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<ul id="toc-Open_cover_definition-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Compactness_of_subsets" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Compactness_of_subsets"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>Compactness of subsets</span> </div> </a> <ul id="toc-Compactness_of_subsets-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Characterization" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Characterization"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.3</span> <span>Characterization</span> </div> </a> <ul id="toc-Characterization-sublist" class="vector-toc-list"> <li id="toc-Euclidean_space" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Euclidean_space"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.3.1</span> <span>Euclidean space</span> </div> </a> <ul id="toc-Euclidean_space-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Metric_spaces" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Metric_spaces"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.3.2</span> <span>Metric spaces</span> </div> </a> <ul id="toc-Metric_spaces-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Ordered_spaces" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Ordered_spaces"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.3.3</span> <span>Ordered spaces</span> </div> </a> <ul id="toc-Ordered_spaces-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Characterization_by_continuous_functions" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Characterization_by_continuous_functions"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.3.4</span> <span>Characterization by continuous functions</span> </div> </a> <ul id="toc-Characterization_by_continuous_functions-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Hyperreal_definition" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Hyperreal_definition"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.3.5</span> <span>Hyperreal definition</span> </div> </a> <ul id="toc-Hyperreal_definition-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> </ul> </li> <li id="toc-Sufficient_conditions" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Sufficient_conditions"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Sufficient conditions</span> </div> </a> <ul id="toc-Sufficient_conditions-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Properties_of_compact_spaces" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Properties_of_compact_spaces"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Properties of compact spaces</span> </div> </a> <button aria-controls="toc-Properties_of_compact_spaces-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 Properties of compact spaces subsection</span> </button> <ul id="toc-Properties_of_compact_spaces-sublist" class="vector-toc-list"> <li id="toc-Functions_and_compact_spaces" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Functions_and_compact_spaces"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.1</span> <span>Functions and compact spaces</span> </div> </a> <ul id="toc-Functions_and_compact_spaces-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Compactifications" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Compactifications"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.2</span> <span>Compactifications</span> </div> </a> <ul id="toc-Compactifications-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Ordered_compact_spaces" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Ordered_compact_spaces"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.3</span> <span>Ordered compact spaces</span> </div> </a> <ul id="toc-Ordered_compact_spaces-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Examples" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Examples"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Examples</span> </div> </a> <button aria-controls="toc-Examples-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 Examples subsection</span> </button> <ul id="toc-Examples-sublist" class="vector-toc-list"> <li id="toc-Algebraic_examples" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Algebraic_examples"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.1</span> <span>Algebraic examples</span> </div> </a> <ul id="toc-Algebraic_examples-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Notes" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Notes"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>Notes</span> </div> </a> <ul id="toc-Notes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Bibliography" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Bibliography"> <div class="vector-toc-text"> <span class="vector-toc-numb">10</span> <span>Bibliography</span> </div> </a> <ul id="toc-Bibliography-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">11</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">Compact space</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 38 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-38" 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">38 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/%D9%81%D8%B6%D8%A7%D8%A1_%D9%85%D8%AA%D8%B1%D8%A7%D8%B5" title="فضاء متراص – Arabic" lang="ar" hreflang="ar" data-title="فضاء متراص" 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/Espaciu_compautu" title="Espaciu compautu – Asturian" lang="ast" hreflang="ast" data-title="Espaciu compautu" data-language-autonym="Asturianu" data-language-local-name="Asturian" class="interlanguage-link-target"><span>Asturianu</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Espai_compacte" title="Espai compacte – Catalan" lang="ca" hreflang="ca" data-title="Espai compacte" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Kompaktn%C3%AD_mno%C5%BEina" title="Kompaktní množina – Czech" lang="cs" hreflang="cs" data-title="Kompaktní množina" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Kompakter_Raum" title="Kompakter Raum – German" lang="de" hreflang="de" data-title="Kompakter Raum" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%A3%CF%85%CE%BC%CF%80%CE%B1%CE%B3%CE%AE%CF%82_%CF%87%CF%8E%CF%81%CE%BF%CF%82" title="Συμπαγής χώρος – Greek" lang="el" hreflang="el" data-title="Συμπαγής χώρος" 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/Espacio_compacto" title="Espacio compacto – Spanish" lang="es" hreflang="es" data-title="Espacio compacto" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Kompakta_spaco" title="Kompakta spaco – Esperanto" lang="eo" hreflang="eo" data-title="Kompakta spaco" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Espazio_trinko" title="Espazio trinko – Basque" lang="eu" hreflang="eu" data-title="Espazio trinko" 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/%D9%81%D8%B6%D8%A7%DB%8C_%D9%81%D8%B4%D8%B1%D8%AF%D9%87" title="فضای فشرده – Persian" lang="fa" hreflang="fa" data-title="فضای فشرده" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Compacit%C3%A9_(math%C3%A9matiques)" title="Compacité (mathématiques) – French" lang="fr" hreflang="fr" data-title="Compacité (mathématiques)" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Espazo_compacto" title="Espazo compacto – Galician" lang="gl" hreflang="gl" data-title="Espazo compacto" 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%BD%A4%ED%8C%A9%ED%8A%B8_%EA%B3%B5%EA%B0%84" title="콤팩트 공간 – Korean" lang="ko" hreflang="ko" data-title="콤팩트 공간" 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/Ruang_kompak" title="Ruang kompak – Indonesian" lang="id" hreflang="id" data-title="Ruang kompak" 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-is mw-list-item"><a href="https://is.wikipedia.org/wiki/%C3%9Ejappa%C3%B0_mengi" title="Þjappað mengi – Icelandic" lang="is" hreflang="is" data-title="Þjappað mengi" data-language-autonym="Íslenska" data-language-local-name="Icelandic" class="interlanguage-link-target"><span>Íslenska</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Spazio_compatto" title="Spazio compatto – Italian" lang="it" hreflang="it" data-title="Spazio compatto" 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%A7%D7%91%D7%95%D7%A6%D7%94_%D7%A7%D7%95%D7%9E%D7%A4%D7%A7%D7%98%D7%99%D7%AA" title="קבוצה קומפקטית – Hebrew" lang="he" hreflang="he" data-title="קבוצה קומפקטית" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%9A%D0%BE%D0%BC%D0%BF%D0%B0%D0%BA%D1%82" title="Компакт – Kazakh" lang="kk" hreflang="kk" data-title="Компакт" data-language-autonym="Қазақша" data-language-local-name="Kazakh" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-ky mw-list-item"><a href="https://ky.wikipedia.org/wiki/%D0%9A%D0%BE%D0%BC%D0%BF%D0%B0%D0%BA%D1%82%D1%82%D1%83%D1%83%D0%BB%D1%83%D0%BA" title="Компакттуулук – Kyrgyz" lang="ky" hreflang="ky" data-title="Компакттуулук" data-language-autonym="Кыргызча" data-language-local-name="Kyrgyz" class="interlanguage-link-target"><span>Кыргызча</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Kompakts%C3%A1g" title="Kompaktság – Hungarian" lang="hu" hreflang="hu" data-title="Kompaktság" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Compacte_ruimte" title="Compacte ruimte – Dutch" lang="nl" hreflang="nl" data-title="Compacte ruimte" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E3%82%B3%E3%83%B3%E3%83%91%E3%82%AF%E3%83%88%E7%A9%BA%E9%96%93" title="コンパクト空間 – Japanese" lang="ja" hreflang="ja" data-title="コンパクト空間" 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-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Przestrze%C5%84_zwarta" title="Przestrzeń zwarta – Polish" lang="pl" hreflang="pl" data-title="Przestrzeń zwarta" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Espa%C3%A7o_compacto" title="Espaço compacto – Portuguese" lang="pt" hreflang="pt" data-title="Espaço compacto" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Spa%C8%9Biu_compact" title="Spațiu compact – Romanian" lang="ro" hreflang="ro" data-title="Spațiu compact" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%9A%D0%BE%D0%BC%D0%BF%D0%B0%D0%BA%D1%82%D0%BD%D0%BE%D0%B5_%D0%BF%D1%80%D0%BE%D1%81%D1%82%D1%80%D0%B0%D0%BD%D1%81%D1%82%D0%B2%D0%BE" title="Компактное пространство – Russian" lang="ru" hreflang="ru" data-title="Компактное пространство" 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/Compact_space" title="Compact space – Simple English" lang="en-simple" hreflang="en-simple" data-title="Compact space" 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-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Kompaktn%C3%A1_mno%C5%BEina" title="Kompaktná množina – Slovak" lang="sk" hreflang="sk" data-title="Kompaktná množina" data-language-autonym="Slovenčina" data-language-local-name="Slovak" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/Kompaktan_prostor" title="Kompaktan prostor – Serbian" lang="sr" hreflang="sr" data-title="Kompaktan prostor" 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/Kompaktius" title="Kompaktius – Finnish" lang="fi" hreflang="fi" data-title="Kompaktius" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Kompakthet" title="Kompakthet – Swedish" lang="sv" hreflang="sv" data-title="Kompakthet" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/T%C4%B1k%C4%B1zl%C4%B1k" title="Tıkızlık – Turkish" lang="tr" hreflang="tr" data-title="Tıkızlık" data-language-autonym="Türkçe" data-language-local-name="Turkish" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%9A%D0%BE%D0%BC%D0%BF%D0%B0%D0%BA%D1%82%D0%BD%D0%B8%D0%B9_%D0%BF%D1%80%D0%BE%D1%81%D1%82%D1%96%D1%80" title="Компактний простір – Ukrainian" lang="uk" hreflang="uk" data-title="Компактний простір" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/Compact" title="Compact – Vietnamese" lang="vi" hreflang="vi" data-title="Compact" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamese" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-zh-classical mw-list-item"><a href="https://zh-classical.wikipedia.org/wiki/%E7%B7%8A%E9%9B%86" title="緊集 – Literary Chinese" lang="lzh" hreflang="lzh" data-title="緊集" data-language-autonym="文言" data-language-local-name="Literary Chinese" 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id="siteSub" class="noprint">From Wikipedia, the free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"><span class="mw-redirectedfrom">(Redirected from <a href="/w/index.php?title=Compactness&redirect=no" class="mw-redirect" title="Compactness">Compactness</a>)</span></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><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">"Compactness" redirects here. For other uses, see <a href="/wiki/Compactness_(disambiguation)" class="mw-disambig" title="Compactness (disambiguation)">Compactness (disambiguation)</a>.</div> <div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Type of mathematical space</div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Compact.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/7c/Compact.svg/350px-Compact.svg.png" decoding="async" width="350" height="163" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/7c/Compact.svg/525px-Compact.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/7c/Compact.svg/700px-Compact.svg.png 2x" data-file-width="512" data-file-height="238" /></a><figcaption> Per the compactness criteria for Euclidean space as stated in the <a href="/wiki/Heine%E2%80%93Borel_theorem" title="Heine–Borel theorem">Heine–Borel theorem</a>, the interval <span class="texhtml"><i>A</i> = (−∞, −2]</span> is not compact because it is not bounded. The interval <span class="texhtml"><i>C</i> = (2, 4)</span> is not compact because it is not closed (but bounded). The interval <span class="texhtml"><i>B</i> = [0, 1]</span> is compact because it is both closed and bounded.</figcaption></figure> <p>In <a href="/wiki/Mathematics" title="Mathematics">mathematics</a>, specifically <a href="/wiki/General_topology" title="General topology">general topology</a>, <b>compactness</b> is a property that seeks to generalize the notion of a <a href="/wiki/Closed_set" title="Closed set">closed</a> and <a href="/wiki/Bounded_set" title="Bounded set">bounded</a> subset of <a href="/wiki/Euclidean_space" title="Euclidean space">Euclidean space</a>.<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> The idea is that a compact space has no "punctures" or "missing endpoints", i.e., it includes all <a href="/wiki/Limit_(mathematics)" title="Limit (mathematics)"><i>limiting values</i></a> of points. For example, the open <a href="/wiki/Interval_(mathematics)" title="Interval (mathematics)">interval</a> (0,1) would not be compact because it excludes the limiting values of 0 and 1, whereas the closed interval [0,1] would be compact. Similarly, the space of <a href="/wiki/Rational_number" title="Rational number">rational numbers</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 \mathbb {Q} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5909f0b54e4718fa24d5fd34d54189d24a66e9a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.808ex; height:2.509ex;" alt="{\displaystyle \mathbb {Q} }"></span> is not compact, because it has infinitely many "punctures" corresponding to the <a href="/wiki/Irrational_number" title="Irrational number">irrational numbers</a>, and the space of <a href="/wiki/Real_number" title="Real number">real numbers</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 \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }"></span> is not compact either, because it excludes the two limiting values <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 +\infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>+</mo> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle +\infty }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bddbb0e4420a7e744cf71bd71216e11b0bf88831" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:4.132ex; height:2.176ex;" alt="{\displaystyle +\infty }"></span> 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 -\infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -\infty }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ca2608c4b5fd3bffc73585f8c67e379b4e99b6f1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:4.132ex; height:2.176ex;" alt="{\displaystyle -\infty }"></span>. However, the <a href="/wiki/Extended_real_numbers" class="mw-redirect" title="Extended real numbers"><i>extended</i> real number line</a> <i>would</i> be compact, since it contains both infinities. There are many ways to make this heuristic notion precise. These ways usually agree in a <a href="/wiki/Metric_space" title="Metric space">metric space</a>, but may not be <a href="/wiki/Logical_equivalence" title="Logical equivalence">equivalent</a> in other <a href="/wiki/Topological_space" title="Topological space">topological spaces</a>. </p><p>One such generalization is that a topological space is <a href="/wiki/Sequentially_compact" class="mw-redirect" title="Sequentially compact"><i>sequentially</i> compact</a> if every <a href="/wiki/Infinite_sequence" class="mw-redirect" title="Infinite sequence">infinite sequence</a> of points sampled from the space has an infinite <a href="/wiki/Subsequence" title="Subsequence">subsequence</a> that converges to some point of the space.<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> The <a href="/wiki/Bolzano%E2%80%93Weierstrass_theorem" title="Bolzano–Weierstrass theorem">Bolzano–Weierstrass theorem</a> states that a subset of Euclidean space is compact in this sequential sense if and only if it is closed and bounded. Thus, if one chooses an infinite number of points in the closed <a href="/wiki/Unit_interval" title="Unit interval">unit interval</a> <span class="texhtml">[0, 1]</span>, some of those points will get arbitrarily close to some real number in that space. For instance, some of the numbers in the sequence <span class="nowrap"> <style data-mw-deduplicate="TemplateStyles:r1214402035">.mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num{display:block;line-height:1em;margin:0.0em 0.1em;border-bottom:1px solid}.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0.1em 0.1em}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}</style><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span>, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">4</span><span class="sr-only">/</span><span class="den">5</span></span>⁠</span>, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">3</span></span>⁠</span>, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">5</span><span class="sr-only">/</span><span class="den">6</span></span>⁠</span>, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">4</span></span>⁠</span>, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">6</span><span class="sr-only">/</span><span class="den">7</span></span>⁠</span>, ...</span> accumulate to 0 (while others accumulate to 1). Since neither 0 nor 1 are members of the open unit interval <span class="texhtml">(0, 1)</span>, those same sets of points would not accumulate to any point of it, so the open unit interval is not compact. Although subsets (subspaces) of Euclidean space can be compact, the entire space itself is not compact, since it is not bounded. For example, considering <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 \mathbb {R} ^{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec2e282911e406fc800fb1095093667d66f18c7f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.732ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{1}}"></span> (the real number line), the sequence of points <span class="nowrap"> 0, 1, 2, 3, ...</span> has no subsequence that converges to any real number. </p><p>Compactness was formally introduced by <a href="/wiki/Maurice_Fr%C3%A9chet" class="mw-redirect" title="Maurice Fréchet">Maurice Fréchet</a> in 1906 to generalize the Bolzano–Weierstrass theorem from spaces of geometrical points to <a href="/wiki/Function_space" title="Function space">spaces of functions</a>. The <a href="/wiki/Arzel%C3%A0%E2%80%93Ascoli_theorem" title="Arzelà–Ascoli theorem">Arzelà–Ascoli theorem</a> and the <a href="/wiki/Peano_existence_theorem" title="Peano existence theorem">Peano existence theorem</a> exemplify applications of this notion of compactness to classical analysis. Following its initial introduction, various equivalent notions of compactness, including <a href="/wiki/Sequentially_compact_space" title="Sequentially compact space">sequential compactness</a> and <a href="/wiki/Limit_point_compact" title="Limit point compact">limit point compactness</a>, were developed in general <a href="/wiki/Metric_space" title="Metric space">metric spaces</a>.<sup id="cite_ref-:0_3-0" class="reference"><a href="#cite_note-:0-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> In general topological spaces, however, these notions of compactness are not necessarily equivalent. The most useful notion — and the standard definition of the unqualified term <i>compactness</i> — is phrased in terms of the existence of finite families of <a href="/wiki/Open_set" title="Open set">open sets</a> that "<a href="/wiki/Cover_(topology)" title="Cover (topology)">cover</a>" the space, in the sense that each point of the space lies in some set contained in the family. This more subtle notion, introduced by <a href="/wiki/Pavel_Alexandrov" title="Pavel Alexandrov">Pavel Alexandrov</a> and <a href="/wiki/Pavel_Urysohn" title="Pavel Urysohn">Pavel Urysohn</a> in 1929, exhibits compact spaces as generalizations of <a href="/wiki/Finite_set" title="Finite set">finite sets</a>. In spaces that are compact in this sense, it is often possible to patch together information that holds <a href="/wiki/Local_property" title="Local property">locally</a> – that is, in a neighborhood of each point – into corresponding statements that hold throughout the space, and many theorems are of this character. </p><p>The term <b>compact set</b> is sometimes used as a synonym for compact space, but also often refers to a <a href="#Compactness_of_subsets">compact subspace</a> of a <a href="/wiki/Topological_space" title="Topological space">topological space</a>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Historical_development">Historical development</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Compact_space&action=edit&section=1" title="Edit section: Historical development"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In the 19th century, several disparate mathematical properties were understood that would later be seen as consequences of compactness. On the one hand, <a href="/wiki/Bernard_Bolzano" title="Bernard Bolzano">Bernard Bolzano</a> (<a href="#CITEREFBolzano1817">1817</a>) had been aware that any bounded sequence of points (in the line or plane, for instance) has a subsequence that must eventually get arbitrarily close to some other point, called a <a href="/wiki/Limit_point_of_a_sequence" class="mw-redirect" title="Limit point of a sequence">limit point</a>. Bolzano's proof relied on the <a href="/wiki/Method_of_bisection" class="mw-redirect" title="Method of bisection">method of bisection</a>: the sequence was placed into an interval that was then divided into two equal parts, and a part containing infinitely many terms of the sequence was selected. The process could then be repeated by dividing the resulting smaller interval into smaller and smaller parts – until it closes down on the desired limit point. The full significance of <a href="/wiki/Bolzano%E2%80%93Weierstrass_theorem" title="Bolzano–Weierstrass theorem">Bolzano's theorem</a>, and its method of proof, would not emerge until almost 50 years later when it was rediscovered by <a href="/wiki/Karl_Weierstrass" title="Karl Weierstrass">Karl Weierstrass</a>.<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> </p><p>In the 1880s, it became clear that results similar to the Bolzano–Weierstrass theorem could be formulated for <a href="/wiki/Function_space" title="Function space">spaces of functions</a> rather than just numbers or geometrical points. The idea of regarding functions as themselves points of a generalized space dates back to the investigations of <a href="/wiki/Giulio_Ascoli" title="Giulio Ascoli">Giulio Ascoli</a> and <a href="/wiki/Cesare_Arzel%C3%A0" title="Cesare Arzelà">Cesare Arzelà</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> The culmination of their investigations, the <a href="/wiki/Arzel%C3%A0%E2%80%93Ascoli_theorem" title="Arzelà–Ascoli theorem">Arzelà–Ascoli theorem</a>, was a generalization of the Bolzano–Weierstrass theorem to families of <a href="/wiki/Continuous_function" title="Continuous function">continuous functions</a>, the precise conclusion of which was that it was possible to extract a <a href="/wiki/Uniform_convergence" title="Uniform convergence">uniformly convergent</a> sequence of functions from a suitable family of functions. The uniform limit of this sequence then played precisely the same role as Bolzano's "limit point". Towards the beginning of the twentieth century, results similar to that of Arzelà and Ascoli began to accumulate in the area of <a href="/wiki/Integral_equation" title="Integral equation">integral equations</a>, as investigated by <a href="/wiki/David_Hilbert" title="David Hilbert">David Hilbert</a> and <a href="/wiki/Erhard_Schmidt" title="Erhard Schmidt">Erhard Schmidt</a>. For a certain class of <a href="/wiki/Green%27s_functions" class="mw-redirect" title="Green's functions">Green's functions</a> coming from solutions of integral equations, Schmidt had shown that a property analogous to the Arzelà–Ascoli theorem held in the sense of <a href="/wiki/Mean_convergence" class="mw-redirect" title="Mean convergence">mean convergence</a> – or convergence in what would later be dubbed a <a href="/wiki/Hilbert_space" title="Hilbert space">Hilbert space</a>. This ultimately led to the notion of a <a href="/wiki/Compact_operator" title="Compact operator">compact operator</a> as an offshoot of the general notion of a compact space. It was <a href="/wiki/Maurice_Ren%C3%A9_Fr%C3%A9chet" class="mw-redirect" title="Maurice René Fréchet">Maurice Fréchet</a> who, in <a href="#CITEREFFréchet1906">1906</a>, had distilled the essence of the Bolzano–Weierstrass property and coined the term <i>compactness</i> to refer to this general phenomenon (he used the term already in his 1904 paper<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> which led to the famous 1906 thesis). </p><p>However, a different notion of compactness altogether had also slowly emerged at the end of the 19th century from the study of the <a href="/wiki/Linear_continuum" title="Linear continuum">continuum</a>, which was seen as fundamental for the rigorous formulation of analysis. In 1870, <a href="/wiki/Eduard_Heine" title="Eduard Heine">Eduard Heine</a> showed that a <a href="/wiki/Continuous_function" title="Continuous function">continuous function</a> defined on a closed and bounded interval was in fact <a href="/wiki/Uniformly_continuous" class="mw-redirect" title="Uniformly continuous">uniformly continuous</a>. In the course of the proof, he made use of a lemma that from any countable cover of the interval by smaller open intervals, it was possible to select a finite number of these that also covered it. The significance of this lemma was recognized by <a href="/wiki/%C3%89mile_Borel" title="Émile Borel">Émile Borel</a> (<a href="#CITEREFBorel1895">1895</a>), and it was generalized to arbitrary collections of intervals by <a href="/w/index.php?title=Pierre_Cousin_(mathematician)&action=edit&redlink=1" class="new" title="Pierre Cousin (mathematician) (page does not exist)">Pierre Cousin</a> (1895) and <a href="/wiki/Henri_Lebesgue" title="Henri Lebesgue">Henri Lebesgue</a> (<a href="#CITEREFLebesgue1904">1904</a>). The <a href="/wiki/Heine%E2%80%93Borel_theorem" title="Heine–Borel theorem">Heine–Borel theorem</a>, as the result is now known, is another special property possessed by closed and bounded sets of real numbers. </p><p>This property was significant because it allowed for the passage from <a href="/wiki/Local_property" title="Local property">local information</a> about a set (such as the continuity of a function) to global information about the set (such as the uniform continuity of a function). This sentiment was expressed by <a href="#CITEREFLebesgue1904">Lebesgue (1904)</a>, who also exploited it in the development of the <a href="/wiki/Lebesgue_integral" title="Lebesgue integral">integral now bearing his name</a>. Ultimately, the Russian school of <a href="/wiki/Point-set_topology" class="mw-redirect" title="Point-set topology">point-set topology</a>, under the direction of <a href="/wiki/Pavel_Alexandrov" title="Pavel Alexandrov">Pavel Alexandrov</a> and <a href="/wiki/Pavel_Urysohn" title="Pavel Urysohn">Pavel Urysohn</a>, formulated Heine–Borel compactness in a way that could be applied to the modern notion of a <a href="/wiki/Topological_space" title="Topological space">topological space</a>. <a href="#CITEREFAlexandrovUrysohn1929">Alexandrov & Urysohn (1929)</a> showed that the earlier version of compactness due to Fréchet, now called (relative) <a href="/wiki/Sequential_compactness" class="mw-redirect" title="Sequential compactness">sequential compactness</a>, under appropriate conditions followed from the version of compactness that was formulated in terms of the existence of finite subcovers. It was this notion of compactness that became the dominant one, because it was not only a stronger property, but it could be formulated in a more general setting with a minimum of additional technical machinery, as it relied only on the structure of the open sets in a space. </p> <div class="mw-heading mw-heading2"><h2 id="Basic_examples">Basic examples</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Compact_space&action=edit&section=2" title="Edit section: Basic examples"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Any <a href="/wiki/Finite_topological_space" title="Finite topological space">finite space</a> is compact; a finite subcover can be obtained by selecting, for each point, an open set containing it. A nontrivial example of a compact space is the (closed) <a href="/wiki/Unit_interval" title="Unit interval">unit interval</a> <span class="texhtml">[0,1]</span> of <a href="/wiki/Real_number" title="Real number">real numbers</a>. If one chooses an infinite number of distinct points in the unit interval, then there must be some <a href="/wiki/Accumulation_point" title="Accumulation point">accumulation point</a> among these points in that interval. For instance, the odd-numbered terms of the sequence <span class="nowrap">1, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span>, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">3</span></span>⁠</span>, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">3</span><span class="sr-only">/</span><span class="den">4</span></span>⁠</span>, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">5</span></span>⁠</span>, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">5</span><span class="sr-only">/</span><span class="den">6</span></span>⁠</span>, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">7</span></span>⁠</span>, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">7</span><span class="sr-only">/</span><span class="den">8</span></span>⁠</span>, ...</span> get arbitrarily close to 0, while the even-numbered ones get arbitrarily close to 1. The given example sequence shows the importance of including the <a href="/wiki/Boundary_(topology)" title="Boundary (topology)">boundary</a> points of the interval, since the <a href="/wiki/Limit_of_a_sequence" title="Limit of a sequence">limit points</a> must be in the space itself — an open (or half-open) interval of the real numbers is not compact. It is also crucial that the interval be <a href="/wiki/Bounded_set" title="Bounded set">bounded</a>, since in the interval <span class="texhtml">[0,∞)</span>, one could choose the sequence of points <span class="nowrap">0, 1, 2, 3, ...</span>, of which no sub-sequence ultimately gets arbitrarily close to any given real number. </p><p>In two dimensions, closed <a href="/wiki/Disk_(mathematics)" title="Disk (mathematics)">disks</a> are compact since for any infinite number of points sampled from a disk, some subset of those points must get arbitrarily close either to a point within the disc, or to a point on the boundary. However, an open disk is not compact, because a sequence of points can tend to the boundary – without getting arbitrarily close to any point in the interior. Likewise, spheres are compact, but a sphere missing a point is not since a sequence of points can still tend to the missing point, thereby not getting arbitrarily close to any point <i>within</i> the space. Lines and planes are not compact, since one can take a set of equally-spaced points in any given direction without approaching any point. </p> <div class="mw-heading mw-heading2"><h2 id="Definitions">Definitions</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Compact_space&action=edit&section=3" title="Edit section: Definitions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Various definitions of compactness may apply, depending on the level of generality. A subset of <a href="/wiki/Euclidean_space" title="Euclidean space">Euclidean space</a> in particular is called compact if it is <a href="/wiki/Closed_set" title="Closed set">closed</a> and <a href="/wiki/Bounded_set" title="Bounded set">bounded</a>. This implies, by the <a href="/wiki/Bolzano%E2%80%93Weierstrass_theorem" title="Bolzano–Weierstrass theorem">Bolzano–Weierstrass theorem</a>, that any infinite <a href="/wiki/Sequence_(mathematics)" class="mw-redirect" title="Sequence (mathematics)">sequence</a> from the set has a <a href="/wiki/Subsequence" title="Subsequence">subsequence</a> that converges to a point in the set. Various equivalent notions of compactness, such as <a href="/wiki/Sequential_compactness" class="mw-redirect" title="Sequential compactness">sequential compactness</a> and <a href="/wiki/Limit_point_compact" title="Limit point compact">limit point compactness</a>, can be developed in general <a href="/wiki/Metric_space" title="Metric space">metric spaces</a>.<sup id="cite_ref-:0_3-1" class="reference"><a href="#cite_note-:0-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p><p>In contrast, the different notions of compactness are not equivalent in general <a href="/wiki/Topological_space" title="Topological space">topological spaces</a>, and the most useful notion of compactness – originally called <i>bicompactness</i> – is defined using <a href="/wiki/Cover_(topology)" title="Cover (topology)">covers</a> consisting of <a href="/wiki/Open_set" title="Open set">open sets</a> (see <i>Open cover definition</i> below). That this form of compactness holds for closed and bounded subsets of Euclidean space is known as the <a href="/wiki/Heine%E2%80%93Borel_theorem" title="Heine–Borel theorem">Heine–Borel theorem</a>. Compactness, when defined in this manner, often allows one to take information that is known <a href="/wiki/Local_property" title="Local property">locally</a> – in a <a href="/wiki/Neighbourhood_(mathematics)" title="Neighbourhood (mathematics)">neighbourhood</a> of each point of the space – and to extend it to information that holds globally throughout the space. An example of this phenomenon is Dirichlet's theorem, to which it was originally applied by Heine, that a continuous function on a compact interval is <a href="/wiki/Uniformly_continuous" class="mw-redirect" title="Uniformly continuous">uniformly continuous</a>; here, continuity is a local property of the function, and uniform continuity the corresponding global property. </p> <div class="mw-heading mw-heading3"><h3 id="Open_cover_definition">Open cover definition</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Compact_space&action=edit&section=4" title="Edit section: Open cover definition"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Formally, a <a href="/wiki/Topological_space" title="Topological space">topological space</a> <span class="texhtml mvar" style="font-style:italic;">X</span> is called <i>compact</i> if every <a href="/wiki/Open_cover" class="mw-redirect" title="Open cover">open cover</a> of <span class="texhtml mvar" style="font-style:italic;">X</span> has a <a href="/wiki/Finite_set" title="Finite set">finite</a> <a href="/wiki/Subcover" class="mw-redirect" title="Subcover">subcover</a>.<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> That is, <span class="texhtml mvar" style="font-style:italic;">X</span> is compact if for every collection <span class="texhtml mvar" style="font-style:italic;">C</span> of open subsets<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> of <span class="texhtml mvar" style="font-style:italic;">X</span> such that </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X=\bigcup _{S\in C}S\ ,}"> <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>S</mi> <mo>∈</mo> <mi>C</mi> </mrow> </munder> <mi>S</mi> <mtext> </mtext> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X=\bigcup _{S\in C}S\ ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d92107fc5d99ad6da70aa8e3a3b7c1ef8aab0028" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:11.598ex; height:5.676ex;" alt="{\displaystyle X=\bigcup _{S\in C}S\ ,}"></span> </p><p>there is a <b>finite</b> subcollection <span class="texhtml mvar" style="font-style:italic;">F</span> ⊆ <span class="texhtml mvar" style="font-style:italic;">C</span> such that </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X=\bigcup _{S\in F}S\ .}"> <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>S</mi> <mo>∈</mo> <mi>F</mi> </mrow> </munder> <mi>S</mi> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X=\bigcup _{S\in F}S\ .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7c5a64504a64f698540213012db7709a3d039395" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:11.58ex; height:5.676ex;" alt="{\displaystyle X=\bigcup _{S\in F}S\ .}"></span> </p><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_space" title="Hausdorff space">Hausdorff</a> and <i>quasi-compact</i>. A compact set is sometimes referred to as a <i>compactum</i>, plural <i>compacta</i>. </p> <div class="mw-heading mw-heading3"><h3 id="Compactness_of_subsets">Compactness of subsets</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Compact_space&action=edit&section=5" title="Edit section: Compactness of subsets"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A subset <span class="texhtml mvar" style="font-style:italic;">K</span> of a topological space <span class="texhtml mvar" style="font-style:italic;">X</span> is said to be compact if it is compact as a subspace (in the <a href="/wiki/Subspace_topology" title="Subspace topology">subspace topology</a>). That is, <span class="texhtml mvar" style="font-style:italic;">K</span> is compact if for every arbitrary collection <span class="texhtml mvar" style="font-style:italic;">C</span> of open subsets of <span class="texhtml mvar" style="font-style:italic;">X</span> such that </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K\subseteq \bigcup _{S\in C}S\ ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> <mo>⊆</mo> <munder> <mo>⋃</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> <mo>∈</mo> <mi>C</mi> </mrow> </munder> <mi>S</mi> <mtext> </mtext> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K\subseteq \bigcup _{S\in C}S\ ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0dbf680eb4ef91af13172505c35018e965bba8cd" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:11.684ex; height:5.676ex;" alt="{\displaystyle K\subseteq \bigcup _{S\in C}S\ ,}"></span> </p><p>there is a <b>finite</b> subcollection <span class="texhtml mvar" style="font-style:italic;">F</span> ⊆ <span class="texhtml mvar" style="font-style:italic;">C</span> such that </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K\subseteq \bigcup _{S\in F}S\ .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> <mo>⊆</mo> <munder> <mo>⋃</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> <mo>∈</mo> <mi>F</mi> </mrow> </munder> <mi>S</mi> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K\subseteq \bigcup _{S\in F}S\ .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f0482b85a60713832bcfe43e178ad8415198d0" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:11.666ex; height:5.676ex;" alt="{\displaystyle K\subseteq \bigcup _{S\in F}S\ .}"></span> </p><p>Because compactness is a <a href="/wiki/Topological_property" title="Topological property">topological property</a>, the compactness of a subset depends only on the subspace topology induced on it. It follows that, if <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 K\subset Z\subset Y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> <mo>⊂</mo> <mi>Z</mi> <mo>⊂</mo> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K\subset Z\subset Y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ba6330afdad305c81ed6b60f7db5b062f20044e9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.717ex; height:2.176ex;" alt="{\displaystyle K\subset Z\subset Y}"></span>, with subset <span class="texhtml mvar" style="font-style:italic;">Z</span> equipped with the subspace topology, then <span class="texhtml mvar" style="font-style:italic;">K</span> is compact in <span class="texhtml mvar" style="font-style:italic;">Z</span> if and only if <span class="texhtml mvar" style="font-style:italic;">K</span> is compact in <span class="texhtml mvar" style="font-style:italic;">Y</span>. </p> <div class="mw-heading mw-heading3"><h3 id="Characterization">Characterization</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Compact_space&action=edit&section=6" title="Edit section: Characterization"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>If <span class="texhtml mvar" style="font-style:italic;">X</span> is a topological space then the following are equivalent: </p> <ol><li><span class="texhtml mvar" style="font-style:italic;">X</span> is compact; i.e., every <a href="/wiki/Open_cover" class="mw-redirect" title="Open cover">open cover</a> of <span class="texhtml mvar" style="font-style:italic;">X</span> has a finite <a href="/wiki/Subcover" class="mw-redirect" title="Subcover">subcover</a>.</li> <li><span class="texhtml mvar" style="font-style:italic;">X</span> has a sub-base such that every cover of the space, by members of the sub-base, has a finite subcover (<a href="/wiki/Alexander%27s_sub-base_theorem" class="mw-redirect" title="Alexander's sub-base theorem">Alexander's sub-base theorem</a>).</li> <li><span class="texhtml mvar" style="font-style:italic;">X</span> is <a href="/wiki/Lindel%C3%B6f_space" title="Lindelöf space">Lindelöf</a> and <a href="/wiki/Countably_compact" class="mw-redirect" title="Countably compact">countably compact</a>.<sup id="cite_ref-FOOTNOTEHowes1995xxvi–xxviii_9-0" class="reference"><a href="#cite_note-FOOTNOTEHowes1995xxvi–xxviii-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup></li> <li>Any collection of closed subsets of <span class="texhtml mvar" style="font-style:italic;">X</span> with the <a href="/wiki/Finite_intersection_property" title="Finite intersection property">finite intersection property</a> has nonempty intersection.</li> <li>Every <a href="/wiki/Net_(mathematics)" title="Net (mathematics)">net</a> on <span class="texhtml mvar" style="font-style:italic;">X</span> has a convergent subnet (see the article on <a href="/wiki/Net_(mathematics)" title="Net (mathematics)">nets</a> for a proof).</li> <li>Every <a href="/wiki/Filters_in_topology" title="Filters in topology">filter</a> on <span class="texhtml mvar" style="font-style:italic;">X</span> has a convergent refinement.</li> <li>Every net on <span class="texhtml mvar" style="font-style:italic;">X</span> has a cluster point.</li> <li>Every filter on <span class="texhtml mvar" style="font-style:italic;">X</span> has a cluster point.</li> <li>Every <a href="/wiki/Ultrafilter_(set_theory)" class="mw-redirect" title="Ultrafilter (set theory)">ultrafilter</a> on <span class="texhtml mvar" style="font-style:italic;">X</span> converges to at least one point.</li> <li>Every infinite subset of <span class="texhtml mvar" style="font-style:italic;">X</span> has a <a href="/wiki/Complete_accumulation_point" class="mw-redirect" title="Complete accumulation point">complete accumulation point</a>.<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup></li> <li>For every topological space <span class="texhtml mvar" style="font-style:italic;">Y</span>, the projection <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\times Y\to Y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>×</mo> <mi>Y</mi> <mo stretchy="false">→</mo> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X\times Y\to Y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/66e9d4154e92a7e0f561725fe46b479499743d82" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.981ex; height:2.176ex;" alt="{\displaystyle X\times Y\to Y}"></span> is a <a href="/wiki/Closed_mapping" class="mw-redirect" title="Closed mapping">closed mapping</a><sup id="cite_ref-Bourbaki_11-0" class="reference"><a href="#cite_note-Bourbaki-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> (see <a href="/wiki/Proper_map" title="Proper map">proper map</a>).</li> <li>Every open cover linearly ordered by subset inclusion contains <span class="texhtml mvar" style="font-style:italic;">X</span>.<sup id="cite_ref-FOOTNOTEMack1967_12-0" class="reference"><a href="#cite_note-FOOTNOTEMack1967-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup></li></ol> <p>Bourbaki defines a compact space (quasi-compact space) as a topological space where each filter has a cluster point (i.e., 8. in the above).<sup id="cite_ref-BourbakiDefinition_13-0" class="reference"><a href="#cite_note-BourbakiDefinition-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading4"><h4 id="Euclidean_space">Euclidean space</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Compact_space&action=edit&section=7" title="Edit section: Euclidean space"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>For any <a href="/wiki/Subset" title="Subset">subset</a> <span class="texhtml mvar" style="font-style:italic;">A</span> of <a href="/wiki/Euclidean_space" title="Euclidean space">Euclidean space</a>, <span class="texhtml mvar" style="font-style:italic;">A</span> is compact if and only if it is <a href="/wiki/Closed_set" title="Closed set">closed</a> and <a href="/wiki/Bounded_set" title="Bounded set">bounded</a>; this is the <a href="/wiki/Heine%E2%80%93Borel_theorem" title="Heine–Borel theorem">Heine–Borel theorem</a>. </p><p>As a <a href="/wiki/Euclidean_space" title="Euclidean space">Euclidean space</a> is a metric space, the conditions in the next subsection also apply to all of its subsets. Of all of the equivalent conditions, it is in practice easiest to verify that a subset is closed and bounded, for example, for a closed <a href="/wiki/Interval_(mathematics)" title="Interval (mathematics)">interval</a> or closed <span class="texhtml mvar" style="font-style:italic;">n</span>-ball. </p> <div class="mw-heading mw-heading4"><h4 id="Metric_spaces">Metric spaces</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Compact_space&action=edit&section=8" title="Edit section: Metric spaces"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>For any metric space <span class="texhtml">(<i>X</i>, <i>d</i>)</span>, the following are equivalent (assuming <a href="/wiki/Countable_choice" class="mw-redirect" title="Countable choice">countable choice</a>): </p> <ol><li><span class="texhtml">(<i>X</i>, <i>d</i>)</span> is compact.</li> <li><span class="texhtml">(<i>X</i>, <i>d</i>)</span> is <a href="/wiki/Completeness_(topology)" class="mw-redirect" title="Completeness (topology)">complete</a> and <a href="/wiki/Totally_bounded" class="mw-redirect" title="Totally bounded">totally bounded</a> (this is also equivalent to compactness for <a href="/wiki/Uniform_space" title="Uniform space">uniform spaces</a>).<sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup></li> <li><span class="texhtml">(<i>X</i>, <i>d</i>)</span> is sequentially compact; that is, every <a href="/wiki/Sequence" title="Sequence">sequence</a> in <span class="texhtml mvar" style="font-style:italic;">X</span> has a convergent subsequence whose limit is in <span class="texhtml mvar" style="font-style:italic;">X</span> (this is also equivalent to compactness for <a href="/wiki/First-countable" class="mw-redirect" title="First-countable">first-countable</a> <a href="/wiki/Uniform_space" title="Uniform space">uniform spaces</a>).</li> <li><span class="texhtml">(<i>X</i>, <i>d</i>)</span> is <a href="/wiki/Limit_point_compact" title="Limit point compact">limit point compact</a> (also called weakly countably compact); that is, every infinite subset of <span class="texhtml mvar" style="font-style:italic;">X</span> has at least one <a href="/wiki/Limit_point_of_a_set" class="mw-redirect" title="Limit point of a set">limit point</a> in <span class="texhtml mvar" style="font-style:italic;">X</span>.</li> <li><span class="texhtml">(<i>X</i>, <i>d</i>)</span> is <a href="/wiki/Countably_compact" class="mw-redirect" title="Countably compact">countably compact</a>; that is, every countable open cover of <span class="texhtml mvar" style="font-style:italic;">X</span> has a finite subcover.</li> <li><span class="texhtml">(<i>X</i>, <i>d</i>)</span> is an image of a continuous function from the <a href="/wiki/Cantor_set" title="Cantor set">Cantor set</a>.<sup id="cite_ref-15" class="reference"><a href="#cite_note-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup></li> <li>Every decreasing nested sequence of nonempty closed subsets <span class="texhtml"><i>S</i><sub>1</sub> ⊇ <i>S</i><sub>2</sub> ⊇ ...</span> in <span class="texhtml">(<i>X</i>, <i>d</i>)</span> has a nonempty intersection.</li> <li>Every increasing nested sequence of proper open subsets <span class="texhtml"><i>S</i><sub>1</sub> ⊆ <i>S</i><sub>2</sub> ⊆ ...</span> in <span class="texhtml">(<i>X</i>, <i>d</i>)</span> fails to cover <span class="texhtml mvar" style="font-style:italic;">X</span>.</li></ol> <p>A compact metric space <span class="texhtml">(<i>X</i>, <i>d</i>)</span> also satisfies the following properties: </p> <ol><li><a href="/wiki/Lebesgue%27s_number_lemma" title="Lebesgue's number lemma">Lebesgue's number lemma</a>: For every open cover of <span class="texhtml mvar" style="font-style:italic;">X</span>, there exists a number <span class="nowrap"><i>δ</i> > 0</span> such that every subset of <span class="texhtml mvar" style="font-style:italic;">X</span> of diameter < <span class="texhtml mvar" style="font-style:italic;">δ</span> is contained in some member of the cover.</li> <li><span class="texhtml">(<i>X</i>, <i>d</i>)</span> is <a href="/wiki/Second-countable_space" title="Second-countable space">second-countable</a>, <a href="/wiki/Separable_space" title="Separable space">separable</a> and <a href="/wiki/Lindel%C3%B6f_space" title="Lindelöf space">Lindelöf</a> – these three conditions are equivalent for metric spaces. The converse is not true; e.g., a countable discrete space satisfies these three conditions, but is not compact.</li> <li><span class="texhtml mvar" style="font-style:italic;">X</span> is closed and bounded (as a subset of any metric space whose restricted metric is <span class="texhtml mvar" style="font-style:italic;">d</span>). The converse may fail for a non-Euclidean space; e.g. the <a href="/wiki/Real_line" class="mw-redirect" title="Real line">real line</a> equipped with the <a href="/wiki/Discrete_metric" class="mw-redirect" title="Discrete metric">discrete metric</a> is closed and bounded but not compact, as the collection of all <a href="/wiki/Singleton_(mathematics)" title="Singleton (mathematics)">singletons</a> of the space is an open cover which admits no finite subcover. It is complete but not totally bounded.</li></ol> <div class="mw-heading mw-heading4"><h4 id="Ordered_spaces">Ordered spaces</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Compact_space&action=edit&section=9" title="Edit section: Ordered spaces"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>For an ordered space <span class="texhtml">(<i>X</i>, <)</span> (i.e. a totally ordered set equipped with the order topology), the following are equivalent: </p> <ol><li><span class="texhtml">(<i>X</i>, <)</span> is compact.</li> <li>Every subset of <span class="texhtml mvar" style="font-style:italic;">X</span> has a supremum (i.e. a least upper bound) in <span class="texhtml mvar" style="font-style:italic;">X</span>.</li> <li>Every subset of <span class="texhtml mvar" style="font-style:italic;">X</span> has an infimum (i.e. a greatest lower bound) in <span class="texhtml mvar" style="font-style:italic;">X</span>.</li> <li>Every nonempty closed subset of <span class="texhtml mvar" style="font-style:italic;">X</span> has a maximum and a minimum element.</li></ol> <p>An ordered space satisfying (any one of) these conditions is called a complete lattice. </p><p>In addition, the following are equivalent for all ordered spaces <span class="texhtml">(<i>X</i>, <)</span>, and (assuming <a href="/wiki/Countable_choice" class="mw-redirect" title="Countable choice">countable choice</a>) are true whenever <span class="texhtml">(<i>X</i>, <)</span> is compact. (The converse in general fails if <span class="texhtml">(<i>X</i>, <)</span> is not also metrizable.): </p> <ol><li>Every sequence in <span class="texhtml">(<i>X</i>, <)</span> has a subsequence that converges in <span class="texhtml">(<i>X</i>, <)</span>.</li> <li>Every monotone increasing sequence in <span class="texhtml mvar" style="font-style:italic;">X</span> converges to a unique limit in <span class="texhtml mvar" style="font-style:italic;">X</span>.</li> <li>Every monotone decreasing sequence in <span class="texhtml mvar" style="font-style:italic;">X</span> converges to a unique limit in <span class="texhtml mvar" style="font-style:italic;">X</span>.</li> <li>Every decreasing nested sequence of nonempty closed subsets <span class="texhtml mvar" style="font-style:italic;">S</span><sub>1</sub> ⊇ <span class="texhtml mvar" style="font-style:italic;">S</span><sub>2</sub> ⊇ ... in <span class="texhtml">(<i>X</i>, <)</span> has a nonempty intersection.</li> <li>Every increasing nested sequence of proper open subsets <span class="texhtml mvar" style="font-style:italic;">S</span><sub>1</sub> ⊆ <span class="texhtml mvar" style="font-style:italic;">S</span><sub>2</sub> ⊆ ... in <span class="texhtml">(<i>X</i>, <)</span> fails to cover <span class="texhtml mvar" style="font-style:italic;">X</span>.</li></ol> <div class="mw-heading mw-heading4"><h4 id="Characterization_by_continuous_functions">Characterization by continuous functions</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Compact_space&action=edit&section=10" title="Edit section: Characterization by continuous functions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Let <span class="texhtml mvar" style="font-style:italic;">X</span> be a topological space and <span class="texhtml">C(<i>X</i>)</span> the ring of real continuous functions on <span class="texhtml mvar" style="font-style:italic;">X</span>. For each <span class="texhtml"><i>p</i> ∈ <i>X</i></span>, the evaluation map <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 \operatorname {ev} _{p}\colon C(X)\to \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ev</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo>:</mo> <mi>C</mi> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {ev} _{p}\colon C(X)\to \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7e8866d64e224786e38571ce6adabbc5fd3923bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:15.201ex; height:3.009ex;" alt="{\displaystyle \operatorname {ev} _{p}\colon C(X)\to \mathbb {R} }"></span> given by <span class="texhtml">ev<sub><i>p</i></sub>(<i>f</i>) = <i>f</i>(<i>p</i>)</span> is a ring homomorphism. The <a href="/wiki/Kernel_(algebra)" title="Kernel (algebra)">kernel</a> of <span class="texhtml">ev<sub><i>p</i></sub></span> is a <a href="/wiki/Maximal_ideal" title="Maximal ideal">maximal ideal</a>, since the <a href="/wiki/Residue_field" title="Residue field">residue field</a> <span class="nowrap"><span class="texhtml">C(<i>X</i>)/ker ev<sub><i>p</i></sub></span></span> is the field of real numbers, by the <a href="/wiki/First_isomorphism_theorem" class="mw-redirect" title="First isomorphism theorem">first isomorphism theorem</a>. A topological space <span class="texhtml mvar" style="font-style:italic;">X</span> is <a href="/wiki/Pseudocompact_space" title="Pseudocompact space">pseudocompact</a> if and only if every maximal ideal in <span class="texhtml">C(<i>X</i>)</span> has residue field the real numbers. For <a href="/wiki/Completely_regular_space" class="mw-redirect" title="Completely regular space">completely regular spaces</a>, this is equivalent to every maximal ideal being the kernel of an evaluation homomorphism.<sup id="cite_ref-16" class="reference"><a href="#cite_note-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> There are pseudocompact spaces that are not compact, though. </p><p>In general, for non-pseudocompact spaces there are always maximal ideals <span class="texhtml mvar" style="font-style:italic;">m</span> in <span class="texhtml">C(<i>X</i>)</span> such that the residue field <span class="texhtml">C(<i>X</i>)/<i>m</i></span> is a (<a href="/wiki/Non-archimedean_field" class="mw-redirect" title="Non-archimedean field">non-Archimedean</a>) <a href="/wiki/Hyperreal_field" class="mw-redirect" title="Hyperreal field">hyperreal field</a>. The framework of <a href="/wiki/Non-standard_analysis" class="mw-redirect" title="Non-standard analysis">non-standard analysis</a> allows for the following alternative characterization of compactness:<sup id="cite_ref-17" class="reference"><a href="#cite_note-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> a topological space <span class="texhtml mvar" style="font-style:italic;">X</span> is compact if and only if every point <span class="texhtml mvar" style="font-style:italic;">x</span> of the natural extension <span class="texhtml"><i>*X</i></span> is <a href="/wiki/Infinitesimal" title="Infinitesimal">infinitely close</a> to a point <span class="texhtml"><i>x</i><sub>0</sub></span> of <span class="texhtml mvar" style="font-style:italic;">X</span> (more precisely, <span class="texhtml mvar" style="font-style:italic;">x</span> is contained in the <a href="/wiki/Monad_(non-standard_analysis)" class="mw-redirect" title="Monad (non-standard analysis)">monad</a> of <span class="texhtml"><i>x</i><sub>0</sub></span>). </p> <div class="mw-heading mw-heading4"><h4 id="Hyperreal_definition">Hyperreal definition</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Compact_space&action=edit&section=11" title="Edit section: Hyperreal definition"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A space <span class="texhtml mvar" style="font-style:italic;">X</span> is compact if its <a href="/wiki/Hyperreal_number" title="Hyperreal number">hyperreal extension</a> <span class="texhtml"><i>*X</i></span> (constructed, for example, by the <a href="/wiki/Ultrapower_construction" class="mw-redirect" title="Ultrapower construction">ultrapower construction</a>) has the property that every point of <span class="texhtml"><i>*X</i></span> is infinitely close to some point of <span class="texhtml"><i>X</i> ⊂ <i>*X</i></span>. For example, an open real interval <span class="nowrap"><span class="texhtml"><i>X</i> = (0, 1)</span></span> is not compact because its hyperreal extension <span class="texhtml">*(0,1)</span> contains infinitesimals, which are infinitely close to 0, which is not a point of <span class="texhtml mvar" style="font-style:italic;">X</span>. </p> <div class="mw-heading mw-heading2"><h2 id="Sufficient_conditions">Sufficient conditions</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Compact_space&action=edit&section=12" title="Edit section: Sufficient conditions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>A closed subset of a compact space is compact.<sup id="cite_ref-18" class="reference"><a href="#cite_note-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup></li> <li>A finite <a href="/wiki/Union_(set_theory)" title="Union (set theory)">union</a> of compact sets is compact.</li> <li>A <a href="/wiki/Continuous_function_(topology)" class="mw-redirect" title="Continuous function (topology)">continuous</a> image of a compact space is compact.<sup id="cite_ref-19" class="reference"><a href="#cite_note-19"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup></li> <li>The intersection of any non-empty collection of compact subsets of a Hausdorff space is compact (and closed); <ul><li>If <span class="texhtml mvar" style="font-style:italic;">X</span> is not Hausdorff then the intersection of two compact subsets may fail to be compact (see footnote for example).<sup id="cite_ref-20" class="reference"><a href="#cite_note-20"><span class="cite-bracket">[</span>a<span class="cite-bracket">]</span></a></sup></li></ul></li> <li>The <a href="/wiki/Product_topology" title="Product topology">product</a> of any collection of compact spaces is compact. (This is <a href="/wiki/Tychonoff%27s_theorem" title="Tychonoff's theorem">Tychonoff's theorem</a>, which is equivalent to the <a href="/wiki/Axiom_of_choice" title="Axiom of choice">axiom of choice</a>.)</li> <li>In a <a href="/wiki/Metrizable_space" title="Metrizable space">metrizable space</a>, a subset is compact if and only if it is <a href="/wiki/Sequentially_compact" class="mw-redirect" title="Sequentially compact">sequentially compact</a> (assuming <a href="/wiki/Axiom_of_countable_choice" title="Axiom of countable choice">countable choice</a>)</li> <li>A finite set endowed with any topology is compact.</li></ul> <div class="mw-heading mw-heading2"><h2 id="Properties_of_compact_spaces">Properties of compact spaces</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Compact_space&action=edit&section=13" title="Edit section: Properties of compact spaces"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>A compact subset of a <a href="/wiki/Hausdorff_space" title="Hausdorff space">Hausdorff space</a> <span class="texhtml mvar" style="font-style:italic;">X</span> is closed. <ul><li>If <span class="texhtml mvar" style="font-style:italic;">X</span> is not Hausdorff then a compact subset of <span class="texhtml mvar" style="font-style:italic;">X</span> may fail to be a closed subset of <span class="texhtml mvar" style="font-style:italic;">X</span> (see footnote for example).<sup id="cite_ref-21" class="reference"><a href="#cite_note-21"><span class="cite-bracket">[</span>b<span class="cite-bracket">]</span></a></sup></li> <li>If <span class="texhtml mvar" style="font-style:italic;">X</span> is not Hausdorff then the closure of a compact set may fail to be compact (see footnote for example).<sup id="cite_ref-22" class="reference"><a href="#cite_note-22"><span class="cite-bracket">[</span>c<span class="cite-bracket">]</span></a></sup></li></ul></li> <li>In any <a href="/wiki/Topological_vector_space" title="Topological vector space">topological vector space</a> (TVS), a compact subset is <a href="/wiki/Complete_space" class="mw-redirect" title="Complete space">complete</a>. However, every non-Hausdorff TVS contains compact (and thus complete) subsets that are <i>not</i> closed.</li> <li>If <span class="texhtml mvar" style="font-style:italic;">A</span> and <span class="texhtml mvar" style="font-style:italic;">B</span> are disjoint compact subsets of a Hausdorff space <span class="texhtml mvar" style="font-style:italic;">X</span>, then there exist disjoint open sets <span class="texhtml mvar" style="font-style:italic;">U</span> and <span class="texhtml mvar" style="font-style:italic;">V</span> in <span class="texhtml mvar" style="font-style:italic;">X</span> such that <span class="texhtml"><i>A</i> ⊆ <i>U</i></span> and <span class="texhtml"><i>B</i> ⊆ <i>V</i></span>.</li> <li>A continuous bijection from a compact space into a Hausdorff space is a <a href="/wiki/Homeomorphism" title="Homeomorphism">homeomorphism</a>.</li> <li>A compact Hausdorff space is <a href="/wiki/Normal_space" title="Normal space">normal</a> and <a href="/wiki/Regular_space" title="Regular space">regular</a>.</li> <li>If a space <span class="texhtml mvar" style="font-style:italic;">X</span> is compact and Hausdorff, then no finer topology on <span class="texhtml mvar" style="font-style:italic;">X</span> is compact and no coarser topology on <span class="texhtml mvar" style="font-style:italic;">X</span> is Hausdorff.</li> <li>If a subset of a metric space <span class="texhtml">(<i>X</i>, <i>d</i>)</span> is compact then it is <span class="texhtml mvar" style="font-style:italic;">d</span>-bounded.</li></ul> <div class="mw-heading mw-heading3"><h3 id="Functions_and_compact_spaces">Functions and compact spaces</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Compact_space&action=edit&section=14" title="Edit section: Functions and compact spaces"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Since a <a href="/wiki/Continuous_function_(topology)" class="mw-redirect" title="Continuous function (topology)">continuous</a> image of a compact space is compact, the <a href="/wiki/Extreme_value_theorem" title="Extreme value theorem">extreme value theorem</a> holds for such spaces: a continuous real-valued function on a nonempty compact space is bounded above and attains its supremum.<sup id="cite_ref-23" class="reference"><a href="#cite_note-23"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup> (Slightly more generally, this is true for an upper semicontinuous function.) As a sort of converse to the above statements, the pre-image of a compact space under a <a href="/wiki/Proper_map" title="Proper map">proper map</a> is compact. </p> <div class="mw-heading mw-heading3"><h3 id="Compactifications">Compactifications</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Compact_space&action=edit&section=15" title="Edit section: Compactifications"><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/Compactification_(mathematics)" title="Compactification (mathematics)">Compactification (mathematics)</a></div> <p>Every topological space <span class="texhtml mvar" style="font-style:italic;">X</span> is an open <a href="/wiki/Dense_topological_subspace" class="mw-redirect" title="Dense topological subspace">dense subspace</a> of a compact space having at most one point more than <span class="texhtml mvar" style="font-style:italic;">X</span>, by the <a href="/wiki/Alexandroff_one-point_compactification" class="mw-redirect" title="Alexandroff one-point compactification">Alexandroff one-point compactification</a>. By the same construction, every <a href="/wiki/Locally_compact" class="mw-redirect" title="Locally compact">locally compact</a> Hausdorff space <span class="texhtml mvar" style="font-style:italic;">X</span> is an open dense subspace of a compact Hausdorff space having at most one point more than <span class="texhtml mvar" style="font-style:italic;">X</span>. </p> <div class="mw-heading mw-heading3"><h3 id="Ordered_compact_spaces">Ordered compact spaces</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Compact_space&action=edit&section=16" title="Edit section: Ordered compact spaces"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A nonempty compact subset of the <a href="/wiki/Real_number" title="Real number">real numbers</a> has a greatest element and a least element. </p><p>Let <span class="texhtml mvar" style="font-style:italic;">X</span> be a <a href="/wiki/Total_order" title="Total order">simply ordered</a> set endowed with the <a href="/wiki/Order_topology" title="Order topology">order topology</a>. Then <span class="texhtml mvar" style="font-style:italic;">X</span> is compact if and only if <span class="texhtml mvar" style="font-style:italic;">X</span> is a <a href="/wiki/Complete_lattice" title="Complete lattice">complete lattice</a> (i.e. all subsets have suprema and infima).<sup id="cite_ref-24" class="reference"><a href="#cite_note-24"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Examples">Examples</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Compact_space&action=edit&section=17" title="Edit section: Examples"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>Any <a href="/wiki/Finite_topological_space" title="Finite topological space">finite topological space</a>, including the <a href="/wiki/Empty_set" title="Empty set">empty set</a>, is compact. More generally, any space with a <a href="/wiki/Finite_topology" title="Finite topology">finite topology</a> (only finitely many open sets) is compact; this includes in particular the <a href="/wiki/Trivial_topology" title="Trivial topology">trivial topology</a>.</li> <li>Any space carrying the <a href="/wiki/Cofinite_topology" class="mw-redirect" title="Cofinite topology">cofinite topology</a> is compact.</li> <li>Any <a href="/wiki/Locally_compact" class="mw-redirect" title="Locally compact">locally compact</a> Hausdorff space can be turned into a compact space by adding a single point to it, by means of <a href="/wiki/Alexandroff_one-point_compactification" class="mw-redirect" title="Alexandroff one-point compactification">Alexandroff one-point compactification</a>. The one-point compactification 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 \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }"></span> is homeomorphic to the circle <span class="texhtml"><b>S</b><sup>1</sup></span>; the one-point compactification 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 \mathbb {R} ^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e150115ab9f63023215109595b76686a1ff890fd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.732ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{2}}"></span> is homeomorphic to the sphere <span class="texhtml"><b>S</b><sup>2</sup></span>. Using the one-point compactification, one can also easily construct compact spaces which are not Hausdorff, by starting with a non-Hausdorff space.</li> <li>The <a href="/wiki/Right_order_topology" class="mw-redirect" title="Right order topology">right order topology</a> or <a href="/wiki/Left_order_topology" class="mw-redirect" title="Left order topology">left order topology</a> on any bounded <a href="/wiki/Totally_ordered_set" class="mw-redirect" title="Totally ordered set">totally ordered set</a> is compact. In particular, <a href="/wiki/Sierpi%C5%84ski_space" title="Sierpiński space">Sierpiński space</a> is compact.</li> <li>No <a href="/wiki/Discrete_space" title="Discrete space">discrete space</a> with an infinite number of points is compact. The collection of all <a href="/wiki/Singleton_(mathematics)" title="Singleton (mathematics)">singletons</a> of the space is an open cover which admits no finite subcover. Finite discrete spaces are compact.</li> <li>In <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 \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }"></span> carrying the <a href="/wiki/Lower_limit_topology" title="Lower limit topology">lower limit topology</a>, no uncountable set is compact.</li> <li>In the <a href="/wiki/Cocountable_topology" title="Cocountable topology">cocountable topology</a> on an uncountable set, no infinite set is compact. Like the previous example, the space as a whole is not <a href="/wiki/Locally_compact" class="mw-redirect" title="Locally compact">locally compact</a> but is still <a href="/wiki/Lindel%C3%B6f_space" title="Lindelöf space">Lindelöf</a>.</li> <li>The closed <a href="/wiki/Unit_interval" title="Unit interval">unit interval</a> <span class="texhtml"><span class="texhtml">[0, 1]</span></span> is compact. This follows from the <a href="/wiki/Heine%E2%80%93Borel_theorem" title="Heine–Borel theorem">Heine–Borel theorem</a>. The open interval <span class="texhtml">(0, 1)</span> is not compact: the <a href="/wiki/Open_cover" class="mw-redirect" title="Open cover">open cover</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="{\textstyle \left({\frac {1}{n}},1-{\frac {1}{n}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>n</mi> </mfrac> </mrow> <mo>,</mo> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>n</mi> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \left({\frac {1}{n}},1-{\frac {1}{n}}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c2b7b7ae8290bf973c0d47bfc9e99105a9e194ea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:10.811ex; height:3.343ex;" alt="{\textstyle \left({\frac {1}{n}},1-{\frac {1}{n}}\right)}"></span> for <span class="texhtml"><span class="texhtml mvar" style="font-style:italic;">n</span> = 3, 4, ... </span> does not have a finite subcover. Similarly, the set of <i><a href="/wiki/Rational_numbers" class="mw-redirect" title="Rational numbers">rational numbers</a></i> in the closed interval <span class="texhtml">[0,1]</span> is not compact: the sets of rational numbers in the intervals <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="{\textstyle \left[0,{\frac {1}{\pi }}-{\frac {1}{n}}\right]{\text{ and }}\left[{\frac {1}{\pi }}+{\frac {1}{n}},1\right]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow> <mo>[</mo> <mrow> <mn>0</mn> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>π</mi> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>n</mi> </mfrac> </mrow> </mrow> <mo>]</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext> and </mtext> </mrow> <mrow> <mo>[</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>π</mi> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>n</mi> </mfrac> </mrow> <mo>,</mo> <mn>1</mn> </mrow> <mo>]</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \left[0,{\frac {1}{\pi }}-{\frac {1}{n}}\right]{\text{ and }}\left[{\frac {1}{\pi }}+{\frac {1}{n}},1\right]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cd15c0d260a029851c6acad5e36712fbb0f88c71" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:26.836ex; height:3.343ex;" alt="{\textstyle \left[0,{\frac {1}{\pi }}-{\frac {1}{n}}\right]{\text{ and }}\left[{\frac {1}{\pi }}+{\frac {1}{n}},1\right]}"></span> cover all the rationals in [0, 1] for <span class="texhtml"><span class="texhtml mvar" style="font-style:italic;">n</span> = 4, 5, ... </span> but this cover does not have a finite subcover. Here, the sets are open in the subspace topology even though they are not open as subsets 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 \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }"></span>.</li> <li>The set <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 \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }"></span> of all real numbers is not compact as there is a cover of open intervals that does not have a finite subcover. For example, intervals <span class="texhtml"><span class="texhtml">(<span class="texhtml mvar" style="font-style:italic;">n</span> − 1, <span class="texhtml mvar" style="font-style:italic;">n</span> + 1)</span> </span>, where <span class="texhtml mvar" style="font-style:italic;">n</span> takes all integer values in <span class="texhtml"><b>Z</b></span>, cover <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 \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }"></span> but there is no finite subcover.</li> <li>On the other hand, the <a href="/wiki/Extended_real_number_line" title="Extended real number line">extended real number line</a> carrying the analogous topology <i>is</i> compact; note that the cover described above would never reach the points at infinity and thus would <i>not</i> cover the extended real line. In fact, the set has the <a href="/wiki/Homeomorphism" title="Homeomorphism">homeomorphism</a> to [−1, 1] of mapping each infinity to its corresponding unit and every real number to its sign multiplied by the unique number in the positive part of interval that results in its absolute value when divided by one minus itself, and since homeomorphisms preserve covers, the Heine-Borel property can be inferred.</li> <li>For every <a href="/wiki/Natural_number" title="Natural number">natural number</a> <span class="texhtml mvar" style="font-style:italic;">n</span>, the <a href="/wiki/N-sphere" title="N-sphere"><span class="texhtml mvar" style="font-style:italic;">n</span>-sphere</a> is compact. Again from the Heine–Borel theorem, the closed unit ball of any finite-dimensional <a href="/wiki/Normed_vector_space" title="Normed vector space">normed vector space</a> is compact. This is not true for infinite dimensions; in fact, a normed vector space is finite-dimensional if and only if its <a href="/wiki/Closed_unit_ball" class="mw-redirect" title="Closed unit ball">closed unit ball</a> is compact.</li> <li>On the other hand, the closed unit ball of the dual of a normed space is compact for the weak-* topology. (<a href="/wiki/Alaoglu%27s_theorem" class="mw-redirect" title="Alaoglu's theorem">Alaoglu's theorem</a>)</li> <li>The <a href="/wiki/Cantor_set" title="Cantor set">Cantor set</a> is compact. In fact, every compact metric space is a continuous image of the Cantor set.</li> <li>Consider the set <span class="texhtml mvar" style="font-style:italic;">K</span> of all functions <span class="texhtml"><i>f</i> : <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 \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }"></span> → [0, 1]</span> from the real number line to the closed unit interval, and define a topology on <span class="texhtml mvar" style="font-style:italic;">K</span> so that a sequence <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_{n}\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{f_{n}\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/009619e33115347a277b099ff493347bdd5776aa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.683ex; height:2.843ex;" alt="{\displaystyle \{f_{n}\}}"></span> in <span class="texhtml mvar" style="font-style:italic;">K</span> converges towards <span class="texhtml"><i>f</i> ∈ <i>K</i></span> if and only if <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_{n}(x)\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{f_{n}(x)\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/db082772469c125b07c34485abb261915262e57f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.822ex; height:2.843ex;" alt="{\displaystyle \{f_{n}(x)\}}"></span> converges towards <span class="texhtml"><i>f</i>(<i>x</i>)</span> for all real numbers <span class="texhtml mvar" style="font-style:italic;">x</span>. There is only one such topology; it is called the topology of <a href="/wiki/Pointwise_convergence" title="Pointwise convergence">pointwise convergence</a> or the <a href="/wiki/Product_topology" title="Product topology">product topology</a>. Then <span class="texhtml mvar" style="font-style:italic;">K</span> is a compact topological space; this follows from the <a href="/wiki/Tychonoff_theorem" class="mw-redirect" title="Tychonoff theorem">Tychonoff theorem</a>.</li> <li>A subset of the Banach space of real-valued continuous functions on a compact Hausdorff space is relatively compact if and only if it is equicontinuous and pointwise bounded (<a href="/wiki/Arzel%C3%A0%E2%80%93Ascoli_theorem" title="Arzelà–Ascoli theorem">Arzelà–Ascoli theorem</a>).</li> <li>Consider the set <span class="texhtml mvar" style="font-style:italic;">K</span> of all functions <span class="texhtml"><i>f</i> : <span class="texhtml">[0, 1]</span> → <span class="texhtml">[0, 1]</span></span> satisfying the <a href="/wiki/Lipschitz_condition" class="mw-redirect" title="Lipschitz condition">Lipschitz condition</a> <span class="texhtml">|<span class="nowrap" style="padding-left:0.1em; padding-right:0.1em;"><i>f</i>(<i>x</i>) − <i>f</i>(<i>y</i>)</span>| ≤ |<span class="nowrap" style="padding-left:0.1em; padding-right:0.1em;"><i>x</i> − <i>y</i></span>|</span> for all <span class="texhtml"><i>x</i>, <i>y</i> ∈ <span class="texhtml">[0,1]</span></span>. Consider on <span class="texhtml mvar" style="font-style:italic;">K</span> the metric induced by the <a href="/wiki/Uniform_convergence" title="Uniform convergence">uniform distance</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 d(f,g)=\sup _{x\in [0,1]}|f(x)-g(x)|.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo stretchy="false">(</mo> <mi>f</mi> <mo>,</mo> <mi>g</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munder> <mo movablelimits="true" form="prefix">sup</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo>∈</mo> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">]</mo> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d(f,g)=\sup _{x\in [0,1]}|f(x)-g(x)|.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d4968cb9f2028481e49bb5c4a09a337ae1840ce7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:28.445ex; height:5.009ex;" alt="{\displaystyle d(f,g)=\sup _{x\in [0,1]}|f(x)-g(x)|.}"></span> Then by the Arzelà–Ascoli theorem the space <span class="texhtml mvar" style="font-style:italic;">K</span> is compact.</li> <li>The <a href="/wiki/Spectrum_of_an_operator" class="mw-redirect" title="Spectrum of an operator">spectrum</a> of any <a href="/wiki/Bounded_linear_operator" class="mw-redirect" title="Bounded linear operator">bounded linear operator</a> on a <a href="/wiki/Banach_space" title="Banach space">Banach space</a> is a nonempty compact subset of the <a href="/wiki/Complex_number" title="Complex number">complex numbers</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 \mathbb {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9add4085095b9b6d28d045fd9c92c2c09f549a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {C} }"></span>. Conversely, any compact subset 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 \mathbb {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9add4085095b9b6d28d045fd9c92c2c09f549a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {C} }"></span> arises in this manner, as the spectrum of some bounded linear operator. For instance, a diagonal operator on the Hilbert space <a href="/wiki/Sequence_spaces#ℓp_spaces" class="mw-redirect" title="Sequence spaces"><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 \ell ^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>ℓ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ell ^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/91f1f909abd70bd3d8fff0f7ae1ac23052387e18" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.024ex; height:2.676ex;" alt="{\displaystyle \ell ^{2}}"></span></a> may have any compact nonempty subset 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 \mathbb {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9add4085095b9b6d28d045fd9c92c2c09f549a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {C} }"></span> as spectrum.</li> <li>The space of Borel <a href="/wiki/Probability_measure" title="Probability measure">probability measures</a> on a compact Hausdorff space is compact for the <a href="/wiki/Vague_topology" title="Vague topology">vague topology</a>, by the Alaoglu theorem.</li> <li>A collection of probability measures on the Borel sets of Euclidean space is called <i>tight</i> if, for any positive epsilon, there exists a compact subset containing all but at most epsilon of the mass of each of the measures. Helly's theorem then asserts that a collection of probability measures is relatively compact for the vague topology if and only if it is tight.</li></ul> <div class="mw-heading mw-heading3"><h3 id="Algebraic_examples">Algebraic examples</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Compact_space&action=edit&section=18" title="Edit section: Algebraic examples"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Topological_group" title="Topological group">Topological groups</a> such as an <a href="/wiki/Orthogonal_group" title="Orthogonal group">orthogonal group</a> are compact, while groups such as a <a href="/wiki/General_linear_group" title="General linear group">general linear group</a> are not.</li> <li>Since the <a href="/wiki/P-adic_numbers" class="mw-redirect" title="P-adic numbers"><span class="texhtml mvar" style="font-style:italic;">p</span>-adic integers</a> are <a href="/wiki/Homeomorphic" class="mw-redirect" title="Homeomorphic">homeomorphic</a> to the Cantor set, they form a compact set.</li> <li>Any <a href="/wiki/Global_field" title="Global field">global field</a> <i>K</i> is a discrete additive subgroup of its <a href="/wiki/Adele_ring" title="Adele ring">adele ring</a>, and the quotient space is compact. This was used in <a href="/wiki/John_Tate_(mathematician)" title="John Tate (mathematician)">John Tate</a>'s <a href="/wiki/Tate%27s_thesis" title="Tate's thesis">thesis</a> to allow <a href="/wiki/Harmonic_analysis" title="Harmonic analysis">harmonic analysis</a> to be used in <a href="/wiki/Number_theory" title="Number theory">number theory</a>.</li> <li>The <a href="/wiki/Spectrum_of_a_ring" title="Spectrum of a ring">spectrum</a> of any <a href="/wiki/Commutative_ring" title="Commutative ring">commutative ring</a> with the <a href="/wiki/Zariski_topology" title="Zariski topology">Zariski topology</a> (that is, the set of all prime ideals) is compact, but never <a href="/wiki/Hausdorff_space" title="Hausdorff space">Hausdorff</a> (except in trivial cases). In algebraic geometry, such topological spaces are examples of quasi-compact <a href="/wiki/Scheme_(mathematics)" title="Scheme (mathematics)">schemes</a>, "quasi" referring to the non-Hausdorff nature of the topology.</li> <li>The spectrum of a <a href="/wiki/Boolean_algebra" title="Boolean algebra">Boolean algebra</a> is compact, a fact which is part of the <a href="/wiki/Stone_representation_theorem" class="mw-redirect" title="Stone representation theorem">Stone representation theorem</a>. <a href="/wiki/Stone_space" title="Stone space">Stone spaces</a>, compact <a href="/wiki/Totally_disconnected_space" title="Totally disconnected space">totally disconnected</a> Hausdorff spaces, form the abstract framework in which these spectra are studied. Such spaces are also useful in the study of <a href="/wiki/Profinite_group" title="Profinite group">profinite groups</a>.</li> <li>The <a href="/wiki/Structure_space" class="mw-redirect" title="Structure space">structure space</a> of a commutative unital <a href="/wiki/Banach_algebra" title="Banach algebra">Banach algebra</a> is a compact Hausdorff space.</li> <li>The <a href="/wiki/Hilbert_cube" title="Hilbert cube">Hilbert cube</a> is compact, again a consequence of Tychonoff's theorem.</li> <li>A <a href="/wiki/Profinite_group" title="Profinite group">profinite group</a> (e.g. <a href="/wiki/Galois_group" title="Galois group">Galois group</a>) is compact.</li></ul> <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=Compact_space&action=edit&section=19" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1184024115">.mw-parser-output .div-col{margin-top:0.3em;column-width:30em}.mw-parser-output .div-col-small{font-size:90%}.mw-parser-output .div-col-rules{column-rule:1px solid #aaa}.mw-parser-output .div-col dl,.mw-parser-output .div-col ol,.mw-parser-output .div-col ul{margin-top:0}.mw-parser-output .div-col li,.mw-parser-output .div-col dd{page-break-inside:avoid;break-inside:avoid-column}</style><div class="div-col" style="column-width: 16em;"> <ul><li><a href="/wiki/Compactly_generated_space" title="Compactly generated space">Compactly generated space</a></li> <li><a href="/wiki/Compactness_theorem" title="Compactness theorem">Compactness theorem</a></li> <li><a href="/wiki/Eberlein_compactum" title="Eberlein compactum">Eberlein compactum</a></li> <li><a href="/wiki/Exhaustion_by_compact_sets" title="Exhaustion by compact sets">Exhaustion by compact sets</a></li> <li><a href="/wiki/Lindel%C3%B6f_space" title="Lindelöf space">Lindelöf space</a></li> <li><a href="/wiki/Metacompact_space" title="Metacompact space">Metacompact space</a></li> <li><a href="/wiki/Noetherian_topological_space" title="Noetherian topological space">Noetherian topological space</a></li> <li><a href="/wiki/Orthocompact_space" title="Orthocompact space">Orthocompact space</a></li> <li><a href="/wiki/Paracompact_space" title="Paracompact space">Paracompact space</a></li> <li><a href="/wiki/Quasi-compact_morphism" title="Quasi-compact morphism">Quasi-compact morphism</a></li> <li><a href="/wiki/Totally_bounded_space" title="Totally bounded space">Precompact set</a> - also called <i><a href="/wiki/Totally_bounded" class="mw-redirect" title="Totally bounded">totally bounded</a></i></li> <li><a href="/wiki/Relatively_compact_subspace" title="Relatively compact subspace">Relatively compact subspace</a></li> <li><a href="/wiki/Totally_bounded" class="mw-redirect" title="Totally bounded">Totally bounded</a></li></ul> </div> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Compact_space&action=edit&section=20" title="Edit section: Notes"><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 reflist-lower-alpha"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-20"><span class="mw-cite-backlink"><b><a href="#cite_ref-20">^</a></b></span> <span class="reference-text"> Let <span class="texhtml"><i>X</i> = {<i>a</i>, <i>b</i>} ∪ <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 \mathbb {N} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {N} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fdf9a96b565ea202d0f4322e9195613fb26a9bed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {N} }"></span></span>, <span class="texhtml"><i>U</i> = {<i>a</i>} ∪ <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 \mathbb {N} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {N} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fdf9a96b565ea202d0f4322e9195613fb26a9bed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {N} }"></span></span>, and <span class="texhtml"><i>V</i> = {<i>b</i>} ∪ <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 \mathbb {N} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {N} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fdf9a96b565ea202d0f4322e9195613fb26a9bed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {N} }"></span></span>. Endow <span class="texhtml">X</span> with the topology generated by the following basic open sets: every subset 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 \mathbb {N} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {N} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fdf9a96b565ea202d0f4322e9195613fb26a9bed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {N} }"></span> is open; the only open sets containing <span class="texhtml mvar" style="font-style:italic;">a</span> are <span class="texhtml mvar" style="font-style:italic;">X</span> and <span class="texhtml mvar" style="font-style:italic;">U</span>; and the only open sets containing <span class="texhtml mvar" style="font-style:italic;">b</span> are <span class="texhtml mvar" style="font-style:italic;">X</span> and <span class="texhtml mvar" style="font-style:italic;">V</span>. Then <span class="texhtml mvar" style="font-style:italic;">U</span> and <span class="texhtml mvar" style="font-style:italic;">V</span> are both compact subsets but their intersection, which 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 \mathbb {N} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {N} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fdf9a96b565ea202d0f4322e9195613fb26a9bed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {N} }"></span>, is not compact. Note that both <span class="texhtml mvar" style="font-style:italic;">U</span> and <span class="texhtml mvar" style="font-style:italic;">V</span> are compact open subsets, neither one of which is closed.</span> </li> <li id="cite_note-21"><span class="mw-cite-backlink"><b><a href="#cite_ref-21">^</a></b></span> <span class="reference-text"> Let <span class="texhtml"><i>X</i> = {<i>a</i>, <i>b</i>}</span> and endow <span class="texhtml mvar" style="font-style:italic;">X</span> with the topology <span class="texhtml">{<i>X</i>, ∅, {<i>a</i>}}</span>. Then <span class="texhtml">{<i>a</i>}</span> is a compact set but it is not closed.</span> </li> <li id="cite_note-22"><span class="mw-cite-backlink"><b><a href="#cite_ref-22">^</a></b></span> <span class="reference-text"> Let <span class="texhtml mvar" style="font-style:italic;">X</span> be the set of non-negative integers. We endow <span class="texhtml mvar" style="font-style:italic;">X</span> with the <a href="/wiki/Particular_point_topology" title="Particular point topology">particular point topology</a> by defining a subset <span class="texhtml"><i>U</i> ⊆ <i>X</i></span> to be open if and only if <span class="texhtml">0 ∈ <i>U</i></span>. Then <span class="texhtml"><i>S</i> := {0}</span> is compact, the closure of <span class="texhtml mvar" style="font-style:italic;">S</span> is all of <span class="texhtml mvar" style="font-style:italic;">X</span>, but <span class="texhtml mvar" style="font-style:italic;">X</span> is not compact since the collection of open subsets <span class="texhtml">{{0, <i>x</i>} : <i>x</i> ∈ <i>X</i>}</span> does not have a finite subcover.</span> </li> </ol></div></div> <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=Compact_space&action=edit&section=21" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239543626"><div class="reflist reflist-columns references-column-width" style="column-width: 25em;"> <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"><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 class="citation encyclopaedia cs1"><a rel="nofollow" class="external text" href="https://www.britannica.com/science/compactness">"Compactness"</a>. <i><a href="/wiki/Encyclopaedia_Britannica" class="mw-redirect" title="Encyclopaedia Britannica">Encyclopaedia Britannica</a></i>. mathematics<span class="reference-accessdate">. Retrieved <span class="nowrap">2019-11-25</span></span> – via britannica.com.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Compactness&rft.btitle=Encyclopaedia+Britannica&rft_id=https%3A%2F%2Fwww.britannica.com%2Fscience%2Fcompactness&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACompact+space" class="Z3988"></span></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"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFEngelking1977" class="citation book cs1">Engelking, Ryszard (1977). <i>General Topology</i>. Warsaw, PL: PWN. p. 266.</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.place=Warsaw%2C+PL&rft.pages=266&rft.pub=PWN&rft.date=1977&rft.aulast=Engelking&rft.aufirst=Ryszard&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACompact+space" class="Z3988"></span></span> </li> <li id="cite_note-:0-3"><span class="mw-cite-backlink">^ <a href="#cite_ref-:0_3-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-:0_3-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://www-groups.mcs.st-andrews.ac.uk/~john/MT4522/Lectures/L22.html">"Sequential compactness"</a>. <i>www-groups.mcs.st-andrews.ac.uk</i>. MT 4522 course lectures<span class="reference-accessdate">. Retrieved <span class="nowrap">2019-11-25</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=www-groups.mcs.st-andrews.ac.uk&rft.atitle=Sequential+compactness&rft_id=http%3A%2F%2Fwww-groups.mcs.st-andrews.ac.uk%2F~john%2FMT4522%2FLectures%2FL22.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACompact+space" 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"><a href="#CITEREFKline1990">Kline 1990</a>, pp. 952–953; <a href="#CITEREFBoyerMerzbach1991">Boyer & Merzbach 1991</a>, p. 561</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"><a href="#CITEREFKline1990">Kline 1990</a>, Chapter 46, §2</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">Frechet, M. 1904. <span title="French-language text"><span lang="fr" style="font-style: normal;">"Generalisation d'un theorem de Weierstrass"</span></span>. <span title="French-language text"><i lang="fr">Analyse Mathematique</i></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"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWeisstein" class="citation web cs1">Weisstein, Eric W. <a rel="nofollow" class="external text" href="http://mathworld.wolfram.com/CompactSpace.html">"Compact Space"</a>. <i>Wolfram MathWorld</i><span class="reference-accessdate">. Retrieved <span class="nowrap">2019-11-25</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Wolfram+MathWorld&rft.atitle=Compact+Space&rft.aulast=Weisstein&rft.aufirst=Eric+W.&rft_id=http%3A%2F%2Fmathworld.wolfram.com%2FCompactSpace.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACompact+space" class="Z3988"></span></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">Here, "collection" means "<a href="/wiki/Set_(mathematics)" title="Set (mathematics)">set</a>" but is used because "collection of open subsets" is less awkward than "set of open subsets". Similarly, "subcollection" means "subset".</span> </li> <li id="cite_note-FOOTNOTEHowes1995xxvi–xxviii-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEHowes1995xxvi–xxviii_9-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFHowes1995">Howes 1995</a>, pp. xxvi–xxviii.</span> </li> <li id="cite_note-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-10">^</a></b></span> <span class="reference-text"><a href="#CITEREFKelley1955">Kelley 1955</a>, p. 163</span> </li> <li id="cite_note-Bourbaki-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-Bourbaki_11-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFBourbaki2007">Bourbaki 2007</a>, § 10.2. Theorem 1, Corollary 1.</span> </li> <li id="cite_note-FOOTNOTEMack1967-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEMack1967_12-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFMack1967">Mack 1967</a>.</span> </li> <li id="cite_note-BourbakiDefinition-13"><span class="mw-cite-backlink"><b><a href="#cite_ref-BourbakiDefinition_13-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFBourbaki2007">Bourbaki 2007</a>, § 9.1. Definition 1.</span> </li> <li id="cite_note-14"><span class="mw-cite-backlink"><b><a href="#cite_ref-14">^</a></b></span> <span class="reference-text"><a href="#CITEREFArkhangel'skiiFedorchuk1990">Arkhangel'skii & Fedorchuk 1990</a>, Theorem 5.3.7</span> </li> <li id="cite_note-15"><span class="mw-cite-backlink"><b><a href="#cite_ref-15">^</a></b></span> <span class="reference-text"><a href="#CITEREFWillard1970">Willard 1970</a> Theorem 30.7.</span> </li> <li id="cite_note-16"><span class="mw-cite-backlink"><b><a href="#cite_ref-16">^</a></b></span> <span class="reference-text"><a href="#CITEREFGillmanJerison1976">Gillman & Jerison 1976</a>, §5.6</span> </li> <li id="cite_note-17"><span class="mw-cite-backlink"><b><a href="#cite_ref-17">^</a></b></span> <span class="reference-text"><a href="#CITEREFRobinson1996">Robinson 1996</a>, Theorem 4.1.13</span> </li> <li id="cite_note-18"><span class="mw-cite-backlink"><b><a href="#cite_ref-18">^</a></b></span> <span class="reference-text"><a href="#CITEREFArkhangel'skiiFedorchuk1990">Arkhangel'skii & Fedorchuk 1990</a>, Theorem 5.2.3</span> </li> <li id="cite_note-19"><span class="mw-cite-backlink"><b><a href="#cite_ref-19">^</a></b></span> <span class="reference-text"><a href="#CITEREFArkhangel'skiiFedorchuk1990">Arkhangel'skii & Fedorchuk 1990</a>, Theorem 5.2.2</span> </li> <li id="cite_note-23"><span class="mw-cite-backlink"><b><a href="#cite_ref-23">^</a></b></span> <span class="reference-text"><a href="#CITEREFArkhangel'skiiFedorchuk1990">Arkhangel'skii & Fedorchuk 1990</a>, Corollary 5.2.1</span> </li> <li id="cite_note-24"><span class="mw-cite-backlink"><b><a href="#cite_ref-24">^</a></b></span> <span class="reference-text"><a href="#CITEREFSteenSeebach1995">Steen & Seebach 1995</a>, p. 67</span> </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="Bibliography">Bibliography</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Compact_space&action=edit&section=22" title="Edit section: Bibliography"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239549316">.mw-parser-output .refbegin{margin-bottom:0.5em}.mw-parser-output .refbegin-hanging-indents>ul{margin-left:0}.mw-parser-output .refbegin-hanging-indents>ul>li{margin-left:0;padding-left:3.2em;text-indent:-3.2em}.mw-parser-output .refbegin-hanging-indents ul,.mw-parser-output .refbegin-hanging-indents ul li{list-style:none}@media(max-width:720px){.mw-parser-output .refbegin-hanging-indents>ul>li{padding-left:1.6em;text-indent:-1.6em}}.mw-parser-output .refbegin-columns{margin-top:0.3em}.mw-parser-output .refbegin-columns ul{margin-top:0}.mw-parser-output .refbegin-columns li{page-break-inside:avoid;break-inside:avoid-column}@media screen{.mw-parser-output .refbegin{font-size:90%}}</style><div class="refbegin refbegin-columns references-column-width" style="column-width: 25em"> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAlexandrovUrysohn1929" class="citation journal cs1"><a href="/wiki/Pavel_Alexandrov" title="Pavel Alexandrov">Alexandrov, Pavel</a>; <a href="/wiki/Pavel_Urysohn" title="Pavel Urysohn">Urysohn, Pavel</a> (1929). 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Wilhelm Engelmann.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Rein+analytischer+Beweis+des+Lehrsatzes%2C+dass+zwischen+je+zwey+Werthen%2C+die+ein+entgegengesetzes+Resultat+gew%C3%A4hren%2C+wenigstens+eine+reele+Wurzel+der+Gleichung+liege&rft.pub=Wilhelm+Engelmann&rft.date=1817&rft.aulast=Bolzano&rft.aufirst=Bernard&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DEoW4AAAAIAAJ%26q%3D%2522Rein%2520analytischer%2520Beweis%2520des%2520Lehrsatzes%2522%26pg%3DPA2-IA3&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACompact+space" class="Z3988"></span> (<i>Purely analytic proof of the theorem that between any two values which give results of opposite sign, there lies at least one real root of the equation</i>).</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBorel1895" class="citation journal cs1"><a href="/wiki/%C3%89mile_Borel" title="Émile Borel">Borel, Émile</a> (1895). <a rel="nofollow" class="external text" href="https://doi.org/10.24033%2Fasens.406">"Sur quelques points de la théorie des fonctions"</a>. <i><a href="/wiki/Annales_Scientifiques_de_l%27%C3%89cole_Normale_Sup%C3%A9rieure" title="Annales Scientifiques de l'École Normale Supérieure">Annales Scientifiques de l'École Normale Supérieure</a></i>. 3. <b>12</b>: 9–55. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.24033%2Fasens.406">10.24033/asens.406</a></span>. <a href="/wiki/JFM_(identifier)" class="mw-redirect" title="JFM (identifier)">JFM</a> <a rel="nofollow" class="external text" href="https://zbmath.org/?format=complete&q=an:26.0429.03">26.0429.03</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Annales+Scientifiques+de+l%27%C3%89cole+Normale+Sup%C3%A9rieure&rft.atitle=Sur+quelques+points+de+la+th%C3%A9orie+des+fonctions&rft.volume=12&rft.pages=9-55&rft.date=1895&rft_id=https%3A%2F%2Fzbmath.org%2F%3Fformat%3Dcomplete%26q%3Dan%3A26.0429.03%23id-name%3DJFM&rft_id=info%3Adoi%2F10.24033%2Fasens.406&rft.aulast=Borel&rft.aufirst=%C3%89mile&rft_id=https%3A%2F%2Fdoi.org%2F10.24033%252Fasens.406&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACompact+space" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBourbaki2007" class="citation book cs1">Bourbaki, Nicolas (2007). <a rel="nofollow" class="external text" href="https://link.springer.com/book/10.1007/978-3-540-33982-3"><i>Topologie générale. 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New York: <a href="/wiki/Springer_Publishing" title="Springer Publishing">Springer-Verlag</a> Science & Business Media. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-97986-1" title="Special:BookSources/978-0-387-97986-1"><bdi>978-0-387-97986-1</bdi></a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/31969970">31969970</a>. <a href="/wiki/OL_(identifier)" class="mw-redirect" title="OL (identifier)">OL</a> <a rel="nofollow" class="external text" href="https://openlibrary.org/books/OL1272666M">1272666M</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Modern+Analysis+and+Topology&rft.place=New+York&rft.series=Graduate+Texts+in+Mathematics&rft.pub=Springer-Verlag+Science+%26+Business+Media&rft.date=1995-06-23&rft_id=info%3Aoclcnum%2F31969970&rft_id=https%3A%2F%2Fopenlibrary.org%2Fbooks%2FOL1272666M%23id-name%3DOL&rft.isbn=978-0-387-97986-1&rft.aulast=Howes&rft.aufirst=Norman+R.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACompact+space" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKelley1955" class="citation book cs1">Kelley, John (1955). <i>General topology</i>. Graduate Texts in Mathematics. Vol. 27. Springer-Verlag.</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.series=Graduate+Texts+in+Mathematics&rft.pub=Springer-Verlag&rft.date=1955&rft.aulast=Kelley&rft.aufirst=John&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACompact+space" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKline1990" class="citation book cs1"><a href="/wiki/Morris_Kline" title="Morris Kline">Kline, Morris</a> (1990) [1972]. <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/mathematicalthou00klin"><i>Mathematical thought from ancient to modern times</i></a></span> (3rd ed.). Oxford University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-19-506136-9" title="Special:BookSources/978-0-19-506136-9"><bdi>978-0-19-506136-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Mathematical+thought+from+ancient+to+modern+times&rft.edition=3rd&rft.pub=Oxford+University+Press&rft.date=1990&rft.isbn=978-0-19-506136-9&rft.aulast=Kline&rft.aufirst=Morris&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fmathematicalthou00klin&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACompact+space" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLebesgue1904" class="citation book cs1">Lebesgue, Henri (1904). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=VfUKAAAAYAAJ&q=%22Lebesgue%22%20%22Le%C3%A7ons%20sur%20l'int%C3%A9gration%20et%20la%20recherche%20des%20fonctions%20...%22&pg=PA1"><i>Leçons sur l'intégration et la recherche des fonctions primitives</i></a>. Gauthier-Villars.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Le%C3%A7ons+sur+l%27int%C3%A9gration+et+la+recherche+des+fonctions+primitives&rft.pub=Gauthier-Villars&rft.date=1904&rft.aulast=Lebesgue&rft.aufirst=Henri&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DVfUKAAAAYAAJ%26q%3D%2522Lebesgue%2522%2520%2522Le%25C3%25A7ons%2520sur%2520l%27int%25C3%25A9gration%2520et%2520la%2520recherche%2520des%2520fonctions%2520...%2522%26pg%3DPA1&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACompact+space" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMack1967" class="citation journal cs1">Mack, John (1967). "Directed covers and paracompact spaces". <i>Canadian Journal of Mathematics</i>. <b>19</b>: 649–654. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.4153%2FCJM-1967-059-0">10.4153/CJM-1967-059-0</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=0211382">0211382</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Canadian+Journal+of+Mathematics&rft.atitle=Directed+covers+and+paracompact+spaces&rft.volume=19&rft.pages=649-654&rft.date=1967&rft_id=info%3Adoi%2F10.4153%2FCJM-1967-059-0&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D211382%23id-name%3DMR&rft.aulast=Mack&rft.aufirst=John&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACompact+space" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRobinson1996" class="citation book cs1"><a href="/wiki/Abraham_Robinson" title="Abraham Robinson">Robinson, Abraham</a> (1996). <i>Non-standard analysis</i>. Princeton University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-691-04490-3" title="Special:BookSources/978-0-691-04490-3"><bdi>978-0-691-04490-3</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=0205854">0205854</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Non-standard+analysis&rft.pub=Princeton+University+Press&rft.date=1996&rft.isbn=978-0-691-04490-3&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D0205854%23id-name%3DMR&rft.aulast=Robinson&rft.aufirst=Abraham&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACompact+space" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFScarboroughStone1966" class="citation journal cs1">Scarborough, C.T.; Stone, A.H. (1966). <a rel="nofollow" class="external text" href="https://www.ams.org/tran/1966-124-01/S0002-9947-1966-0203679-7/S0002-9947-1966-0203679-7.pdf">"Products of nearly compact spaces"</a> <span class="cs1-format">(PDF)</span>. <i>Transactions of the American Mathematical Society</i>. <b>124</b> (1): 131–147. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F1994440">10.2307/1994440</a></span>. <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/1994440">1994440</a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20170816203049/http://www.ams.org/tran/1966-124-01/S0002-9947-1966-0203679-7/S0002-9947-1966-0203679-7.pdf">Archived</a> <span class="cs1-format">(PDF)</span> from the original on 2017-08-16.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Transactions+of+the+American+Mathematical+Society&rft.atitle=Products+of+nearly+compact+spaces&rft.volume=124&rft.issue=1&rft.pages=131-147&rft.date=1966&rft_id=info%3Adoi%2F10.2307%2F1994440&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F1994440%23id-name%3DJSTOR&rft.aulast=Scarborough&rft.aufirst=C.T.&rft.au=Stone%2C+A.H.&rft_id=https%3A%2F%2Fwww.ams.org%2Ftran%2F1966-124-01%2FS0002-9947-1966-0203679-7%2FS0002-9947-1966-0203679-7.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACompact+space" class="Z3988"></span>.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSteenSeebach1995" class="citation book cs1"><a href="/wiki/Lynn_Arthur_Steen" class="mw-redirect" title="Lynn Arthur Steen">Steen, Lynn Arthur</a>; <a href="/wiki/J._Arthur_Seebach,_Jr." class="mw-redirect" 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> (Dover Publications reprint of 1978 ed.). Berlin, New York: Springer-Verlag. <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+Publications+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%3ACompact+space" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWillard1970" class="citation book cs1">Willard, Stephen (1970). <i>General Topology</i>. Dover publications. <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"><bdi>0-486-43479-6</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.pub=Dover+publications&rft.date=1970&rft.isbn=0-486-43479-6&rft.aulast=Willard&rft.aufirst=Stephen&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACompact+space" class="Z3988"></span></li></ul> </div> <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=Compact_space&action=edit&section=23" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSundström2010" class="citation arxiv cs1">Sundström, Manya Raman (2010). "A pedagogical history of compactness". <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/1006.4131v1">1006.4131v1</a></span> [<a rel="nofollow" class="external text" href="https://arxiv.org/archive/math.HO">math.HO</a>].</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=preprint&rft.jtitle=arXiv&rft.atitle=A+pedagogical+history+of+compactness&rft.date=2010&rft_id=info%3Aarxiv%2F1006.4131v1&rft.aulast=Sundstr%C3%B6m&rft.aufirst=Manya+Raman&rfr_id=info%3Asid%2Fen.wikipedia.org%3ACompact+space" class="Z3988"></span></li></ul> <hr /> <p><i>This article incorporates material from <a rel="nofollow" class="external text" href="https://planetmath.org/examplesofcompactspaces">Examples of compact spaces</a> on <a href="/wiki/PlanetMath" title="PlanetMath">PlanetMath</a>, which is licensed under the <a href="/wiki/Wikipedia:CC-BY-SA" class="mw-redirect" title="Wikipedia:CC-BY-SA">Creative Commons Attribution/Share-Alike License</a>.</i> </p> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl dl,.mw-parser-output .hlist dl ol,.mw-parser-output .hlist dl ul,.mw-parser-output .hlist ol dl,.mw-parser-output .hlist ol ol,.mw-parser-output .hlist ol ul,.mw-parser-output .hlist ul dl,.mw-parser-output .hlist ul ol,.mw-parser-output .hlist ul ul{display:inline}.mw-parser-output .hlist .mw-empty-li{display:none}.mw-parser-output .hlist dt::after{content:": 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navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="3"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1239400231">.mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output 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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 href="/wiki/General_topology" title="General topology">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 class="mw-selflink selflink">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> <ul><li><span class="noviewer" typeof="mw:File"><span title="Category"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/16px-Symbol_category_class.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/23px-Symbol_category_class.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/31px-Symbol_category_class.svg.png 2x" data-file-width="180" data-file-height="185" /></span></span> <a href="/wiki/Category:Topology" title="Category:Topology">Category</a></li> <li><span class="nowrap"><span class="noviewer" typeof="mw:File"><a href="/wiki/File:Nuvola_apps_edu_mathematics_blue-p.svg" class="mw-file-description"><img alt="icon" src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/28px-Nuvola_apps_edu_mathematics_blue-p.svg.png" decoding="async" width="28" height="28" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/42px-Nuvola_apps_edu_mathematics_blue-p.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/56px-Nuvola_apps_edu_mathematics_blue-p.svg.png 2x" data-file-width="128" data-file-height="128" /></a></span> </span><a href="/wiki/Portal:Mathematics" title="Portal:Mathematics">Mathematics portal</a></li> <li><span class="noviewer" typeof="mw:File"><a href="/wiki/File:Wikibooks-logo.svg" class="mw-file-description" title="Wikibooks page"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Wikibooks-logo.svg/16px-Wikibooks-logo.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Wikibooks-logo.svg/24px-Wikibooks-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Wikibooks-logo.svg/32px-Wikibooks-logo.svg.png 2x" data-file-width="300" data-file-height="300" /></a></span> <a href="https://en.wikibooks.org/wiki/Topology" class="extiw" title="wikibooks:Topology">Wikibook</a></li> <li><span class="noviewer" typeof="mw:File"><a href="/wiki/File:Wikiversity_logo_2017.svg" class="mw-file-description" title="Wikiversity page"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/0/0b/Wikiversity_logo_2017.svg/16px-Wikiversity_logo_2017.svg.png" decoding="async" width="16" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/0b/Wikiversity_logo_2017.svg/24px-Wikiversity_logo_2017.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/0b/Wikiversity_logo_2017.svg/32px-Wikiversity_logo_2017.svg.png 2x" data-file-width="626" data-file-height="512" /></a></span> <a href="https://en.wikiversity.org/wiki/Topology" class="extiw" title="wikiversity:Topology">Wikiversity</a></li> <li><span class="noviewer" typeof="mw:File"><span title="List-Class article"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/d/db/Symbol_list_class.svg/16px-Symbol_list_class.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/d/db/Symbol_list_class.svg/23px-Symbol_list_class.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/d/db/Symbol_list_class.svg/31px-Symbol_list_class.svg.png 2x" data-file-width="180" data-file-height="185" /></span></span> <a href="/wiki/List_of_topology_topics" title="List of topology topics">Topics</a> <ul><li><a href="/wiki/List_of_general_topology_topics" title="List of general topology topics">general</a></li> <li><a href="/wiki/List_of_algebraic_topology_topics" title="List of algebraic topology topics">algebraic</a></li> <li><a href="/wiki/List_of_geometric_topology_topics" title="List of geometric topology topics">geometric</a></li></ul></li> <li><span class="noviewer" typeof="mw:File"><span title="List-Class article"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/d/db/Symbol_list_class.svg/16px-Symbol_list_class.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/d/db/Symbol_list_class.svg/23px-Symbol_list_class.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/d/db/Symbol_list_class.svg/31px-Symbol_list_class.svg.png 2x" data-file-width="180" data-file-height="185" /></span></span> <a href="/wiki/List_of_important_publications_in_mathematics#Topology" title="List of important publications in mathematics">Publications</a></li></ul> </div></td></tr></tbody></table></div>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" href="https://en.wikipedia.org/w/index.php?title=Compact_space&oldid=1256991238">https://en.wikipedia.org/w/index.php?title=Compact_space&oldid=1256991238</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Help:Category" title="Help:Category">Categories</a>: <ul><li><a href="/wiki/Category:Compactness_(mathematics)" title="Category:Compactness (mathematics)">Compactness (mathematics)</a></li><li><a href="/wiki/Category:General_topology" 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