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<li id="toc-Examples_of_topological_spaces" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Examples_of_topological_spaces"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3</span> <span>Examples of topological spaces</span> </div> </a> <ul id="toc-Examples_of_topological_spaces-sublist" class="vector-toc-list"> <li id="toc-Discrete_and_trivial_topologies" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Discrete_and_trivial_topologies"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3.1</span> <span>Discrete and trivial topologies</span> </div> </a> <ul id="toc-Discrete_and_trivial_topologies-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Cofinite_and_cocountable_topologies" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Cofinite_and_cocountable_topologies"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3.2</span> <span>Cofinite and cocountable topologies</span> </div> </a> <ul id="toc-Cofinite_and_cocountable_topologies-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Topologies_on_the_real_and_complex_numbers" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Topologies_on_the_real_and_complex_numbers"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3.3</span> <span>Topologies on the real and complex numbers</span> </div> </a> <ul id="toc-Topologies_on_the_real_and_complex_numbers-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-The_metric_topology" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#The_metric_topology"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3.4</span> <span>The metric topology</span> </div> </a> <ul id="toc-The_metric_topology-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Further_examples" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Further_examples"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3.5</span> <span>Further examples</span> </div> </a> <ul id="toc-Further_examples-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> </ul> </li> <li id="toc-Continuous_functions" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Continuous_functions"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Continuous functions</span> </div> </a> <button aria-controls="toc-Continuous_functions-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Continuous functions subsection</span> </button> <ul id="toc-Continuous_functions-sublist" class="vector-toc-list"> <li id="toc-Alternative_definitions" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Alternative_definitions"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>Alternative definitions</span> </div> </a> <ul id="toc-Alternative_definitions-sublist" class="vector-toc-list"> <li id="toc-Neighborhood_definition" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Neighborhood_definition"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1.1</span> <span>Neighborhood definition</span> </div> </a> <ul id="toc-Neighborhood_definition-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Sequences_and_nets" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Sequences_and_nets"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1.2</span> <span>Sequences and nets</span> </div> </a> <ul id="toc-Sequences_and_nets-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Closure_operator_definition" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Closure_operator_definition"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1.3</span> <span>Closure operator definition</span> </div> </a> <ul id="toc-Closure_operator_definition-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Properties" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Properties"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>Properties</span> </div> </a> <ul id="toc-Properties-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Homeomorphisms" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Homeomorphisms"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.3</span> <span>Homeomorphisms</span> </div> </a> <ul id="toc-Homeomorphisms-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Defining_topologies_via_continuous_functions" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Defining_topologies_via_continuous_functions"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.4</span> <span>Defining topologies via continuous functions</span> </div> </a> <ul id="toc-Defining_topologies_via_continuous_functions-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Compact_sets" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Compact_sets"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Compact sets</span> </div> </a> <ul id="toc-Compact_sets-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Connected_sets" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Connected_sets"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Connected sets</span> </div> </a> <button aria-controls="toc-Connected_sets-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Connected sets subsection</span> </button> <ul id="toc-Connected_sets-sublist" class="vector-toc-list"> <li id="toc-Connected_components" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Connected_components"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.1</span> <span>Connected components</span> </div> </a> <ul id="toc-Connected_components-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Disconnected_spaces" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Disconnected_spaces"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.2</span> <span>Disconnected spaces</span> </div> </a> <ul id="toc-Disconnected_spaces-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Path-connected_sets" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Path-connected_sets"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.3</span> <span>Path-connected sets</span> </div> </a> <ul id="toc-Path-connected_sets-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Products_of_spaces" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Products_of_spaces"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Products of spaces</span> </div> </a> <ul id="toc-Products_of_spaces-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Separation_axioms" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Separation_axioms"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Separation axioms</span> </div> </a> <ul id="toc-Separation_axioms-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Countability_axioms" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Countability_axioms"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>Countability axioms</span> </div> </a> <ul id="toc-Countability_axioms-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Metric_spaces" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Metric_spaces"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>Metric spaces</span> </div> </a> <ul id="toc-Metric_spaces-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Baire_category_theorem" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Baire_category_theorem"> <div class="vector-toc-text"> <span class="vector-toc-numb">10</span> <span>Baire category theorem</span> </div> </a> <ul id="toc-Baire_category_theorem-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Main_areas_of_research" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Main_areas_of_research"> <div class="vector-toc-text"> <span class="vector-toc-numb">11</span> <span>Main areas of research</span> </div> </a> <button aria-controls="toc-Main_areas_of_research-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Main areas of research subsection</span> </button> <ul id="toc-Main_areas_of_research-sublist" class="vector-toc-list"> <li id="toc-Continuum_theory" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Continuum_theory"> <div class="vector-toc-text"> <span class="vector-toc-numb">11.1</span> <span>Continuum theory</span> </div> </a> <ul id="toc-Continuum_theory-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Dynamical_systems" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Dynamical_systems"> <div class="vector-toc-text"> <span class="vector-toc-numb">11.2</span> <span>Dynamical systems</span> </div> </a> <ul id="toc-Dynamical_systems-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Pointless_topology" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Pointless_topology"> <div class="vector-toc-text"> <span class="vector-toc-numb">11.3</span> <span>Pointless topology</span> </div> </a> <ul id="toc-Pointless_topology-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Dimension_theory" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Dimension_theory"> <div class="vector-toc-text"> <span class="vector-toc-numb">11.4</span> <span>Dimension theory</span> </div> </a> <ul id="toc-Dimension_theory-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Topological_algebras" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Topological_algebras"> <div class="vector-toc-text"> <span class="vector-toc-numb">11.5</span> <span>Topological algebras</span> </div> </a> <ul id="toc-Topological_algebras-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Metrizability_theory" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Metrizability_theory"> <div class="vector-toc-text"> <span class="vector-toc-numb">11.6</span> <span>Metrizability theory</span> </div> </a> <ul id="toc-Metrizability_theory-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Set-theoretic_topology" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Set-theoretic_topology"> <div class="vector-toc-text"> <span class="vector-toc-numb">11.7</span> <span>Set-theoretic topology</span> </div> </a> <ul id="toc-Set-theoretic_topology-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"> <span class="vector-toc-numb">12</span> <span>See also</span> </div> </a> <ul id="toc-See_also-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"> <span class="vector-toc-numb">13</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Further_reading" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Further_reading"> <div class="vector-toc-text"> <span class="vector-toc-numb">14</span> <span>Further reading</span> </div> </a> <ul id="toc-Further_reading-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"> <span class="vector-toc-numb">15</span> <span>External links</span> </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" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Toggle the table of contents</span> </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"><span class="mw-page-title-main">General topology</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Go to an article in another language. Available in 29 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-29" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">29 languages</span> </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/%D8%B7%D9%88%D8%A8%D9%88%D9%84%D9%88%D8%AC%D9%8A%D8%A7_%D8%B9%D8%A7%D9%85%D8%A9" title="طوبولوجيا عامة – Arabic" lang="ar" hreflang="ar" data-title="طوبولوجيا عامة" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://ast.wikipedia.org/wiki/Topolox%C3%ADa_xeneral" title="Topoloxía xeneral – Asturian" lang="ast" hreflang="ast" data-title="Topoloxía xeneral" data-language-autonym="Asturianu" data-language-local-name="Asturian" class="interlanguage-link-target"><span>Asturianu</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%9E%D0%B1%D1%89%D0%B0_%D1%82%D0%BE%D0%BF%D0%BE%D0%BB%D0%BE%D0%B3%D0%B8%D1%8F" title="Обща топология – Bulgarian" lang="bg" hreflang="bg" data-title="Обща топология" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Topologia_general" title="Topologia general – Catalan" lang="ca" hreflang="ca" data-title="Topologia general" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-de badge-Q70894304 mw-list-item" title=""><a href="https://de.wikipedia.org/wiki/Mengentheoretische_Topologie" title="Mengentheoretische Topologie – German" lang="de" hreflang="de" data-title="Mengentheoretische Topologie" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/%C3%9Cldtopoloogia" title="Üldtopoloogia – Estonian" lang="et" hreflang="et" data-title="Üldtopoloogia" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%93%CE%B5%CE%BD%CE%B9%CE%BA%CE%AE_%CF%84%CE%BF%CF%80%CE%BF%CE%BB%CE%BF%CE%B3%CE%AF%CE%B1" title="Γενική τοπολογία – Greek" lang="el" hreflang="el" data-title="Γενική τοπολογία" data-language-autonym="Ελληνικά" data-language-local-name="Greek" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Topolog%C3%ADa_general" title="Topología general – Spanish" lang="es" hreflang="es" data-title="Topología general" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Topologia_orokor" title="Topologia orokor – Basque" lang="eu" hreflang="eu" data-title="Topologia orokor" data-language-autonym="Euskara" data-language-local-name="Basque" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%AA%D9%88%D9%BE%D9%88%D9%84%D9%88%DA%98%DB%8C_%D8%B9%D9%85%D9%88%D9%85%DB%8C" title="توپولوژی عمومی – Persian" lang="fa" hreflang="fa" data-title="توپولوژی عمومی" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Topolox%C3%ADa_xeral" title="Topoloxía xeral – Galician" lang="gl" hreflang="gl" data-title="Topoloxía xeral" data-language-autonym="Galego" data-language-local-name="Galician" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%9D%BC%EB%B0%98%EC%9C%84%EC%83%81%EC%88%98%ED%95%99" title="일반위상수학 – Korean" lang="ko" hreflang="ko" data-title="일반위상수학" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Topologi_umum" title="Topologi umum – Indonesian" lang="id" hreflang="id" data-title="Topologi umum" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Topologia_generale" title="Topologia generale – Italian" lang="it" hreflang="it" data-title="Topologia generale" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%98%D7%95%D7%A4%D7%95%D7%9C%D7%95%D7%92%D7%99%D7%94_%D7%A7%D7%91%D7%95%D7%A6%D7%AA%D7%99%D7%AA" title="טופולוגיה קבוצתית – Hebrew" lang="he" hreflang="he" data-title="טופולוגיה קבוצתית" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Topologi_am" title="Topologi am – Malay" lang="ms" hreflang="ms" data-title="Topologi am" data-language-autonym="Bahasa Melayu" data-language-local-name="Malay" class="interlanguage-link-target"><span>Bahasa Melayu</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E4%BD%8D%E7%9B%B8%E7%A9%BA%E9%96%93%E8%AB%96" title="位相空間論 – Japanese" lang="ja" hreflang="ja" data-title="位相空間論" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-pa mw-list-item"><a href="https://pa.wikipedia.org/wiki/%E0%A8%86%E0%A8%AE_%E0%A8%9F%E0%A9%8C%E0%A8%AA%E0%A9%8C%E0%A8%B2%E0%A9%8C%E0%A8%9C%E0%A9%80" title="ਆਮ ਟੌਪੌਲੌਜੀ – Punjabi" lang="pa" hreflang="pa" data-title="ਆਮ ਟੌਪੌਲੌਜੀ" data-language-autonym="ਪੰਜਾਬੀ" data-language-local-name="Punjabi" class="interlanguage-link-target"><span>ਪੰਜਾਬੀ</span></a></li><li class="interlanguage-link interwiki-pnb mw-list-item"><a href="https://pnb.wikipedia.org/wiki/%D8%AC%D9%86%D8%B1%D9%84_%D9%B9%D9%88%D9%BE%D9%88%D9%84%D9%88%D8%AC%DB%8C" title="جنرل ٹوپولوجی – Western Punjabi" lang="pnb" hreflang="pnb" data-title="جنرل ٹوپولوجی" data-language-autonym="پنجابی" data-language-local-name="Western Punjabi" class="interlanguage-link-target"><span>پنجابی</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Topologia_geral" title="Topologia geral – Portuguese" lang="pt" hreflang="pt" data-title="Topologia geral" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%9E%D0%B1%D1%89%D0%B0%D1%8F_%D1%82%D0%BE%D0%BF%D0%BE%D0%BB%D0%BE%D0%B3%D0%B8%D1%8F" title="Общая топология – Russian" lang="ru" hreflang="ru" data-title="Общая топология" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/General_topology" title="General topology – Simple English" lang="en-simple" hreflang="en-simple" data-title="General topology" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/Op%C5%A1ta_topologija" title="Opšta topologija – Serbian" lang="sr" hreflang="sr" data-title="Opšta topologija" data-language-autonym="Српски / srpski" data-language-local-name="Serbian" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Yleinen_topologia" title="Yleinen topologia – Finnish" lang="fi" hreflang="fi" data-title="Yleinen topologia" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Genel_topoloji" title="Genel topoloji – Turkish" lang="tr" hreflang="tr" data-title="Genel topoloji" data-language-autonym="Türkçe" data-language-local-name="Turkish" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-uk 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decoding="async" width="420" height="300" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d2/Topologist%27s_sine_curve.svg/630px-Topologist%27s_sine_curve.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/d2/Topologist%27s_sine_curve.svg/840px-Topologist%27s_sine_curve.svg.png 2x" data-file-width="630" data-file-height="450" /></a><figcaption>The <a href="/wiki/Topologist%27s_sine_curve" title="Topologist's sine curve">Topologist's sine curve</a>, a useful example in point-set topology. It is connected but not path-connected.</figcaption></figure> <p>In <a href="/wiki/Mathematics" title="Mathematics">mathematics</a>, <b>general topology</b> (or <b>point set topology</b>) is the branch of <a href="/wiki/Topology" title="Topology">topology</a> that deals with the basic <a href="/wiki/Set_theory" title="Set theory">set-theoretic</a> definitions and constructions used in topology. It is the foundation of most other branches of topology, including <a href="/wiki/Differential_topology" title="Differential topology">differential topology</a>, <a href="/wiki/Geometric_topology" title="Geometric topology">geometric topology</a>, and <a href="/wiki/Algebraic_topology" title="Algebraic topology">algebraic topology</a>. </p><p>The fundamental concepts in point-set topology are <i>continuity</i>, <i>compactness</i>, and <i>connectedness</i>: </p> <ul><li><a href="/wiki/Continuous_function" title="Continuous function">Continuous functions</a>, intuitively, take nearby points to nearby points.</li> <li><a href="/wiki/Compact_set" class="mw-redirect" title="Compact set">Compact sets</a> are those that can be covered by finitely many sets of arbitrarily small size.</li> <li><a href="/wiki/Connected_set" class="mw-redirect" title="Connected set">Connected sets</a> are sets that cannot be divided into two pieces that are far apart.</li></ul> <p>The terms 'nearby', 'arbitrarily small', and 'far apart' can all be made precise by using the concept of <a href="/wiki/Open_set" title="Open set">open sets</a>. If we change the definition of 'open set', we change what continuous functions, compact sets, and connected sets are. Each choice of definition for 'open set' is called a <i>topology</i>. A set with a topology is called a <i><a href="/wiki/Topological_space" title="Topological space">topological space</a></i>. </p><p><i><a href="/wiki/Metric_space" title="Metric space">Metric spaces</a></i> are an important class of topological spaces where a real, non-negative distance, also called a <i><a href="/wiki/Metric_(mathematics)" class="mw-redirect" title="Metric (mathematics)">metric</a></i>, can be defined on pairs of points in the set. Having a metric simplifies many proofs, and many of the most common topological spaces are metric spaces. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="History">History</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=General_topology&action=edit&section=1" title="Edit section: History"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>General topology grew out of a number of areas, most importantly the following: </p> <ul><li>the detailed study of subsets of the <a href="/wiki/Real_line" class="mw-redirect" title="Real line">real line</a> (once known as the <i>topology of point sets</i>; this usage is now obsolete)</li> <li>the introduction of the <a href="/wiki/Manifold" title="Manifold">manifold</a> concept</li> <li>the study of <a href="/wiki/Metric_space" title="Metric space">metric spaces</a>, especially <a href="/wiki/Normed_linear_space" class="mw-redirect" title="Normed linear space">normed linear spaces</a>, in the early days of <a href="/wiki/Functional_analysis" title="Functional analysis">functional analysis</a>.</li></ul> <p>General topology assumed its present form around 1940. It captures, one might say, almost everything in the intuition of <a href="/wiki/Continuous_function" title="Continuous function">continuity</a>, in a technically adequate form that can be applied in any area of mathematics. </p> <div class="mw-heading mw-heading2"><h2 id="A_topology_on_a_set">A topology on a set</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=General_topology&action=edit&section=2" title="Edit section: A topology on a set"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></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/Topological_space" title="Topological space">Topological space</a></div> <p>Let <i>X</i> be a set and let <i>τ</i> be a <a href="/wiki/Family_of_sets" title="Family of sets">family</a> of <a href="/wiki/Subset" title="Subset">subsets</a> of <i>X</i>. Then <i>τ</i> is called a <i>topology on X</i> if:<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p> <ol><li>Both the <a href="/wiki/Empty_set" title="Empty set">empty set</a> and <i>X</i> are elements of <i>τ</i></li> <li>Any <a href="/wiki/Union_(set_theory)" title="Union (set theory)">union</a> of elements of <i>τ</i> is an element of <i>τ</i></li> <li>Any <a href="/wiki/Intersection_(set_theory)" title="Intersection (set theory)">intersection</a> of finitely many elements of <i>τ</i> is an element of <i>τ</i></li></ol> <p>If <i>τ</i> is a topology on <i>X</i>, then the pair (<i>X</i>, <i>τ</i>) is called a <i>topological space</i>. The notation <i>X<sub>τ</sub></i> may be used to denote a set <i>X</i> endowed with the particular topology <i>τ</i>. </p><p>The members of <i>τ</i> are called <i><a href="/wiki/Open_set" title="Open set">open sets</a></i> in <i>X</i>. A subset of <i>X</i> is said to be <a href="/wiki/Closed_set" title="Closed set">closed</a> if its <a href="/wiki/Complement_(set_theory)" title="Complement (set theory)">complement</a> is in <i>τ</i> (i.e., its complement is open). A subset of <i>X</i> may be open, closed, both (<a href="/wiki/Clopen_set" title="Clopen set">clopen set</a>), or neither. The empty set and <i>X</i> itself are always both closed and open. </p> <div class="mw-heading mw-heading3"><h3 id="Basis_for_a_topology">Basis for a topology</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=General_topology&action=edit&section=3" title="Edit section: Basis for a topology"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></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/Basis_(topology)" class="mw-redirect" title="Basis (topology)">Basis (topology)</a></div> <p>A <b>base</b> (or <b>basis</b>) <i>B</i> for a <a href="/wiki/Topological_space" title="Topological space">topological space</a> <i>X</i> with <a href="/wiki/Topological_space" title="Topological space">topology</a> <i>T</i> is a collection of <a href="/wiki/Open_set" title="Open set">open sets</a> in <i>T</i> such that every open set in <i>T</i> can be written as a union of elements of <i>B</i>.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> We say that the base <i>generates</i> the topology <i>T</i>. Bases are useful because many properties of topologies can be reduced to statements about a base that generates that topology—and because many topologies are most easily defined in terms of a base that generates them. </p> <div class="mw-heading mw-heading3"><h3 id="Subspace_and_quotient">Subspace and quotient</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=General_topology&action=edit&section=4" title="Edit section: Subspace and quotient"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Every subset of a topological space can be given the <a href="/wiki/Subspace_topology" title="Subspace topology">subspace topology</a> in which the open sets are the intersections of the open sets of the larger space with the subset. For any <a href="/wiki/Indexed_family" title="Indexed family">indexed family</a> of topological spaces, the product can be given the <a href="/wiki/Product_topology" title="Product topology">product topology</a>, which is generated by the inverse images of open sets of the factors under the <a href="/wiki/Projection_(mathematics)" title="Projection (mathematics)">projection</a> mappings. For example, in finite products, a basis for the product topology consists of all products of open sets. For infinite products, there is the additional requirement that in a basic open set, all but finitely many of its projections are the entire space. </p><p>A <a href="/wiki/Quotient_space_(topology)" title="Quotient space (topology)">quotient space</a> is defined as follows: if <i>X</i> is a topological space and <i>Y</i> is a set, and if <i>f</i> : <i>X</i>→ <i>Y</i> is a <a href="/wiki/Surjection" class="mw-redirect" title="Surjection">surjective</a> <a href="/wiki/Function_(mathematics)" title="Function (mathematics)">function</a>, then the <a href="/wiki/Quotient_topology" class="mw-redirect" title="Quotient topology">quotient topology</a> on <i>Y</i> is the collection of subsets of <i>Y</i> that have open <a href="/wiki/Inverse_image" class="mw-redirect" title="Inverse image">inverse images</a> under <i>f</i>. In other words, the quotient topology is the finest topology on <i>Y</i> for which <i>f</i> is continuous. A common example of a quotient topology is when an <a href="/wiki/Equivalence_relation" title="Equivalence relation">equivalence relation</a> is defined on the topological space <i>X</i>. The map <i>f</i> is then the natural projection onto the set of <a href="/wiki/Equivalence_class" title="Equivalence class">equivalence classes</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Examples_of_topological_spaces">Examples of topological spaces</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=General_topology&action=edit&section=5" title="Edit section: Examples of topological spaces"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A given set may have many different topologies. If a set is given a different topology, it is viewed as a different topological space. </p> <div class="mw-heading mw-heading4"><h4 id="Discrete_and_trivial_topologies">Discrete and trivial topologies</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=General_topology&action=edit&section=6" title="Edit section: Discrete and trivial topologies"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Any set can be given the <a href="/wiki/Discrete_space" title="Discrete space">discrete topology</a>, in which every subset is open. The only convergent sequences or nets in this topology are those that are eventually constant. Also, any set can be given the <a href="/wiki/Trivial_topology" title="Trivial topology">trivial topology</a> (also called the indiscrete topology), in which only the empty set and the whole space are open. Every sequence and net in this topology converges to every point of the space. This example shows that in general topological spaces, limits of sequences need not be unique. However, often topological spaces must be <a href="/wiki/Hausdorff_space" title="Hausdorff space">Hausdorff spaces</a> where limit points are unique. </p> <div class="mw-heading mw-heading4"><h4 id="Cofinite_and_cocountable_topologies">Cofinite and cocountable topologies</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=General_topology&action=edit&section=7" title="Edit section: Cofinite and cocountable topologies"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Any set can be given the <a href="/wiki/Cofinite_topology" class="mw-redirect" title="Cofinite topology">cofinite topology</a> in which the open sets are the empty set and the sets whose complement is finite. This is the smallest <a href="/wiki/T1_space" title="T1 space">T<sub>1</sub></a> topology on any infinite set. </p><p>Any set can be given the <a href="/wiki/Cocountable_topology" title="Cocountable topology">cocountable topology</a>, in which a set is defined as open if it is either empty or its complement is countable. When the set is uncountable, this topology serves as a counterexample in many situations. </p> <div class="mw-heading mw-heading4"><h4 id="Topologies_on_the_real_and_complex_numbers">Topologies on the real and complex numbers</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=General_topology&action=edit&section=8" title="Edit section: Topologies on the real and complex numbers"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>There are many ways to define a topology on <b>R</b>, the set of <a href="/wiki/Real_number" title="Real number">real numbers</a>. The standard topology on <b>R</b> is generated by the <a href="/wiki/Interval_(mathematics)#Definitions" title="Interval (mathematics)">open intervals</a>. The set of all open intervals forms a <a href="/wiki/Base_(topology)" title="Base (topology)">base</a> or basis for the topology, meaning that every open set is a union of some collection of sets from the base. In particular, this means that a set is open if there exists an open interval of non zero radius about every point in the set. More generally, the <a href="/wiki/Euclidean_space" title="Euclidean space">Euclidean spaces</a> <b>R</b><sup><i>n</i></sup> can be given a topology. In the usual topology on <b>R</b><sup><i>n</i></sup> the basic open sets are the open <a href="/wiki/Ball_(mathematics)" title="Ball (mathematics)">balls</a>. Similarly, <b>C</b>, the set of <a href="/wiki/Complex_number" title="Complex number">complex numbers</a>, and <b>C</b><sup><i>n</i></sup> have a standard topology in which the basic open sets are open balls. </p><p>The real line can also be given the <a href="/wiki/Lower_limit_topology" title="Lower limit topology">lower limit topology</a>. Here, the basic open sets are the half open intervals [<i>a</i>, <i>b</i>). This topology on <b>R</b> is strictly finer than the Euclidean topology defined above; a sequence converges to a point in this topology if and only if it converges from above in the Euclidean topology. This example shows that a set may have many distinct topologies defined on it. </p> <div class="mw-heading mw-heading4"><h4 id="The_metric_topology">The metric topology</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=General_topology&action=edit&section=9" title="Edit section: The metric topology"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Every <a href="/wiki/Metric_space" title="Metric space">metric space</a> can be given a metric topology, in which the basic open sets are open balls defined by the metric. This is the standard topology on any <a href="/wiki/Normed_vector_space" title="Normed vector space">normed vector space</a>. On a finite-dimensional <a href="/wiki/Vector_space" title="Vector space">vector space</a> this topology is the same for all norms. </p> <div class="mw-heading mw-heading4"><h4 id="Further_examples">Further examples</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=General_topology&action=edit&section=10" title="Edit section: Further examples"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>There exist numerous topologies on any given <a href="/wiki/Finite_set" title="Finite set">finite set</a>. Such spaces are called <a href="/wiki/Finite_topological_space" title="Finite topological space">finite topological spaces</a>. Finite spaces are sometimes used to provide examples or counterexamples to conjectures about topological spaces in general.</li> <li>Every <a href="/wiki/Manifold" title="Manifold">manifold</a> has a <a href="/wiki/Natural_topology" title="Natural topology">natural topology</a>, since it is locally Euclidean. Similarly, every <a href="/wiki/Simplex" title="Simplex">simplex</a> and every <a href="/wiki/Simplicial_complex" title="Simplicial complex">simplicial complex</a> inherits a natural topology from <b>R</b><sup>n</sup>.</li> <li>The <a href="/wiki/Zariski_topology" title="Zariski topology">Zariski topology</a> is defined algebraically on the <a href="/wiki/Spectrum_of_a_ring" title="Spectrum of a ring">spectrum of a ring</a> or an <a href="/wiki/Algebraic_variety" title="Algebraic variety">algebraic variety</a>. On <b>R</b><sup><i>n</i></sup> or <b>C</b><sup><i>n</i></sup>, the closed sets of the Zariski topology are the <a href="/wiki/Solution_set" title="Solution set">solution sets</a> of systems of <a href="/wiki/Polynomial" title="Polynomial">polynomial</a> equations.</li> <li>A <a href="/wiki/Linear_graph" class="mw-redirect" title="Linear graph">linear graph</a> has a natural topology that generalises many of the geometric aspects of <a href="/wiki/Graph_theory" title="Graph theory">graphs</a> with <a href="/wiki/Vertex_(graph_theory)" title="Vertex (graph theory)">vertices</a> and <a href="/wiki/Graph_(discrete_mathematics)#Graph" title="Graph (discrete mathematics)">edges</a>.</li> <li>Many sets of <a href="/wiki/Linear_operator" class="mw-redirect" title="Linear operator">linear operators</a> in <a href="/wiki/Functional_analysis" title="Functional analysis">functional analysis</a> are endowed with topologies that are defined by specifying when a particular sequence of functions converges to the zero function.</li> <li>Any <a href="/wiki/Local_field" title="Local field">local field</a> has a topology native to it, and this can be extended to vector spaces over that field.</li> <li>The <a href="/wiki/Sierpi%C5%84ski_space" title="Sierpiński space">Sierpiński space</a> is the simplest non-discrete topological space. It has important relations to the <a href="/wiki/Theory_of_computation" title="Theory of computation">theory of computation</a> and semantics.</li> <li>If Γ is an <a href="/wiki/Ordinal_number" title="Ordinal number">ordinal number</a>, then the set Γ = [0, Γ) may be endowed with the <a href="/wiki/Order_topology" title="Order topology">order topology</a> generated by the intervals (<i>a</i>, <i>b</i>), [0, <i>b</i>) and (<i>a</i>, Γ) where <i>a</i> and <i>b</i> are elements of Γ.</li></ul> <div class="mw-heading mw-heading2"><h2 id="Continuous_functions">Continuous functions</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=General_topology&action=edit&section=11" title="Edit section: Continuous functions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></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" title="Continuous function">Continuous function</a></div> <p>Continuity is expressed in terms of <a href="/wiki/Neighborhood_(topology)" class="mw-redirect" title="Neighborhood (topology)">neighborhoods</a>: <span class="texhtml"><i>f</i></span> is continuous at some point <span class="texhtml"><i>x</i> ∈ <i>X</i></span> if and only if for any neighborhood <span class="texhtml"><i>V</i></span> of <span class="texhtml"><i>f</i>(<i>x</i>)</span>, there is a neighborhood <span class="texhtml"><i>U</i></span> of <span class="texhtml"><i>x</i></span> such that <span class="texhtml"><i>f</i>(<i>U</i>) ⊆ <i>V</i></span>. Intuitively, continuity means no matter how "small" <span class="texhtml"><i>V</i></span> becomes, there is always a <span class="texhtml"><i>U</i></span> containing <span class="texhtml"><i>x</i></span> that maps inside <span class="texhtml"><i>V</i></span> and whose image under <span class="texhtml"><i>f</i></span> contains <span class="texhtml"><i>f</i>(<i>x</i>)</span>. This is equivalent to the condition that the <a href="/wiki/Image_(mathematics)#Inverse_image" title="Image (mathematics)">preimages</a> of the open (closed) sets in <span class="texhtml"><i>Y</i></span> are open (closed) in <span class="texhtml"><i>X</i></span>. In metric spaces, this definition is equivalent to the <a href="/wiki/Epsilon-delta_definition" class="mw-redirect" title="Epsilon-delta definition">ε–δ-definition</a> that is often used in analysis. </p><p>An extreme example: if a set <span class="texhtml"><i>X</i></span> is given the <a href="/wiki/Discrete_topology" class="mw-redirect" title="Discrete topology">discrete topology</a>, all functions </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\colon X\rightarrow T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>:</mo> <mi>X</mi> <mo stretchy="false">→</mo> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f\colon X\rightarrow T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aa0b7ef294ac2e2036b953c216fc6325f364f4bd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.543ex; height:2.509ex;" alt="{\displaystyle f\colon X\rightarrow T}"></span></dd></dl> <p>to any topological space <span class="texhtml"><i>T</i></span> are continuous. On the other hand, if <span class="texhtml"><i>X</i></span> is equipped with the <a href="/wiki/Indiscrete_topology" class="mw-redirect" title="Indiscrete topology">indiscrete topology</a> and the space <span class="texhtml"><i>T</i></span> set is at least <a href="/wiki/T0_space" class="mw-redirect" title="T0 space">T<sub>0</sub></a>, then the only continuous functions are the constant functions. Conversely, any function whose range is indiscrete is continuous. </p> <div class="mw-heading mw-heading3"><h3 id="Alternative_definitions">Alternative definitions</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=General_topology&action=edit&section=12" title="Edit section: Alternative definitions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Several <a href="/wiki/Characterizations_of_the_category_of_topological_spaces" class="mw-redirect" title="Characterizations of the category of topological spaces">equivalent definitions for a topological structure</a> exist and thus there are several equivalent ways to define a continuous function. </p> <div class="mw-heading mw-heading4"><h4 id="Neighborhood_definition">Neighborhood definition</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=General_topology&action=edit&section=13" title="Edit section: Neighborhood definition"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Definitions based on preimages are often difficult to use directly. The following criterion expresses continuity in terms of <a href="/wiki/Neighborhood_(topology)" class="mw-redirect" title="Neighborhood (topology)">neighborhoods</a>: <i>f</i> is continuous at some point <i>x</i> ∈ <i>X</i> if and only if for any neighborhood <i>V</i> of <i>f</i>(<i>x</i>), there is a neighborhood <i>U</i> of <i>x</i> such that <i>f</i>(<i>U</i>) ⊆ <i>V</i>. Intuitively, continuity means no matter how "small" <i>V</i> becomes, there is always a <i>U</i> containing <i>x</i> that maps inside <i>V</i>. </p><p>If <i>X</i> and <i>Y</i> are metric spaces, it is equivalent to consider the <a href="/wiki/Neighborhood_system" class="mw-redirect" title="Neighborhood system">neighborhood system</a> of <a href="/wiki/Open_ball" class="mw-redirect" title="Open ball">open balls</a> centered at <i>x</i> and <i>f</i>(<i>x</i>) instead of all neighborhoods. This gives back the above δ-ε definition of continuity in the context of metric spaces. However, in general topological spaces, there is no notion of nearness or distance. </p><p>Note, however, that if the target space is <a href="/wiki/Hausdorff_space" title="Hausdorff space">Hausdorff</a>, it is still true that <i>f</i> is continuous at <i>a</i> if and only if the limit of <i>f</i> as <i>x</i> approaches <i>a</i> is <i>f</i>(<i>a</i>). At an <a href="/wiki/Isolated_point" title="Isolated point">isolated point</a>, every function is continuous. </p> <div class="mw-heading mw-heading4"><h4 id="Sequences_and_nets">Sequences and nets <span class="anchor" id="Heine_definition_of_continuity"></span></h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=General_topology&action=edit&section=14" title="Edit section: Sequences and nets"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In several contexts, the topology of a space is conveniently specified in terms of <a href="/wiki/Limit_points" class="mw-redirect" title="Limit points">limit points</a>. In many instances, this is accomplished by specifying when a point is the <a href="/wiki/Limit_of_a_sequence" title="Limit of a sequence">limit of a sequence</a>, but for some spaces that are too large in some sense, one specifies also when a point is the limit of more general sets of points indexed by a <a href="/wiki/Directed_set" title="Directed set">directed set</a>, known as <a href="/wiki/Net_(mathematics)" title="Net (mathematics)">nets</a>.<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> A function is continuous only if it takes limits of sequences to limits of sequences. In the former case, preservation of limits is also sufficient; in the latter, a function may preserve all limits of sequences yet still fail to be continuous, and preservation of nets is a necessary and sufficient condition. </p><p>In detail, a function <i>f</i>: <i>X</i> → <i>Y</i> is <b>sequentially continuous</b> if whenever a sequence (<i>x</i><sub><i>n</i></sub>) in <i>X</i> converges to a limit <i>x</i>, the sequence (<i>f</i>(<i>x</i><sub><i>n</i></sub>)) converges to <i>f</i>(<i>x</i>).<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> Thus sequentially continuous functions "preserve sequential limits". Every continuous function is sequentially continuous. If <i>X</i> is a <a href="/wiki/First-countable_space" title="First-countable space">first-countable space</a> and <a href="/wiki/Axiom_of_countable_choice" title="Axiom of countable choice">countable choice</a> holds, then the converse also holds: any function preserving sequential limits is continuous. In particular, if <i>X</i> is a metric space, sequential continuity and continuity are equivalent. For non first-countable spaces, sequential continuity might be strictly weaker than continuity. (The spaces for which the two properties are equivalent are called <a href="/wiki/Sequential_space" title="Sequential space">sequential spaces</a>.) This motivates the consideration of nets instead of sequences in general topological spaces. Continuous functions preserve limits of nets, and in fact this property characterizes continuous functions. </p> <div class="mw-heading mw-heading4"><h4 id="Closure_operator_definition">Closure operator definition</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=General_topology&action=edit&section=15" title="Edit section: Closure operator definition"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Instead of specifying the open subsets of a topological space, the topology can also be determined by a <a href="/wiki/Kuratowski_closure_operator" class="mw-redirect" title="Kuratowski closure operator">closure operator</a> (denoted cl), which assigns to any subset <i>A</i> ⊆ <i>X</i> its <a href="/wiki/Closure_(topology)" title="Closure (topology)">closure</a>, or an <a href="/wiki/Interior_operator" class="mw-redirect" title="Interior operator">interior operator</a> (denoted int), which assigns to any subset <i>A</i> of <i>X</i> its <a href="/wiki/Interior_(topology)" title="Interior (topology)">interior</a>. In these terms, a function </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\colon (X,\mathrm {cl} )\to (X',\mathrm {cl} ')\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>:</mo> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">c</mi> <mi mathvariant="normal">l</mi> </mrow> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mo stretchy="false">(</mo> <msup> <mi>X</mi> <mo>′</mo> </msup> <mo>,</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">c</mi> <mi mathvariant="normal">l</mi> </mrow> <mo>′</mo> </msup> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f\colon (X,\mathrm {cl} )\to (X',\mathrm {cl} ')\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fc15d00de8e149cd4fef3bbdb8811951de513e73" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.705ex; height:3.009ex;" alt="{\displaystyle f\colon (X,\mathrm {cl} )\to (X',\mathrm {cl} ')\,}"></span></dd></dl> <p>between topological spaces is continuous in the sense above if and only if for all subsets <i>A</i> of <i>X</i> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(\mathrm {cl} (A))\subseteq \mathrm {cl} '(f(A)).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">c</mi> <mi mathvariant="normal">l</mi> </mrow> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>⊆</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">c</mi> <mi mathvariant="normal">l</mi> </mrow> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(\mathrm {cl} (A))\subseteq \mathrm {cl} '(f(A)).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b73fc640d91223a950e8ae0b236b9574dc96587b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.069ex; height:3.009ex;" alt="{\displaystyle f(\mathrm {cl} (A))\subseteq \mathrm {cl} '(f(A)).}"></span></dd></dl> <p>That is to say, given any element <i>x</i> of <i>X</i> that is in the closure of any subset <i>A</i>, <i>f</i>(<i>x</i>) belongs to the closure of <i>f</i>(<i>A</i>). This is equivalent to the requirement that for all subsets <i>A</i>' of <i>X</i>' </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f^{-1}(\mathrm {cl} '(A'))\supseteq \mathrm {cl} (f^{-1}(A')).}"> <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> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">c</mi> <mi mathvariant="normal">l</mi> </mrow> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <msup> <mi>A</mi> <mo>′</mo> </msup> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>⊇</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">c</mi> <mi mathvariant="normal">l</mi> </mrow> <mo stretchy="false">(</mo> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">(</mo> <msup> <mi>A</mi> <mo>′</mo> </msup> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f^{-1}(\mathrm {cl} '(A'))\supseteq \mathrm {cl} (f^{-1}(A')).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b33c6479b71e9564adf954b9bed018d831990949" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.188ex; height:3.176ex;" alt="{\displaystyle f^{-1}(\mathrm {cl} '(A'))\supseteq \mathrm {cl} (f^{-1}(A')).}"></span></dd></dl> <p>Moreover, </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\colon (X,\mathrm {int} )\to (X',\mathrm {int} ')\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>:</mo> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">t</mi> </mrow> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mo stretchy="false">(</mo> <msup> <mi>X</mi> <mo>′</mo> </msup> <mo>,</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">t</mi> </mrow> <mo>′</mo> </msup> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f\colon (X,\mathrm {int} )\to (X',\mathrm {int} ')\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ea9e53dad52f1479e44b72d45dd85a3919788a22" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.034ex; height:3.009ex;" alt="{\displaystyle f\colon (X,\mathrm {int} )\to (X',\mathrm {int} ')\,}"></span></dd></dl> <p>is continuous if and only if </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f^{-1}(\mathrm {int} '(A))\subseteq \mathrm {int} (f^{-1}(A))}"> <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> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">t</mi> </mrow> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>⊆</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">i</mi> <mi mathvariant="normal">n</mi> <mi mathvariant="normal">t</mi> </mrow> <mo stretchy="false">(</mo> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f^{-1}(\mathrm {int} '(A))\subseteq \mathrm {int} (f^{-1}(A))}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d15e649764e56ae4ff9247ea6f736086943e53e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.501ex; height:3.176ex;" alt="{\displaystyle f^{-1}(\mathrm {int} '(A))\subseteq \mathrm {int} (f^{-1}(A))}"></span></dd></dl> <p>for any subset <i>A</i> of <i>X</i>. </p> <div class="mw-heading mw-heading3"><h3 id="Properties">Properties</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=General_topology&action=edit&section=16" title="Edit section: Properties"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>If <i>f</i>: <i>X</i> → <i>Y</i> and <i>g</i>: <i>Y</i> → <i>Z</i> are continuous, then so is the composition <i>g</i> ∘ <i>f</i>: <i>X</i> → <i>Z</i>. If <i>f</i>: <i>X</i> → <i>Y</i> is continuous and </p> <ul><li><i>X</i> is <a href="/wiki/Compact_space" title="Compact space">compact</a>, then <i>f</i>(<i>X</i>) is compact.</li> <li><i>X</i> is <a href="/wiki/Connected_space" title="Connected space">connected</a>, then <i>f</i>(<i>X</i>) is connected.</li> <li><i>X</i> is <a href="/wiki/Path-connected" class="mw-redirect" title="Path-connected">path-connected</a>, then <i>f</i>(<i>X</i>) is path-connected.</li> <li><i>X</i> is <a href="/wiki/Lindel%C3%B6f_space" title="Lindelöf space">Lindelöf</a>, then <i>f</i>(<i>X</i>) is Lindelöf.</li> <li><i>X</i> is <a href="/wiki/Separable_space" title="Separable space">separable</a>, then <i>f</i>(<i>X</i>) is separable.</li></ul> <p>The possible topologies on a fixed set <i>X</i> are <a href="/wiki/Partial_ordering" class="mw-redirect" title="Partial ordering">partially ordered</a>: a topology τ<sub>1</sub> is said to be <a href="/wiki/Comparison_of_topologies" title="Comparison of topologies">coarser</a> than another topology τ<sub>2</sub> (notation: τ<sub>1</sub> ⊆ τ<sub>2</sub>) if every open subset with respect to τ<sub>1</sub> is also open with respect to τ<sub>2</sub>. Then, the <a href="/wiki/Identity_function" title="Identity function">identity map</a> </p> <dl><dd>id<sub>X</sub>: (<i>X</i>, τ<sub>2</sub>) → (<i>X</i>, τ<sub>1</sub>)</dd></dl> <p>is continuous if and only if τ<sub>1</sub> ⊆ τ<sub>2</sub> (see also <a href="/wiki/Comparison_of_topologies" title="Comparison of topologies">comparison of topologies</a>). More generally, a continuous function </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (X,\tau _{X})\rightarrow (Y,\tau _{Y})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <msub> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mo stretchy="false">(</mo> <mi>Y</mi> <mo>,</mo> <msub> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>Y</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (X,\tau _{X})\rightarrow (Y,\tau _{Y})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/37bfe71aab47edad54afabea6166c30c815d5007" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.205ex; height:2.843ex;" alt="{\displaystyle (X,\tau _{X})\rightarrow (Y,\tau _{Y})}"></span></dd></dl> <p>stays continuous if the topology τ<sub><i>Y</i></sub> is replaced by a <a href="/wiki/Comparison_of_topologies" title="Comparison of topologies">coarser topology</a> and/or τ<sub><i>X</i></sub> is replaced by a <a href="/wiki/Comparison_of_topologies" title="Comparison of topologies">finer topology</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Homeomorphisms">Homeomorphisms</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=General_topology&action=edit&section=17" title="Edit section: Homeomorphisms"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Symmetric to the concept of a continuous map is an <a href="/wiki/Open_map" class="mw-redirect" title="Open map">open map</a>, for which <i>images</i> of open sets are open. In fact, if an open map <i>f</i> has an <a href="/wiki/Inverse_function" title="Inverse function">inverse function</a>, that inverse is continuous, and if a continuous map <i>g</i> has an inverse, that inverse is open. Given a <a href="/wiki/Bijective" class="mw-redirect" title="Bijective">bijective</a> function <i>f</i> between two topological spaces, the inverse function <i>f</i><sup>−1</sup> need not be continuous. A bijective continuous function with continuous inverse function is called a <i><a href="/wiki/Homeomorphism" title="Homeomorphism">homeomorphism</a></i>. </p><p>If a continuous bijection has as its <a href="/wiki/Domain_of_a_function" title="Domain of a function">domain</a> a <a href="/wiki/Compact_space" title="Compact space">compact space</a> and its <a href="/wiki/Codomain" title="Codomain">codomain</a> is <a href="/wiki/Hausdorff_space" title="Hausdorff space">Hausdorff</a>, then it is a homeomorphism. </p> <div class="mw-heading mw-heading3"><h3 id="Defining_topologies_via_continuous_functions">Defining topologies via continuous functions</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=General_topology&action=edit&section=18" title="Edit section: Defining topologies via continuous functions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Given a function </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\colon X\rightarrow S,\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>:</mo> <mi>X</mi> <mo stretchy="false">→</mo> <mi>S</mi> <mo>,</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f\colon X\rightarrow S,\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5dc80594b71ae436e006381986a2364bfe35f326" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.44ex; height:2.509ex;" alt="{\displaystyle f\colon X\rightarrow S,\,}"></span></dd></dl> <p>where <i>X</i> is a topological space and <i>S</i> is a set (without a specified topology), the <a href="/wiki/Final_topology" title="Final topology">final topology</a> on <i>S</i> is defined by letting the open sets of <i>S</i> be those subsets <i>A</i> of <i>S</i> for which <i>f</i><sup>−1</sup>(<i>A</i>) is open in <i>X</i>. If <i>S</i> has an existing topology, <i>f</i> is continuous with respect to this topology if and only if the existing topology is <a href="/wiki/Comparison_of_topologies" title="Comparison of topologies">coarser</a> than the final topology on <i>S</i>. Thus the final topology can be characterized as the finest topology on <i>S</i> that makes <i>f</i> continuous. If <i>f</i> is <a href="/wiki/Surjective" class="mw-redirect" title="Surjective">surjective</a>, this topology is canonically identified with the <a href="/wiki/Quotient_topology" class="mw-redirect" title="Quotient topology">quotient topology</a> under the <a href="/wiki/Equivalence_relation" title="Equivalence relation">equivalence relation</a> defined by <i>f</i>. </p><p>Dually, for a function <i>f</i> from a set <i>S</i> to a topological space <i>X</i>, the <a href="/wiki/Initial_topology" title="Initial topology">initial topology</a> on <i>S</i> has a basis of open sets given by those sets of the form <i>f^(-1)</i>(<i>U</i>) where <i>U</i> is open in <i>X</i> . If <i>S</i> has an existing topology, <i>f</i> is continuous with respect to this topology if and only if the existing topology is finer than the initial topology on <i>S</i>. Thus the initial topology can be characterized as the coarsest topology on <i>S</i> that makes <i>f</i> continuous. If <i>f</i> is injective, this topology is canonically identified with the <a href="/wiki/Subspace_topology" title="Subspace topology">subspace topology</a> of <i>S</i>, viewed as a subset of <i>X</i>. </p><p>A topology on a set <i>S</i> is uniquely determined by the class of all continuous functions <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S\rightarrow X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo stretchy="false">→</mo> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S\rightarrow X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f0e189edef0d01e9ababdf5a3ccb7f28bc2e782e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.093ex; height:2.176ex;" alt="{\displaystyle S\rightarrow X}"></span> into all topological spaces <i>X</i>. <a href="/wiki/Duality_(mathematics)" title="Duality (mathematics)">Dually</a>, a similar idea can be applied to maps <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X\rightarrow S.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo stretchy="false">→</mo> <mi>S</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X\rightarrow S.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/144f6836a8a2d4a0bc49b834ba658f8e12c37a68" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.74ex; height:2.176ex;" alt="{\displaystyle X\rightarrow S.}"></span> </p> <div class="mw-heading mw-heading2"><h2 id="Compact_sets">Compact sets</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=General_topology&action=edit&section=19" title="Edit section: Compact sets"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></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_(mathematics)" class="mw-redirect" title="Compact (mathematics)">Compact (mathematics)</a></div> <p>Formally, a <a href="/wiki/Topological_space" title="Topological space">topological space</a> <i>X</i> is called <i>compact</i> if each of its <a href="/wiki/Open_cover" class="mw-redirect" title="Open cover">open covers</a> has a <a href="/wiki/Finite_set" title="Finite set">finite</a> <a href="/wiki/Subcover" class="mw-redirect" title="Subcover">subcover</a>. Otherwise it is called <i>non-compact</i>. Explicitly, this means that for every arbitrary collection </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{U_{\alpha }\}_{\alpha \in A}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msub> <msub> <mo fence="false" stretchy="false">}</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mo>∈</mo> <mi>A</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{U_{\alpha }\}_{\alpha \in A}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8ee60af71f3a9be7713c75ee88fa354228de34fe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.81ex; height:2.843ex;" alt="{\displaystyle \{U_{\alpha }\}_{\alpha \in A}}"></span></dd></dl> <p>of open subsets of <span class="texhtml mvar" style="font-style:italic;">X</span> such that </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X=\bigcup _{\alpha \in A}U_{\alpha },}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>=</mo> <munder> <mo>⋃</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mo>∈</mo> <mi>A</mi> </mrow> </munder> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X=\bigcup _{\alpha \in A}U_{\alpha },}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/85a890b537be3458d6da8c577bec2f8b474a3b82" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:12.365ex; height:5.676ex;" alt="{\displaystyle X=\bigcup _{\alpha \in A}U_{\alpha },}"></span></dd></dl> <p>there is a finite subset <span class="texhtml mvar" style="font-style:italic;">J</span> of <span class="texhtml mvar" style="font-style:italic;">A</span> such that </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X=\bigcup _{i\in J}U_{i}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>=</mo> <munder> <mo>⋃</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>∈</mo> <mi>J</mi> </mrow> </munder> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X=\bigcup _{i\in J}U_{i}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e62aa0efde28d5358605eaedcd208219ca8c908" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:11.204ex; height:5.676ex;" alt="{\displaystyle X=\bigcup _{i\in J}U_{i}.}"></span></dd></dl> <p>Some branches of mathematics such as <a href="/wiki/Algebraic_geometry" title="Algebraic geometry">algebraic geometry</a>, typically influenced by the French school of <a href="/wiki/Nicolas_Bourbaki" title="Nicolas Bourbaki">Bourbaki</a>, use the term <i>quasi-compact</i> for the general notion, and reserve the term <i>compact</i> for topological spaces that are both <a href="/wiki/Hausdorff_spaces" class="mw-redirect" title="Hausdorff spaces">Hausdorff</a> and <i>quasi-compact</i>. A compact set is sometimes referred to as a <i>compactum</i>, plural <i>compacta</i>. </p><p>Every closed <a href="/wiki/Interval_(mathematics)" title="Interval (mathematics)">interval</a> in <b><a href="/wiki/Real_number" title="Real number">R</a></b> of finite length is <a href="/wiki/Compact_space" title="Compact space">compact</a>. More is true: In <b>R</b><sup><var>n</var></sup>, a set is compact <a href="/wiki/If_and_only_if" title="If and only if">if and only if</a> it is <a href="/wiki/Closed_set" title="Closed set">closed</a> and bounded. (See <a href="/wiki/Heine%E2%80%93Borel_theorem" title="Heine–Borel theorem">Heine–Borel theorem</a>). </p><p>Every continuous image of a compact space is compact. </p><p>A compact subset of a Hausdorff space is closed. </p><p>Every continuous <a href="/wiki/Bijection" title="Bijection">bijection</a> from a compact space to a Hausdorff space is necessarily a <a href="/wiki/Homeomorphism" title="Homeomorphism">homeomorphism</a>. </p><p>Every <a href="/wiki/Sequence" title="Sequence">sequence</a> of points in a compact metric space has a convergent subsequence. </p><p>Every compact finite-dimensional <a href="/wiki/Manifold" title="Manifold">manifold</a> can be embedded in some Euclidean space <b>R</b><sup><var>n</var></sup>. </p> <div class="mw-heading mw-heading2"><h2 id="Connected_sets">Connected sets</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=General_topology&action=edit&section=20" title="Edit section: Connected sets"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></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/Connected_space" title="Connected space">connected space</a></div> <p>A <a href="/wiki/Topological_space" title="Topological space">topological space</a> <i>X</i> is said to be <b>disconnected</b> if it is the <a href="/wiki/Union_(set_theory)" title="Union (set theory)">union</a> of two <a href="/wiki/Disjoint_sets" title="Disjoint sets">disjoint</a> <a href="/wiki/Nonempty" class="mw-redirect" title="Nonempty">nonempty</a> <a href="/wiki/Open_set" title="Open set">open sets</a>. Otherwise, <i>X</i> is said to be <b>connected</b>. A <a href="/wiki/Subset" title="Subset">subset</a> of a topological space is said to be connected if it is connected under its <a href="/wiki/Subspace_(topology)" class="mw-redirect" title="Subspace (topology)">subspace topology</a>. Some authors exclude the <a href="/wiki/Empty_set" title="Empty set">empty set</a> (with its unique topology) as a connected space, but this article does not follow that practice. </p><p>For a topological space <i>X</i> the following conditions are equivalent: </p> <ol><li><i>X</i> is connected.</li> <li><i>X</i> cannot be divided into two disjoint nonempty <a href="/wiki/Closed_set" title="Closed set">closed sets</a>.</li> <li>The only subsets of <i>X</i> that are both open and closed (<a href="/wiki/Clopen_set" title="Clopen set">clopen sets</a>) are <i>X</i> and the empty set.</li> <li>The only subsets of <i>X</i> with empty <a href="/wiki/Boundary_(topology)" title="Boundary (topology)">boundary</a> are <i>X</i> and the empty set.</li> <li><i>X</i> cannot be written as the union of two nonempty <a href="/wiki/Separated_sets" title="Separated sets">separated sets</a>.</li> <li>The only continuous functions from <i>X</i> to {0,1}, the two-point space endowed with the discrete topology, are constant.</li></ol> <p>Every interval in <b>R</b> is <a href="/wiki/Connected_space" title="Connected space">connected</a>. </p><p>The continuous image of a <a href="/wiki/Connectedness" title="Connectedness">connected</a> space is connected. </p> <div class="mw-heading mw-heading3"><h3 id="Connected_components">Connected components</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=General_topology&action=edit&section=21" title="Edit section: Connected components"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <a href="/wiki/Maximal_element" class="mw-redirect" title="Maximal element">maximal</a> connected subsets (ordered by <a href="/wiki/Subset" title="Subset">inclusion</a>) of a nonempty topological space are called the <b>connected components</b> of the space. The components of any topological space <i>X</i> form a <a href="/wiki/Partition_of_a_set" title="Partition of a set">partition</a> of <i>X</i>: they are <a href="/wiki/Disjoint_sets" title="Disjoint sets">disjoint</a>, nonempty, and their union is the whole space. Every component is a <a href="/wiki/Closed_subset" class="mw-redirect" title="Closed subset">closed subset</a> of the original space. It follows that, in the case where their number is finite, each component is also an open subset. However, if their number is infinite, this might not be the case; for instance, the connected components of the set of the <a href="/wiki/Rational_number" title="Rational number">rational numbers</a> are the one-point sets, which are not open. </p><p>Let <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Gamma _{x}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">Γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Gamma _{x}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/49c230e1b875ae363247ad684861c5a1471f8b66" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.625ex; height:2.509ex;" alt="{\displaystyle \Gamma _{x}}"></span> be the connected component of <i>x</i> in a topological space <i>X</i>, and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Gamma _{x}'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi mathvariant="normal">Γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> <mo>′</mo> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Gamma _{x}'}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8569e28816687dc688008907c538c1a37a3f060a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.625ex; height:2.509ex;" alt="{\displaystyle \Gamma _{x}'}"></span> be the intersection of all open-closed sets containing <i>x</i> (called <a href="/wiki/Locally_connected_space" title="Locally connected space">quasi-component</a> of <i>x</i>.) Then <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Gamma _{x}\subset \Gamma '_{x}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">Γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>⊂</mo> <msubsup> <mi mathvariant="normal">Γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> <mo>′</mo> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Gamma _{x}\subset \Gamma '_{x}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b043d906f731389de3888b1fb5654a977d3661dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.349ex; height:2.509ex;" alt="{\displaystyle \Gamma _{x}\subset \Gamma '_{x}}"></span> where the equality holds if <i>X</i> is compact Hausdorff or locally connected. </p> <div class="mw-heading mw-heading3"><h3 id="Disconnected_spaces">Disconnected spaces</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=General_topology&action=edit&section=22" title="Edit section: Disconnected spaces"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A space in which all components are one-point sets is called <a href="/wiki/Totally_disconnected" class="mw-redirect" title="Totally disconnected">totally disconnected</a>. Related to this property, a space <i>X</i> is called <b>totally separated</b> if, for any two distinct elements <i>x</i> and <i>y</i> of <i>X</i>, there exist disjoint <a href="/wiki/Neighborhood_(topology)" class="mw-redirect" title="Neighborhood (topology)">open neighborhoods</a> <i>U</i> of <i>x</i> and <i>V</i> of <i>y</i> such that <i>X</i> is the union of <i>U</i> and <i>V</i>. Clearly any totally separated space is totally disconnected, but the converse does not hold. For example, take two copies of the rational numbers <b>Q</b>, and identify them at every point except zero. The resulting space, with the quotient topology, is totally disconnected. However, by considering the two copies of zero, one sees that the space is not totally separated. In fact, it is not even <a href="/wiki/Hausdorff_space" title="Hausdorff space">Hausdorff</a>, and the condition of being totally separated is strictly stronger than the condition of being Hausdorff. </p> <div class="mw-heading mw-heading3"><h3 id="Path-connected_sets">Path-connected sets</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=General_topology&action=edit&section=23" title="Edit section: Path-connected sets"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Path-connected_space.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b8/Path-connected_space.svg/220px-Path-connected_space.svg.png" decoding="async" width="220" height="143" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b8/Path-connected_space.svg/330px-Path-connected_space.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/b8/Path-connected_space.svg/440px-Path-connected_space.svg.png 2x" data-file-width="125" data-file-height="81" /></a><figcaption>This subspace of <b>R</b>² is path-connected, because a path can be drawn between any two points in the space.</figcaption></figure> <p>A <i><a href="/wiki/Path_(topology)" title="Path (topology)">path</a></i> from a point <i>x</i> to a point <i>y</i> in a <a href="/wiki/Topological_space" title="Topological space">topological space</a> <i>X</i> is a <a href="/wiki/Continuous_function_(topology)" class="mw-redirect" title="Continuous function (topology)">continuous function</a> <i>f</i> from the <a href="/wiki/Unit_interval" title="Unit interval">unit interval</a> [0,1] to <i>X</i> with <i>f</i>(0) = <i>x</i> and <i>f</i>(1) = <i>y</i>. A <i><a href="/wiki/Path_component" class="mw-redirect" title="Path component">path-component</a></i> of <i>X</i> is an <a href="/wiki/Equivalence_class" title="Equivalence class">equivalence class</a> of <i>X</i> under the <a href="/wiki/Equivalence_relation" title="Equivalence relation">equivalence relation</a>, which makes <i>x</i> equivalent to <i>y</i> if there is a path from <i>x</i> to <i>y</i>. The space <i>X</i> is said to be <i><a href="/wiki/Path-connected_space" class="mw-redirect" title="Path-connected space">path-connected</a></i> (or <i>pathwise connected</i> or <i>0-connected</i>) if there is at most one path-component; that is, if there is a path joining any two points in <i>X</i>. Again, many authors exclude the empty space. </p><p>Every path-connected space is connected. The converse is not always true: examples of connected spaces that are not path-connected include the extended <a href="/wiki/Long_line_(topology)" title="Long line (topology)">long line</a> <i>L</i>* and the <i><a href="/wiki/Topologist%27s_sine_curve" title="Topologist's sine curve">topologist's sine curve</a></i>. </p><p>However, subsets of the <a href="/wiki/Real_line" class="mw-redirect" title="Real line">real line</a> <b>R</b> are connected <a href="/wiki/If_and_only_if" title="If and only if">if and only if</a> they are path-connected; these subsets are the <a href="/wiki/Interval_(mathematics)" title="Interval (mathematics)">intervals</a> of <b>R</b>. Also, <a href="/wiki/Open_subset" class="mw-redirect" title="Open subset">open subsets</a> of <b>R</b><sup><i>n</i></sup> or <b>C</b><sup><i>n</i></sup> are connected if and only if they are path-connected. Additionally, connectedness and path-connectedness are the same for <a href="/wiki/Finite_topological_space" title="Finite topological space">finite topological spaces</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Products_of_spaces">Products of spaces</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=General_topology&action=edit&section=24" title="Edit section: Products of spaces"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></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_topology" title="Product topology">Product topology</a></div> <p>Given <i>X</i> such that </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X:=\prod _{i\in I}X_{i},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>:=</mo> <munder> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>∈</mo> <mi>I</mi> </mrow> </munder> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X:=\prod _{i\in I}X_{i},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/455a090918ae6122bd6a620b8503f89c3ae735dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:12.453ex; height:5.676ex;" alt="{\displaystyle X:=\prod _{i\in I}X_{i},}"></span></dd></dl> <p>is the Cartesian product of the topological spaces <i>X<sub>i</sub></i>, <a href="/wiki/Index_set" title="Index set">indexed</a> by <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i\in I}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <mo>∈</mo> <mi>I</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i\in I}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2d740fe587228ce31b71c9628e089d1a9b37c6be" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.815ex; height:2.176ex;" alt="{\displaystyle i\in I}"></span>, and the <b><a href="/wiki/Projection_(set_theory)" title="Projection (set theory)">canonical projections</a></b> <i>p<sub>i</sub></i> : <i>X</i> → <i>X<sub>i</sub></i>, the <b>product topology</b> on <i>X</i> is defined as the <a href="/wiki/Coarsest_topology" class="mw-redirect" title="Coarsest topology">coarsest topology</a> (i.e. the topology with the fewest open sets) for which all the projections <i>p<sub>i</sub></i> are <a href="/wiki/Continuous_(topology)" class="mw-redirect" title="Continuous (topology)">continuous</a>. The product topology is sometimes called the <b>Tychonoff topology</b>. </p><p>The open sets in the product topology are unions (finite or infinite) of sets of the form <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \prod _{i\in I}U_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>∈</mo> <mi>I</mi> </mrow> </munder> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \prod _{i\in I}U_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3607aee6e64a23efec7be0e931e6c5b16ddd5c44" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:5.744ex; height:5.676ex;" alt="{\displaystyle \prod _{i\in I}U_{i}}"></span>, where each <i>U<sub>i</sub></i> is open in <i>X<sub>i</sub></i> and <i>U</i><sub><i>i</i></sub> ≠ <i>X</i><sub><i>i</i></sub> only finitely many times. In particular, for a finite product (in particular, for the product of two topological spaces), the products of base elements of the <i>X<sub>i</sub></i> gives a basis for the product <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \prod _{i\in I}X_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>∈</mo> <mi>I</mi> </mrow> </munder> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \prod _{i\in I}X_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e4cf2ee0af61dd971a31005ff6d10fb3fe3267f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:6.08ex; height:5.676ex;" alt="{\displaystyle \prod _{i\in I}X_{i}}"></span>. </p><p>The product topology on <i>X</i> is the topology generated by sets of the form <i>p<sub>i</sub></i><sup>−1</sup>(<i>U</i>), where <i>i</i> is in <i>I </i> and <i>U</i> is an open subset of <i>X<sub>i</sub></i>. In other words, the sets {<i>p<sub>i</sub></i><sup>−1</sup>(<i>U</i>)} form a <a href="/wiki/Subbase" title="Subbase">subbase</a> for the topology on <i>X</i>. A <a href="/wiki/Subset" title="Subset">subset</a> of <i>X</i> is open if and only if it is a (possibly infinite) <a href="/wiki/Union_(set_theory)" title="Union (set theory)">union</a> of <a href="/wiki/Intersection_(set_theory)" title="Intersection (set theory)">intersections</a> of finitely many sets of the form <i>p<sub>i</sub></i><sup>−1</sup>(<i>U</i>). The <i>p<sub>i</sub></i><sup>−1</sup>(<i>U</i>) are sometimes called <a href="/wiki/Open_cylinder" class="mw-redirect" title="Open cylinder">open cylinders</a>, and their intersections are <a href="/wiki/Cylinder_set" title="Cylinder set">cylinder sets</a>. </p><p>In general, the product of the topologies of each <i>X<sub>i</sub></i> forms a basis for what is called the <a href="/wiki/Box_topology" title="Box topology">box topology</a> on <i>X</i>. In general, the box topology is <a href="/wiki/Finer_topology" class="mw-redirect" title="Finer topology">finer</a> than the product topology, but for finite products they coincide. </p><p>Related to compactness is <a href="/wiki/Tychonoff%27s_theorem" title="Tychonoff's theorem">Tychonoff's theorem</a>: the (arbitrary) <a href="/wiki/Product_topology" title="Product topology">product</a> of compact spaces is compact. </p> <div class="mw-heading mw-heading2"><h2 id="Separation_axioms">Separation axioms</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=General_topology&action=edit&section=25" title="Edit section: Separation axioms"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></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/Separation_axiom" title="Separation axiom">Separation axiom</a></div> <p>Many of these names have alternative meanings in some of mathematical literature, as explained on <a href="/wiki/History_of_the_separation_axioms" title="History of the separation axioms">History of the separation axioms</a>; for example, the meanings of "normal" and "T<sub>4</sub>" are sometimes interchanged, similarly "regular" and "T<sub>3</sub>", etc. Many of the concepts also have several names; however, the one listed first is always least likely to be ambiguous. </p><p>Most of these axioms have alternative definitions with the same meaning; the definitions given here fall into a consistent pattern that relates the various notions of separation defined in the previous section. Other possible definitions can be found in the individual articles. </p><p>In all of the following definitions, <i>X</i> is again a <a href="/wiki/Topological_space" title="Topological space">topological space</a>. </p> <ul><li><i>X</i> is <i><a href="/wiki/T0_space" class="mw-redirect" title="T0 space">T<sub>0</sub></a></i>, or <i>Kolmogorov</i>, if any two distinct points in <i>X</i> are <a href="/wiki/Topological_distinguishability" class="mw-redirect" title="Topological distinguishability">topologically distinguishable</a>. (It is a common theme among the separation axioms to have one version of an axiom that requires T<sub>0</sub> and one version that doesn't.)</li> <li><i>X</i> is <i><a href="/wiki/T1_space" title="T1 space">T<sub>1</sub></a></i>, or <i>accessible</i> or <i>Fréchet</i>, if any two distinct points in <i>X</i> are separated. Thus, <i>X</i> is T<sub>1</sub> if and only if it is both T<sub>0</sub> and R<sub>0</sub>. (Though you may say such things as <i>T<sub>1</sub> space</i>, <i>Fréchet topology</i>, and <i>Suppose that the topological space </i>X<i> is Fréchet</i>, avoid saying <i>Fréchet space</i> in this context, since there is another entirely different notion of <a href="/wiki/Fr%C3%A9chet_space" title="Fréchet space">Fréchet space</a> in <a href="/wiki/Functional_analysis" title="Functional analysis">functional analysis</a>.)</li> <li><i>X</i> is <i><a href="/wiki/Hausdorff_space" title="Hausdorff space">Hausdorff</a></i>, or <i>T<sub>2</sub></i> or <i>separated</i>, if any two distinct points in <i>X</i> are separated by neighbourhoods. Thus, <i>X</i> is Hausdorff if and only if it is both T<sub>0</sub> and R<sub>1</sub>. A Hausdorff space must also be T<sub>1</sub>.</li> <li><i>X</i> is <i><a href="/wiki/Urysohn_and_completely_Hausdorff_spaces" title="Urysohn and completely Hausdorff spaces">T<sub>2½</sub></a></i>, or <i>Urysohn</i>, if any two distinct points in <i>X</i> are separated by closed neighbourhoods. A T<sub>2½</sub> space must also be Hausdorff.</li> <li><i>X</i> is <i><a href="/wiki/Regular_space" title="Regular space">regular</a></i>, or <i>T<sub>3</sub></i>, if it is T<sub>0</sub> and if given any point <i>x</i> and closed set <i>F</i> in <i>X</i> such that <i>x</i> does not belong to <i>F</i>, they are separated by neighbourhoods. (In fact, in a regular space, any such <i>x</i> and <i>F</i> is also separated by closed neighbourhoods.)</li> <li><i>X</i> is <i><a href="/wiki/Tychonoff_space" title="Tychonoff space">Tychonoff</a></i>, or <i>T<sub>3½</sub></i>, <i>completely T<sub>3</sub></i>, or <i>completely regular</i>, if it is T<sub>0</sub> and if f, given any point <i>x</i> and closed set <i>F</i> in <i>X</i> such that <i>x</i> does not belong to <i>F</i>, they are separated by a continuous function.</li> <li><i>X</i> is <i><a href="/wiki/Normal_space" title="Normal space">normal</a></i>, or <i>T<sub>4</sub></i>, if it is Hausdorff and if any two disjoint closed subsets of <i>X</i> are separated by neighbourhoods. (In fact, a space is normal if and only if any two disjoint closed sets can be separated by a continuous function; this is <a href="/wiki/Urysohn%27s_lemma" title="Urysohn's lemma">Urysohn's lemma</a>.)</li> <li><i>X</i> is <i><a href="/wiki/Completely_normal_space" class="mw-redirect" title="Completely normal space">completely normal</a></i>, or <i>T<sub>5</sub></i> or <i>completely T<sub>4</sub></i>, if it is T<sub>1</sub> and if any two separated sets are separated by neighbourhoods. A completely normal space must also be normal.</li> <li><i>X</i> is <i><a href="/wiki/Perfectly_normal_space" class="mw-redirect" title="Perfectly normal space">perfectly normal</a></i>, or <i>T<sub>6</sub></i> or <i>perfectly T<sub>4</sub></i>, if it is T<sub>1</sub> and if any two disjoint closed sets are precisely separated by a continuous function. A perfectly normal Hausdorff space must also be completely normal Hausdorff.</li></ul> <p>The <a href="/wiki/Tietze_extension_theorem" title="Tietze extension theorem">Tietze extension theorem</a>: In a normal space, every continuous real-valued function defined on a closed subspace can be extended to a continuous map defined on the whole space. </p> <div class="mw-heading mw-heading2"><h2 id="Countability_axioms">Countability axioms</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=General_topology&action=edit&section=26" title="Edit section: Countability axioms"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></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/Axiom_of_countability" title="Axiom of countability">axiom of countability</a></div> <p>An <b>axiom of countability</b> is a <a href="/wiki/Property" title="Property">property</a> of certain <a href="/wiki/Mathematical_object" title="Mathematical object">mathematical objects</a> (usually in a <a href="/wiki/Category_(mathematics)" title="Category (mathematics)">category</a>) that requires the existence of a <a href="/wiki/Countable" class="mw-redirect" title="Countable">countable set</a> with certain properties, while without it such sets might not exist. </p><p>Important countability axioms for <a href="/wiki/Topological_space" title="Topological space">topological spaces</a>: </p> <ul><li><a href="/wiki/Sequential_space" title="Sequential space">sequential space</a>: a set is open if every <a href="/wiki/Sequence" title="Sequence">sequence</a> <a href="/wiki/Limit_of_a_sequence" title="Limit of a sequence">convergent</a> to a <a href="/wiki/Point_(geometry)" title="Point (geometry)">point</a> in the set is eventually in the set</li> <li><a href="/wiki/First-countable_space" title="First-countable space">first-countable space</a>: every point has a countable <a href="/wiki/Neighbourhood_system" title="Neighbourhood system">neighbourhood basis</a> (local base)</li> <li><a href="/wiki/Second-countable_space" title="Second-countable space">second-countable space</a>: the topology has a countable <a href="/wiki/Base_(topology)" title="Base (topology)">base</a></li> <li><a href="/wiki/Separable_space" title="Separable space">separable space</a>: there exists a countable <a href="/wiki/Dense_(topology)" class="mw-redirect" title="Dense (topology)">dense subspace</a></li> <li><a href="/wiki/Lindel%C3%B6f_space" title="Lindelöf space">Lindelöf space</a>: every <a href="/wiki/Open_cover" class="mw-redirect" title="Open cover">open cover</a> has a countable subcover</li> <li><a href="/wiki/%CE%A3-compact_space" title="Σ-compact space">σ-compact space</a>: there exists a countable cover by compact spaces</li></ul> <p>Relations: </p> <ul><li>Every first countable space is sequential.</li> <li>Every second-countable space is first-countable, separable, and Lindelöf.</li> <li>Every σ-compact space is Lindelöf.</li> <li>A <a href="/wiki/Metric_space" title="Metric space">metric space</a> is first-countable.</li> <li>For metric spaces second-countability, separability, and the Lindelöf property are all equivalent.</li></ul> <div class="mw-heading mw-heading2"><h2 id="Metric_spaces">Metric spaces</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=General_topology&action=edit&section=27" title="Edit section: Metric spaces"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></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/Metric_space" title="Metric space">Metric space</a></div> <p>A <b>metric space</b><sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> is an <a href="/wiki/Ordered_pair" title="Ordered pair">ordered pair</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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)}"></span> where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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}"></span> is a set and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e85ff03cbe0c7341af6b982e47e9f90d235c66ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.216ex; height:2.176ex;" alt="{\displaystyle d}"></span> is a <a href="/wiki/Metric_(mathematics)" class="mw-redirect" title="Metric (mathematics)">metric</a> on <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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}"></span>, i.e., a <a href="/wiki/Function_(mathematics)" title="Function (mathematics)">function</a> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d\colon M\times M\rightarrow \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <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\rightarrow \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fcf5a1da6629cc651072dab329ed2e4874d19e34" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:15.267ex; height:2.176ex;" alt="{\displaystyle d\colon M\times M\rightarrow \mathbb {R} }"></span></dd></dl> <p>such that for any <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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}"></span>, the following holds: </p> <ol><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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}"></span>     (<i>non-negative</i>),</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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"> <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="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d(x,y)=0\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6164232933841ec58e8a3470e02ef3110d288aee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.192ex; height:2.843ex;" alt="{\displaystyle d(x,y)=0\,}"></span> <a href="/wiki/If_and_only_if" title="If and only if">iff</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=y\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>=</mo> <mi>y</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x=y\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f8def71aa31e67583ef8e0eda1392b3dbd596dbe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.971ex; height:2.009ex;" alt="{\displaystyle x=y\,}"></span>     (<i><a href="/wiki/Identity_of_indiscernibles" title="Identity of indiscernibles">identity of indiscernibles</a></i>),</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d(x,y)=d(y,x)\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5567121952cd4602c1851b5fe7c62f6fdb6842d2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.574ex; height:2.843ex;" alt="{\displaystyle d(x,y)=d(y,x)\,}"></span>     (<i>symmetry</i>) and</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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)}"></span>     (<i><a href="/wiki/Triangle_inequality" title="Triangle inequality">triangle inequality</a></i>) .</li></ol> <p>The function <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e85ff03cbe0c7341af6b982e47e9f90d235c66ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.216ex; height:2.176ex;" alt="{\displaystyle d}"></span> is also called <i>distance function</i> or simply <i>distance</i>. Often, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e85ff03cbe0c7341af6b982e47e9f90d235c66ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.216ex; height:2.176ex;" alt="{\displaystyle d}"></span> is omitted and one just writes <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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}"></span> for a metric space if it is clear from the context what metric is used. </p><p>Every <a href="/wiki/Metric_space" title="Metric space">metric space</a> is <a href="/wiki/Paracompact" class="mw-redirect" title="Paracompact">paracompact</a> and <a href="/wiki/Hausdorff_space" title="Hausdorff space">Hausdorff</a>, and thus <a href="/wiki/Normal_space" title="Normal space">normal</a>. </p><p>The <a href="/wiki/Metrization_theorems" class="mw-redirect" title="Metrization theorems">metrization theorems</a> provide necessary and sufficient conditions for a topology to come from a metric. </p> <div class="mw-heading mw-heading2"><h2 id="Baire_category_theorem">Baire category theorem</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=General_topology&action=edit&section=28" title="Edit section: Baire category theorem"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></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/Baire_category_theorem" title="Baire category theorem">Baire category theorem</a></div> <p>The <a href="/wiki/Baire_category_theorem" title="Baire category theorem">Baire category theorem</a> says: If <i>X</i> is a <a href="/wiki/Completeness_(topology)" class="mw-redirect" title="Completeness (topology)">complete</a> metric space or a <a href="/wiki/Locally_compact" class="mw-redirect" title="Locally compact">locally compact</a> Hausdorff space, then the interior of every union of <a href="/wiki/Countable" class="mw-redirect" title="Countable">countably many</a> <a href="/wiki/Nowhere_dense" class="mw-redirect" title="Nowhere dense">nowhere dense</a> sets is empty.<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> </p><p>Any open subspace of a <a href="/wiki/Baire_space" title="Baire space">Baire space</a> is itself a Baire space. </p> <div class="mw-heading mw-heading2"><h2 id="Main_areas_of_research">Main areas of research</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=General_topology&action=edit&section=29" title="Edit section: Main areas of research"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Peanocurve.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/64/Peanocurve.svg/400px-Peanocurve.svg.png" decoding="async" width="400" height="122" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/64/Peanocurve.svg/600px-Peanocurve.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/64/Peanocurve.svg/800px-Peanocurve.svg.png 2x" data-file-width="930" data-file-height="284" /></a><figcaption>Three iterations of a Peano curve construction, whose limit is a space-filling curve. The Peano curve is studied in <a href="/wiki/Continuum_theory" class="mw-redirect" title="Continuum theory">continuum theory</a>, a branch of <b>general topology</b>.</figcaption></figure> <div class="mw-heading mw-heading3"><h3 id="Continuum_theory">Continuum theory</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=General_topology&action=edit&section=30" title="Edit section: Continuum theory"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></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/Continuum_(topology)" title="Continuum (topology)">Continuum (topology)</a></div> <p>A <b>continuum</b> (pl <i>continua</i>) is a nonempty <a href="/wiki/Compact_space" title="Compact space">compact</a> <a href="/wiki/Connected_space" title="Connected space">connected</a> <a href="/wiki/Metric_space" title="Metric space">metric space</a>, or less frequently, a <a href="/wiki/Compact_space" title="Compact space">compact</a> <a href="/wiki/Connected_space" title="Connected space">connected</a> <a href="/wiki/Hausdorff_space" title="Hausdorff space">Hausdorff space</a>. <b>Continuum theory</b> is the branch of topology devoted to the study of continua. These objects arise frequently in nearly all areas of topology and <a href="/wiki/Mathematical_analysis" title="Mathematical analysis">analysis</a>, and their properties are strong enough to yield many 'geometric' features. </p> <div class="mw-heading mw-heading3"><h3 id="Dynamical_systems">Dynamical systems</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=General_topology&action=edit&section=31" title="Edit section: Dynamical systems"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></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/Topological_dynamics" title="Topological dynamics">Topological dynamics</a></div> <p>Topological dynamics concerns the behavior of a space and its subspaces over time when subjected to continuous change. Many examples with applications to physics and other areas of math include <a href="/wiki/Fluid_dynamics" title="Fluid dynamics">fluid dynamics</a>, <a href="/wiki/Dynamical_billiards" title="Dynamical billiards">billiards</a> and <a href="/wiki/Geometric_flow" title="Geometric flow">flows</a> on manifolds. The topological characteristics of <a href="/wiki/Fractal" title="Fractal">fractals</a> in fractal geometry, of <a href="/wiki/Julia_set" title="Julia set">Julia sets</a> and the <a href="/wiki/Mandelbrot_set" title="Mandelbrot set">Mandelbrot set</a> arising in <a href="/wiki/Complex_dynamics" title="Complex dynamics">complex dynamics</a>, and of <a href="/wiki/Attractor" title="Attractor">attractors</a> in differential equations are often critical to understanding these systems.<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (December 2019)">citation needed</span></a></i>]</sup> </p> <div class="mw-heading mw-heading3"><h3 id="Pointless_topology">Pointless topology</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=General_topology&action=edit&section=32" title="Edit section: Pointless topology"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></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/Pointless_topology" title="Pointless topology">Pointless topology</a></div> <p><b>Pointless topology</b> (also called <b>point-free</b> or <b>pointfree topology</b>) is an approach to <a href="/wiki/Topology" title="Topology">topology</a> that avoids mentioning points. The name 'pointless topology' is due to <a href="/wiki/John_von_Neumann" title="John von Neumann">John von Neumann</a>.<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> The ideas of pointless topology are closely related to <a href="/wiki/Mereotopology" title="Mereotopology">mereotopologies</a>, in which regions (sets) are treated as foundational without explicit reference to underlying point sets. </p> <div class="mw-heading mw-heading3"><h3 id="Dimension_theory">Dimension theory</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=General_topology&action=edit&section=33" title="Edit section: Dimension theory"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></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/Dimension_theory" class="mw-redirect" title="Dimension theory">Dimension theory</a></div> <p><b>Dimension theory</b> is a branch of general topology dealing with <a href="/w/index.php?title=Dimensional_invariant&action=edit&redlink=1" class="new" title="Dimensional invariant (page does not exist)">dimensional invariants</a> of <a href="/wiki/Topological_space" title="Topological space">topological spaces</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Topological_algebras">Topological algebras</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=General_topology&action=edit&section=34" title="Edit section: Topological algebras"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></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/Topological_algebra" title="Topological algebra">Topological algebra</a></div> <p>A <b>topological algebra</b> <i>A</i> over a <a href="/wiki/Topological_field" class="mw-redirect" title="Topological field">topological field</a> <b>K</b> is a <a href="/wiki/Topological_vector_space" title="Topological vector space">topological vector space</a> together with a continuous multiplication </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cdot :A\times A\longrightarrow A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>⋅</mo> <mo>:</mo> <mi>A</mi> <mo>×</mo> <mi>A</mi> <mo stretchy="false">⟶</mo> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cdot :A\times A\longrightarrow A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/24659af2ca07029b75f338fde5908fcc1315f78d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:15.75ex; height:2.176ex;" alt="{\displaystyle \cdot :A\times A\longrightarrow A}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (a,b)\longmapsto a\cdot b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo stretchy="false">⟼</mo> <mi>a</mi> <mo>⋅</mo> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (a,b)\longmapsto a\cdot b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/84a8fc6b7e2a098917f60f31170b08c2101bace5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.073ex; height:2.843ex;" alt="{\displaystyle (a,b)\longmapsto a\cdot b}"></span></dd></dl> <p>that makes it an <a href="/wiki/Algebra_over_a_field" title="Algebra over a field">algebra</a> over <b>K</b>. A unital <a href="/wiki/Associative_algebra" title="Associative algebra">associative</a> topological algebra is a <a href="/wiki/Topological_ring" title="Topological ring">topological ring</a>. </p><p>The term was coined by <a href="/wiki/David_van_Dantzig" title="David van Dantzig">David van Dantzig</a>; it appears in the title of his <a href="/wiki/Thesis" title="Thesis">doctoral dissertation</a> (1931). </p> <div class="mw-heading mw-heading3"><h3 id="Metrizability_theory">Metrizability theory</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=General_topology&action=edit&section=35" title="Edit section: Metrizability theory"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></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/Metrization_theorem" class="mw-redirect" title="Metrization theorem">Metrization theorem</a></div> <p>In <a href="/wiki/Topology" title="Topology">topology</a> and related areas of <a href="/wiki/Mathematics" title="Mathematics">mathematics</a>, a <b>metrizable space</b> is a <a href="/wiki/Topological_space" title="Topological space">topological space</a> that is <a href="/wiki/Homeomorphism" title="Homeomorphism">homeomorphic</a> to a <a href="/wiki/Metric_space" title="Metric space">metric space</a>. That is, a topological space <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (X,\tau )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mi>τ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (X,\tau )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dede4b8004c6222d11e9b1a9802dc0496ad7e1fc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.025ex; height:2.843ex;" alt="{\displaystyle (X,\tau )}"></span> is said to be metrizable if there is a metric </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d\colon X\times X\to [0,\infty )}"> <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> <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 d\colon X\times X\to [0,\infty )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/96fd1e6761c77af90cb76c9994fd10b52c9621e1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.736ex; height:2.843ex;" alt="{\displaystyle d\colon X\times X\to [0,\infty )}"></span></dd></dl> <p>such that the topology induced by <i>d</i> is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tau }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>τ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tau }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/38a7dcde9730ef0853809fefc18d88771f95206c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.202ex; height:1.676ex;" alt="{\displaystyle \tau }"></span>. <b>Metrization theorems</b> are <a href="/wiki/Theorem" title="Theorem">theorems</a> that give <a href="/wiki/Sufficient_condition" class="mw-redirect" title="Sufficient condition">sufficient conditions</a> for a topological space to be metrizable. </p> <div class="mw-heading mw-heading3"><h3 id="Set-theoretic_topology">Set-theoretic topology</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=General_topology&action=edit&section=36" title="Edit section: Set-theoretic topology"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></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/Set-theoretic_topology" title="Set-theoretic topology">Set-theoretic topology</a></div> <p>Set-theoretic topology is a subject that combines set theory and general topology. It focuses on topological questions that are independent of <a href="/wiki/Zermelo%E2%80%93Fraenkel_set_theory" title="Zermelo–Fraenkel set theory">Zermelo–Fraenkel set theory</a> (ZFC). A famous problem is <a href="/wiki/Moore_space_(topology)#Normal_Moore_space_conjecture" title="Moore space (topology)">the normal Moore space question</a>, a question in general topology that was the subject of intense research. The answer to the normal Moore space question was eventually proved to be independent of ZFC. </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=General_topology&action=edit&section=37" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/List_of_examples_in_general_topology" title="List of examples in general topology">List of examples in general topology</a></li> <li><a href="/wiki/Glossary_of_general_topology" title="Glossary of general topology">Glossary of general topology</a> for detailed definitions</li> <li><a href="/wiki/List_of_general_topology_topics" title="List of general topology topics">List of general topology topics</a> for related articles</li> <li><a href="/wiki/Category_of_topological_spaces" title="Category of topological spaces">Category of topological spaces</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=General_topology&action=edit&section=38" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></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"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text">Munkres, James R. Topology. Vol. 2. Upper Saddle River: Prentice Hall, 2000.</span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text">Adams, Colin Conrad, and Robert David Franzosa. Introduction to topology: pure and applied. Pearson Prentice Hall, 2008.</span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><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="CITEREFMerrifieldSimmons1989" class="citation book cs1">Merrifield, Richard E.; <a href="/wiki/Howard_Ensign_Simmons_Jr." title="Howard Ensign Simmons Jr.">Simmons, Howard E.</a> (1989). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/topologicalmetho00merr/page/16"><i>Topological Methods in Chemistry</i></a></span>. New York: John Wiley & Sons. pp. <a rel="nofollow" class="external text" href="https://archive.org/details/topologicalmetho00merr/page/16">16</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-471-83817-9" title="Special:BookSources/0-471-83817-9"><bdi>0-471-83817-9</bdi></a><span class="reference-accessdate">. Retrieved <span class="nowrap">27 July</span> 2012</span>. <q><b>Definition.</b> A collection <i>B</i> of subsets of a topological space <i>(X,T)</i> is called a <i>basis</i> for <i>T</i> if every open set can be expressed as a union of members of <i>B</i>.</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Topological+Methods+in+Chemistry&rft.place=New+York&rft.pages=16&rft.pub=John+Wiley+%26+Sons&rft.date=1989&rft.isbn=0-471-83817-9&rft.aulast=Merrifield&rft.aufirst=Richard+E.&rft.au=Simmons%2C+Howard+E.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Ftopologicalmetho00merr%2Fpage%2F16&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGeneral+topology" class="Z3988"></span></span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFArmstrong1983" class="citation book cs1">Armstrong, M. A. (1983). <a rel="nofollow" class="external text" href="https://www.springer.com/mathematics/geometry/book/978-0-387-90839-7"><i>Basic Topology</i></a>. Springer. p. 30. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-387-90839-0" title="Special:BookSources/0-387-90839-0"><bdi>0-387-90839-0</bdi></a><span class="reference-accessdate">. Retrieved <span class="nowrap">13 June</span> 2013</span>. <q>Suppose we have a topology on a set <i>X</i>, and a collection <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>β</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ed48a5e36207156fb792fa79d29925d2f7901e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.332ex; height:2.509ex;" alt="{\displaystyle \beta }"></span> of open sets such that every open set is a union of members of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>β</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ed48a5e36207156fb792fa79d29925d2f7901e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.332ex; height:2.509ex;" alt="{\displaystyle \beta }"></span>. Then <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>β</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ed48a5e36207156fb792fa79d29925d2f7901e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.332ex; height:2.509ex;" alt="{\displaystyle \beta }"></span> is called a <i>base</i> for the topology...</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Basic+Topology&rft.pages=30&rft.pub=Springer&rft.date=1983&rft.isbn=0-387-90839-0&rft.aulast=Armstrong&rft.aufirst=M.+A.&rft_id=https%3A%2F%2Fwww.springer.com%2Fmathematics%2Fgeometry%2Fbook%2F978-0-387-90839-7&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGeneral+topology" class="Z3988"></span></span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMooreSmith1922" class="citation journal cs1"><a href="/wiki/E._H._Moore" title="E. H. Moore">Moore, E. H.</a>; <a href="/wiki/Herman_L._Smith" title="Herman L. Smith">Smith, H. L.</a> (1922). "A General Theory of Limits". <i>American Journal of Mathematics</i>. <b>44</b> (2): 102–121. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F2370388">10.2307/2370388</a>. <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2370388">2370388</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=American+Journal+of+Mathematics&rft.atitle=A+General+Theory+of+Limits&rft.volume=44&rft.issue=2&rft.pages=102-121&rft.date=1922&rft_id=info%3Adoi%2F10.2307%2F2370388&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2370388%23id-name%3DJSTOR&rft.aulast=Moore&rft.aufirst=E.+H.&rft.au=Smith%2C+H.+L.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGeneral+topology" class="Z3988"></span></span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHeine1872" class="citation journal cs1">Heine, E. (1872). <a rel="nofollow" class="external text" href="http://eudml.org/doc/148175">"Die Elemente der Functionenlehre."</a> <i>Journal für die reine und angewandte Mathematik</i>. <b>74</b>: 172–188.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+f%C3%BCr+die+reine+und+angewandte+Mathematik&rft.atitle=Die+Elemente+der+Functionenlehre..&rft.volume=74&rft.pages=172-188&rft.date=1872&rft.aulast=Heine&rft.aufirst=E.&rft_id=http%3A%2F%2Feudml.org%2Fdoc%2F148175&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGeneral+topology" class="Z3988"></span></span> </li> <li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text"><a href="/wiki/Maurice_Fr%C3%A9chet" class="mw-redirect" title="Maurice Fréchet">Maurice Fréchet</a> introduced metric spaces in his work <i>Sur quelques points du calcul fonctionnel</i>, Rendic. Circ. Mat. Palermo 22 (1906) 1–74.</span> </li> <li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text">R. Baire. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=cS4LAAAAYAAJ">Sur les fonctions de variables réelles.</a> Ann. di Mat., 3:1–123, 1899.</span> </li> <li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text">Garrett Birkhoff, <i>VON NEUMANN AND LATTICE THEORY</i>, <i>John Von Neumann 1903-1957</i>, J. C. Oxtoley, B. J. Pettis, American Mathematical Soc., 1958, page 50-5</span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=General_topology&action=edit&section=39" title="Edit section: Further reading"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Some standard books on general topology include: </p> <ul><li><a href="/wiki/Nicolas_Bourbaki" title="Nicolas Bourbaki">Bourbaki</a>, <cite>Topologie Générale</cite> (<cite>General Topology</cite>), <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-387-19374-X" title="Special:BookSources/0-387-19374-X">0-387-19374-X</a>.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKelley1975" class="citation book cs1"><a href="/wiki/John_L._Kelley" title="John L. Kelley">Kelley, John L.</a> (1975). <i>General Topology</i> (2nd ed.). <a href="/wiki/Springer_Science%2BBusiness_Media" title="Springer Science+Business Media">Springer-Verlag</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-90125-1" title="Special:BookSources/978-0-387-90125-1"><bdi>978-0-387-90125-1</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&rft.edition=2nd&rft.pub=Springer-Verlag&rft.date=1975&rft.isbn=978-0-387-90125-1&rft.aulast=Kelley&rft.aufirst=John+L.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGeneral+topology" class="Z3988"></span> (<a rel="nofollow" class="external text" href="https://archive.org/details/GeneralTopologyJohnL.Kelley">1st ed., 1955</a>)</li> <li>Stephen Willard, <cite>General Topology</cite>, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-486-43479-6" title="Special:BookSources/0-486-43479-6">0-486-43479-6</a>.</li> <li><a href="/wiki/James_Munkres" title="James Munkres">James Munkres</a>, <cite>Topology</cite>, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-13-181629-2" title="Special:BookSources/0-13-181629-2">0-13-181629-2</a>.</li> <li><a href="/wiki/George_F._Simmons" title="George F. Simmons">George F. Simmons</a>, <cite>Introduction to Topology and Modern Analysis</cite>, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/1-575-24238-9" title="Special:BookSources/1-575-24238-9">1-575-24238-9</a>.</li> <li><a href="/w/index.php?title=Paul_L._Shick&action=edit&redlink=1" class="new" title="Paul L. Shick (page does not exist)">Paul L. Shick</a>, <cite>Topology: Point-Set and Geometric</cite>, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-470-09605-5" title="Special:BookSources/0-470-09605-5">0-470-09605-5</a>.</li> <li><a href="/wiki/Ryszard_Engelking" title="Ryszard Engelking">Ryszard Engelking</a>, <cite>General Topology</cite>, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/3-88538-006-4" title="Special:BookSources/3-88538-006-4">3-88538-006-4</a>.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSteenSeebach1995" class="citation cs2"><a href="/wiki/Lynn_Arthur_Steen" class="mw-redirect" title="Lynn Arthur Steen">Steen, Lynn Arthur</a>; <a href="/wiki/J._Arthur_Seebach_Jr." title="J. Arthur Seebach Jr.">Seebach, J. Arthur Jr.</a> (1995) [1978], <a href="/wiki/Counterexamples_in_Topology" title="Counterexamples in Topology"><i>Counterexamples in Topology</i></a> (<a href="/wiki/Dover_Publications" title="Dover Publications">Dover</a> reprint of 1978 ed.), Berlin, New York: <a href="/wiki/Springer-Verlag" class="mw-redirect" title="Springer-Verlag">Springer-Verlag</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-486-68735-3" title="Special:BookSources/978-0-486-68735-3"><bdi>978-0-486-68735-3</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=0507446">0507446</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Counterexamples+in+Topology&rft.place=Berlin%2C+New+York&rft.edition=Dover+reprint+of+1978&rft.pub=Springer-Verlag&rft.date=1995&rft.isbn=978-0-486-68735-3&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D507446%23id-name%3DMR&rft.aulast=Steen&rft.aufirst=Lynn+Arthur&rft.au=Seebach%2C+J.+Arthur+Jr.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGeneral+topology" class="Z3988"></span></li> <li>O.Ya. Viro, O.A. Ivanov, V.M. Kharlamov and N.Yu. Netsvetaev, <a rel="nofollow" class="external text" href="https://www.ams.org/bookstore-getitem/item=mbk-54"><cite>Elementary Topology: Textbook in Problems</cite></a>, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-8218-4506-6" title="Special:BookSources/978-0-8218-4506-6">978-0-8218-4506-6</a>.</li></ul> <p>The <a href="/wiki/ArXiv" title="ArXiv">arXiv</a> subject code is <a rel="nofollow" class="external text" href="https://arxiv.org/list/math.GN/recent">math.GN</a>. </p> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=General_topology&action=edit&section=40" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><span 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href="/wiki/Template:Topology" title="Template:Topology"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Topology" title="Template talk:Topology"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Topology" title="Special:EditPage/Template:Topology"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Topology" style="font-size:114%;margin:0 4em"><a href="/wiki/Topology" title="Topology">Topology</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%;background:#e5e5ff;">Fields</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a class="mw-selflink selflink">General (point-set)</a></li> <li><a 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><span typeof="mw:File"><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></span></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 href="/wiki/Metric_space" title="Metric space">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> <li><a href="/wiki/Polyhedral_complex" title="Polyhedral complex">Polyhedral complex</a></li> <li><a href="/wiki/Manifold" title="Manifold">Manifold</a></li> <li><a href="/wiki/Bundle_(mathematics)" title="Bundle (mathematics)">Bundle (mathematics)</a></li> <li><a href="/wiki/Second-countable_space" title="Second-countable space">Second-countable space</a></li> <li><a href="/wiki/Cobordism" title="Cobordism">Cobordism</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;background:#e5e5ff;">Metrics and properties</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Euler_characteristic" title="Euler characteristic">Euler characteristic</a></li> <li><a href="/wiki/Betti_number" title="Betti number">Betti number</a></li> <li><a href="/wiki/Winding_number" title="Winding number">Winding number</a></li> <li><a href="/wiki/Chern_class" title="Chern class">Chern 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> 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class="navbox" aria-labelledby="Major_mathematics_areas" style="padding:3px"><table class="nowraplinks hlist mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Areas_of_mathematics" title="Template:Areas of mathematics"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Areas_of_mathematics" title="Template talk:Areas of mathematics"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Areas_of_mathematics" title="Special:EditPage/Template:Areas of mathematics"><abbr title="Edit this 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style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Category_theory" title="Category theory">Category theory</a></li> <li><a href="/wiki/Information_theory" title="Information theory">Information theory</a></li> <li><a href="/wiki/Mathematical_logic" title="Mathematical logic">Mathematical logic</a></li> <li><a href="/wiki/Philosophy_of_mathematics" title="Philosophy of mathematics">Philosophy of mathematics</a></li> <li><a href="/wiki/Set_theory" title="Set theory">Set theory</a></li> <li><a href="/wiki/Type_theory" title="Type theory">Type theory</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Algebra" title="Algebra">Algebra</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/Abstract_algebra" title="Abstract algebra">Abstract</a></li> <li><a href="/wiki/Commutative_algebra" title="Commutative 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style="width:1%"><a href="/wiki/Number_theory" title="Number theory">Number theory</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/Arithmetic" title="Arithmetic">Arithmetic</a></li> <li><a href="/wiki/Algebraic_number_theory" title="Algebraic number theory">Algebraic number theory</a></li> <li><a href="/wiki/Analytic_number_theory" title="Analytic number theory">Analytic number theory</a></li> <li><a href="/wiki/Diophantine_geometry" title="Diophantine geometry">Diophantine geometry</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Topology" title="Topology">Topology</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a class="mw-selflink selflink">General</a></li> <li><a href="/wiki/Algebraic_topology" title="Algebraic topology">Algebraic</a></li> 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