theory">coding theory</a>. Edit distances attempt to measure the number of changes necessary to get from one string to another. For example, the <a href="/wiki/Hamming_distance" title="Hamming distance">Hamming distance</a> measures the minimal number of substitutions needed, while the <a href="/wiki/Levenshtein_distance" title="Levenshtein distance">Levenshtein distance</a> measures the minimal number of deletions, insertions, and substitutions; both of these can be thought of as distances in an appropriate graph.</li> <li><a href="/wiki/Graph_edit_distance" title="Graph edit distance">Graph edit distance</a> is a measure of dissimilarity between two <a href="/wiki/Graph_(discrete_mathematics)" title="Graph (discrete mathematics)">graphs</a>, defined as the minimal number of <a href="/wiki/Graph_operations" title="Graph operations">graph edit operations</a> required to transform one graph into another.</li> <li><a href="/wiki/Wasserstein_metric" title="Wasserstein metric">Wasserstein metrics</a> measure the distance between two <a href="/wiki/Measure_(mathematics)" title="Measure (mathematics)">measures</a> on the same metric space. The Wasserstein distance between two measures is, roughly speaking, the <a href="/wiki/Optimal_transport" class="mw-redirect" title="Optimal transport">cost of transporting</a> one to the other.</li> <li>The set of all m by n <a href="/wiki/Matrix_(mathematics)" title="Matrix (mathematics)">matrices</a> over some <a href="/wiki/Field_(mathematics)" title="Field (mathematics)">field</a> is a metric space with respect to the <a href="/wiki/Rank_(linear_algebra)" title="Rank (linear algebra)">rank</a> distance <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d(A,B)=\mathrm {rank} (B-A)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo stretchy="false">(</mo> <mi>A</mi> <mo>,</mo> <mi>B</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">r</mi> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">k</mi> </mrow> <mo stretchy="false">(</mo> <mi>B</mi> <mo>−</mo> <mi>A</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d(A,B)=\mathrm {rank} (B-A)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cb439fd4e5e09a4306d1c8840918ba4e0cc19175" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.416ex; height:2.843ex;" alt="{\displaystyle d(A,B)=\mathrm {rank} (B-A)}">.</li> <li>The <a href="/wiki/Helly_metric" title="Helly metric">Helly metric</a> in <a href="/wiki/Game_theory" title="Game theory">game theory</a> measures the difference between <a href="/wiki/Strategy_(game_theory)" title="Strategy (game theory)">strategies</a> in a game.</li></ul> <div class="mw-heading mw-heading3"><h3 id="Hausdorff_and_Gromov–Hausdorff_distance">Hausdorff and Gromov–Hausdorff distance</h3>[<a href="/w/index.php?title=Metric_space&action=edit&section=30" title="Edit section: Hausdorff and Gromov–Hausdorff distance">edit</a>]</div> The idea of spaces of mathematical objects can also be applied to subsets of a metric space, as well as metric spaces themselves. <a href="/wiki/Hausdorff_distance" title="Hausdorff distance">Hausdorff</a> and <a href="/wiki/Gromov%E2%80%93Hausdorff_convergence" title="Gromov–Hausdorff convergence">Gromov–Hausdorff distance</a> define metrics on the set of compact subsets of a metric space and the set of compact metric spaces, respectively. Suppose (M, d) is a metric space, and let S be a subset of M. The distance from S to a point x of M is, informally, the distance from x to the closest point of S. However, since there may not be a single closest point, it is defined via an <a href="/wiki/Infimum" class="mw-redirect" title="Infimum">infimum</a>: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d(x,S)=\inf\{d(x,s):s\in S\}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>S</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo movablelimits="true" form="prefix">inf</mo> <mo fence="false" stretchy="false">{</mo> <mi>d</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>:</mo> <mi>s</mi> <mo>∈</mo> <mi>S</mi> <mo fence="false" stretchy="false">}</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d(x,S)=\inf\{d(x,s):s\in S\}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c7c78d84b87b3cf79cd3426d990c4e625822f42e" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.456ex; height:2.843ex;" alt="{\displaystyle d(x,S)=\inf\{d(x,s):s\in S\}.}"> In particular, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d(x,S)=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>S</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d(x,S)=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7357226d0d09b20f4b272e471a5ea3399b2603ef" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.149ex; height:2.843ex;" alt="{\displaystyle d(x,S)=0}"> if and only if x belongs to the <a href="/wiki/Closure_(topology)" title="Closure (topology)">closure</a> of S. Furthermore, distances between points and sets satisfy a version of the triangle inequality: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d(x,S)\leq d(x,y)+d(y,S),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>S</mi> <mo stretchy="false">)</mo> <mo>≤</mo> <mi>d</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>d</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo>,</mo> <mi>S</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d(x,S)\leq d(x,y)+d(y,S),}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8ba425307ff730dd52b36f126ee6676d41d026ad" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.732ex; height:2.843ex;" alt="{\displaystyle d(x,S)\leq d(x,y)+d(y,S),}"> and therefore the map <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d_{S}:M\to \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> </msub> <mo>:</mo> <mi>M</mi> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d_{S}:M\to \mathbb {R} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a8df407542bad947427e70703c6028999759187" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.173ex; height:2.509ex;" alt="{\displaystyle d_{S}:M\to \mathbb {R} }"> defined by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d_{S}(x)=d(x,S)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>d</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>S</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d_{S}(x)=d(x,S)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e00ad46069facc42cde980592537671262b7dd65" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.627ex; height:2.843ex;" alt="{\displaystyle d_{S}(x)=d(x,S)}"> is continuous. Incidentally, this shows that metric spaces are <a href="/wiki/Completely_regular" class="mw-redirect" title="Completely regular">completely regular</a>. Given two subsets S and T of M, their Hausdorff distance is <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d_{H}(S,T)=\max\{\sup\{d(s,T):s\in S\},\sup\{d(t,S):t\in T\}\}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>H</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>S</mi> <mo>,</mo> <mi>T</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo movablelimits="true" form="prefix">max</mo> <mo fence="false" stretchy="false">{</mo> <mo movablelimits="true" form="prefix">sup</mo> <mo fence="false" stretchy="false">{</mo> <mi>d</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo>,</mo> <mi>T</mi> <mo stretchy="false">)</mo> <mo>:</mo> <mi>s</mi> <mo>∈</mo> <mi>S</mi> <mo fence="false" stretchy="false">}</mo> <mo>,</mo> <mo movablelimits="true" form="prefix">sup</mo> <mo fence="false" stretchy="false">{</mo> <mi>d</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo>,</mo> <mi>S</mi> <mo stretchy="false">)</mo> <mo>:</mo> <mi>t</mi> <mo>∈</mo> <mi>T</mi> <mo fence="false" stretchy="false">}</mo> <mo fence="false" stretchy="false">}</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d_{H}(S,T)=\max\{\sup\{d(s,T):s\in S\},\sup\{d(t,S):t\in T\}\}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2eb121a92a05c5a675982e5faf4a8fe14ba42f8d" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:59.767ex; height:2.843ex;" alt="{\displaystyle d_{H}(S,T)=\max\{\sup\{d(s,T):s\in S\},\sup\{d(t,S):t\in T\}\}.}"> Informally, two sets S and T are close to each other in the Hausdorff distance if no element of S is too far from T and vice versa. For example, if S is an open set in Euclidean space T is an <a href="/wiki/Delone_set" title="Delone set">ε-net</a> inside S, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d_{H}(S,T)<\varepsilon }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>H</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>S</mi> <mo>,</mo> <mi>T</mi> <mo stretchy="false">)</mo> <mo><</mo> <mi>ε</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d_{H}(S,T)<\varepsilon }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b65dfd5e4eff2be6eb7c2beec013433876581277" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.061ex; height:2.843ex;" alt="{\displaystyle d_{H}(S,T)<\varepsilon }">. In general, the Hausdorff distance <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d_{H}(S,T)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>H</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>S</mi> <mo>,</mo> <mi>T</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d_{H}(S,T)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/29ec1b199a5dd18b8b7e005111a5ab00583a899e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.879ex; height:2.843ex;" alt="{\displaystyle d_{H}(S,T)}"> can be infinite or zero. However, the Hausdorff distance between two distinct compact sets is always positive and finite. Thus the Hausdorff distance defines a metric on the set of compact subsets of M. The Gromov–Hausdorff metric defines a distance between (isometry classes of) compact metric spaces. The Gromov–Hausdorff distance between compact spaces X and Y is the infimum of the Hausdorff distance over all metric spaces Z that contain X and Y as subspaces. While the exact value of the Gromov–Hausdorff distance is rarely useful to know, the resulting topology has found many applications. <div class="mw-heading mw-heading3"><h3 id="Miscellaneous_examples">Miscellaneous examples</h3>[<a href="/w/index.php?title=Metric_space&action=edit&section=31" title="Edit section: Miscellaneous examples">edit</a>]</div> <ul><li>Given a metric space (X, d) and an increasing <a href="/wiki/Concave_function" title="Concave function">concave function</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\colon [0,\infty )\to [0,\infty )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>:</mo> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mi mathvariant="normal">∞</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mi mathvariant="normal">∞</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f\colon [0,\infty )\to [0,\infty )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f22e0e3813ce78c9a62957e81da67e89a05b6b47" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.07ex; height:2.843ex;" alt="{\displaystyle f\colon [0,\infty )\to [0,\infty )}"> such that f(t) = 0 if and only if t = 0, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d_{f}(x,y)=f(d(x,y))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>d</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d_{f}(x,y)=f(d(x,y))}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f3b5fce607ab2bbad59ca435efa8e8bdc89529ef" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:20.404ex; height:3.009ex;" alt="{\displaystyle d_{f}(x,y)=f(d(x,y))}"> is also a metric on X. If f(t) = tα for some real number α < 1, such a metric is known as a snowflake of d.<a href="#cite_note-30">[27]</a></li> <li>The <a href="/wiki/Tight_span" title="Tight span">tight span</a> of a metric space is another metric space which can be thought of as an abstract version of the <a href="/wiki/Convex_hull" title="Convex hull">convex hull</a>.</li> <li>The knight's move metric, the minimal number of knight's moves to reach one point in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} ^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} ^{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a807ab4cb3de13a66771b5a303aca31e0391e6aa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.605ex; height:2.676ex;" alt="{\displaystyle \mathbb {Z} ^{2}}"> from another, is a metric on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} ^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} ^{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a807ab4cb3de13a66771b5a303aca31e0391e6aa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.605ex; height:2.676ex;" alt="{\displaystyle \mathbb {Z} ^{2}}">.</li> <li>The <a href="/wiki/British_Rail" title="British Rail">British Rail</a> metric (also called the "post office metric" or the "<a href="/wiki/SNCF" title="SNCF">SNCF</a> metric") on a <a href="/wiki/Normed_vector_space" title="Normed vector space">normed vector space</a> is given by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d(x,y)=\lVert x\rVert +\lVert y\rVert }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo fence="false" stretchy="false">‖</mo> <mi>x</mi> <mo fence="false" stretchy="false">‖</mo> <mo>+</mo> <mo fence="false" stretchy="false">‖</mo> <mi>y</mi> <mo fence="false" stretchy="false">‖</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d(x,y)=\lVert x\rVert +\lVert y\rVert }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/13e1e365ad25bd9facf08c903c6e867928f24649" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.618ex; height:2.843ex;" alt="{\displaystyle d(x,y)=\lVert x\rVert +\lVert y\rVert }"> for distinct points <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}">, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d(x,x)=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d(x,x)=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f8b0b392db0a4abf6c47dcd5b35f90592401e756" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.979ex; height:2.843ex;" alt="{\displaystyle d(x,x)=0}">. More generally <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lVert \cdot \rVert }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">‖</mo> <mo>⋅</mo> <mo fence="false" stretchy="false">‖</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lVert \cdot \rVert }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e52a20a813a9cfbc68eae56c73cad60c81deac43" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.972ex; height:2.843ex;" alt="{\displaystyle \lVert \cdot \rVert }"> can be replaced with a function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"> taking an arbitrary set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}"> to non-negative reals and taking the value <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2aae8864a3c1fec9585261791a809ddec1489950" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 0}"> at most once: then the metric is defined on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}"> by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d(x,y)=f(x)+f(y)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d(x,y)=f(x)+f(y)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e44b8ac740a7bb352af64910a98e844862b901a9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.144ex; height:2.843ex;" alt="{\displaystyle d(x,y)=f(x)+f(y)}"> for distinct points <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}">, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d(x,x)=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d(x,x)=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f8b0b392db0a4abf6c47dcd5b35f90592401e756" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.979ex; height:2.843ex;" alt="{\displaystyle d(x,x)=0}">. The name alludes to the tendency of railway journeys to proceed via London (or Paris) irrespective of their final destination.</li> <li>The <a href="/wiki/Robinson%E2%80%93Foulds_metric" title="Robinson–Foulds metric">Robinson–Foulds metric</a> used for calculating the distances between <a href="/wiki/Phylogenetic_tree" title="Phylogenetic tree">Phylogenetic trees</a> in <a href="/wiki/Phylogenetics" title="Phylogenetics">Phylogenetics</a><a href="#cite_note-31">[28]</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Constructions">Constructions</h2>[<a href="/w/index.php?title=Metric_space&action=edit&section=32" title="Edit section: Constructions">edit</a>]</div> <div class="mw-heading mw-heading3"><h3 id="Product_metric_spaces">Product metric spaces</h3>[<a href="/w/index.php?title=Metric_space&action=edit&section=33" title="Edit section: Product metric spaces">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Product_metric" title="Product metric">Product metric</a></div> If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (M_{1},d_{1}),\ldots ,(M_{n},d_{n})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mo stretchy="false">(</mo> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (M_{1},d_{1}),\ldots ,(M_{n},d_{n})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0dc0196b8ce0cbd450496c2ae526bfb1dbec9932" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.336ex; height:2.843ex;" alt="{\displaystyle (M_{1},d_{1}),\ldots ,(M_{n},d_{n})}"> are metric spaces, and N is the <a href="/wiki/Euclidean_norm" class="mw-redirect" title="Euclidean norm">Euclidean norm</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c510b63578322050121fe966f2e5770bea43308d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.897ex; height:2.343ex;" alt="{\displaystyle \mathbb {R} ^{n}}">, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bigl (}M_{1}\times \cdots \times M_{n},d_{\times }{\bigr )}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>×</mo> <mo>⋯</mo> <mo>×</mo> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>×</mo> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\bigl (}M_{1}\times \cdots \times M_{n},d_{\times }{\bigr )}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/581f2973a4e207cf31674934f26a9d61dd4f569f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:21.068ex; height:3.176ex;" alt="{\displaystyle {\bigl (}M_{1}\times \cdots \times M_{n},d_{\times }{\bigr )}}"> is a metric space, where the <a href="/wiki/Product_metric" title="Product metric">product metric</a> is defined by <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d_{\times }{\bigl (}(x_{1},\ldots ,x_{n}),(y_{1},\ldots ,y_{n}){\bigr )}=N{\bigl (}d_{1}(x_{1},y_{1}),\ldots ,d_{n}(x_{n},y_{n}){\bigr )},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>×</mo> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>,</mo> <mo stretchy="false">(</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mo>=</mo> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d_{\times }{\bigl (}(x_{1},\ldots ,x_{n}),(y_{1},\ldots ,y_{n}){\bigr )}=N{\bigl (}d_{1}(x_{1},y_{1}),\ldots ,d_{n}(x_{n},y_{n}){\bigr )},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/185b237e4eae21a0037bfa59b3deddef95f0d915" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:62.319ex; height:3.176ex;" alt="{\displaystyle d_{\times }{\bigl (}(x_{1},\ldots ,x_{n}),(y_{1},\ldots ,y_{n}){\bigr )}=N{\bigl (}d_{1}(x_{1},y_{1}),\ldots ,d_{n}(x_{n},y_{n}){\bigr )},}"> and the induced topology agrees with the <a href="/wiki/Product_topology" title="Product topology">product topology</a>. By the equivalence of norms in finite dimensions, a topologically equivalent metric is obtained if N is the <a href="/wiki/Taxicab_norm" class="mw-redirect" title="Taxicab norm">taxicab norm</a>, a <a href="/wiki/Norm_(mathematics)#p-norm" title="Norm (mathematics)">p-norm</a>, the <a href="/wiki/Maximum_norm" class="mw-redirect" title="Maximum norm">maximum norm</a>, or any other norm which is non-decreasing as the coordinates of a positive n-tuple increase (yielding the triangle inequality). Similarly, a metric on the topological product of countably many metric spaces can be obtained using the metric <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d(x,y)=\sum _{i=1}^{\infty }{\frac {1}{2^{i}}}{\frac {d_{i}(x_{i},y_{i})}{1+d_{i}(x_{i},y_{i})}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mrow> <mn>1</mn> <mo>+</mo> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d(x,y)=\sum _{i=1}^{\infty }{\frac {1}{2^{i}}}{\frac {d_{i}(x_{i},y_{i})}{1+d_{i}(x_{i},y_{i})}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/56a5f182da6193316b6f30d0a233e308e75b77ae" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:30.589ex; height:6.843ex;" alt="{\displaystyle d(x,y)=\sum _{i=1}^{\infty }{\frac {1}{2^{i}}}{\frac {d_{i}(x_{i},y_{i})}{1+d_{i}(x_{i},y_{i})}}.}"> The topological product of uncountably many metric spaces need not be metrizable. For example, an uncountable product of copies of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }"> is not <a href="/wiki/First-countable_space" title="First-countable space">first-countable</a> and thus is not metrizable. <div class="mw-heading mw-heading3"><h3 id="Quotient_metric_spaces">Quotient metric spaces</h3>[<a href="/w/index.php?title=Metric_space&action=edit&section=34" title="Edit section: Quotient metric spaces">edit</a>]</div> If M is a metric space with metric d, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sim }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∼</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sim }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/afcc42adfcfdc24d5c4c474869e5d8eaa78d1173" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.307ex; margin-bottom: -0.478ex; width:1.808ex; height:1.343ex;" alt="{\displaystyle \sim }"> is an <a href="/wiki/Equivalence_relation" title="Equivalence relation">equivalence relation</a> on M, then we can endow the quotient set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M/\!\sim }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mspace width="negativethinmathspace" /> <mo>∼</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M/\!\sim }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ff61e9b8e0301cd76db0a7491c2ef7ce60377d26" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.671ex; height:2.843ex;" alt="{\displaystyle M/\!\sim }"> with a pseudometric. The distance between two equivalence classes <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [x]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mi>x</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [x]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/07548563c21e128890501e14eb7c80ee2d6fda4d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.623ex; height:2.843ex;" alt="{\displaystyle [x]}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [y]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mi>y</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [y]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f637a17dac262ee0fc8d58e31d08ca3ebe5a0fed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.449ex; height:2.843ex;" alt="{\displaystyle [y]}"> is defined as <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d'([x],[y])=\inf\{d(p_{1},q_{1})+d(p_{2},q_{2})+\dotsb +d(p_{n},q_{n})\},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>d</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mo stretchy="false">[</mo> <mi>x</mi> <mo stretchy="false">]</mo> <mo>,</mo> <mo stretchy="false">[</mo> <mi>y</mi> <mo stretchy="false">]</mo> <mo stretchy="false">)</mo> <mo>=</mo> <mo movablelimits="true" form="prefix">inf</mo> <mo fence="false" stretchy="false">{</mo> <mi>d</mi> <mo stretchy="false">(</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>+</mo> <mi>d</mi> <mo stretchy="false">(</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <mi>d</mi> <mo stretchy="false">(</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">}</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d'([x],[y])=\inf\{d(p_{1},q_{1})+d(p_{2},q_{2})+\dotsb +d(p_{n},q_{n})\},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c12948e3aa74a8510139fedcafd35cc2844a73d4" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:55.235ex; height:3.009ex;" alt="{\displaystyle d'([x],[y])=\inf\{d(p_{1},q_{1})+d(p_{2},q_{2})+\dotsb +d(p_{n},q_{n})\},}"> where the <a href="/wiki/Infimum" class="mw-redirect" title="Infimum">infimum</a> is taken over all finite sequences <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (p_{1},p_{2},\dots ,p_{n})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (p_{1},p_{2},\dots ,p_{n})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ad0bb27c095bc998293f9ef21c05a574c16b7100" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.857ex; height:2.843ex;" alt="{\displaystyle (p_{1},p_{2},\dots ,p_{n})}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (q_{1},q_{2},\dots ,q_{n})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (q_{1},q_{2},\dots ,q_{n})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/18cf72f1465acdf17cda70842b2b98f6e2a586b7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.46ex; height:2.843ex;" alt="{\displaystyle (q_{1},q_{2},\dots ,q_{n})}"> with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{1}\sim x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>∼</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p_{1}\sim x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5d74c5d4ca8283557ba3b1d7a1148249f74342fc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:6.741ex; height:2.009ex;" alt="{\displaystyle p_{1}\sim x}">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q_{n}\sim y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>∼</mo> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q_{n}\sim y}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c64a72e4d52f27cb0394f85c8640e129130ba8a2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.509ex; height:2.009ex;" alt="{\displaystyle q_{n}\sim y}">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q_{i}\sim p_{i+1},i=1,2,\dots ,n-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>∼</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mi>i</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q_{i}\sim p_{i+1},i=1,2,\dots ,n-1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8b0d2b1e0e9a088d4bdea5565a8d3d4b2d348c1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:27.874ex; height:2.509ex;" alt="{\displaystyle q_{i}\sim p_{i+1},i=1,2,\dots ,n-1}">.<a href="#cite_note-FOOTNOTEBuragoBuragoIvanov2001Definition_3.1.12-32">[29]</a> In general this will only define a <a href="/wiki/Pseudometric_space" title="Pseudometric space">pseudometric</a>, i.e. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d'([x],[y])=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>d</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mo stretchy="false">[</mo> <mi>x</mi> <mo stretchy="false">]</mo> <mo>,</mo> <mo stretchy="false">[</mo> <mi>y</mi> <mo stretchy="false">]</mo> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d'([x],[y])=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4f0686af91a21fdf1ed41059fa39ffbbe883bd4f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.079ex; height:3.009ex;" alt="{\displaystyle d'([x],[y])=0}"> does not necessarily imply that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [x]=[y]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mi>x</mi> <mo stretchy="false">]</mo> <mo>=</mo> <mo stretchy="false">[</mo> <mi>y</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [x]=[y]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9c34866e2a1216182b168dc1e272b0b02f1aab3a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.171ex; height:2.843ex;" alt="{\displaystyle [x]=[y]}">. However, for some equivalence relations (e.g., those given by gluing together polyhedra along faces), <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>d</mi> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d'}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f310a68106a9e308bdaf887ff8f7171c4cb9d96" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.903ex; height:2.509ex;" alt="{\displaystyle d'}"> is a metric. The quotient metric <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>d</mi> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d'}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f310a68106a9e308bdaf887ff8f7171c4cb9d96" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.903ex; height:2.509ex;" alt="{\displaystyle d'}"> is characterized by the following <a href="/wiki/Universal_property" title="Universal property">universal property</a>. If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\,\colon (M,d)\to (X,\delta )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mspace width="thinmathspace" /> <mo>:</mo> <mo stretchy="false">(</mo> <mi>M</mi> <mo>,</mo> <mi>d</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mi>δ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f\,\colon (M,d)\to (X,\delta )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/482f907df9256fc335a774b302ca80749a5acbc3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.687ex; height:2.843ex;" alt="{\displaystyle f\,\colon (M,d)\to (X,\delta )}"> is a metric (i.e. 1-Lipschitz) map between metric spaces satisfying f(x) = f(y) whenever <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\sim y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>∼</mo> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\sim y}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bbd1014d850b7c883eb76301dd58c643e3c7e4eb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.584ex; height:2.009ex;" alt="{\displaystyle x\sim y}">, then the induced function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {f}}\,\colon {M/\sim }\to X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo accent="false">¯</mo> </mover> </mrow> <mspace width="thinmathspace" /> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mo>∼</mo> </mrow> <mo stretchy="false">→</mo> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {f}}\,\colon {M/\sim }\to X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4d963c71b05c8393d1d331379f1811830d39748b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.557ex; height:3.509ex;" alt="{\displaystyle {\overline {f}}\,\colon {M/\sim }\to X}">, given by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {f}}([x])=f(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo accent="false">¯</mo> </mover> </mrow> <mo stretchy="false">(</mo> <mo stretchy="false">[</mo> <mi>x</mi> <mo stretchy="false">]</mo> <mo stretchy="false">)</mo> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {f}}([x])=f(x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aa40e344823d6033e5b82eaf6f31f3a562c6c9c6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.433ex; height:3.509ex;" alt="{\displaystyle {\overline {f}}([x])=f(x)}">, is a metric map <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {f}}\,\colon (M/\sim ,d')\to (X,\delta ).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo accent="false">¯</mo> </mover> </mrow> <mspace width="thinmathspace" /> <mo>:</mo> <mo stretchy="false">(</mo> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mo>∼</mo> <mo>,</mo> <msup> <mi>d</mi> <mo>′</mo> </msup> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mi>δ</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {f}}\,\colon (M/\sim ,d')\to (X,\delta ).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bea19e71e4a1f532560303795db7d528f88fd24b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.842ex; height:3.509ex;" alt="{\displaystyle {\overline {f}}\,\colon (M/\sim ,d')\to (X,\delta ).}"> The quotient metric does not always induce the <a href="/wiki/Quotient_topology" class="mw-redirect" title="Quotient topology">quotient topology</a>. For example, the topological quotient of the metric space <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {N} \times [0,1]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> <mo>×</mo> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {N} \times [0,1]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0c5ec3356c30166cf0b83c98cb406ee5517cb93c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.171ex; height:2.843ex;" alt="{\displaystyle \mathbb {N} \times [0,1]}"> identifying all points of the form <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (n,0)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>n</mi> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (n,0)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5fd8a7c3a302914ba5ae7cac4d8df11b59943934" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.4ex; height:2.843ex;" alt="{\displaystyle (n,0)}"> is not metrizable since it is not <a href="/wiki/First-countable_space" title="First-countable space">first-countable</a>, but the quotient metric is a well-defined metric on the same set which induces a <a href="/wiki/Comparison_of_topologies" title="Comparison of topologies">coarser topology</a>. Moreover, different metrics on the original topological space (a disjoint union of countably many intervals) lead to different topologies on the quotient.<a href="#cite_note-33">[30]</a> A topological space is <a href="/wiki/Sequential_space" title="Sequential space">sequential</a> if and only if it is a (topological) quotient of a metric space.<a href="#cite_note-34">[31]</a> <div class="mw-heading mw-heading2"><h2 id="Generalizations_of_metric_spaces">Generalizations of metric spaces</h2>[<a href="/w/index.php?title=Metric_space&action=edit&section=35" title="Edit section: Generalizations of metric spaces">edit</a>]</div> There are several notions of spaces which have less structure than a metric space, but more than a topological space. <ul><li><a href="/wiki/Uniform_space" title="Uniform space">Uniform spaces</a> are spaces in which distances are not defined, but uniform continuity is.</li> <li><a href="/wiki/Approach_space" title="Approach space">Approach spaces</a> are spaces in which point-to-set distances are defined, instead of point-to-point distances. They have particularly good properties from the point of view of <a href="/wiki/Category_theory" title="Category theory">category theory</a>.</li> <li><a href="/wiki/Continuity_space" class="mw-redirect" title="Continuity space">Continuity spaces</a> are a generalization of metric spaces and <a href="/wiki/Poset" class="mw-redirect" title="Poset">posets</a> that can be used to unify the notions of metric spaces and <a href="/wiki/Domain_theory" title="Domain theory">domains</a>.</li></ul> There are also numerous ways of relaxing the axioms for a metric, giving rise to various notions of generalized metric spaces. These generalizations can also be combined. The terminology used to describe them is not completely standardized. Most notably, in <a href="/wiki/Functional_analysis" title="Functional analysis">functional analysis</a> pseudometrics often come from <a href="/wiki/Seminorm" title="Seminorm">seminorms</a> on vector spaces, and so it is natural to call them "semimetrics". This conflicts with the use of the term in <a href="/wiki/Topology" title="Topology">topology</a>. <div class="mw-heading mw-heading3"><h3 id="Extended_metrics">Extended metrics</h3>[<a href="/w/index.php?title=Metric_space&action=edit&section=36" title="Edit section: Extended metrics">edit</a>]</div> Some authors define metrics so as to allow the distance function d to attain the value ∞, i.e. distances are non-negative numbers on the <a href="/wiki/Extended_real_number_line" title="Extended real number line">extended real number line</a>.<a href="#cite_note-FOOTNOTEBuragoBuragoIvanov20011-4">[4]</a> Such a function is also called an extended metric or "∞-metric". Every extended metric can be replaced by a real-valued metric that is topologically equivalent. This can be done using a <a href="/wiki/Subadditive_function" class="mw-redirect" title="Subadditive function">subadditive</a> monotonically increasing bounded function which is zero at zero, e.g. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d'(x,y)=d(x,y)/(1+d(x,y))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>d</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>d</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mi>d</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d'(x,y)=d(x,y)/(1+d(x,y))}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d15fa446f7d2925af23c4c3b6c0fdd77b794d863" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:30.393ex; height:3.009ex;" alt="{\displaystyle d'(x,y)=d(x,y)/(1+d(x,y))}"> or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d''(x,y)=\min(1,d(x,y))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>d</mi> <mo>″</mo> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo movablelimits="true" form="prefix">min</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mi>d</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d''(x,y)=\min(1,d(x,y))}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f898bd846d889bbc8f4806bdf9baa57a55fbe07" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.207ex; height:3.009ex;" alt="{\displaystyle d''(x,y)=\min(1,d(x,y))}">. <div class="mw-heading mw-heading3"><h3 id="Metrics_valued_in_structures_other_than_the_real_numbers">Metrics valued in structures other than the real numbers</h3>[<a href="/w/index.php?title=Metric_space&action=edit&section=37" title="Edit section: Metrics valued in structures other than the real numbers">edit</a>]</div> The requirement that the metric take values in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [0,\infty )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mi mathvariant="normal">∞</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [0,\infty )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8dc2d914c2df66bc0f7893bfb8da36766650fe47" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.072ex; height:2.843ex;" alt="{\displaystyle [0,\infty )}"> can be relaxed to consider metrics with values in other structures, including: <ul><li><a href="/wiki/Ordered_field" title="Ordered field">Ordered fields</a>, yielding the notion of a <a href="/wiki/Generalised_metric" title="Generalised metric">generalised metric</a>.</li> <li>More general <a href="/wiki/Directed_set" title="Directed set">directed sets</a>. In the absence of an addition operation, the triangle inequality does not make sense and is replaced with an <a href="/wiki/Ultrametric_space" title="Ultrametric space">ultrametric inequality</a>. This leads to the notion of a generalized ultrametric.<a href="#cite_note-FOOTNOTEHitzlerSeda2016Definition_4.3.1-35">[32]</a></li></ul> These generalizations still induce a <a href="/wiki/Uniform_space" title="Uniform space">uniform structure</a> on the space. <div class="mw-heading mw-heading3"><h3 id="Pseudometrics">Pseudometrics</h3>[<a href="/w/index.php?title=Metric_space&action=edit&section=38" title="Edit section: Pseudometrics">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Pseudometric_space" title="Pseudometric space">Pseudometric space</a></div> A pseudometric on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"> is a function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d:X\times X\to \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo>:</mo> <mi>X</mi> <mo>×</mo> <mi>X</mi> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d:X\times X\to \mathbb {R} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cb82929b490b4868d886af0b7fd5c2bb00e13da0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:15.246ex; height:2.176ex;" alt="{\displaystyle d:X\times X\to \mathbb {R} }"> which satisfies the axioms for a metric, except that instead of the second (identity of indiscernibles) only <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d(x,x)=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d(x,x)=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f8b0b392db0a4abf6c47dcd5b35f90592401e756" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.979ex; height:2.843ex;" alt="{\displaystyle d(x,x)=0}"> for all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"> is required.<a href="#cite_note-FOOTNOTEHitzlerSeda2016Definition_4.2.1-36">[33]</a> In other words, the axioms for a pseudometric are: <ol><li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d(x,y)\geq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>≥</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d(x,y)\geq 0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1309cec878e6d9effccd8a2d2ed065a8e82bfa82" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.805ex; height:2.843ex;" alt="{\displaystyle d(x,y)\geq 0}"></li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d(x,x)=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d(x,x)=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f8b0b392db0a4abf6c47dcd5b35f90592401e756" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.979ex; height:2.843ex;" alt="{\displaystyle d(x,x)=0}"></li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d(x,y)=d(y,x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>d</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo>,</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d(x,y)=d(y,x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c7fea33d0e60116abd16287351eb6bf142a61fdd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.187ex; height:2.843ex;" alt="{\displaystyle d(x,y)=d(y,x)}"></li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d(x,z)\leq d(x,y)+d(y,z)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>≤</mo> <mi>d</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>d</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d(x,z)\leq d(x,y)+d(y,z)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/43ae751284c2944886e1effbfe4e0c1293f98419" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.263ex; height:2.843ex;" alt="{\displaystyle d(x,z)\leq d(x,y)+d(y,z)}">.</li></ol> In some contexts, pseudometrics are referred to as semimetrics<a href="#cite_note-FOOTNOTEBuragoBuragoIvanov2001Definition_1.1.4-37">[34]</a> because of their relation to <a href="/wiki/Seminorm" title="Seminorm">seminorms</a>. <div class="mw-heading mw-heading3"><h3 id="Quasimetrics">Quasimetrics</h3>[<a href="/w/index.php?title=Metric_space&action=edit&section=39" title="Edit section: Quasimetrics">edit</a>]</div> Occasionally, a quasimetric is defined as a function that satisfies all axioms for a metric with the possible exception of symmetry.<a href="#cite_note-38">[35]</a> The name of this generalisation is not entirely standardized.<a href="#cite_note-39">[36]</a> <ol><li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d(x,y)\geq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>≥</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d(x,y)\geq 0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1309cec878e6d9effccd8a2d2ed065a8e82bfa82" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.805ex; height:2.843ex;" alt="{\displaystyle d(x,y)\geq 0}"></li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d(x,y)=0\iff x=y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> <mspace width="thickmathspace" /> <mo stretchy="false">⟺</mo> <mspace width="thickmathspace" /> <mi>x</mi> <mo>=</mo> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d(x,y)=0\iff x=y}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/32fd7ce57db3ea42e1bbb4aabb703cb709c6bfec" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.286ex; height:2.843ex;" alt="{\displaystyle d(x,y)=0\iff x=y}"></li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d(x,z)\leq d(x,y)+d(y,z)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>≤</mo> <mi>d</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>d</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d(x,z)\leq d(x,y)+d(y,z)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/43ae751284c2944886e1effbfe4e0c1293f98419" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.263ex; height:2.843ex;" alt="{\displaystyle d(x,z)\leq d(x,y)+d(y,z)}"></li></ol> Quasimetrics are common in real life. For example, given a set X of mountain villages, the typical walking times between elements of X form a quasimetric because travel uphill takes longer than travel downhill. Another example is the <a href="/wiki/Taxicab_geometry" title="Taxicab geometry">length of car rides</a> in a city with one-way streets: here, a shortest path from point A to point B goes along a different set of streets than a shortest path from B to A and may have a different length. A quasimetric on the reals can be defined by setting <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d(x,y)={\begin{cases}x-y&{\text{if }}x\geq y,\\1&{\text{otherwise.}}\end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mi>x</mi> <mo>−</mo> <mi>y</mi> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>if </mtext> </mrow> <mi>x</mi> <mo>≥</mo> <mi>y</mi> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>otherwise.</mtext> </mrow> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d(x,y)={\begin{cases}x-y&{\text{if }}x\geq y,\\1&{\text{otherwise.}}\end{cases}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ae2c899c4739466aaa1c2e9aef2f0cd38f581fd5" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:30.01ex; height:6.176ex;" alt="{\displaystyle d(x,y)={\begin{cases}x-y&{\text{if }}x\geq y,\\1&{\text{otherwise.}}\end{cases}}}"> The 1 may be replaced, for example, by infinity or by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1+{\sqrt {y-x}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>y</mi> <mo>−</mo> <mi>x</mi> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1+{\sqrt {y-x}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/46c0a8f19b8e4a2bbcec11b99f3873001e41c1e7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:11.652ex; height:3.509ex;" alt="{\displaystyle 1+{\sqrt {y-x}}}"> or any other <a href="/wiki/Subadditivity" title="Subadditivity">subadditive</a> function of y-x. This quasimetric describes the cost of modifying a metal stick: it is easy to reduce its size by <a href="/wiki/Filing_(metalworking)" title="Filing (metalworking)">filing it down</a>, but it is difficult or impossible to grow it. Given a quasimetric on X, one can define an R-ball around x to be the set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{y\in X|d(x,y)\leq R\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mi>y</mi> <mo>∈</mo> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>d</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>≤</mo> <mi>R</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{y\in X|d(x,y)\leq R\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1871b0f2881bfac168b70b5e0674bf62d5f1382e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.355ex; height:2.843ex;" alt="{\displaystyle \{y\in X|d(x,y)\leq R\}}">. As in the case of a metric, such balls form a basis for a topology on X, but this topology need not be metrizable. For example, the topology induced by the quasimetric on the reals described above is the (reversed) <a href="/wiki/Sorgenfrey_line" class="mw-redirect" title="Sorgenfrey line">Sorgenfrey line</a>. <div class="mw-heading mw-heading3"><h3 id="Metametrics_or_partial_metrics">Metametrics or partial metrics</h3>[<a href="/w/index.php?title=Metric_space&action=edit&section=40" title="Edit section: Metametrics or partial metrics">edit</a>]</div> In a metametric, all the axioms of a metric are satisfied except that the distance between identical points is not necessarily zero. In other words, the axioms for a metametric are: <ol><li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d(x,y)\geq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>≥</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d(x,y)\geq 0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1309cec878e6d9effccd8a2d2ed065a8e82bfa82" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.805ex; height:2.843ex;" alt="{\displaystyle d(x,y)\geq 0}"></li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d(x,y)=0\implies x=y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> <mspace width="thickmathspace" /> <mo stretchy="false">⟹</mo> <mspace width="thickmathspace" /> <mi>x</mi> <mo>=</mo> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d(x,y)=0\implies x=y}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e9cb5aefcc6aef0b554176319ffde03fcc806576" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.775ex; height:2.843ex;" alt="{\displaystyle d(x,y)=0\implies x=y}"></li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d(x,y)=d(y,x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>d</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo>,</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d(x,y)=d(y,x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c7fea33d0e60116abd16287351eb6bf142a61fdd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.187ex; height:2.843ex;" alt="{\displaystyle d(x,y)=d(y,x)}"></li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d(x,z)\leq d(x,y)+d(y,z).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>≤</mo> <mi>d</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>d</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d(x,z)\leq d(x,y)+d(y,z).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1e7072502a79725a25151d5c96b7310e7dbb97a3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.91ex; height:2.843ex;" alt="{\displaystyle d(x,z)\leq d(x,y)+d(y,z).}"></li></ol> Metametrics appear in the study of <a href="/wiki/%CE%94-hyperbolic_space" class="mw-redirect" title="Δ-hyperbolic space">Gromov hyperbolic metric spaces</a> and their boundaries. The visual metametric on such a space satisfies <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d(x,x)=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d(x,x)=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f8b0b392db0a4abf6c47dcd5b35f90592401e756" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.979ex; height:2.843ex;" alt="{\displaystyle d(x,x)=0}"> for points <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"> on the boundary, but otherwise <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d(x,x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d(x,x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b16cf40c69ad9e073e7a5acd8fc8cc19d85e4e9f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.718ex; height:2.843ex;" alt="{\displaystyle d(x,x)}"> is approximately the distance from <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"> to the boundary. Metametrics were first defined by Jussi Väisälä.<a href="#cite_note-FOOTNOTEVäisälä2005-40">[37]</a> In other work, a function satisfying these axioms is called a partial metric<a href="#cite_note-41">[38]</a><a href="#cite_note-42">[39]</a> or a dislocated metric.<a href="#cite_note-FOOTNOTEHitzlerSeda2016Definition_4.2.1-36">[33]</a> <div class="mw-heading mw-heading3"><h3 id="Semimetrics">Semimetrics</h3>[<a href="/w/index.php?title=Metric_space&action=edit&section=41" title="Edit section: Semimetrics">edit</a>]</div> A semimetric on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"> is a function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d:X\times X\to \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo>:</mo> <mi>X</mi> <mo>×</mo> <mi>X</mi> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d:X\times X\to \mathbb {R} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cb82929b490b4868d886af0b7fd5c2bb00e13da0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:15.246ex; height:2.176ex;" alt="{\displaystyle d:X\times X\to \mathbb {R} }"> that satisfies the first three axioms, but not necessarily the triangle inequality: <ol><li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d(x,y)\geq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>≥</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d(x,y)\geq 0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1309cec878e6d9effccd8a2d2ed065a8e82bfa82" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.805ex; height:2.843ex;" alt="{\displaystyle d(x,y)\geq 0}"></li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d(x,y)=0\iff x=y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> <mspace width="thickmathspace" /> <mo stretchy="false">⟺</mo> <mspace width="thickmathspace" /> <mi>x</mi> <mo>=</mo> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d(x,y)=0\iff x=y}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/32fd7ce57db3ea42e1bbb4aabb703cb709c6bfec" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.286ex; height:2.843ex;" alt="{\displaystyle d(x,y)=0\iff x=y}"></li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d(x,y)=d(y,x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>d</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo>,</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d(x,y)=d(y,x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c7fea33d0e60116abd16287351eb6bf142a61fdd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.187ex; height:2.843ex;" alt="{\displaystyle d(x,y)=d(y,x)}"></li></ol> Some authors work with a weaker form of the triangle inequality, such as: <dl><dd><table> <tbody><tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d(x,z)\leq \rho \,(d(x,y)+d(y,z))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>≤</mo> <mi>ρ</mi> <mspace width="thinmathspace" /> <mo stretchy="false">(</mo> <mi>d</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>d</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d(x,z)\leq \rho \,(d(x,y)+d(y,z))}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/49a778e6ae464194322fa243bae534354c804d1a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.661ex; height:2.843ex;" alt="{\displaystyle d(x,z)\leq \rho \,(d(x,y)+d(y,z))}"> </td> <td>ρ-relaxed triangle inequality </td></tr> <tr> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d(x,z)\leq \rho \,\max\{d(x,y),d(y,z)\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>≤</mo> <mi>ρ</mi> <mspace width="thinmathspace" /> <mo movablelimits="true" form="prefix">max</mo> <mo fence="false" stretchy="false">{</mo> <mi>d</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mi>d</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d(x,z)\leq \rho \,\max\{d(x,y),d(y,z)\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f065a63a4edf244f4187c3ddf8457d343abeb207" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:32.083ex; height:2.843ex;" alt="{\displaystyle d(x,z)\leq \rho \,\max\{d(x,y),d(y,z)\}}"> </td> <td>ρ-inframetric inequality </td></tr></tbody></table></dd></dl> The ρ-inframetric inequality implies the ρ-relaxed triangle inequality (assuming the first axiom), and the ρ-relaxed triangle inequality implies the 2ρ-inframetric inequality. Semimetrics satisfying these equivalent conditions have sometimes been referred to as quasimetrics,<a href="#cite_note-FOOTNOTEXia2009-43">[40]</a> nearmetrics<a href="#cite_note-FOOTNOTEXia2008-44">[41]</a> or inframetrics.<a href="#cite_note-FOOTNOTEFraigniaudLebharViennot2008-45">[42]</a> The ρ-inframetric inequalities were introduced to model <a href="/wiki/Round-trip_delay_time" class="mw-redirect" title="Round-trip delay time">round-trip delay times</a> in the <a href="/wiki/Internet" title="Internet">internet</a>.<a href="#cite_note-FOOTNOTEFraigniaudLebharViennot2008-45">[42]</a> The triangle inequality implies the 2-inframetric inequality, and the <a href="/wiki/Ultrametric_inequality" class="mw-redirect" title="Ultrametric inequality">ultrametric inequality</a> is exactly the 1-inframetric inequality. <div class="mw-heading mw-heading3"><h3 id="Premetrics">Premetrics</h3>[<a href="/w/index.php?title=Metric_space&action=edit&section=42" title="Edit section: Premetrics">edit</a>]</div> Relaxing the last three axioms leads to the notion of a premetric, i.e. a function satisfying the following conditions: <ol><li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d(x,y)\geq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>≥</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d(x,y)\geq 0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1309cec878e6d9effccd8a2d2ed065a8e82bfa82" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.805ex; height:2.843ex;" alt="{\displaystyle d(x,y)\geq 0}"></li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d(x,x)=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d(x,x)=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f8b0b392db0a4abf6c47dcd5b35f90592401e756" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.979ex; height:2.843ex;" alt="{\displaystyle d(x,x)=0}"></li></ol> This is not a standard term. Sometimes it is used to refer to other generalizations of metrics such as pseudosemimetrics<a href="#cite_note-FOOTNOTEBuldyginKozachenko2000-46">[43]</a> or pseudometrics;<a href="#cite_note-FOOTNOTEHelemskii2006-47">[44]</a> in translations of Russian books it sometimes appears as "prametric".<a href="#cite_note-48">[45]</a> A premetric that satisfies symmetry, i.e. a pseudosemimetric, is also called a distance.<a href="#cite_note-FOOTNOTEDezaLaurent1997-49">[46]</a> Any premetric gives rise to a topology as follows. For a positive real <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}">, the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}">-ball centered at a point <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"> is defined as <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B_{r}(p)=\{x|d(x,p)<r\}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>p</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>d</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>p</mi> <mo stretchy="false">)</mo> <mo><</mo> <mi>r</mi> <mo fence="false" stretchy="false">}</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B_{r}(p)=\{x|d(x,p)<r\}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a49056c2e48220a355f7943439027d76d834c5e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.469ex; height:2.843ex;" alt="{\displaystyle B_{r}(p)=\{x|d(x,p)<r\}.}"></dd></dl> A set is called open if for any point <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"> in the set there is an <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}">-ball centered at <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"> which is contained in the set. Every premetric space is a topological space, and in fact a <a href="/wiki/Sequential_space" title="Sequential space">sequential space</a>. In general, the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}">-balls themselves need not be open sets with respect to this topology. As for metrics, the distance between two sets <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}">, is defined as <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d(A,B)={\underset {x\in A,y\in B}{\inf }}d(x,y).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo stretchy="false">(</mo> <mi>A</mi> <mo>,</mo> <mi>B</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <munder> <mo form="prefix">inf</mo> <mrow> <mi>x</mi> <mo>∈</mo> <mi>A</mi> <mo>,</mo> <mi>y</mi> <mo>∈</mo> <mi>B</mi> </mrow> </munder> </mrow> <mi>d</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d(A,B)={\underset {x\in A,y\in B}{\inf }}d(x,y).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/df6af580f98b86475bd0e1e89dcdb3400a777204" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:24.743ex; height:4.343ex;" alt="{\displaystyle d(A,B)={\underset {x\in A,y\in B}{\inf }}d(x,y).}"></dd></dl> This defines a premetric on the <a href="/wiki/Power_set" title="Power set">power set</a> of a premetric space. If we start with a (pseudosemi-)metric space, we get a pseudosemimetric, i.e. a symmetric premetric. Any premetric gives rise to a <a href="/wiki/Preclosure_operator" title="Preclosure operator">preclosure operator</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle cl}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> <mi>l</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle cl}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/85b5926255332eae23b75cddefdb4ebdebcf8549" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.7ex; height:2.176ex;" alt="{\displaystyle cl}"> as follows: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle cl(A)=\{x|d(x,A)=0\}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> <mi>l</mi> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>d</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>A</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> <mo fence="false" stretchy="false">}</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle cl(A)=\{x|d(x,A)=0\}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/212250be7f636eb5f88c49060865102aeb22382d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.692ex; height:2.843ex;" alt="{\displaystyle cl(A)=\{x|d(x,A)=0\}.}"></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Pseudoquasimetrics">Pseudoquasimetrics</h3>[<a href="/w/index.php?title=Metric_space&action=edit&section=43" title="Edit section: Pseudoquasimetrics">edit</a>]</div> The prefixes pseudo-, quasi- and semi- can also be combined, e.g., a pseudoquasimetric (sometimes called hemimetric) relaxes both the indiscernibility axiom and the symmetry axiom and is simply a premetric satisfying the triangle inequality. For pseudoquasimetric spaces the open <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}">-balls form a basis of open sets. A very basic example of a pseudoquasimetric space is the set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{0,1\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{0,1\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/28de5781698336d21c9c560fb1cbb3fb406923eb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.684ex; height:2.843ex;" alt="{\displaystyle \{0,1\}}"> with the premetric given by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d(0,1)=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d(0,1)=1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a0431794d4df3e6afdf278f8c68ba11013fb83e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.645ex; height:2.843ex;" alt="{\displaystyle d(0,1)=1}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d(1,0)=0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d(1,0)=0.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cc226aba7f6599af4c67aee18615edbc6a0d4c5d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.292ex; height:2.843ex;" alt="{\displaystyle d(1,0)=0.}"> The associated topological space is the <a href="/wiki/Sierpi%C5%84ski_space" title="Sierpiński space">Sierpiński space</a>. Sets equipped with an extended pseudoquasimetric were studied by <a href="/wiki/William_Lawvere" title="William Lawvere">William Lawvere</a> as "generalized metric spaces".<a href="#cite_note-50">[47]</a> From a <a href="/wiki/Category_theory" title="Category theory">categorical</a> point of view, the extended pseudometric spaces and the extended pseudoquasimetric spaces, along with their corresponding nonexpansive maps, are the best behaved of the <a href="/wiki/Category_of_metric_spaces" title="Category of metric spaces">metric space categories</a>. One can take arbitrary products and coproducts and form quotient objects within the given category. If one drops "extended", one can only take finite products and coproducts. If one drops "pseudo", one cannot take quotients. Lawvere also gave an alternate definition of such spaces as <a href="/wiki/Enriched_category" title="Enriched category">enriched categories</a>. The ordered set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\mathbb {R} ,\geq )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo>,</mo> <mo>≥</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\mathbb {R} ,\geq )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/17c6ead871e941d70ad70f69def2e5133bca8590" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.329ex; height:2.843ex;" alt="{\displaystyle (\mathbb {R} ,\geq )}"> can be seen as a <a href="/wiki/Category_(mathematics)" title="Category (mathematics)">category</a> with one <a href="/wiki/Morphism" title="Morphism">morphism</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\to b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo stretchy="false">→</mo> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\to b}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7989b7da84c001dc910a97023b8fcdcfd8f97353" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.841ex; height:2.176ex;" alt="{\displaystyle a\to b}"> if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\geq b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>≥</mo> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\geq b}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ed5d3957d5f94566507526017e4ebb67c02efe81" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.326ex; height:2.343ex;" alt="{\displaystyle a\geq b}"> and none otherwise. Using + as the <a href="/wiki/Tensor_product" title="Tensor product">tensor product</a> and 0 as the <a href="/wiki/Identity_element" title="Identity element">identity</a> makes this category into a <a href="/wiki/Monoidal_category" title="Monoidal category">monoidal category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R^{*}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R^{*}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7a74b1bc0fa98794b6460254044f8e7b75e6d84f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.818ex; height:2.343ex;" alt="{\displaystyle R^{*}}">. Every (extended pseudoquasi-)metric space <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (M,d)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>M</mi> <mo>,</mo> <mi>d</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (M,d)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d78e6f2ddf5baee227ee2a9f164726ba0c23c263" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.501ex; height:2.843ex;" alt="{\displaystyle (M,d)}"> can now be viewed as a category <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M^{*}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M^{*}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b6d760e72a9f5f578f8ba166127f0713d56dc589" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.553ex; height:2.343ex;" alt="{\displaystyle M^{*}}"> enriched over <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R^{*}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R^{*}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7a74b1bc0fa98794b6460254044f8e7b75e6d84f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.818ex; height:2.343ex;" alt="{\displaystyle R^{*}}">: <ul><li>The objects of the category are the points of M.</li> <li>For every pair of points x and y such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d(x,y)<\infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo><</mo> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d(x,y)<\infty }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/55dbdc92c82bde3972f218e6c6d346250d58438b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.966ex; height:2.843ex;" alt="{\displaystyle d(x,y)<\infty }">, there is a single morphism which is assigned the object <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d(x,y)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d(x,y)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3772957879a8bbf7946bddf5743c508a1d5072c0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.544ex; height:2.843ex;" alt="{\displaystyle d(x,y)}"> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R^{*}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R^{*}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7a74b1bc0fa98794b6460254044f8e7b75e6d84f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.818ex; height:2.343ex;" alt="{\displaystyle R^{*}}">.</li> <li>The triangle inequality and the fact that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d(x,x)=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d(x,x)=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f8b0b392db0a4abf6c47dcd5b35f90592401e756" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.979ex; height:2.843ex;" alt="{\displaystyle d(x,x)=0}"> for all points x derive from the properties of composition and identity in an enriched category.</li> <li>Since <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R^{*}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R^{*}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7a74b1bc0fa98794b6460254044f8e7b75e6d84f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.818ex; height:2.343ex;" alt="{\displaystyle R^{*}}"> is a poset, all <a href="/wiki/Diagram_(category_theory)" title="Diagram (category theory)">diagrams</a> that are required for an enriched category commute automatically.</li></ul> <div class="mw-heading mw-heading3"><h3 id="Metrics_on_multisets">Metrics on multisets</h3>[<a href="/w/index.php?title=Metric_space&action=edit&section=44" title="Edit section: Metrics on multisets">edit</a>]</div> The notion of a metric can be generalized from a distance between two elements to a number assigned to a multiset of elements. A <a href="/wiki/Multiset" title="Multiset">multiset</a> is a generalization of the notion of a <a href="/wiki/Set_(mathematics)" title="Set (mathematics)">set</a> in which an element can occur more than once. Define the multiset union <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U=XY}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> <mo>=</mo> <mi>X</mi> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U=XY}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/49a53edef3399ec8e4fff563c04f6a1a96f84563" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.634ex; height:2.176ex;" alt="{\displaystyle U=XY}"> as follows: if an element x occurs m times in X and n times in Y then it occurs m + n times in U. A function d on the set of nonempty finite multisets of elements of a set M is a metric<a href="#cite_note-FOOTNOTEVitányi2011-51">[48]</a> if <ol><li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d(X)=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d(X)=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1cb585e11d376b2037d9fd710ba5419653a5a8db" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.266ex; height:2.843ex;" alt="{\displaystyle d(X)=0}"> if all elements of X are equal and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d(X)>0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d(X)>0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c8521704ffbbd4390e7bb5eb7bc4452569a345f5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.266ex; height:2.843ex;" alt="{\displaystyle d(X)>0}"> otherwise (<a href="/wiki/Positive_definiteness" title="Positive definiteness">positive definiteness</a>)</li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d(X)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d(X)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5220998dbbd96cbcc2e4fa85026a1925313ff55e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.005ex; height:2.843ex;" alt="{\displaystyle d(X)}"> depends only on the (unordered) multiset X (<a href="/wiki/Symmetry" title="Symmetry">symmetry</a>)</li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d(XY)\leq d(XZ)+d(ZY)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mi>Y</mi> <mo stretchy="false">)</mo> <mo>≤</mo> <mi>d</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mi>Z</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>d</mi> <mo stretchy="false">(</mo> <mi>Z</mi> <mi>Y</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d(XY)\leq d(XZ)+d(ZY)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7b62bb85435fe2c989eca6f9ed62683c0040f054" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.882ex; height:2.843ex;" alt="{\displaystyle d(XY)\leq d(XZ)+d(ZY)}"> (<a href="/wiki/Triangle_inequality" title="Triangle inequality">triangle inequality</a>)</li></ol> By considering the cases of axioms 1 and 2 in which the multiset X has two elements and the case of axiom 3 in which the multisets X, Y, and Z have one element each, one recovers the usual axioms for a metric. That is, every multiset metric yields an ordinary metric when restricted to sets of two elements. A simple example is the set of all nonempty finite multisets <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"> of integers with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d(X)=\max(X)-\min(X)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo movablelimits="true" form="prefix">max</mo> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mo movablelimits="true" form="prefix">min</mo> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d(X)=\max(X)-\min(X)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/232bf79cf04a896880e6ffad329137068c930a60" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.724ex; height:2.843ex;" alt="{\displaystyle d(X)=\max(X)-\min(X)}">. More complex examples are <a href="/wiki/Information_distance" title="Information distance">information distance</a> in multisets;<a href="#cite_note-FOOTNOTEVitányi2011-51">[48]</a> and <a href="/wiki/Normalized_compression_distance" title="Normalized compression distance">normalized compression distance</a> (NCD) in multisets.<a href="#cite_note-FOOTNOTECohenVitányi2012-52">[49]</a> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2>[<a href="/w/index.php?title=Metric_space&action=edit&section=45" title="Edit section: See also">edit</a>]</div> <ul><li><a href="/wiki/Acoustic_metric" title="Acoustic metric">Acoustic metric</a> – Tensor characterizing signal-carrying properties in a medium</li> <li><a href="/wiki/Complete_metric_space" title="Complete metric space">Complete metric space</a> – Metric geometry</li> <li><a href="/wiki/Diversity_(mathematics)" title="Diversity (mathematics)">Diversity (mathematics)</a> – Generalization of metric spaces</li> <li><a href="/wiki/Generalized_metric_space" title="Generalized metric space">Generalized metric space</a></li> <li><a href="/wiki/Hilbert%27s_fourth_problem" title="Hilbert's fourth problem">Hilbert's fourth problem</a> – Construct all metric spaces where lines resemble those on a sphere</li> <li><a href="/wiki/Metric_tree" title="Metric tree">Metric tree</a></li> <li><a href="/wiki/Minkowski_distance" title="Minkowski distance">Minkowski distance</a> – Mathematical metric in normed vector space</li> <li><a href="/wiki/Signed_distance_function" title="Signed distance function">Signed distance function</a> – Distance from a point to the boundary of a set</li> <li><a href="/wiki/Similarity_measure" title="Similarity measure">Similarity measure</a> – Real-valued function that quantifies similarity between two objects</li> <li><a href="/wiki/Space_(mathematics)" title="Space (mathematics)">Space (mathematics)</a> – Mathematical set with some added structure</li> <li><a href="/wiki/Ultrametric_space" title="Ultrametric space">Ultrametric space</a> – Type of metric space</li></ul> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2>[<a href="/w/index.php?title=Metric_space&action=edit&section=46" title="Edit section: Notes">edit</a>]</div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist reflist-lower-alpha"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-12"><a href="#cite_ref-12">^</a> Balls with rational radius around a point x form a <a href="/wiki/Neighborhood_basis" class="mw-redirect" title="Neighborhood basis">neighborhood basis</a> for that point. </li> <li id="cite_note-13"><a href="#cite_ref-13">^</a> In the context of <a href="/wiki/Interval_(mathematics)" title="Interval (mathematics)">intervals</a> in the real line, or more generally regions in Euclidean space, bounded sets are sometimes referred to as "finite intervals" or "finite regions". However, they do not typically have a finite number of elements, and while they all have finite <a href="/wiki/Lebesgue_measure" title="Lebesgue measure">volume</a>, so do many unbounded sets. Therefore this terminology is imprecise. </li> <li id="cite_note-22"><a href="#cite_ref-22">^</a> This differs from usage in <a href="/wiki/Riemannian_geometry" title="Riemannian geometry">Riemannian geometry</a>, where geodesics are only locally shortest paths. Some authors define geodesics in metric spaces in the same way.<a href="#cite_note-FOOTNOTEBuragoBuragoIvanov2001Definition_2.5.27-20">[18]</a><a href="#cite_note-FOOTNOTEGromov2007Definition_1.9-21">[19]</a> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="Citations">Citations</h2>[<a href="/w/index.php?title=Metric_space&action=edit&section=47" title="Edit section: Citations">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239543626"><div class="reflist"> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-FOOTNOTEČech196942-1"><a href="#cite_ref-FOOTNOTEČech196942_1-0">^</a> <a href="#CITEREFČech1969">Čech 1969</a>, p. 42. </li> <li id="cite_note-FOOTNOTEBuragoBuragoIvanov2001-2"><a href="#cite_ref-FOOTNOTEBuragoBuragoIvanov2001_2-0">^</a> <a href="#CITEREFBuragoBuragoIvanov2001">Burago, Burago & Ivanov 2001</a>. </li> <li id="cite_note-FOOTNOTEHeinonen2001-3"><a href="#cite_ref-FOOTNOTEHeinonen2001_3-0">^</a> <a href="#CITEREFHeinonen2001">Heinonen 2001</a>. </li> <li id="cite_note-FOOTNOTEBuragoBuragoIvanov20011-4">^ <a href="#cite_ref-FOOTNOTEBuragoBuragoIvanov20011_4-0">a</a> <a href="#cite_ref-FOOTNOTEBuragoBuragoIvanov20011_4-1">b</a> <a href="#CITEREFBuragoBuragoIvanov2001">Burago, Burago & Ivanov 2001</a>, p. 1. </li> <li id="cite_note-FOOTNOTEGromov2007xv-5"><a href="#cite_ref-FOOTNOTEGromov2007xv_5-0">^</a> <a href="#CITEREFGromov2007">Gromov 2007</a>, p. xv. </li> <li id="cite_note-6"><a href="#cite_ref-6">^</a> <style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFGleason1991" class="citation book cs1">Gleason, Andrew (1991). Fundamentals of Abstract Analysis (1st ed.). <a href="/wiki/Taylor_%26_Francis" title="Taylor & Francis">Taylor & Francis</a>. p. 223. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1201%2F9781315275444">10.1201/9781315275444</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9781315275444" title="Special:BookSources/9781315275444"><bdi>9781315275444</bdi></a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:62222843">62222843</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Fundamentals+of+Abstract+Analysis&rft.pages=223&rft.edition=1st&rft.pub=Taylor+%26+Francis&rft.date=1991&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A62222843%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1201%2F9781315275444&rft.isbn=9781315275444&rft.aulast=Gleason&rft.aufirst=Andrew&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMetric+space" class="Z3988"> </li> <li id="cite_note-7"><a href="#cite_ref-7">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFréchet1906" class="citation journal cs1">Fréchet, M. 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Rendiconti del Circolo Matematico di Palermo. 22 (1): 1–72. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF03018603">10.1007/BF03018603</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:123251660">123251660</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Rendiconti+del+Circolo+Matematico+di+Palermo&rft.atitle=Sur+quelques+points+du+calcul+fonctionnel&rft.volume=22&rft.issue=1&rft.pages=1-72&rft.date=1906-12&rft_id=info%3Adoi%2F10.1007%2FBF03018603&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A123251660%23id-name%3DS2CID&rft.aulast=Fr%C3%A9chet&rft.aufirst=M.&rft_id=https%3A%2F%2Fzenodo.org%2Frecord%2F1428464&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMetric+space" class="Z3988"> </li> <li id="cite_note-8"><a href="#cite_ref-8">^</a> F. Hausdorff (1914) Grundzuge der Mengenlehre </li> <li id="cite_note-9"><a href="#cite_ref-9">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBlumberg1927" class="citation journal cs1">Blumberg, Henry (1927). <a rel="nofollow" class="external text" href="https://doi.org/10.1090%2FS0002-9904-1920-03378-1">"Hausdorff's Grundzüge der Mengenlehre"</a>. Bulletin of the American Mathematical Society. 6: 778–781. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1090%2FS0002-9904-1920-03378-1">10.1090/S0002-9904-1920-03378-1</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Bulletin+of+the+American+Mathematical+Society&rft.atitle=Hausdorff%27s+Grundz%C3%BCge+der+Mengenlehre&rft.volume=6&rft.pages=778-781&rft.date=1927&rft_id=info%3Adoi%2F10.1090%2FS0002-9904-1920-03378-1&rft.aulast=Blumberg&rft.aufirst=Henry&rft_id=https%3A%2F%2Fdoi.org%2F10.1090%252FS0002-9904-1920-03378-1&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMetric+space" class="Z3988"> </li> <li id="cite_note-10"><a href="#cite_ref-10">^</a> Mohamed A. Khamsi & William A. Kirk (2001) Introduction to Metric Spaces and Fixed Point Theory, page 14, <a href="/wiki/John_Wiley_%26_Sons" class="mw-redirect" title="John Wiley & Sons">John Wiley & Sons</a> </li> <li id="cite_note-11"><a href="#cite_ref-11">^</a> Rudin, Mary Ellen. <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2035708">A new proof that metric spaces are paracompact</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20160412015215/http://www.jstor.org/stable/2035708">Archived</a> 2016-04-12 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a>. Proceedings of the American Mathematical Society, Vol. 20, No. 2. (Feb., 1969), p. 603. </li> <li id="cite_note-FOOTNOTEBuragoBuragoIvanov20012-14"><a href="#cite_ref-FOOTNOTEBuragoBuragoIvanov20012_14-0">^</a> <a href="#CITEREFBuragoBuragoIvanov2001">Burago, Burago & Ivanov 2001</a>, p. 2. </li> <li id="cite_note-15"><a href="#cite_ref-15">^</a> <a href="#CITEREFBuragoBuragoIvanov2001">Burago, Burago & Ivanov 2001</a>, p. 2. Some authors refer to any distance-preserving function as an isometry, e.g. <a href="#CITEREFMunkres2000">Munkres 2000</a>, p. 181. </li> <li id="cite_note-FOOTNOTEGromov2007xvii-16"><a href="#cite_ref-FOOTNOTEGromov2007xvii_16-0">^</a> <a href="#CITEREFGromov2007">Gromov 2007</a>, p. xvii. </li> <li id="cite_note-FOOTNOTEMargalitThomas2017-17">^ <a href="#cite_ref-FOOTNOTEMargalitThomas2017_17-0">a</a> <a href="#cite_ref-FOOTNOTEMargalitThomas2017_17-1">b</a> <a href="#cite_ref-FOOTNOTEMargalitThomas2017_17-2">c</a> <a href="#CITEREFMargalitThomas2017">Margalit & Thomas 2017</a>. </li> <li id="cite_note-FOOTNOTENariciBeckenstein201147–66-18"><a href="#cite_ref-FOOTNOTENariciBeckenstein201147–66_18-0">^</a> <a href="#CITEREFNariciBeckenstein2011">Narici & Beckenstein 2011</a>, pp. 47–66. </li> <li id="cite_note-FOOTNOTEBuragoBuragoIvanov2001Definition_2.3.1-19"><a href="#cite_ref-FOOTNOTEBuragoBuragoIvanov2001Definition_2.3.1_19-0">^</a> <a href="#CITEREFBuragoBuragoIvanov2001">Burago, Burago & Ivanov 2001</a>, Definition 2.3.1. </li> <li id="cite_note-FOOTNOTEBuragoBuragoIvanov2001Definition_2.5.27-20"><a href="#cite_ref-FOOTNOTEBuragoBuragoIvanov2001Definition_2.5.27_20-0">^</a> <a href="#CITEREFBuragoBuragoIvanov2001">Burago, Burago & Ivanov 2001</a>, Definition 2.5.27. </li> <li id="cite_note-FOOTNOTEGromov2007Definition_1.9-21"><a href="#cite_ref-FOOTNOTEGromov2007Definition_1.9_21-0">^</a> <a href="#CITEREFGromov2007">Gromov 2007</a>, Definition 1.9. </li> <li id="cite_note-FOOTNOTEBuragoBuragoIvanov2001127-23"><a href="#cite_ref-FOOTNOTEBuragoBuragoIvanov2001127_23-0">^</a> <a href="#CITEREFBuragoBuragoIvanov2001">Burago, Burago & Ivanov 2001</a>, p. 127. </li> <li id="cite_note-FOOTNOTEHeinonen2007191-24"><a href="#cite_ref-FOOTNOTEHeinonen2007191_24-0">^</a> <a href="#CITEREFHeinonen2007">Heinonen 2007</a>, p. 191. </li> <li id="cite_note-25"><a href="#cite_ref-25">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGigli2018" class="citation journal cs1">Gigli, Nicola (2018-10-18). "Lecture notes on differential calculus on RCD spaces". 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"On lipschitz embedding of finite metric spaces in Hilbert space". <a href="/wiki/Israel_Journal_of_Mathematics" title="Israel Journal of Mathematics">Israel Journal of Mathematics</a>. 52 (1–2): 46–52. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF02776078">10.1007/BF02776078</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:121649019">121649019</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Israel+Journal+of+Mathematics&rft.atitle=On+lipschitz+embedding+of+finite+metric+spaces+in+Hilbert+space&rft.volume=52&rft.issue=1%E2%80%932&rft.pages=46-52&rft.date=1985&rft_id=info%3Adoi%2F10.1007%2FBF02776078&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A121649019%23id-name%3DS2CID&rft.aulast=Bourgain&rft.aufirst=J.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMetric+space" class="Z3988"> </li> <li id="cite_note-28"><a href="#cite_ref-28">^</a> <a href="/wiki/Ji%C5%99%C3%AD_Matou%C5%A1ek_(mathematician)" title="Jiří Matoušek (mathematician)">Jiří Matoušek</a> and <a href="/wiki/Assaf_Naor" title="Assaf Naor">Assaf Naor</a>, ed. <a rel="nofollow" class="external text" href="http://kam.mff.cuni.cz/~matousek/metrop.ps">"Open problems on embeddings of finite metric spaces"</a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20101226232112/http://kam.mff.cuni.cz/~matousek/metrop.ps">Archived</a> 2010-12-26 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a>. </li> <li id="cite_note-FOOTNOTEÓ_Searcóid2006107-29"><a href="#cite_ref-FOOTNOTEÓ_Searcóid2006107_29-0">^</a> <a href="#CITEREFÓ_Searcóid2006">Ó Searcóid 2006</a>, p. 107. </li> <li id="cite_note-30"><a href="#cite_ref-30">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGottliebSolomon2014" class="citation conference cs1">Gottlieb, Lee-Ad; Solomon, Shay (2014-06-08). Light spanners for snowflake metrics. SOCG '14: Proceedings of the thirtieth annual symposium on Computational geometry. pp. 387–395. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/1401.5014">1401.5014</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1145%2F2582112.2582140">10.1145/2582112.2582140</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=conference&rft.btitle=Light+spanners+for+snowflake+metrics&rft.pages=387-395&rft.date=2014-06-08&rft_id=info%3Aarxiv%2F1401.5014&rft_id=info%3Adoi%2F10.1145%2F2582112.2582140&rft.aulast=Gottlieb&rft.aufirst=Lee-Ad&rft.au=Solomon%2C+Shay&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMetric+space" class="Z3988"> </li> <li id="cite_note-31"><a href="#cite_ref-31">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRobinsonFoulds1981" class="citation journal cs1">Robinson, D.F.; Foulds, L.R. (February 1981). <a rel="nofollow" class="external text" href="https://linkinghub.elsevier.com/retrieve/pii/0025556481900432">"Comparison of phylogenetic trees"</a>. Mathematical Biosciences. 53 (1–2): 131–147. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2F0025-5564%2881%2990043-2">10.1016/0025-5564(81)90043-2</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:121156920">121156920</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Mathematical+Biosciences&rft.atitle=Comparison+of+phylogenetic+trees&rft.volume=53&rft.issue=1%E2%80%932&rft.pages=131-147&rft.date=1981-02&rft_id=info%3Adoi%2F10.1016%2F0025-5564%2881%2990043-2&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A121156920%23id-name%3DS2CID&rft.aulast=Robinson&rft.aufirst=D.F.&rft.au=Foulds%2C+L.R.&rft_id=https%3A%2F%2Flinkinghub.elsevier.com%2Fretrieve%2Fpii%2F0025556481900432&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMetric+space" class="Z3988"> </li> <li id="cite_note-FOOTNOTEBuragoBuragoIvanov2001Definition_3.1.12-32"><a href="#cite_ref-FOOTNOTEBuragoBuragoIvanov2001Definition_3.1.12_32-0">^</a> <a href="#CITEREFBuragoBuragoIvanov2001">Burago, Burago & Ivanov 2001</a>, Definition 3.1.12. </li> <li id="cite_note-33"><a href="#cite_ref-33">^</a> See <a href="#CITEREFBuragoBuragoIvanov2001">Burago, Burago & Ivanov 2001</a>, Example 3.1.17, although in this book the quotient <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {N} \times [0,1]/\mathbb {N} \times \{0\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> <mo>×</mo> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> <mo>×</mo> <mo fence="false" stretchy="false">{</mo> <mn>0</mn> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {N} \times [0,1]/\mathbb {N} \times \{0\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/164e745a9cdb6d724c627cfb69a25fc21f4c7902" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.339ex; height:2.843ex;" alt="{\displaystyle \mathbb {N} \times [0,1]/\mathbb {N} \times \{0\}}"> is incorrectly claimed to be homeomorphic to the topological quotient. </li> <li id="cite_note-34"><a href="#cite_ref-34">^</a> Goreham, Anthony. <a rel="nofollow" class="external text" href="http://at.yorku.ca/p/a/a/o/51.pdf">Sequential convergence in Topological Spaces</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20110604232111/http://at.yorku.ca/p/a/a/o/51.pdf">Archived</a> 2011-06-04 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a>. Honours' Dissertation, Queen's College, Oxford (April, 2001), p. 14 </li> <li id="cite_note-FOOTNOTEHitzlerSeda2016Definition_4.3.1-35"><a href="#cite_ref-FOOTNOTEHitzlerSeda2016Definition_4.3.1_35-0">^</a> <a href="#CITEREFHitzlerSeda2016">Hitzler & Seda 2016</a>, Definition 4.3.1. </li> <li id="cite_note-FOOTNOTEHitzlerSeda2016Definition_4.2.1-36">^ <a href="#cite_ref-FOOTNOTEHitzlerSeda2016Definition_4.2.1_36-0">a</a> <a href="#cite_ref-FOOTNOTEHitzlerSeda2016Definition_4.2.1_36-1">b</a> <a href="#CITEREFHitzlerSeda2016">Hitzler & Seda 2016</a>, Definition 4.2.1. </li> <li id="cite_note-FOOTNOTEBuragoBuragoIvanov2001Definition_1.1.4-37"><a href="#cite_ref-FOOTNOTEBuragoBuragoIvanov2001Definition_1.1.4_37-0">^</a> <a href="#CITEREFBuragoBuragoIvanov2001">Burago, Burago & Ivanov 2001</a>, Definition 1.1.4. </li> <li id="cite_note-38"><a href="#cite_ref-38">^</a> <a href="#CITEREFSteenSeebach1995">Steen & Seebach (1995)</a>; <a href="#CITEREFSmyth1988">Smyth (1988)</a> </li> <li id="cite_note-39"><a href="#cite_ref-39">^</a> <a href="#CITEREFRolewicz1987">Rolewicz (1987)</a> calls them "semimetrics". That same term is also frequently used for two other generalizations of metrics. </li> <li id="cite_note-FOOTNOTEVäisälä2005-40"><a href="#cite_ref-FOOTNOTEVäisälä2005_40-0">^</a> <a href="#CITEREFVäisälä2005">Väisälä 2005</a>. </li> <li id="cite_note-41"><a href="#cite_ref-41">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://www.dcs.warwick.ac.uk/pmetric/">"Partial metrics: welcome"</a>. www.dcs.warwick.ac.uk. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20170727003912/http://www.dcs.warwick.ac.uk/pmetric/">Archived</a> from the original on 2017-07-27. Retrieved 2018-05-02.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=www.dcs.warwick.ac.uk&rft.atitle=Partial+metrics%3A+welcome&rft_id=http%3A%2F%2Fwww.dcs.warwick.ac.uk%2Fpmetric%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMetric+space" class="Z3988"> </li> <li id="cite_note-42"><a href="#cite_ref-42">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBukatinKoppermanMatthewsPajoohesh2009" class="citation journal cs1">Bukatin, Michael; Kopperman, Ralph; Matthews, Steve; Pajoohesh, Homeira (2009-10-01). <a rel="nofollow" class="external text" href="https://www.dcs.warwick.ac.uk/pmetric/monthly708-718.pdf">"Partial Metric Spaces"</a> (PDF). American Mathematical Monthly. 116 (8): 708–718. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.4169%2F193009709X460831">10.4169/193009709X460831</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:13969183">13969183</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=American+Mathematical+Monthly&rft.atitle=Partial+Metric+Spaces&rft.volume=116&rft.issue=8&rft.pages=708-718&rft.date=2009-10-01&rft_id=info%3Adoi%2F10.4169%2F193009709X460831&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A13969183%23id-name%3DS2CID&rft.aulast=Bukatin&rft.aufirst=Michael&rft.au=Kopperman%2C+Ralph&rft.au=Matthews%2C+Steve&rft.au=Pajoohesh%2C+Homeira&rft_id=https%3A%2F%2Fwww.dcs.warwick.ac.uk%2Fpmetric%2Fmonthly708-718.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMetric+space" class="Z3988"> </li> <li id="cite_note-FOOTNOTEXia2009-43"><a href="#cite_ref-FOOTNOTEXia2009_43-0">^</a> <a href="#CITEREFXia2009">Xia 2009</a>. </li> <li id="cite_note-FOOTNOTEXia2008-44"><a href="#cite_ref-FOOTNOTEXia2008_44-0">^</a> <a href="#CITEREFXia2008">Xia 2008</a>. </li> <li id="cite_note-FOOTNOTEFraigniaudLebharViennot2008-45">^ <a href="#cite_ref-FOOTNOTEFraigniaudLebharViennot2008_45-0">a</a> <a href="#cite_ref-FOOTNOTEFraigniaudLebharViennot2008_45-1">b</a> <a href="#CITEREFFraigniaudLebharViennot2008">Fraigniaud, Lebhar & Viennot 2008</a>. </li> <li id="cite_note-FOOTNOTEBuldyginKozachenko2000-46"><a href="#cite_ref-FOOTNOTEBuldyginKozachenko2000_46-0">^</a> <a href="#CITEREFBuldyginKozachenko2000">Buldygin & Kozachenko 2000</a>. </li> <li id="cite_note-FOOTNOTEHelemskii2006-47"><a href="#cite_ref-FOOTNOTEHelemskii2006_47-0">^</a> <a href="#CITEREFHelemskii2006">Helemskii 2006</a>. </li> <li id="cite_note-48"><a href="#cite_ref-48">^</a> <a href="#CITEREFArkhangel'skiiPontryagin1990">Arkhangel'skii & Pontryagin (1990)</a>; <a href="#CITEREFAldrovandiPereira2017">Aldrovandi & Pereira (2017)</a> </li> <li id="cite_note-FOOTNOTEDezaLaurent1997-49"><a href="#cite_ref-FOOTNOTEDezaLaurent1997_49-0">^</a> <a href="#CITEREFDezaLaurent1997">Deza & Laurent 1997</a>. </li> <li id="cite_note-50"><a href="#cite_ref-50">^</a> <a href="#CITEREFLawvere1973">Lawvere (1973)</a>; <a href="#CITEREFVickers2005">Vickers (2005)</a> </li> <li id="cite_note-FOOTNOTEVitányi2011-51">^ <a href="#cite_ref-FOOTNOTEVitányi2011_51-0">a</a> <a href="#cite_ref-FOOTNOTEVitányi2011_51-1">b</a> <a href="#CITEREFVitányi2011">Vitányi 2011</a>. </li> <li id="cite_note-FOOTNOTECohenVitányi2012-52"><a href="#cite_ref-FOOTNOTECohenVitányi2012_52-0">^</a> <a href="#CITEREFCohenVitányi2012">Cohen & Vitányi 2012</a>. </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2>[<a href="/w/index.php?title=Metric_space&action=edit&section=48" title="Edit section: References">edit</a>]</div> <style data-mw-deduplicate="TemplateStyles:r1239549316">.mw-parser-output .refbegin{margin-bottom:0.5em}.mw-parser-output .refbegin-hanging-indents>ul{margin-left:0}.mw-parser-output .refbegin-hanging-indents>ul>li{margin-left:0;padding-left:3.2em;text-indent:-3.2em}.mw-parser-output .refbegin-hanging-indents ul,.mw-parser-output .refbegin-hanging-indents ul li{list-style:none}@media(max-width:720px){.mw-parser-output .refbegin-hanging-indents>ul>li{padding-left:1.6em;text-indent:-1.6em}}.mw-parser-output .refbegin-columns{margin-top:0.3em}.mw-parser-output .refbegin-columns ul{margin-top:0}.mw-parser-output .refbegin-columns li{page-break-inside:avoid;break-inside:avoid-column}@media screen{.mw-parser-output .refbegin{font-size:90%}}</style><div class="refbegin" style=""> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAldrovandiPereira2017" class="citation cs2">Aldrovandi, Ruben; Pereira, José Geraldo (2017), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=NWhtDQAAQBAJ&pg=PA20">An Introduction to Geometrical Physics</a> (2nd ed.), Hackensack, New Jersey: World Scientific, p. 20, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-981-3146-81-5" title="Special:BookSources/978-981-3146-81-5"><bdi>978-981-3146-81-5</bdi></a>, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=3561561">3561561</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=An+Introduction+to+Geometrical+Physics&rft.place=Hackensack%2C+New+Jersey&rft.pages=20&rft.edition=2nd&rft.pub=World+Scientific&rft.date=2017&rft.isbn=978-981-3146-81-5&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D3561561%23id-name%3DMR&rft.aulast=Aldrovandi&rft.aufirst=Ruben&rft.au=Pereira%2C+Jos%C3%A9+Geraldo&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DNWhtDQAAQBAJ%26pg%3DPA20&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMetric+space" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFArkhangel'skiiPontryagin1990" class="citation cs2"><a href="/wiki/Alexander_Arhangelskii" title="Alexander Arhangelskii">Arkhangel'skii, A. V.</a>; <a href="/wiki/Lev_Pontryagin" title="Lev Pontryagin">Pontryagin, L. S.</a> (1990), General Topology I: Basic Concepts and Constructions Dimension Theory, Encyclopaedia of Mathematical Sciences, <a href="/wiki/Springer_Science%2BBusiness_Media" title="Springer Science+Business Media">Springer</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/3-540-18178-4" title="Special:BookSources/3-540-18178-4"><bdi>3-540-18178-4</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=General+Topology+I%3A+Basic+Concepts+and+Constructions+Dimension+Theory&rft.series=Encyclopaedia+of+Mathematical+Sciences&rft.pub=Springer&rft.date=1990&rft.isbn=3-540-18178-4&rft.aulast=Arkhangel%27skii&rft.aufirst=A.+V.&rft.au=Pontryagin%2C+L.+S.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMetric+space" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBryant1985" class="citation book cs1">Bryant, Victor (1985). 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Topology (Second ed.). <a href="/wiki/Upper_Saddle_River,_NJ" class="mw-redirect" title="Upper Saddle River, NJ">Upper Saddle River, NJ</a>: <a href="/wiki/Prentice_Hall,_Inc" class="mw-redirect" title="Prentice Hall, Inc">Prentice Hall, Inc</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-13-181629-9" title="Special:BookSources/978-0-13-181629-9"><bdi>978-0-13-181629-9</bdi></a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/42683260">42683260</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Topology&rft.place=Upper+Saddle+River%2C+NJ&rft.edition=Second&rft.pub=Prentice+Hall%2C+Inc&rft.date=2000&rft_id=info%3Aoclcnum%2F42683260&rft.isbn=978-0-13-181629-9&rft.aulast=Munkres&rft.aufirst=James+R.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMetric+space" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFNariciBeckenstein2011" class="citation book cs2">Narici, Lawrence; Beckenstein, Edward (2011), Topological Vector Spaces, Pure and applied mathematics (Second ed.), Boca Raton, FL: CRC Press, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1584888666" title="Special:BookSources/978-1584888666"><bdi>978-1584888666</bdi></a>, <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/144216834">144216834</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Topological+Vector+Spaces&rft.place=Boca+Raton%2C+FL&rft.series=Pure+and+applied+mathematics&rft.edition=Second&rft.pub=CRC+Press&rft.date=2011&rft_id=info%3Aoclcnum%2F144216834&rft.isbn=978-1584888666&rft.aulast=Narici&rft.aufirst=Lawrence&rft.au=Beckenstein%2C+Edward&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMetric+space" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFÓ_Searcóid2006" class="citation book cs1">Ó Searcóid, Mícheál (2006). Metric spaces. London: Springer. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/1-84628-369-8" title="Special:BookSources/1-84628-369-8"><bdi>1-84628-369-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Metric+spaces&rft.place=London&rft.pub=Springer&rft.date=2006&rft.isbn=1-84628-369-8&rft.aulast=%C3%93+Searc%C3%B3id&rft.aufirst=M%C3%ADche%C3%A1l&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMetric+space" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPapadopoulos2014" class="citation book cs1">Papadopoulos, Athanase (2014). Metric spaces, convexity, and non-positive curvature (Second ed.). Zürich, Switzerland: <a href="/wiki/European_Mathematical_Society" title="European Mathematical Society">European Mathematical Society</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-03719-132-3" title="Special:BookSources/978-3-03719-132-3"><bdi>978-3-03719-132-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Metric+spaces%2C+convexity%2C+and+non-positive+curvature&rft.place=Z%C3%BCrich%2C+Switzerland&rft.edition=Second&rft.pub=European+Mathematical+Society&rft.date=2014&rft.isbn=978-3-03719-132-3&rft.aulast=Papadopoulos&rft.aufirst=Athanase&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMetric+space" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRolewicz1987" class="citation book cs1">Rolewicz, Stefan (1987). 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IEEE Transactions on Information Theory. 57 (4): 2451–2456. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/0905.3347">0905.3347</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1109%2FTIT.2011.2110130">10.1109/TIT.2011.2110130</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:6302496">6302496</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=IEEE+Transactions+on+Information+Theory&rft.atitle=Information+distance+in+multiples&rft.volume=57&rft.issue=4&rft.pages=2451-2456&rft.date=2011&rft_id=info%3Aarxiv%2F0905.3347&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A6302496%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1109%2FTIT.2011.2110130&rft.aulast=Vit%C3%A1nyi&rft.aufirst=Paul+M.+B.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMetric+space" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFVäisälä2005" class="citation journal cs1">Väisälä, Jussi (2005). <a rel="nofollow" class="external text" href="http://www.helsinki.fi/~jvaisala/grobok.pdf">"Gromov hyperbolic spaces"</a> (PDF). 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Theory and Applications of Categories. 14 (15): 328–356. <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=2182680">2182680</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Theory+and+Applications+of+Categories&rft.atitle=Localic+completion+of+generalized+metric+spaces%2C+I&rft.volume=14&rft.issue=15&rft.pages=328-356&rft.date=2005&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D2182680%23id-name%3DMR&rft.aulast=Vickers&rft.aufirst=Steven&rft_id=https%3A%2F%2Fwww.tac.mta.ca%2Ftac%2Fvolumes%2F14%2F15%2F14-15abs.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMetric+space" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWeisstein" class="citation web cs1"><a href="/wiki/Eric_W._Weisstein" title="Eric W. Weisstein">Weisstein, Eric W.</a> <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/ProductMetric.html">"Product Metric"</a>. <a href="/wiki/MathWorld" title="MathWorld">MathWorld</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=MathWorld&rft.atitle=Product+Metric&rft.au=Weisstein%2C+Eric+W.&rft_id=https%3A%2F%2Fmathworld.wolfram.com%2FProductMetric.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMetric+space" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFXia2008" class="citation journal cs1">Xia, Qinglan (2008). "The geodesic problem in nearmetric spaces". Journal of Geometric Analysis. 19 (2): 452–479. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/0807.3377">0807.3377</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2Fs12220-008-9065-4">10.1007/s12220-008-9065-4</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:17475581">17475581</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Geometric+Analysis&rft.atitle=The+geodesic+problem+in+nearmetric+spaces&rft.volume=19&rft.issue=2&rft.pages=452-479&rft.date=2008&rft_id=info%3Aarxiv%2F0807.3377&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A17475581%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1007%2Fs12220-008-9065-4&rft.aulast=Xia&rft.aufirst=Qinglan&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMetric+space" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFXia2009" class="citation journal cs1">Xia, Q. (2009). "The geodesic problem in quasimetric spaces". Journal of Geometric Analysis. 19 (2): 452–479. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/0807.3377">0807.3377</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2Fs12220-008-9065-4">10.1007/s12220-008-9065-4</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:17475581">17475581</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Geometric+Analysis&rft.atitle=The+geodesic+problem+in+quasimetric+spaces&rft.volume=19&rft.issue=2&rft.pages=452-479&rft.date=2009&rft_id=info%3Aarxiv%2F0807.3377&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A17475581%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1007%2Fs12220-008-9065-4&rft.aulast=Xia&rft.aufirst=Q.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMetric+space" class="Z3988"></li></ul> </div> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2>[<a href="/w/index.php?title=Metric_space&action=edit&section=49" title="Edit section: External links">edit</a>]</div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation cs2"><a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php?title=Metric_space">"Metric space"</a>, <a href="/wiki/Encyclopedia_of_Mathematics" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a>, <a href="/wiki/European_Mathematical_Society" title="European Mathematical Society">EMS Press</a>, 2001 [1994]</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Metric+space&rft.btitle=Encyclopedia+of+Mathematics&rft.pub=EMS+Press&rft.date=2001&rft_id=https%3A%2F%2Fwww.encyclopediaofmath.org%2Findex.php%3Ftitle%3DMetric_space&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMetric+space" class="Z3988"></li> <li><a rel="nofollow" class="external text" href="http://www.cut-the-knot.org/do_you_know/far_near.shtml">Far and near—several examples of distance functions</a> at <a href="/wiki/Cut-the-knot" class="mw-redirect" title="Cut-the-knot">cut-the-knot</a>.</li></ul> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output 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href="/wiki/Algebraic_topology" title="Algebraic topology">Algebraic</a></li> <li><a href="/wiki/Combinatorial_topology" title="Combinatorial topology">Combinatorial</a></li> <li><a href="/wiki/Continuum_(topology)" title="Continuum (topology)">Continuum</a></li> <li><a href="/wiki/Differential_topology" title="Differential topology">Differential</a></li> <li><a href="/wiki/Geometric_topology" title="Geometric topology">Geometric</a> <ul><li><a href="/wiki/Low-dimensional_topology" title="Low-dimensional topology">low-dimensional</a></li></ul></li> <li><a href="/wiki/Homology_(mathematics)" title="Homology (mathematics)">Homology</a> <ul><li><a href="/wiki/Cohomology" title="Cohomology">cohomology</a></li></ul></li> <li><a href="/wiki/Set-theoretic_topology" title="Set-theoretic topology">Set-theoretic</a></li> <li><a href="/wiki/Digital_topology" title="Digital topology">Digital</a></li></ul> </div></td><td class="noviewer navbox-image" rowspan="4" style="width:1px;padding:0 0 0 2px"><div><a href="/wiki/Klein_bottle" title="Klein bottle"><img alt="Computer graphics rendering of a Klein bottle" src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b9/Kleinsche_Flasche.png/60px-Kleinsche_Flasche.png" decoding="async" width="60" height="80" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b9/Kleinsche_Flasche.png/90px-Kleinsche_Flasche.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/b9/Kleinsche_Flasche.png/120px-Kleinsche_Flasche.png 2x" data-file-width="1171" data-file-height="1561" /></a></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;background:#e5e5ff;">Key concepts</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Open_set" title="Open set">Open set</a> / <a href="/wiki/Closed_set" title="Closed set">Closed set</a></li> <li><a href="/wiki/Interior_(topology)" title="Interior (topology)">Interior</a></li> <li><a href="/wiki/Continuity_(topology)" class="mw-redirect" title="Continuity (topology)">Continuity</a></li> <li><a href="/wiki/Topological_space" title="Topological space">Space</a> <ul><li><a href="/wiki/Compact_space" title="Compact space">compact</a></li> <li><a href="/wiki/Connected_space" title="Connected space">connected</a></li> <li><a href="/wiki/Hausdorff_space" title="Hausdorff space">Hausdorff</a></li> <li><a class="mw-selflink selflink">metric</a></li> <li><a href="/wiki/Uniform_space" title="Uniform space">uniform</a></li></ul></li> <li><a href="/wiki/Homotopy" title="Homotopy">Homotopy</a> <ul><li><a href="/wiki/Homotopy_group" title="Homotopy group">homotopy group</a></li> <li><a href="/wiki/Fundamental_group" title="Fundamental group">fundamental group</a></li></ul></li> <li><a href="/wiki/Simplicial_complex" title="Simplicial complex">Simplicial complex</a></li> <li><a href="/wiki/CW_complex" title="CW complex">CW complex</a></li> 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number</a></li> <li><a href="/wiki/Orientability" title="Orientability">Orientability</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;background:#e5e5ff;"><a href="/wiki/Category:Theorems_in_topology" title="Category:Theorems in topology">Key results</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Banach_fixed-point_theorem" title="Banach fixed-point theorem">Banach fixed-point theorem</a></li> <li><a href="/wiki/De_Rham_cohomology" title="De Rham cohomology">De Rham cohomology</a></li> <li><a href="/wiki/Invariance_of_domain" title="Invariance of domain">Invariance of domain</a></li> <li><a href="/wiki/Poincar%C3%A9_conjecture" title="Poincaré conjecture">Poincaré conjecture</a></li> <li><a href="/wiki/Tychonoff%27s_theorem" title="Tychonoff's theorem">Tychonoff's theorem</a></li> <li><a href="/wiki/Urysohn%27s_lemma" title="Urysohn's lemma">Urysohn's lemma</a></li></ul> </div></td></tr><tr><td class="navbox-abovebelow" colspan="3"><div> <ul><li><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/16px-Symbol_category_class.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/23px-Symbol_category_class.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/31px-Symbol_category_class.svg.png 2x" data-file-width="180" data-file-height="185" /> <a href="/wiki/Category:Topology" title="Category:Topology">Category</a></li> <li><a href="/wiki/File:Nuvola_apps_edu_mathematics_blue-p.svg" class="mw-file-description"><img alt="icon" src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/28px-Nuvola_apps_edu_mathematics_blue-p.svg.png" decoding="async" width="28" height="28" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/42px-Nuvola_apps_edu_mathematics_blue-p.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/56px-Nuvola_apps_edu_mathematics_blue-p.svg.png 2x" data-file-width="128" data-file-height="128" /></a> <a href="/wiki/Portal:Mathematics" title="Portal:Mathematics">Mathematics portal</a></li> <li><a href="/wiki/File:Wikibooks-logo.svg" class="mw-file-description" title="Wikibooks page"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Wikibooks-logo.svg/16px-Wikibooks-logo.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Wikibooks-logo.svg/24px-Wikibooks-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Wikibooks-logo.svg/32px-Wikibooks-logo.svg.png 2x" data-file-width="300" data-file-height="300" /></a> <a href="https://en.wikibooks.org/wiki/Topology" class="extiw" title="wikibooks:Topology">Wikibook</a></li> <li><a href="/wiki/File:Wikiversity_logo_2017.svg" class="mw-file-description" title="Wikiversity page"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/0/0b/Wikiversity_logo_2017.svg/16px-Wikiversity_logo_2017.svg.png" decoding="async" width="16" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/0b/Wikiversity_logo_2017.svg/24px-Wikiversity_logo_2017.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/0b/Wikiversity_logo_2017.svg/32px-Wikiversity_logo_2017.svg.png 2x" data-file-width="626" data-file-height="512" /></a> <a href="https://en.wikiversity.org/wiki/Topology" class="extiw" title="wikiversity:Topology">Wikiversity</a></li> <li><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/d/db/Symbol_list_class.svg/16px-Symbol_list_class.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/d/db/Symbol_list_class.svg/23px-Symbol_list_class.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/d/db/Symbol_list_class.svg/31px-Symbol_list_class.svg.png 2x" data-file-width="180" data-file-height="185" /> <a href="/wiki/List_of_topology_topics" title="List of topology topics">Topics</a> <ul><li><a href="/wiki/List_of_general_topology_topics" title="List of general topology topics">general</a></li> <li><a href="/wiki/List_of_algebraic_topology_topics" title="List of algebraic topology topics">algebraic</a></li> <li><a href="/wiki/List_of_geometric_topology_topics" title="List of geometric topology topics">geometric</a></li></ul></li> <li><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/d/db/Symbol_list_class.svg/16px-Symbol_list_class.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/d/db/Symbol_list_class.svg/23px-Symbol_list_class.svg.png 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