CINXE.COM

<!DOCTYPE html> <html class="client-nojs vector-feature-language-in-header-enabled vector-feature-language-in-main-page-header-disabled vector-feature-sticky-header-disabled vector-feature-page-tools-pinned-disabled vector-feature-toc-pinned-clientpref-1 vector-feature-main-menu-pinned-disabled vector-feature-limited-width-clientpref-1 vector-feature-limited-width-content-enabled vector-feature-custom-font-size-clientpref-1 vector-feature-appearance-pinned-clientpref-1 vector-feature-night-mode-enabled skin-theme-clientpref-day vector-toc-available" lang="en" dir="ltr"> <head> <meta charset="UTF-8"> <title>Topological manifold - Wikipedia</title> <script>(function(){var className="client-js vector-feature-language-in-header-enabled vector-feature-language-in-main-page-header-disabled vector-feature-sticky-header-disabled vector-feature-page-tools-pinned-disabled vector-feature-toc-pinned-clientpref-1 vector-feature-main-menu-pinned-disabled vector-feature-limited-width-clientpref-1 vector-feature-limited-width-content-enabled vector-feature-custom-font-size-clientpref-1 vector-feature-appearance-pinned-clientpref-1 vector-feature-night-mode-enabled skin-theme-clientpref-day vector-toc-available";var cookie=document.cookie.match(/(?:^|; )enwikimwclientpreferences=([^;]+)/);if(cookie){cookie[1].split('%2C').forEach(function(pref){className=className.replace(new RegExp('(^| )'+pref.replace(/-clientpref-\w+$|[^\w-]+/g,'')+'-clientpref-\\w+( |$)'),'$1'+pref+'$2');});}document.documentElement.className=className;}());RLCONF={"wgBreakFrames":false,"wgSeparatorTransformTable":["",""],"wgDigitTransformTable":["",""],"wgDefaultDateFormat":"dmy", "wgMonthNames":["","January","February","March","April","May","June","July","August","September","October","November","December"],"wgRequestId":"32c36b67-55ac-4fe2-ab9c-be2aa0a6482d","wgCanonicalNamespace":"","wgCanonicalSpecialPageName":false,"wgNamespaceNumber":0,"wgPageName":"Topological_manifold","wgTitle":"Topological manifold","wgCurRevisionId":1251981624,"wgRevisionId":1251981624,"wgArticleId":2119193,"wgIsArticle":true,"wgIsRedirect":false,"wgAction":"view","wgUserName":null,"wgUserGroups":["*"],"wgCategories":["Articles with short description","Short description is different from Wikidata","Commons category link is on Wikidata","Manifolds","Properties of topological spaces"],"wgPageViewLanguage":"en","wgPageContentLanguage":"en","wgPageContentModel":"wikitext","wgRelevantPageName":"Topological_manifold","wgRelevantArticleId":2119193,"wgIsProbablyEditable":true,"wgRelevantPageIsProbablyEditable":true,"wgRestrictionEdit":[],"wgRestrictionMove":[],"wgNoticeProject":"wikipedia", "wgCiteReferencePreviewsActive":false,"wgFlaggedRevsParams":{"tags":{"status":{"levels":1}}},"wgMediaViewerOnClick":true,"wgMediaViewerEnabledByDefault":true,"wgPopupsFlags":0,"wgVisualEditor":{"pageLanguageCode":"en","pageLanguageDir":"ltr","pageVariantFallbacks":"en"},"wgMFDisplayWikibaseDescriptions":{"search":true,"watchlist":true,"tagline":false,"nearby":true},"wgWMESchemaEditAttemptStepOversample":false,"wgWMEPageLength":20000,"wgRelatedArticlesCompat":[],"wgEditSubmitButtonLabelPublish":true,"wgULSPosition":"interlanguage","wgULSisCompactLinksEnabled":false,"wgVector2022LanguageInHeader":true,"wgULSisLanguageSelectorEmpty":false,"wgWikibaseItemId":"Q2721559","wgCheckUserClientHintsHeadersJsApi":["brands","architecture","bitness","fullVersionList","mobile","model","platform","platformVersion"],"GEHomepageSuggestedEditsEnableTopics":true,"wgGETopicsMatchModeEnabled":false,"wgGEStructuredTaskRejectionReasonTextInputEnabled":false,"wgGELevelingUpEnabledForUser":false};RLSTATE={ "ext.globalCssJs.user.styles":"ready","site.styles":"ready","user.styles":"ready","ext.globalCssJs.user":"ready","user":"ready","user.options":"loading","ext.cite.styles":"ready","ext.math.styles":"ready","skins.vector.search.codex.styles":"ready","skins.vector.styles":"ready","skins.vector.icons":"ready","jquery.makeCollapsible.styles":"ready","ext.wikimediamessages.styles":"ready","ext.visualEditor.desktopArticleTarget.noscript":"ready","ext.uls.interlanguage":"ready","wikibase.client.init":"ready","ext.wikimediaBadges":"ready"};RLPAGEMODULES=["ext.cite.ux-enhancements","mediawiki.page.media","site","mediawiki.page.ready","jquery.makeCollapsible","mediawiki.toc","skins.vector.js","ext.centralNotice.geoIP","ext.centralNotice.startUp","ext.gadget.ReferenceTooltips","ext.gadget.switcher","ext.urlShortener.toolbar","ext.centralauth.centralautologin","mmv.bootstrap","ext.popups","ext.visualEditor.desktopArticleTarget.init","ext.visualEditor.targetLoader","ext.echo.centralauth", "ext.eventLogging","ext.wikimediaEvents","ext.navigationTiming","ext.uls.interface","ext.cx.eventlogging.campaigns","ext.cx.uls.quick.actions","wikibase.client.vector-2022","ext.checkUser.clientHints","ext.quicksurveys.init","ext.growthExperiments.SuggestedEditSession","wikibase.sidebar.tracking"];</script> <script>(RLQ=window.RLQ||[]).push(function(){mw.loader.impl(function(){return["user.options@12s5i",function($,jQuery,require,module){mw.user.tokens.set({"patrolToken":"+\\","watchToken":"+\\","csrfToken":"+\\"}); }];});});</script> <link rel="stylesheet" href="/w/load.php?lang=en&modules=ext.cite.styles%7Cext.math.styles%7Cext.uls.interlanguage%7Cext.visualEditor.desktopArticleTarget.noscript%7Cext.wikimediaBadges%7Cext.wikimediamessages.styles%7Cjquery.makeCollapsible.styles%7Cskins.vector.icons%2Cstyles%7Cskins.vector.search.codex.styles%7Cwikibase.client.init&only=styles&skin=vector-2022"> <script async="" src="/w/load.php?lang=en&modules=startup&only=scripts&raw=1&skin=vector-2022"></script> <meta name="ResourceLoaderDynamicStyles" content=""> <link rel="stylesheet" href="/w/load.php?lang=en&modules=site.styles&only=styles&skin=vector-2022"> <meta name="generator" content="MediaWiki 1.44.0-wmf.5"> <meta name="referrer" content="origin"> <meta name="referrer" content="origin-when-cross-origin"> <meta name="robots" content="max-image-preview:standard"> <meta name="format-detection" content="telephone=no"> <meta name="viewport" content="width=1120"> <meta property="og:title" content="Topological manifold - Wikipedia"> <meta property="og:type" content="website"> <link rel="preconnect" href="//upload.wikimedia.org"> <link rel="alternate" media="only screen and (max-width: 640px)" href="//en.m.wikipedia.org/wiki/Topological_manifold"> <link rel="alternate" type="application/x-wiki" title="Edit this page" href="/w/index.php?title=Topological_manifold&action=edit"> <link rel="apple-touch-icon" href="/static/apple-touch/wikipedia.png"> <link rel="icon" href="/static/favicon/wikipedia.ico"> <link rel="search" type="application/opensearchdescription+xml" href="/w/rest.php/v1/search" title="Wikipedia (en)"> <link rel="EditURI" type="application/rsd+xml" href="//en.wikipedia.org/w/api.php?action=rsd"> <link rel="canonical" href="https://en.wikipedia.org/wiki/Topological_manifold"> <link rel="license" href="https://creativecommons.org/licenses/by-sa/4.0/deed.en"> <link rel="alternate" type="application/atom+xml" title="Wikipedia Atom feed" href="/w/index.php?title=Special:RecentChanges&feed=atom"> <link rel="dns-prefetch" href="//meta.wikimedia.org" /> <link rel="dns-prefetch" href="//login.wikimedia.org"> </head> <body class="skin--responsive skin-vector skin-vector-search-vue mediawiki ltr sitedir-ltr mw-hide-empty-elt ns-0 ns-subject mw-editable page-Topological_manifold rootpage-Topological_manifold skin-vector-2022 action-view"><a class="mw-jump-link" href="#bodyContent">Jump to content</a> <div class="vector-header-container"> <header class="vector-header mw-header"> <div class="vector-header-start"> <nav class="vector-main-menu-landmark" aria-label="Site"> <div id="vector-main-menu-dropdown" class="vector-dropdown vector-main-menu-dropdown vector-button-flush-left vector-button-flush-right" > <input type="checkbox" id="vector-main-menu-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-main-menu-dropdown" class="vector-dropdown-checkbox " aria-label="Main menu" > <label id="vector-main-menu-dropdown-label" for="vector-main-menu-dropdown-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-menu mw-ui-icon-wikimedia-menu"></span> <span class="vector-dropdown-label-text">Main menu</span> </label> <div class="vector-dropdown-content"> <div id="vector-main-menu-unpinned-container" class="vector-unpinned-container"> <div id="vector-main-menu" class="vector-main-menu vector-pinnable-element"> <div class="vector-pinnable-header vector-main-menu-pinnable-header vector-pinnable-header-unpinned" data-feature-name="main-menu-pinned" data-pinnable-element-id="vector-main-menu" data-pinned-container-id="vector-main-menu-pinned-container" data-unpinned-container-id="vector-main-menu-unpinned-container" > <div class="vector-pinnable-header-label">Main menu</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-main-menu.pin">move to sidebar</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-main-menu.unpin">hide</button> </div> <div id="p-navigation" class="vector-menu mw-portlet mw-portlet-navigation" > <div class="vector-menu-heading"> Navigation </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="n-mainpage-description" class="mw-list-item"><a href="/wiki/Main_Page" title="Visit the main page [z]" accesskey="z"><span>Main page</span></a></li><li id="n-contents" class="mw-list-item"><a href="/wiki/Wikipedia:Contents" title="Guides to browsing Wikipedia"><span>Contents</span></a></li><li id="n-currentevents" class="mw-list-item"><a href="/wiki/Portal:Current_events" title="Articles related to current events"><span>Current events</span></a></li><li id="n-randompage" class="mw-list-item"><a href="/wiki/Special:Random" title="Visit a randomly selected article [x]" accesskey="x"><span>Random article</span></a></li><li id="n-aboutsite" class="mw-list-item"><a href="/wiki/Wikipedia:About" title="Learn about Wikipedia and how it works"><span>About Wikipedia</span></a></li><li id="n-contactpage" class="mw-list-item"><a href="//en.wikipedia.org/wiki/Wikipedia:Contact_us" title="How to contact Wikipedia"><span>Contact us</span></a></li> </ul> </div> </div> <div id="p-interaction" class="vector-menu mw-portlet mw-portlet-interaction" > <div class="vector-menu-heading"> Contribute </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="n-help" class="mw-list-item"><a href="/wiki/Help:Contents" title="Guidance on how to use and edit Wikipedia"><span>Help</span></a></li><li id="n-introduction" class="mw-list-item"><a href="/wiki/Help:Introduction" title="Learn how to edit Wikipedia"><span>Learn to edit</span></a></li><li id="n-portal" class="mw-list-item"><a href="/wiki/Wikipedia:Community_portal" title="The hub for editors"><span>Community portal</span></a></li><li id="n-recentchanges" class="mw-list-item"><a href="/wiki/Special:RecentChanges" title="A list of recent changes to Wikipedia [r]" accesskey="r"><span>Recent changes</span></a></li><li id="n-upload" class="mw-list-item"><a href="/wiki/Wikipedia:File_upload_wizard" title="Add images or other media for use on Wikipedia"><span>Upload file</span></a></li> </ul> </div> </div> </div> </div> </div> </div> </nav> <a href="/wiki/Main_Page" class="mw-logo"> <img class="mw-logo-icon" src="/static/images/icons/wikipedia.png" alt="" aria-hidden="true" height="50" width="50"> <span class="mw-logo-container skin-invert"> <img class="mw-logo-wordmark" alt="Wikipedia" src="/static/images/mobile/copyright/wikipedia-wordmark-en.svg" style="width: 7.5em; height: 1.125em;"> <img class="mw-logo-tagline" alt="The Free Encyclopedia" src="/static/images/mobile/copyright/wikipedia-tagline-en.svg" width="117" height="13" style="width: 7.3125em; height: 0.8125em;"> </span> </a> </div> <div class="vector-header-end"> <div id="p-search" role="search" class="vector-search-box-vue vector-search-box-collapses vector-search-box-show-thumbnail vector-search-box-auto-expand-width vector-search-box"> <a href="/wiki/Special:Search" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only search-toggle" title="Search Wikipedia [f]" accesskey="f"><span class="vector-icon mw-ui-icon-search mw-ui-icon-wikimedia-search"></span> <span>Search</span> </a> <div class="vector-typeahead-search-container"> <div class="cdx-typeahead-search cdx-typeahead-search--show-thumbnail cdx-typeahead-search--auto-expand-width"> <form action="/w/index.php" id="searchform" class="cdx-search-input cdx-search-input--has-end-button"> <div id="simpleSearch" class="cdx-search-input__input-wrapper" data-search-loc="header-moved"> <div class="cdx-text-input cdx-text-input--has-start-icon"> <input class="cdx-text-input__input" type="search" name="search" placeholder="Search Wikipedia" aria-label="Search Wikipedia" autocapitalize="sentences" title="Search Wikipedia [f]" accesskey="f" id="searchInput" > <span class="cdx-text-input__icon cdx-text-input__start-icon"></span> </div> <input type="hidden" name="title" value="Special:Search"> </div> <button class="cdx-button cdx-search-input__end-button">Search</button> </form> </div> </div> </div> <nav class="vector-user-links vector-user-links-wide" aria-label="Personal tools"> <div class="vector-user-links-main"> <div id="p-vector-user-menu-preferences" class="vector-menu mw-portlet emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> <div id="p-vector-user-menu-userpage" class="vector-menu mw-portlet emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> <nav class="vector-appearance-landmark" aria-label="Appearance"> <div id="vector-appearance-dropdown" class="vector-dropdown " title="Change the appearance of the page's font size, width, and color" > <input type="checkbox" id="vector-appearance-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-appearance-dropdown" class="vector-dropdown-checkbox " aria-label="Appearance" > <label id="vector-appearance-dropdown-label" for="vector-appearance-dropdown-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-appearance mw-ui-icon-wikimedia-appearance"></span> <span class="vector-dropdown-label-text">Appearance</span> </label> <div class="vector-dropdown-content"> <div id="vector-appearance-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <div id="p-vector-user-menu-notifications" class="vector-menu mw-portlet emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> <div id="p-vector-user-menu-overflow" class="vector-menu mw-portlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-sitesupport-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="https://donate.wikimedia.org/wiki/Special:FundraiserRedirector?utm_source=donate&utm_medium=sidebar&utm_campaign=C13_en.wikipedia.org&uselang=en" class=""><span>Donate</span></a> </li> <li id="pt-createaccount-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="/w/index.php?title=Special:CreateAccount&returnto=Topological+manifold" title="You are encouraged to create an account and log in; however, it is not mandatory" class=""><span>Create account</span></a> </li> <li id="pt-login-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="/w/index.php?title=Special:UserLogin&returnto=Topological+manifold" title="You're encouraged to log in; however, it's not mandatory. [o]" accesskey="o" class=""><span>Log in</span></a> </li> </ul> </div> </div> </div> <div id="vector-user-links-dropdown" class="vector-dropdown vector-user-menu vector-button-flush-right vector-user-menu-logged-out" title="Log in and more options" > <input type="checkbox" id="vector-user-links-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-user-links-dropdown" class="vector-dropdown-checkbox " aria-label="Personal tools" > <label id="vector-user-links-dropdown-label" for="vector-user-links-dropdown-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-ellipsis mw-ui-icon-wikimedia-ellipsis"></span> <span class="vector-dropdown-label-text">Personal tools</span> </label> <div class="vector-dropdown-content"> <div id="p-personal" class="vector-menu mw-portlet mw-portlet-personal user-links-collapsible-item" title="User menu" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-sitesupport" class="user-links-collapsible-item mw-list-item"><a href="https://donate.wikimedia.org/wiki/Special:FundraiserRedirector?utm_source=donate&utm_medium=sidebar&utm_campaign=C13_en.wikipedia.org&uselang=en"><span>Donate</span></a></li><li id="pt-createaccount" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=Special:CreateAccount&returnto=Topological+manifold" title="You are encouraged to create an account and log in; however, it is not mandatory"><span class="vector-icon mw-ui-icon-userAdd mw-ui-icon-wikimedia-userAdd"></span> <span>Create account</span></a></li><li id="pt-login" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=Special:UserLogin&returnto=Topological+manifold" title="You're encouraged to log in; however, it's not mandatory. [o]" accesskey="o"><span class="vector-icon mw-ui-icon-logIn mw-ui-icon-wikimedia-logIn"></span> <span>Log in</span></a></li> </ul> </div> </div> <div id="p-user-menu-anon-editor" class="vector-menu mw-portlet mw-portlet-user-menu-anon-editor" > <div class="vector-menu-heading"> Pages for logged out editors <a href="/wiki/Help:Introduction" aria-label="Learn more about editing"><span>learn more</span></a> </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-anoncontribs" class="mw-list-item"><a href="/wiki/Special:MyContributions" title="A list of edits made from this IP address [y]" accesskey="y"><span>Contributions</span></a></li><li id="pt-anontalk" class="mw-list-item"><a href="/wiki/Special:MyTalk" title="Discussion about edits from this IP address [n]" accesskey="n"><span>Talk</span></a></li> </ul> </div> </div> </div> </div> </nav> </div> </header> </div> <div class="mw-page-container"> <div class="mw-page-container-inner"> <div class="vector-sitenotice-container"> <div id="siteNotice"></div> </div> <div class="vector-column-start"> <div class="vector-main-menu-container"> <div id="mw-navigation"> <nav id="mw-panel" class="vector-main-menu-landmark" aria-label="Site"> <div id="vector-main-menu-pinned-container" class="vector-pinned-container"> </div> </nav> </div> </div> <div class="vector-sticky-pinned-container"> <nav id="mw-panel-toc" aria-label="Contents" data-event-name="ui.sidebar-toc" class="mw-table-of-contents-container vector-toc-landmark"> <div id="vector-toc-pinned-container" class="vector-pinned-container"> <div id="vector-toc" class="vector-toc vector-pinnable-element"> <div class="vector-pinnable-header vector-toc-pinnable-header vector-pinnable-header-pinned" data-feature-name="toc-pinned" data-pinnable-element-id="vector-toc" > <h2 class="vector-pinnable-header-label">Contents</h2> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">move to sidebar</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">hide</button> </div> <ul class="vector-toc-contents" id="mw-panel-toc-list"> <li id="toc-mw-content-text" class="vector-toc-list-item vector-toc-level-1"> <a href="#" class="vector-toc-link"> <div class="vector-toc-text">(Top)</div> </a> </li> <li id="toc-Formal_definition" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Formal_definition"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>Formal definition</span> </div> </a> <ul id="toc-Formal_definition-sublist" class="vector-toc-list"> </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">2</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-n-manifolds" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#n-manifolds"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span><i>n</i>-manifolds</span> </div> </a> <ul id="toc-n-manifolds-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Projective_manifolds" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Projective_manifolds"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>Projective manifolds</span> </div> </a> <ul id="toc-Projective_manifolds-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Other_manifolds" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Other_manifolds"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3</span> <span>Other manifolds</span> </div> </a> <ul id="toc-Other_manifolds-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Properties" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Properties"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Properties</span> </div> </a> <button aria-controls="toc-Properties-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 subsection</span> </button> <ul id="toc-Properties-sublist" class="vector-toc-list"> <li id="toc-The_Hausdorff_axiom" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#The_Hausdorff_axiom"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>The Hausdorff axiom</span> </div> </a> <ul id="toc-The_Hausdorff_axiom-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Compactness_and_countability_axioms" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Compactness_and_countability_axioms"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>Compactness and countability axioms</span> </div> </a> <ul id="toc-Compactness_and_countability_axioms-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Dimensionality" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Dimensionality"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.3</span> <span>Dimensionality</span> </div> </a> <ul id="toc-Dimensionality-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Coordinate_charts" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Coordinate_charts"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Coordinate charts</span> </div> </a> <ul id="toc-Coordinate_charts-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Classification_of_manifolds" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Classification_of_manifolds"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Classification of manifolds</span> </div> </a> <button aria-controls="toc-Classification_of_manifolds-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 Classification of manifolds subsection</span> </button> <ul id="toc-Classification_of_manifolds-sublist" class="vector-toc-list"> <li id="toc-Discrete_spaces_(0-Manifold)" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Discrete_spaces_(0-Manifold)"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.1</span> <span>Discrete spaces (0-Manifold)</span> </div> </a> <ul id="toc-Discrete_spaces_(0-Manifold)-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Curves_(1-Manifold)" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Curves_(1-Manifold)"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.2</span> <span>Curves (1-Manifold)</span> </div> </a> <ul id="toc-Curves_(1-Manifold)-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Surfaces_(2-Manifold)" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Surfaces_(2-Manifold)"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.3</span> <span>Surfaces (2-Manifold)</span> </div> </a> <ul id="toc-Surfaces_(2-Manifold)-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Volumes_(3-Manifold)" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Volumes_(3-Manifold)"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.4</span> <span>Volumes (3-Manifold)</span> </div> </a> <ul id="toc-Volumes_(3-Manifold)-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-General_n-manifold" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#General_n-manifold"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.5</span> <span>General <i>n</i>-manifold</span> </div> </a> <ul id="toc-General_n-manifold-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Manifolds_with_boundary" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Manifolds_with_boundary"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.6</span> <span>Manifolds with boundary</span> </div> </a> <ul id="toc-Manifolds_with_boundary-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Constructions" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Constructions"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Constructions</span> </div> </a> <button aria-controls="toc-Constructions-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 Constructions subsection</span> </button> <ul id="toc-Constructions-sublist" class="vector-toc-list"> <li id="toc-Product_manifolds" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Product_manifolds"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.1</span> <span>Product manifolds</span> </div> </a> <ul id="toc-Product_manifolds-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Disjoint_union" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Disjoint_union"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.2</span> <span>Disjoint union</span> </div> </a> <ul id="toc-Disjoint_union-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Connected_sum" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Connected_sum"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.3</span> <span>Connected sum</span> </div> </a> <ul id="toc-Connected_sum-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Submanifold" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Submanifold"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.4</span> <span>Submanifold</span> </div> </a> <ul id="toc-Submanifold-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Footnotes" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Footnotes"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Footnotes</span> </div> </a> <ul id="toc-Footnotes-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">8</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</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">Topological manifold</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 15 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-15" 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">15 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Varietat_topol%C3%B2gica" title="Varietat topològica – Catalan" lang="ca" hreflang="ca" data-title="Varietat topològica" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Topoloogiline_muutkond" title="Topoloogiline muutkond – Estonian" lang="et" hreflang="et" data-title="Topoloogiline muutkond" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%A4%CE%BF%CF%80%CE%BF%CE%BB%CE%BF%CE%B3%CE%B9%CE%BA%CE%AE_%CF%80%CE%BF%CE%BB%CE%BB%CE%B1%CF%80%CE%BB%CF%8C%CF%84%CE%B7%CF%84%CE%B1" title="Τοπολογική πολλαπλότητα – Greek" lang="el" hreflang="el" data-title="Τοπολογική πολλαπλότητα" 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/Variedad_topol%C3%B3gica" title="Variedad topológica – Spanish" lang="es" hreflang="es" data-title="Variedad topológica" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D9%85%D9%86%DB%8C%D9%81%D9%84%D8%AF_%D8%AA%D9%88%D9%BE%D9%88%D9%84%D9%88%DA%98%DB%8C%DA%A9%DB%8C" title="منیفلد توپولوژیکی – Persian" lang="fa" hreflang="fa" data-title="منیفلد توپولوژیکی" 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/Vari%C3%A9t%C3%A9_topologique" title="Variété topologique – French" lang="fr" hreflang="fr" data-title="Variété topologique" 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/Variedade_topol%C3%B3xica" title="Variedade topolóxica – Galician" lang="gl" hreflang="gl" data-title="Variedade topolóxica" data-language-autonym="Galego" data-language-local-name="Galician" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Keragaman_topologi" title="Keragaman topologi – Indonesian" lang="id" hreflang="id" data-title="Keragaman topologi" 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-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%99%D7%A8%D7%99%D7%A2%D7%94_%D7%98%D7%95%D7%A4%D7%95%D7%9C%D7%95%D7%92%D7%99%D7%AA" title="יריעה טופולוגית – Hebrew" lang="he" hreflang="he" data-title="יריעה טופולוגית" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Topologische_vari%C3%ABteit" title="Topologische variëteit – Dutch" lang="nl" hreflang="nl" data-title="Topologische variëteit" 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/%E4%BD%8D%E7%9B%B8%E5%A4%9A%E6%A7%98%E4%BD%93" title="位相多様体 – Japanese" lang="ja" hreflang="ja" data-title="位相多様体" 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/Rozmaito%C5%9B%C4%87_topologiczna" title="Rozmaitość topologiczna – Polish" lang="pl" hreflang="pl" data-title="Rozmaitość topologiczna" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-ru badge-Q70894304 mw-list-item" title=""><a href="https://ru.wikipedia.org/wiki/%D0%A2%D0%BE%D0%BF%D0%BE%D0%BB%D0%BE%D0%B3%D0%B8%D1%87%D0%B5%D1%81%D0%BA%D0%BE%D0%B5_%D0%BC%D0%BD%D0%BE%D0%B3%D0%BE%D0%BE%D0%B1%D1%80%D0%B0%D0%B7%D0%B8%D0%B5" title="Топологическое многообразие – Russian" lang="ru" hreflang="ru" data-title="Топологическое многообразие" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/P%C3%BCr%C3%BCzs%C3%BCz_%C3%A7okkatl%C4%B1" title="Pürüzsüz çokkatlı – Turkish" lang="tr" hreflang="tr" data-title="Pürüzsüz çokkatlı" data-language-autonym="Türkçe" data-language-local-name="Turkish" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E6%8B%93%E6%89%91%E6%B5%81%E5%BD%A2" title="拓扑流形 – Chinese" lang="zh" hreflang="zh" data-title="拓扑流形" data-language-autonym="中文" data-language-local-name="Chinese" class="interlanguage-link-target"><span>中文</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q2721559#sitelinks-wikipedia" title="Edit interlanguage links" class="wbc-editpage">Edit links</a></span></div> </div> </div> </div> </header> <div class="vector-page-toolbar"> <div class="vector-page-toolbar-container"> <div id="left-navigation"> <nav aria-label="Namespaces"> <div id="p-associated-pages" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-associated-pages" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-nstab-main" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/Topological_manifold" title="View the content page [c]" accesskey="c"><span>Article</span></a></li><li id="ca-talk" class="vector-tab-noicon mw-list-item"><a href="/wiki/Talk:Topological_manifold" rel="discussion" title="Discuss improvements to the content page [t]" accesskey="t"><span>Talk</span></a></li> </ul> </div> </div> <div id="vector-variants-dropdown" class="vector-dropdown emptyPortlet" > <input type="checkbox" id="vector-variants-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-variants-dropdown" class="vector-dropdown-checkbox " aria-label="Change language variant" > <label id="vector-variants-dropdown-label" for="vector-variants-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet" aria-hidden="true" ><span class="vector-dropdown-label-text">English</span> </label> <div class="vector-dropdown-content"> <div id="p-variants" class="vector-menu mw-portlet mw-portlet-variants emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> </div> </div> </nav> </div> <div id="right-navigation" class="vector-collapsible"> <nav aria-label="Views"> <div id="p-views" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-views" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-view" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/Topological_manifold"><span>Read</span></a></li><li id="ca-edit" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Topological_manifold&action=edit" title="Edit this page [e]" accesskey="e"><span>Edit</span></a></li><li id="ca-history" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Topological_manifold&action=history" title="Past revisions of this page [h]" accesskey="h"><span>View history</span></a></li> </ul> </div> </div> </nav> <nav class="vector-page-tools-landmark" aria-label="Page tools"> <div id="vector-page-tools-dropdown" class="vector-dropdown vector-page-tools-dropdown" > <input type="checkbox" id="vector-page-tools-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-tools-dropdown" class="vector-dropdown-checkbox " aria-label="Tools" > <label id="vector-page-tools-dropdown-label" for="vector-page-tools-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet" aria-hidden="true" ><span class="vector-dropdown-label-text">Tools</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-tools-unpinned-container" class="vector-unpinned-container"> <div id="vector-page-tools" class="vector-page-tools vector-pinnable-element"> <div class="vector-pinnable-header vector-page-tools-pinnable-header vector-pinnable-header-unpinned" data-feature-name="page-tools-pinned" data-pinnable-element-id="vector-page-tools" data-pinned-container-id="vector-page-tools-pinned-container" data-unpinned-container-id="vector-page-tools-unpinned-container" > <div class="vector-pinnable-header-label">Tools</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-page-tools.pin">move to sidebar</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-page-tools.unpin">hide</button> </div> <div id="p-cactions" class="vector-menu mw-portlet mw-portlet-cactions emptyPortlet vector-has-collapsible-items" title="More options" > <div class="vector-menu-heading"> Actions </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-more-view" class="selected vector-more-collapsible-item mw-list-item"><a href="/wiki/Topological_manifold"><span>Read</span></a></li><li id="ca-more-edit" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Topological_manifold&action=edit" title="Edit this page [e]" accesskey="e"><span>Edit</span></a></li><li id="ca-more-history" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Topological_manifold&action=history"><span>View history</span></a></li> </ul> </div> </div> <div id="p-tb" class="vector-menu mw-portlet mw-portlet-tb" > <div class="vector-menu-heading"> General </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="t-whatlinkshere" class="mw-list-item"><a href="/wiki/Special:WhatLinksHere/Topological_manifold" title="List of all English Wikipedia pages containing links to this page [j]" accesskey="j"><span>What links here</span></a></li><li id="t-recentchangeslinked" class="mw-list-item"><a href="/wiki/Special:RecentChangesLinked/Topological_manifold" rel="nofollow" title="Recent changes in pages linked from this page [k]" accesskey="k"><span>Related changes</span></a></li><li id="t-upload" class="mw-list-item"><a href="/wiki/Wikipedia:File_Upload_Wizard" title="Upload files [u]" accesskey="u"><span>Upload file</span></a></li><li id="t-specialpages" class="mw-list-item"><a href="/wiki/Special:SpecialPages" title="A list of all special pages [q]" accesskey="q"><span>Special pages</span></a></li><li id="t-permalink" class="mw-list-item"><a href="/w/index.php?title=Topological_manifold&oldid=1251981624" title="Permanent link to this revision of this page"><span>Permanent link</span></a></li><li id="t-info" class="mw-list-item"><a href="/w/index.php?title=Topological_manifold&action=info" title="More information about this page"><span>Page information</span></a></li><li id="t-cite" class="mw-list-item"><a href="/w/index.php?title=Special:CiteThisPage&page=Topological_manifold&id=1251981624&wpFormIdentifier=titleform" title="Information on how to cite this page"><span>Cite this page</span></a></li><li id="t-urlshortener" class="mw-list-item"><a href="/w/index.php?title=Special:UrlShortener&url=https%3A%2F%2Fen.wikipedia.org%2Fwiki%2FTopological_manifold"><span>Get shortened URL</span></a></li><li id="t-urlshortener-qrcode" class="mw-list-item"><a href="/w/index.php?title=Special:QrCode&url=https%3A%2F%2Fen.wikipedia.org%2Fwiki%2FTopological_manifold"><span>Download QR code</span></a></li> </ul> </div> </div> <div id="p-coll-print_export" class="vector-menu mw-portlet mw-portlet-coll-print_export" > <div class="vector-menu-heading"> Print/export </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="coll-download-as-rl" class="mw-list-item"><a href="/w/index.php?title=Special:DownloadAsPdf&page=Topological_manifold&action=show-download-screen" title="Download this page as a PDF file"><span>Download as PDF</span></a></li><li id="t-print" class="mw-list-item"><a href="/w/index.php?title=Topological_manifold&printable=yes" title="Printable version of this page [p]" accesskey="p"><span>Printable version</span></a></li> </ul> </div> </div> <div id="p-wikibase-otherprojects" class="vector-menu mw-portlet mw-portlet-wikibase-otherprojects" > <div class="vector-menu-heading"> In other projects </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="wb-otherproject-link wb-otherproject-commons mw-list-item"><a href="https://commons.wikimedia.org/wiki/Category:Mathematical_manifolds" hreflang="en"><span>Wikimedia Commons</span></a></li><li id="t-wikibase" class="wb-otherproject-link wb-otherproject-wikibase-dataitem mw-list-item"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q2721559" title="Structured data on this page hosted by Wikidata [g]" accesskey="g"><span>Wikidata item</span></a></li> </ul> </div> </div> </div> </div> </div> </div> </nav> </div> </div> </div> <div class="vector-column-end"> <div class="vector-sticky-pinned-container"> <nav class="vector-page-tools-landmark" aria-label="Page tools"> <div id="vector-page-tools-pinned-container" class="vector-pinned-container"> </div> </nav> <nav class="vector-appearance-landmark" aria-label="Appearance"> <div id="vector-appearance-pinned-container" class="vector-pinned-container"> <div id="vector-appearance" class="vector-appearance vector-pinnable-element"> <div class="vector-pinnable-header vector-appearance-pinnable-header vector-pinnable-header-pinned" data-feature-name="appearance-pinned" data-pinnable-element-id="vector-appearance" data-pinned-container-id="vector-appearance-pinned-container" data-unpinned-container-id="vector-appearance-unpinned-container" > <div class="vector-pinnable-header-label">Appearance</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">move to sidebar</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">hide</button> </div> </div> </div> </nav> </div> </div> <div id="bodyContent" class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">From Wikipedia, the free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Type of topological space</div> <p>In <a href="/wiki/Topology" title="Topology">topology</a>, a <b>topological manifold</b> is a <a href="/wiki/Topological_space" title="Topological space">topological space</a> that locally resembles <a href="/wiki/Real_numbers" class="mw-redirect" title="Real numbers">real</a> <i>n</i>-<a href="/wiki/Dimension_(mathematics)" class="mw-redirect" title="Dimension (mathematics)">dimensional</a> <a href="/wiki/Euclidean_space" title="Euclidean space">Euclidean space</a>. Topological manifolds are an important class of topological spaces, with applications throughout mathematics. All <a href="/wiki/Manifold" title="Manifold">manifolds</a> are topological manifolds by definition. Other types of manifolds are formed by adding structure to a topological manifold (e.g. <a href="/wiki/Differentiable_manifold" title="Differentiable manifold">differentiable manifolds</a> are topological manifolds equipped with a <a href="/wiki/Differential_structure" title="Differential structure">differential structure</a>). Every manifold has an "underlying" topological manifold, obtained by simply "forgetting" the added structure.<sup id="cite_ref-Bhatia2011_1-0" class="reference"><a href="#cite_note-Bhatia2011-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> However, not every topological manifold can be endowed with a particular additional structure. For example, the <a href="/wiki/E8_manifold" title="E8 manifold">E8 manifold</a> is a topological manifold which cannot be endowed with a differentiable structure. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Formal_definition">Formal definition</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Topological_manifold&action=edit&section=1" title="Edit section: Formal definition"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A <a href="/wiki/Topological_space" title="Topological space">topological space</a> <i>X</i> is called <b>locally Euclidean</b> if there is a non-negative <a href="/wiki/Integer" title="Integer">integer</a> <i>n</i> such that every point in <i>X</i> has a <a href="/wiki/Neighborhood_(topology)" class="mw-redirect" title="Neighborhood (topology)">neighborhood</a> which is <a href="/wiki/Homeomorphic" class="mw-redirect" title="Homeomorphic">homeomorphic</a> to <a href="/wiki/Real_coordinate_space" title="Real coordinate space">real <i>n</i>-space</a> <b>R</b><sup><i>n</i></sup>.<sup id="cite_ref-Lee2006_2-0" class="reference"><a href="#cite_note-Lee2006-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p><p>A <b>topological manifold</b> is a locally Euclidean <a href="/wiki/Hausdorff_space" title="Hausdorff space">Hausdorff space</a>. It is common to place additional requirements on topological manifolds. In particular, many authors define them to be <a href="/wiki/Paracompact" class="mw-redirect" title="Paracompact">paracompact</a><sup id="cite_ref-Aubin2001_3-0" class="reference"><a href="#cite_note-Aubin2001-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> or <a href="/wiki/Second-countable" class="mw-redirect" title="Second-countable">second-countable</a>.<sup id="cite_ref-Lee2006_2-1" class="reference"><a href="#cite_note-Lee2006-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p><p>In the remainder of this article a <i>manifold</i> will mean a topological manifold. An <i>n-manifold</i> will mean a topological manifold such that every point has a neighborhood homeomorphic to <b>R</b><sup><i>n</i></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=Topological_manifold&action=edit&section=2" title="Edit section: Examples"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/List_of_manifolds" title="List of manifolds">List of manifolds</a></div> <div class="mw-heading mw-heading3"><h3 id="n-manifolds"><i>n</i>-manifolds</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Topological_manifold&action=edit&section=3" title="Edit section: n-manifolds"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>The <a href="/wiki/Real_coordinate_space" title="Real coordinate space">real coordinate space</a> <b>R</b><sup><i>n</i></sup> is an <i>n</i>-manifold.</li> <li>Any <a href="/wiki/Discrete_space" title="Discrete space">discrete space</a> is a 0-dimensional manifold.</li> <li>A <a href="/wiki/Circle" title="Circle">circle</a> is a <a href="/wiki/Compact_space" title="Compact space">compact</a> 1-manifold.</li> <li>A <a href="/wiki/Torus" title="Torus">torus</a> and a <a href="/wiki/Klein_bottle" title="Klein bottle">Klein bottle</a> are compact 2-manifolds (or <a href="/wiki/Surface_(topology)" title="Surface (topology)">surfaces</a>).</li> <li>The <a href="/wiki/N-sphere" title="N-sphere"><i>n</i>-dimensional sphere</a> <i>S</i><sup><i>n</i></sup> is a compact <i>n</i>-manifold.</li> <li>The <a href="/wiki/N-torus" class="mw-redirect" title="N-torus"><i>n</i>-dimensional torus</a> <b>T</b><sup><i>n</i></sup> (the product of <i>n</i> circles) is a compact <i>n</i>-manifold.</li></ul> <div class="mw-heading mw-heading3"><h3 id="Projective_manifolds">Projective manifolds</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Topological_manifold&action=edit&section=4" title="Edit section: Projective manifolds"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Projective_space" title="Projective space">Projective spaces</a> over the <a href="/wiki/Real_number" title="Real number">reals</a>, <a href="/wiki/Complex_number" title="Complex number">complexes</a>, or <a href="/wiki/Quaternion" title="Quaternion">quaternions</a> are compact manifolds. <ul><li><a href="/wiki/Real_projective_space" title="Real projective space">Real projective space</a> <b>RP</b><sup><i>n</i></sup> is a <i>n</i>-dimensional manifold.</li> <li><a href="/wiki/Complex_projective_space" title="Complex projective space">Complex projective space</a> <b>CP</b><sup><i>n</i></sup> is a 2<i>n</i>-dimensional manifold.</li> <li><a href="/wiki/Quaternionic_projective_space" title="Quaternionic projective space">Quaternionic projective space</a> <b>HP</b><sup><i>n</i></sup> is a 4<i>n</i>-dimensional manifold.</li></ul></li> <li>Manifolds related to projective space include <a href="/wiki/Grassmannian" title="Grassmannian">Grassmannians</a>, <a href="/wiki/Flag_manifold" class="mw-redirect" title="Flag manifold">flag manifolds</a>, and <a href="/wiki/Stiefel_manifold" title="Stiefel manifold">Stiefel manifolds</a>.</li></ul> <div class="mw-heading mw-heading3"><h3 id="Other_manifolds">Other manifolds</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Topological_manifold&action=edit&section=5" title="Edit section: Other manifolds"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Differentiable_manifold" title="Differentiable manifold">Differentiable manifolds</a> are a class of topological manifolds equipped with a <a href="/wiki/Differential_structure" title="Differential structure">differential structure</a>.</li> <li><a href="/wiki/Lens_space" title="Lens space">Lens spaces</a> are a class of differentiable manifolds that are <a href="/wiki/Quotient_space_(topology)" title="Quotient space (topology)">quotients</a> of odd-dimensional spheres.</li> <li><a href="/wiki/Lie_group" title="Lie group">Lie groups</a> are a class of differentiable manifolds equipped with a compatible <a href="/wiki/Group_(mathematics)" title="Group (mathematics)">group</a> structure.</li> <li>The <a href="/wiki/E8_manifold" title="E8 manifold">E8 manifold</a> is a topological manifold which cannot be given a differentiable structure.</li></ul> <div class="mw-heading mw-heading2"><h2 id="Properties">Properties</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Topological_manifold&action=edit&section=6" title="Edit section: Properties"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The property of being locally Euclidean is preserved by <a href="/wiki/Local_homeomorphism" title="Local homeomorphism">local homeomorphisms</a>. That is, if <i>X</i> is locally Euclidean of dimension <i>n</i> and <i>f</i> : <i>Y</i> → <i>X</i> is a local homeomorphism, then <i>Y</i> is locally Euclidean of dimension <i>n</i>. In particular, being locally Euclidean is a <a href="/wiki/Topological_property" title="Topological property">topological property</a>. </p><p>Manifolds inherit many of the local properties of Euclidean space. In particular, they are <a href="/wiki/Locally_compact_space" title="Locally compact space">locally compact</a>, <a href="/wiki/Locally_connected_space" title="Locally connected space">locally connected</a>, <a href="/wiki/First-countable_space" title="First-countable space">first countable</a>, <a href="/wiki/Locally_contractible_space" class="mw-redirect" title="Locally contractible space">locally contractible</a>, and <a href="/wiki/Locally_metrizable_space" class="mw-redirect" title="Locally metrizable space">locally metrizable</a>. Being locally compact Hausdorff spaces, manifolds are necessarily <a href="/wiki/Tychonoff_space" title="Tychonoff space">Tychonoff spaces</a>. </p><p>Adding the Hausdorff condition can make several properties become equivalent for a manifold. As an example, we can show that for a Hausdorff manifold, the notions of <a href="/wiki/%CE%A3-compact" class="mw-redirect" title="Σ-compact">σ-compactness</a> and second-countability are the same. Indeed, a <a href="/wiki/Hausdorff_space" title="Hausdorff space">Hausdorff manifold</a> is a locally compact Hausdorff space, hence it is (completely) regular.<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> Assume such a space X is σ-compact. Then it is Lindelöf, and because Lindelöf + regular implies paracompact, X is metrizable. But in a metrizable space, second-countability coincides with being Lindelöf, so X is second-countable. Conversely, if X is a Hausdorff second-countable manifold, it must be σ-compact.<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> </p><p>A manifold need not be connected, but every manifold <i>M</i> is a <a href="/wiki/Disjoint_union_(topology)" title="Disjoint union (topology)">disjoint union</a> of connected manifolds. These are just the <a href="/wiki/Connected_component_(topology)" class="mw-redirect" title="Connected component (topology)">connected components</a> of <i>M</i>, which are <a href="/wiki/Open_set" title="Open set">open sets</a> since manifolds are locally-connected. Being locally path connected, a manifold is path-connected <a href="/wiki/If_and_only_if" title="If and only if">if and only if</a> it is connected. It follows that the path-components are the same as the components. </p> <div class="mw-heading mw-heading3"><h3 id="The_Hausdorff_axiom">The Hausdorff axiom</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Topological_manifold&action=edit&section=7" title="Edit section: The Hausdorff axiom"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The Hausdorff property is not a local one; so even though Euclidean space is Hausdorff, a locally Euclidean space need <a href="/wiki/Non-Hausdorff_manifold" title="Non-Hausdorff manifold"> not be</a>. It is true, however, that every locally Euclidean space is <a href="/wiki/T1_space" title="T1 space">T<sub>1</sub></a>. </p><p>An example of a non-Hausdorff locally Euclidean space is the <a href="/wiki/Line_with_two_origins" class="mw-redirect" title="Line with two origins">line with two origins</a>. This space is created by replacing the origin of the real line with <i>two</i> points, an open neighborhood of either of which includes all nonzero numbers in some open interval centered at zero. This space is not Hausdorff because the two origins cannot be separated. </p> <div class="mw-heading mw-heading3"><h3 id="Compactness_and_countability_axioms">Compactness and countability axioms</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Topological_manifold&action=edit&section=8" title="Edit section: Compactness and countability axioms"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A manifold is <a href="/wiki/Metrizable_space" title="Metrizable space">metrizable</a> if and only if it is <a href="/wiki/Paracompact" class="mw-redirect" title="Paracompact">paracompact</a>. The <a href="/wiki/Long_line_(topology)" title="Long line (topology)">long line</a> is an example a <a href="/wiki/Normal_space" title="Normal space">normal</a> <a href="/wiki/Hausdorff_space" title="Hausdorff space">Hausdorff</a> 1-dimensional topological manifold that is not metrizable nor paracompact. Since metrizability is such a desirable property for a topological space, it is common to add paracompactness to the definition of a manifold. In any case, non-paracompact manifolds are generally regarded as <a href="/wiki/Pathological_(mathematics)" title="Pathological (mathematics)">pathological</a>. An example of a non-paracompact manifold is given by the <a href="/wiki/Long_line_(topology)" title="Long line (topology)">long line</a>. Paracompact manifolds have all the topological properties of metric spaces. In particular, they are <a href="/wiki/Perfectly_normal_Hausdorff_space" class="mw-redirect" title="Perfectly normal Hausdorff space">perfectly normal Hausdorff spaces</a>. </p><p>Manifolds are also commonly required to be <a href="/wiki/Second-countable_space" title="Second-countable space">second-countable</a>. This is precisely the condition required to ensure that the manifold <a href="/wiki/Embedding" title="Embedding">embeds</a> in some finite-dimensional Euclidean space. For any manifold the properties of being second-countable, <a href="/wiki/Lindel%C3%B6f_space" title="Lindelöf space">Lindelöf</a>, and <a href="/wiki/%CE%A3-compact_space" title="Σ-compact space">σ-compact</a> are all equivalent. </p><p>Every second-countable manifold is paracompact, but not vice versa. However, the converse is nearly true: a paracompact manifold is second-countable if and only if it has a <a href="/wiki/Countable_set" title="Countable set">countable</a> number of <a href="/wiki/Connected_component_(topology)" class="mw-redirect" title="Connected component (topology)">connected components</a>. In particular, a connected manifold is paracompact if and only if it is second-countable. Every second-countable manifold is <a href="/wiki/Separable_space" title="Separable space">separable</a> and paracompact. Moreover, if a manifold is separable and paracompact then it is also second-countable. </p><p>Every <a href="/wiki/Compact_space" title="Compact space">compact</a> manifold is second-countable and paracompact. </p> <div class="mw-heading mw-heading3"><h3 id="Dimensionality">Dimensionality</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Topological_manifold&action=edit&section=9" title="Edit section: Dimensionality"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>By <a href="/wiki/Invariance_of_domain" title="Invariance of domain">invariance of domain</a>, a non-empty <i>n</i>-manifold cannot be an <i>m</i>-manifold for <i>n</i> ≠ <i>m</i>.<sup id="cite_ref-Dieck2008_6-0" class="reference"><a href="#cite_note-Dieck2008-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> The dimension of a non-empty <i>n</i>-manifold is <i>n</i>. Being an <i>n</i>-manifold is a <a href="/wiki/Topological_property" title="Topological property">topological property</a>, meaning that any topological space homeomorphic to an <i>n</i>-manifold is also an <i>n</i>-manifold.<sup id="cite_ref-Lee2010_7-0" class="reference"><a href="#cite_note-Lee2010-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Coordinate_charts">Coordinate charts</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Topological_manifold&action=edit&section=10" title="Edit section: Coordinate charts"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>By definition, every point of a locally Euclidean space has a neighborhood homeomorphic to an open 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 {R} ^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c510b63578322050121fe966f2e5770bea43308d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.897ex; height:2.343ex;" alt="{\displaystyle \mathbb {R} ^{n}}"></span>. Such neighborhoods are called <b>Euclidean neighborhoods</b>. It follows from <a href="/wiki/Invariance_of_domain" title="Invariance of domain">invariance of domain</a> that Euclidean neighborhoods are always open sets. One can always find Euclidean neighborhoods that are homeomorphic to "nice" open sets 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} ^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c510b63578322050121fe966f2e5770bea43308d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.897ex; height:2.343ex;" alt="{\displaystyle \mathbb {R} ^{n}}"></span>. Indeed, a space <i>M</i> is locally Euclidean if and only if either of the following equivalent conditions holds: </p> <ul><li>every point of <i>M</i> has a neighborhood homeomorphic to an <a href="/wiki/Open_ball" class="mw-redirect" title="Open ball">open ball</a> 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} ^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c510b63578322050121fe966f2e5770bea43308d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.897ex; height:2.343ex;" alt="{\displaystyle \mathbb {R} ^{n}}"></span>.</li> <li>every point of <i>M</i> has a neighborhood homeomorphic to <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} ^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c510b63578322050121fe966f2e5770bea43308d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.897ex; height:2.343ex;" alt="{\displaystyle \mathbb {R} ^{n}}"></span> itself.</li></ul> <p>A Euclidean neighborhood homeomorphic to an open ball 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} ^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c510b63578322050121fe966f2e5770bea43308d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.897ex; height:2.343ex;" alt="{\displaystyle \mathbb {R} ^{n}}"></span> is called a <b>Euclidean ball</b>. Euclidean balls form a <a href="/wiki/Basis_(topology)" class="mw-redirect" title="Basis (topology)">basis</a> for the topology of a locally Euclidean space. </p><p>For any Euclidean neighborhood <i>U</i>, a homeomorphism <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 \phi :U\rightarrow \phi \left(U\right)\subset \mathbb {R} ^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϕ</mi> <mo>:</mo> <mi>U</mi> <mo stretchy="false">→</mo> <mi>ϕ</mi> <mrow> <mo>(</mo> <mi>U</mi> <mo>)</mo> </mrow> <mo>⊂</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi :U\rightarrow \phi \left(U\right)\subset \mathbb {R} ^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c61a045d6abca1e9b4ec162705fee62029132539" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.079ex; height:2.843ex;" alt="{\displaystyle \phi :U\rightarrow \phi \left(U\right)\subset \mathbb {R} ^{n}}"></span> is called a <b>coordinate chart</b> on <i>U</i> (although the word <i>chart</i> is frequently used to refer to the domain or range of such a map). A space <i>M</i> is locally Euclidean if and only if it can be <a href="/wiki/Cover_(topology)" title="Cover (topology)">covered</a> by Euclidean neighborhoods. A set of Euclidean neighborhoods that cover <i>M</i>, together with their coordinate charts, is called an <b><a href="/wiki/Atlas_(topology)" title="Atlas (topology)">atlas</a></b> on <i>M</i>. (The terminology comes from an analogy with <a href="/wiki/Cartography" title="Cartography">cartography</a> whereby a spherical <a href="/wiki/Globe" title="Globe">globe</a> can be described by an <a href="/wiki/Atlas" title="Atlas">atlas</a> of flat maps or charts). </p><p>Given two charts <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 \phi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϕ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/72b1f30316670aee6270a28334bdf4f5072cdde4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.385ex; height:2.509ex;" alt="{\displaystyle \phi }"></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 \psi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ψ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/45e5789e5d9c8f7c79744f43ecaaf8ba42a8553a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.513ex; height:2.509ex;" alt="{\displaystyle \psi }"></span> with overlapping domains <i>U</i> and <i>V</i>, there is a <b>transition function</b> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi \phi ^{-1}:\phi \left(U\cap V\right)\rightarrow \psi \left(U\cap V\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ψ</mi> <msup> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>:</mo> <mi>ϕ</mi> <mrow> <mo>(</mo> <mrow> <mi>U</mi> <mo>∩</mo> <mi>V</mi> </mrow> <mo>)</mo> </mrow> <mo stretchy="false">→</mo> <mi>ψ</mi> <mrow> <mo>(</mo> <mrow> <mi>U</mi> <mo>∩</mo> <mi>V</mi> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi \phi ^{-1}:\phi \left(U\cap V\right)\rightarrow \psi \left(U\cap V\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/db8118beadcdcd70f5c060921c8e83f988ae668a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:30.379ex; height:3.176ex;" alt="{\displaystyle \psi \phi ^{-1}:\phi \left(U\cap V\right)\rightarrow \psi \left(U\cap V\right)}"></span></dd></dl> <p>Such a map is a homeomorphism between open 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} ^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c510b63578322050121fe966f2e5770bea43308d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.897ex; height:2.343ex;" alt="{\displaystyle \mathbb {R} ^{n}}"></span>. That is, coordinate charts agree on overlaps up to homeomorphism. Different types of manifolds can be defined by placing restrictions on types of transition maps allowed. For example, for <a href="/wiki/Differentiable_manifolds" class="mw-redirect" title="Differentiable manifolds">differentiable manifolds</a> the transition maps are required to be <a href="/wiki/Diffeomorphism" title="Diffeomorphism">smooth</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Classification_of_manifolds">Classification of manifolds</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Topological_manifold&action=edit&section=11" title="Edit section: Classification of manifolds"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Discrete_spaces_(0-Manifold)"><span id="Discrete_spaces_.280-Manifold.29"></span>Discrete spaces (0-Manifold)</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Topological_manifold&action=edit&section=12" title="Edit section: Discrete spaces (0-Manifold)"><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/Discrete_space" title="Discrete space">Discrete space</a></div> <p>A 0-manifold is just a <a href="/wiki/Discrete_space" title="Discrete space">discrete space</a>. A discrete space is second-countable if and only if it is <a href="/wiki/Countable_set" title="Countable set">countable</a>.<sup id="cite_ref-Lee2010_7-1" class="reference"><a href="#cite_note-Lee2010-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Curves_(1-Manifold)"><span id="Curves_.281-Manifold.29"></span>Curves (1-Manifold)</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Topological_manifold&action=edit&section=13" title="Edit section: Curves (1-Manifold)"><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/1-manifold" class="mw-redirect" title="1-manifold">1-manifold</a></div> <p>Every nonempty, paracompact, connected 1-manifold is homeomorphic either to <b>R</b> or the <a href="/wiki/Circle" title="Circle">circle</a>.<sup id="cite_ref-Lee2010_7-2" class="reference"><a href="#cite_note-Lee2010-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Surfaces_(2-Manifold)"><span id="Surfaces_.282-Manifold.29"></span>Surfaces (2-Manifold)</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Topological_manifold&action=edit&section=14" title="Edit section: Surfaces (2-Manifold)"><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 articles: <a href="/wiki/2-manifold" class="mw-redirect" title="2-manifold">2-manifold</a> and <a href="/wiki/Classification_theorem_for_surfaces" class="mw-redirect" title="Classification theorem for surfaces">Classification theorem for surfaces</a></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Sphere_wireframe.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/08/Sphere_wireframe.svg/220px-Sphere_wireframe.svg.png" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/08/Sphere_wireframe.svg/330px-Sphere_wireframe.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/08/Sphere_wireframe.svg/440px-Sphere_wireframe.svg.png 2x" data-file-width="400" data-file-height="400" /></a><figcaption>The <a href="/wiki/Sphere" title="Sphere">sphere</a> is a 2-manifold.</figcaption></figure> <p>Every nonempty, compact, connected 2-manifold (or <a href="/wiki/Surface_(topology)" title="Surface (topology)">surface</a>) is homeomorphic to the <a href="/wiki/Sphere" title="Sphere">sphere</a>, a <a href="/wiki/Connected_sum" title="Connected sum">connected sum</a> of <a href="/wiki/Torus" title="Torus">tori</a>, or a connected sum of <a href="/wiki/Real_projective_plane" title="Real projective plane">projective planes</a>.<sup id="cite_ref-GallierXu2013_8-0" class="reference"><a href="#cite_note-GallierXu2013-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Volumes_(3-Manifold)"><span id="Volumes_.283-Manifold.29"></span>Volumes (3-Manifold)</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Topological_manifold&action=edit&section=15" title="Edit section: Volumes (3-Manifold)"><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/3-manifold" title="3-manifold">3-manifold</a></div> <p>A classification of 3-manifolds results from <a href="/wiki/Thurston%27s_geometrization_conjecture" class="mw-redirect" title="Thurston's geometrization conjecture">Thurston's geometrization conjecture</a>, proven by <a href="/wiki/Grigori_Perelman" title="Grigori Perelman">Grigori Perelman</a> in 2003. More specifically, Perelman's results provide an algorithm for deciding if two three-manifolds are homeomorphic to each other.<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="General_n-manifold">General <i>n</i>-manifold</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Topological_manifold&action=edit&section=16" title="Edit section: General n-manifold"><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 articles: <a href="/wiki/4-manifold" title="4-manifold">4-manifold</a> and <a href="/wiki/5-manifold" title="5-manifold">5-manifold</a></div> <p>The full classification of <i>n</i>-manifolds for <i>n</i> greater than three is known to be impossible; it is at least as hard as the <a href="/wiki/Word_problem_for_groups" title="Word problem for groups">word problem</a> in <a href="/wiki/Group_theory" title="Group theory">group theory</a>, which is known to be <a href="/wiki/Undecidable_problem" title="Undecidable problem">algorithmically undecidable</a>.<sup id="cite_ref-Conlon2013_10-0" class="reference"><a href="#cite_note-Conlon2013-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> </p><p>In fact, there is no <a href="/wiki/Algorithm" title="Algorithm">algorithm</a> for deciding whether a given manifold is <a href="/wiki/Simply_connected" class="mw-redirect" title="Simply connected">simply connected</a>. There is, however, a classification of simply connected manifolds of dimension ≥ 5.<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Manifolds_with_boundary">Manifolds with boundary</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Topological_manifold&action=edit&section=17" title="Edit section: Manifolds with boundary"><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/Manifold#Manifold_with_boundary" title="Manifold">Manifold § Manifold with boundary</a></div> <p>A slightly more general concept is sometimes useful. A <b>topological manifold with boundary</b> is a <a href="/wiki/Hausdorff_space" title="Hausdorff space">Hausdorff space</a> in which every point has a neighborhood homeomorphic to an open subset of Euclidean <a href="/wiki/Half-space_(geometry)" title="Half-space (geometry)">half-space</a> (for a fixed <i>n</i>): </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} _{+}^{n}=\{(x_{1},\ldots ,x_{n})\in \mathbb {R} ^{n}:x_{n}\geq 0\}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msubsup> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>∈</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>:</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>≥</mo> <mn>0</mn> <mo fence="false" stretchy="false">}</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} _{+}^{n}=\{(x_{1},\ldots ,x_{n})\in \mathbb {R} ^{n}:x_{n}\geq 0\}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7e1856c979576a6571c24266874bc4c6f6eaf3b9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:35.662ex; height:3.009ex;" alt="{\displaystyle \mathbb {R} _{+}^{n}=\{(x_{1},\ldots ,x_{n})\in \mathbb {R} ^{n}:x_{n}\geq 0\}.}"></span></dd></dl> <p>Every topological manifold is a topological manifold with boundary, but not vice versa.<sup id="cite_ref-Lee2010_7-3" class="reference"><a href="#cite_note-Lee2010-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Constructions">Constructions</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Topological_manifold&action=edit&section=18" title="Edit section: Constructions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>There are several methods of creating manifolds from other manifolds. </p> <div class="mw-heading mw-heading3"><h3 id="Product_manifolds">Product manifolds</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Topological_manifold&action=edit&section=19" title="Edit section: Product manifolds"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>If <i>M</i> is an <i>m</i>-manifold and <i>N</i> is an <i>n</i>-manifold, the <a href="/wiki/Cartesian_product" title="Cartesian product">Cartesian product</a> <i>M</i>×<i>N</i> is a (<i>m</i>+<i>n</i>)-manifold when given the <a href="/wiki/Product_topology" title="Product topology">product topology</a>.<sup id="cite_ref-LeeLee2009_13-0" class="reference"><a href="#cite_note-LeeLee2009-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Disjoint_union">Disjoint union</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Topological_manifold&action=edit&section=20" title="Edit section: Disjoint union"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <a href="/wiki/Disjoint_union_(topology)" title="Disjoint union (topology)">disjoint union</a> of a countable family of <i>n</i>-manifolds is a <i>n</i>-manifold (the pieces must all have the same dimension).<sup id="cite_ref-Lee2010_7-4" class="reference"><a href="#cite_note-Lee2010-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Connected_sum">Connected sum</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Topological_manifold&action=edit&section=21" title="Edit section: Connected sum"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Connected_sum" title="Connected sum">Connected sum</a></div> <p>The <a href="/wiki/Connected_sum" title="Connected sum">connected sum</a> of two <i>n</i>-manifolds is defined by removing an open ball from each manifold and taking the <a href="/wiki/Quotient_topology" class="mw-redirect" title="Quotient topology">quotient</a> of the disjoint union of the resulting manifolds with boundary, with the quotient taken with regards to a homeomorphism between the boundary spheres of the removed balls. This results in another <i>n</i>-manifold.<sup id="cite_ref-Lee2010_7-5" class="reference"><a href="#cite_note-Lee2010-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Submanifold">Submanifold</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Topological_manifold&action=edit&section=22" title="Edit section: Submanifold"><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/Submanifold" title="Submanifold">Submanifold</a></div> <p>Any open subset of an <i>n</i>-manifold is an <i>n</i>-manifold with the <a href="/wiki/Subspace_topology" title="Subspace topology">subspace topology</a>.<sup id="cite_ref-LeeLee2009_13-1" class="reference"><a href="#cite_note-LeeLee2009-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Footnotes">Footnotes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Topological_manifold&action=edit&section=23" title="Edit section: Footnotes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-Bhatia2011-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-Bhatia2011_1-0">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFRajendra_Bhatia2011" class="citation book cs1">Rajendra Bhatia (6 June 2011). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=GFE1vx2pynMC&pg=PA477"><i>Proceedings of the International Congress of Mathematicians: Hyderabad, August 19-27, 2010</i></a>. World Scientific. pp. 477–. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-981-4324-35-9" title="Special:BookSources/978-981-4324-35-9"><bdi>978-981-4324-35-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Proceedings+of+the+International+Congress+of+Mathematicians%3A+Hyderabad%2C+August+19-27%2C+2010&rft.pages=477-&rft.pub=World+Scientific&rft.date=2011-06-06&rft.isbn=978-981-4324-35-9&rft.au=Rajendra+Bhatia&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DGFE1vx2pynMC%26pg%3DPA477&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATopological+manifold" class="Z3988"></span></span> </li> <li id="cite_note-Lee2006-2"><span class="mw-cite-backlink">^ <a href="#cite_ref-Lee2006_2-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Lee2006_2-1"><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 id="CITEREFJohn_M._Lee2006" class="citation book cs1">John M. Lee (6 April 2006). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=AdIRBwAAQBAJ"><i>Introduction to Topological Manifolds</i></a>. Springer Science & Business Media. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-22727-6" title="Special:BookSources/978-0-387-22727-6"><bdi>978-0-387-22727-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+to+Topological+Manifolds&rft.pub=Springer+Science+%26+Business+Media&rft.date=2006-04-06&rft.isbn=978-0-387-22727-6&rft.au=John+M.+Lee&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DAdIRBwAAQBAJ&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATopological+manifold" class="Z3988"></span></span> </li> <li id="cite_note-Aubin2001-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-Aubin2001_3-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFThierry_Aubin2001" class="citation book cs1">Thierry Aubin (2001). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=q2iS-aSeHK0C&pg=PA25"><i>A Course in Differential Geometry</i></a>. American Mathematical Soc. pp. 25–. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-8218-7214-7" title="Special:BookSources/978-0-8218-7214-7"><bdi>978-0-8218-7214-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+Course+in+Differential+Geometry&rft.pages=25-&rft.pub=American+Mathematical+Soc.&rft.date=2001&rft.isbn=978-0-8218-7214-7&rft.au=Thierry+Aubin&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3Dq2iS-aSeHK0C%26pg%3DPA25&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATopological+manifold" 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">Topospaces subwiki, <a rel="nofollow" class="external text" href="http://topospaces.subwiki.org/wiki/Locally_compact_Hausdorff_implies_completely_regular">Locally compact Hausdorff implies completely regular</a></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">Stack Exchange, <a rel="nofollow" class="external text" href="https://math.stackexchange.com/q/57348">Hausdorff locally compact and second countable is sigma-compact</a></span> </li> <li id="cite_note-Dieck2008-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-Dieck2008_6-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFTammo_tom_Dieck2008" class="citation book cs1">Tammo tom Dieck (2008). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=ruSqmB7LWOcC&pg=PA249"><i>Algebraic Topology</i></a>. European Mathematical Society. pp. 249–. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-03719-048-7" title="Special:BookSources/978-3-03719-048-7"><bdi>978-3-03719-048-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Algebraic+Topology&rft.pages=249-&rft.pub=European+Mathematical+Society&rft.date=2008&rft.isbn=978-3-03719-048-7&rft.au=Tammo+tom+Dieck&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DruSqmB7LWOcC%26pg%3DPA249&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATopological+manifold" class="Z3988"></span></span> </li> <li id="cite_note-Lee2010-7"><span class="mw-cite-backlink">^ <a href="#cite_ref-Lee2010_7-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Lee2010_7-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-Lee2010_7-2"><sup><i><b>c</b></i></sup></a> <a href="#cite_ref-Lee2010_7-3"><sup><i><b>d</b></i></sup></a> <a href="#cite_ref-Lee2010_7-4"><sup><i><b>e</b></i></sup></a> <a href="#cite_ref-Lee2010_7-5"><sup><i><b>f</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJohn_Lee2010" class="citation book cs1">John Lee (25 December 2010). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=ZQVGAAAAQBAJ&pg=PA64"><i>Introduction to Topological Manifolds</i></a>. Springer Science & Business Media. pp. 64–. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4419-7940-7" title="Special:BookSources/978-1-4419-7940-7"><bdi>978-1-4419-7940-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+to+Topological+Manifolds&rft.pages=64-&rft.pub=Springer+Science+%26+Business+Media&rft.date=2010-12-25&rft.isbn=978-1-4419-7940-7&rft.au=John+Lee&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DZQVGAAAAQBAJ%26pg%3DPA64&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATopological+manifold" class="Z3988"></span></span> </li> <li id="cite_note-GallierXu2013-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-GallierXu2013_8-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJean_GallierDianna_Xu2013" class="citation book cs1">Jean Gallier; Dianna Xu (5 February 2013). <a href="/wiki/A_Guide_to_the_Classification_Theorem_for_Compact_Surfaces" title="A Guide to the Classification Theorem for Compact Surfaces"><i>A Guide to the Classification Theorem for Compact Surfaces</i></a>. Springer Science & Business Media. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-642-34364-3" title="Special:BookSources/978-3-642-34364-3"><bdi>978-3-642-34364-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+Guide+to+the+Classification+Theorem+for+Compact+Surfaces&rft.pub=Springer+Science+%26+Business+Media&rft.date=2013-02-05&rft.isbn=978-3-642-34364-3&rft.au=Jean+Gallier&rft.au=Dianna+Xu&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATopological+manifold" class="Z3988"></span></span> </li> <li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation book cs1"><a rel="nofollow" class="external text" href="https://books.google.com/books?id=uZqRpkLwS70C"><i>Geometrisation of 3-manifolds</i></a>. European Mathematical Society. 2010. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-03719-082-1" title="Special:BookSources/978-3-03719-082-1"><bdi>978-3-03719-082-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Geometrisation+of+3-manifolds&rft.pub=European+Mathematical+Society&rft.date=2010&rft.isbn=978-3-03719-082-1&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DuZqRpkLwS70C&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATopological+manifold" class="Z3988"></span></span> </li> <li id="cite_note-Conlon2013-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-Conlon2013_10-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLawrence_Conlon2013" class="citation book cs1">Lawrence Conlon (17 April 2013). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=Sj7rBwAAQBAJ&pg=PA90"><i>Differentiable Manifolds: A First Course</i></a>. Springer Science & Business Media. pp. 90–. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4757-2284-0" title="Special:BookSources/978-1-4757-2284-0"><bdi>978-1-4757-2284-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Differentiable+Manifolds%3A+A+First+Course&rft.pages=90-&rft.pub=Springer+Science+%26+Business+Media&rft.date=2013-04-17&rft.isbn=978-1-4757-2284-0&rft.au=Lawrence+Conlon&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DSj7rBwAAQBAJ%26pg%3DPA90&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATopological+manifold" class="Z3988"></span></span> </li> <li id="cite_note-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-11">^</a></b></span> <span class="reference-text">Žubr A.V. (1988) Classification of simply-connected topological 6-manifolds. In: Viro O.Y., Vershik A.M. (eds) Topology and Geometry — Rohlin Seminar. Lecture Notes in Mathematics, vol 1346. Springer, Berlin, Heidelberg</span> </li> <li id="cite_note-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-12">^</a></b></span> <span class="reference-text">Barden, D. "Simply Connected Five-Manifolds." Annals of Mathematics, vol. 82, no. 3, 1965, pp. 365–385. JSTOR, www.jstor.org/stable/1970702.</span> </li> <li id="cite_note-LeeLee2009-13"><span class="mw-cite-backlink">^ <a href="#cite_ref-LeeLee2009_13-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-LeeLee2009_13-1"><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 id="CITEREFJeffrey_LeeJeffrey_Marc_Lee2009" class="citation book cs1">Jeffrey Lee; Jeffrey Marc Lee (2009). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=QqHdHy9WsEoC&pg=PA7"><i>Manifolds and Differential Geometry</i></a>. American Mathematical Soc. pp. 7–. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-8218-4815-9" title="Special:BookSources/978-0-8218-4815-9"><bdi>978-0-8218-4815-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Manifolds+and+Differential+Geometry&rft.pages=7-&rft.pub=American+Mathematical+Soc.&rft.date=2009&rft.isbn=978-0-8218-4815-9&rft.au=Jeffrey+Lee&rft.au=Jeffrey+Marc+Lee&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DQqHdHy9WsEoC%26pg%3DPA7&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATopological+manifold" class="Z3988"></span></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=Topological_manifold&action=edit&section=24" title="Edit section: References"><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="CITEREFGauld1974" class="citation journal cs1">Gauld, D. B. (1974). "Topological Properties of Manifolds". <i>The American Mathematical Monthly</i>. <b>81</b> (6). Mathematical Association of America: 633–636. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F2319220">10.2307/2319220</a>. <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2319220">2319220</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+American+Mathematical+Monthly&rft.atitle=Topological+Properties+of+Manifolds&rft.volume=81&rft.issue=6&rft.pages=633-636&rft.date=1974&rft_id=info%3Adoi%2F10.2307%2F2319220&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2319220%23id-name%3DJSTOR&rft.aulast=Gauld&rft.aufirst=D.+B.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATopological+manifold" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKirbySiebenmann,_Laurence_C.1977" class="citation book cs1"><a href="/wiki/Robion_Kirby" title="Robion Kirby">Kirby, Robion C.</a>; Siebenmann, Laurence C. (1977). <a rel="nofollow" class="external text" href="http://www.maths.ed.ac.uk/~aar/papers/ks.pdf"><i>Foundational Essays on Topological Manifolds. Smoothings, and Triangulations</i></a> <span class="cs1-format">(PDF)</span>. Princeton: Princeton University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-691-08191-3" title="Special:BookSources/0-691-08191-3"><bdi>0-691-08191-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Foundational+Essays+on+Topological+Manifolds.+Smoothings%2C+and+Triangulations&rft.place=Princeton&rft.pub=Princeton+University+Press&rft.date=1977&rft.isbn=0-691-08191-3&rft.aulast=Kirby&rft.aufirst=Robion+C.&rft.au=Siebenmann%2C+Laurence+C.&rft_id=http%3A%2F%2Fwww.maths.ed.ac.uk%2F~aar%2Fpapers%2Fks.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATopological+manifold" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLee2000" class="citation book cs1">Lee, John M. (2000). <a rel="nofollow" class="external text" href="https://archive.org/details/springer_10.1007-978-0-387-22727-6"><i>Introduction to Topological Manifolds</i></a>. Graduate Texts in Mathematics <b>202</b>. New York: Springer. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-387-98759-2" title="Special:BookSources/0-387-98759-2"><bdi>0-387-98759-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+to+Topological+Manifolds&rft.place=New+York&rft.series=Graduate+Texts+in+Mathematics+%27%27%27202%27%27%27&rft.pub=Springer&rft.date=2000&rft.isbn=0-387-98759-2&rft.aulast=Lee&rft.aufirst=John+M.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fspringer_10.1007-978-0-387-22727-6&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATopological+manifold" class="Z3988"></span></li></ul> <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=Topological_manifold&action=edit&section=25" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><span class="noviewer" typeof="mw:File"><a href="/wiki/File:Commons-logo.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/12px-Commons-logo.svg.png" decoding="async" width="12" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/18px-Commons-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/24px-Commons-logo.svg.png 2x" data-file-width="1024" data-file-height="1376" /></a></span> Media related to <a href="https://commons.wikimedia.org/wiki/Category:Mathematical_manifolds" class="extiw" title="commons:Category:Mathematical manifolds">Mathematical manifolds</a> at Wikimedia Commons</li></ul> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .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:": "}.mw-parser-output .hlist dd::after,.mw-parser-output .hlist li::after{content:" · ";font-weight:bold}.mw-parser-output .hlist dd:last-child::after,.mw-parser-output .hlist dt:last-child::after,.mw-parser-output .hlist li:last-child::after{content:none}.mw-parser-output .hlist dd dd:first-child::before,.mw-parser-output .hlist dd dt:first-child::before,.mw-parser-output .hlist dd li:first-child::before,.mw-parser-output .hlist dt dd:first-child::before,.mw-parser-output .hlist dt dt:first-child::before,.mw-parser-output .hlist dt li:first-child::before,.mw-parser-output .hlist li dd:first-child::before,.mw-parser-output .hlist li dt:first-child::before,.mw-parser-output .hlist li li:first-child::before{content:" (";font-weight:normal}.mw-parser-output .hlist dd dd:last-child::after,.mw-parser-output .hlist dd dt:last-child::after,.mw-parser-output .hlist dd li:last-child::after,.mw-parser-output .hlist dt dd:last-child::after,.mw-parser-output .hlist dt dt:last-child::after,.mw-parser-output .hlist dt li:last-child::after,.mw-parser-output .hlist li dd:last-child::after,.mw-parser-output .hlist li dt:last-child::after,.mw-parser-output .hlist li li:last-child::after{content:")";font-weight:normal}.mw-parser-output .hlist ol{counter-reset:listitem}.mw-parser-output .hlist ol>li{counter-increment:listitem}.mw-parser-output .hlist ol>li::before{content:" "counter(listitem)"\a0 "}.mw-parser-output .hlist dd ol>li:first-child::before,.mw-parser-output .hlist dt ol>li:first-child::before,.mw-parser-output .hlist li ol>li:first-child::before{content:" ("counter(listitem)"\a0 "}</style><style data-mw-deduplicate="TemplateStyles:r1236075235">.mw-parser-output .navbox{box-sizing:border-box;border:1px solid #a2a9b1;width:100%;clear:both;font-size:88%;text-align:center;padding:1px;margin:1em auto 0}.mw-parser-output .navbox .navbox{margin-top:0}.mw-parser-output .navbox+.navbox,.mw-parser-output .navbox+.navbox-styles+.navbox{margin-top:-1px}.mw-parser-output .navbox-inner,.mw-parser-output .navbox-subgroup{width:100%}.mw-parser-output .navbox-group,.mw-parser-output .navbox-title,.mw-parser-output .navbox-abovebelow{padding:0.25em 1em;line-height:1.5em;text-align:center}.mw-parser-output .navbox-group{white-space:nowrap;text-align:right}.mw-parser-output .navbox,.mw-parser-output .navbox-subgroup{background-color:#fdfdfd}.mw-parser-output .navbox-list{line-height:1.5em;border-color:#fdfdfd}.mw-parser-output .navbox-list-with-group{text-align:left;border-left-width:2px;border-left-style:solid}.mw-parser-output tr+tr>.navbox-abovebelow,.mw-parser-output tr+tr>.navbox-group,.mw-parser-output tr+tr>.navbox-image,.mw-parser-output tr+tr>.navbox-list{border-top:2px solid #fdfdfd}.mw-parser-output .navbox-title{background-color:#ccf}.mw-parser-output .navbox-abovebelow,.mw-parser-output .navbox-group,.mw-parser-output .navbox-subgroup .navbox-title{background-color:#ddf}.mw-parser-output .navbox-subgroup .navbox-group,.mw-parser-output .navbox-subgroup .navbox-abovebelow{background-color:#e6e6ff}.mw-parser-output .navbox-even{background-color:#f7f7f7}.mw-parser-output .navbox-odd{background-color:transparent}.mw-parser-output .navbox .hlist td dl,.mw-parser-output .navbox .hlist td ol,.mw-parser-output .navbox .hlist td ul,.mw-parser-output .navbox td.hlist dl,.mw-parser-output .navbox td.hlist ol,.mw-parser-output .navbox td.hlist ul{padding:0.125em 0}.mw-parser-output .navbox .navbar{display:block;font-size:100%}.mw-parser-output .navbox-title .navbar{float:left;text-align:left;margin-right:0.5em}body.skin--responsive .mw-parser-output .navbox-image img{max-width:none!important}@media print{body.ns-0 .mw-parser-output .navbox{display:none!important}}</style></div><div role="navigation" class="navbox" aria-labelledby="Manifolds_(Glossary)" style="padding:3px"><table class="nowraplinks hlist mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><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 .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Manifolds" title="Template:Manifolds"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Manifolds" title="Template talk:Manifolds"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Manifolds" title="Special:EditPage/Template:Manifolds"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Manifolds_(Glossary)" style="font-size:114%;margin:0 4em"><a href="/wiki/Manifold" title="Manifold">Manifolds</a> (<a href="/wiki/Glossary_of_differential_geometry_and_topology" title="Glossary of differential geometry and topology">Glossary</a>)</div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Basic concepts</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a class="mw-selflink selflink">Topological manifold</a> <ul><li><a href="/wiki/Atlas_(topology)" title="Atlas (topology)">Atlas</a></li></ul></li> <li><a href="/wiki/Differentiable_manifold" title="Differentiable manifold">Differentiable/Smooth manifold</a> <ul><li><a href="/wiki/Differential_structure" title="Differential structure">Differential structure</a></li> <li><a href="/wiki/Smooth_structure" title="Smooth structure">Smooth atlas</a></li></ul></li> <li><a href="/wiki/Submanifold" title="Submanifold">Submanifold</a></li> <li><a href="/wiki/Riemannian_manifold" title="Riemannian manifold">Riemannian manifold</a></li> <li><a href="/wiki/Smoothness" title="Smoothness">Smooth map</a></li> <li><a href="/wiki/Submersion_(mathematics)" title="Submersion (mathematics)">Submersion</a></li> <li><a href="/wiki/Pushforward_(differential)" title="Pushforward (differential)">Pushforward</a></li> <li><a href="/wiki/Tangent_space" title="Tangent space">Tangent space</a></li> <li><a href="/wiki/Differential_form" title="Differential form">Differential form</a></li> <li><a href="/wiki/Vector_field" title="Vector field">Vector field</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Main results <span style="font-size:85%;"><a href="/wiki/Category:Theorems_in_differential_geometry" title="Category:Theorems in differential geometry">(list)</a></span></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/Atiyah%E2%80%93Singer_index_theorem" title="Atiyah–Singer index theorem">Atiyah–Singer index</a></li> <li><a href="/wiki/Darboux%27s_theorem" title="Darboux's theorem">Darboux's</a></li> <li><a href="/wiki/De_Rham_cohomology#De_Rham's_theorem" title="De Rham cohomology">De Rham's</a></li> <li><a href="/wiki/Frobenius_theorem_(differential_topology)" title="Frobenius theorem (differential topology)">Frobenius</a></li> <li><a href="/wiki/Generalized_Stokes_theorem" title="Generalized Stokes theorem">Generalized Stokes</a></li> <li><a href="/wiki/Hopf%E2%80%93Rinow_theorem" title="Hopf–Rinow theorem">Hopf–Rinow</a></li> <li><a href="/wiki/Noether%27s_theorem" title="Noether's theorem">Noether's</a></li> <li><a href="/wiki/Sard%27s_theorem" title="Sard's theorem">Sard's</a></li> <li><a href="/wiki/Whitney_embedding_theorem" title="Whitney embedding theorem">Whitney embedding</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Smoothness" title="Smoothness">Maps</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Differentiable_curve" title="Differentiable curve">Curve</a></li> <li><a href="/wiki/Diffeomorphism" title="Diffeomorphism">Diffeomorphism</a> <ul><li><a href="/wiki/Local_diffeomorphism" title="Local diffeomorphism">Local</a></li></ul></li> <li><a href="/wiki/Geodesic" title="Geodesic">Geodesic</a></li> <li><a href="/wiki/Exponential_map_(Riemannian_geometry)" title="Exponential map (Riemannian geometry)">Exponential map</a> <ul><li><a href="/wiki/Exponential_map_(Lie_theory)" title="Exponential map (Lie theory)">in Lie theory</a></li></ul></li> <li><a href="/wiki/Foliation" title="Foliation">Foliation</a></li> <li><a href="/wiki/Immersion_(mathematics)" title="Immersion (mathematics)">Immersion</a></li> <li><a href="/wiki/Integral_curve" title="Integral curve">Integral curve</a></li> <li><a href="/wiki/Lie_derivative" title="Lie derivative">Lie derivative</a></li> <li><a href="/wiki/Section_(fiber_bundle)" title="Section (fiber bundle)">Section</a></li> <li><a href="/wiki/Submersion_(mathematics)" title="Submersion (mathematics)">Submersion</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Types of<br />manifolds</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/Closed_manifold" title="Closed manifold">Closed</a></li> <li>(<a href="/wiki/Almost_complex_manifold" title="Almost complex manifold">Almost</a>) <a href="/wiki/Complex_manifold" title="Complex manifold">Complex</a></li> <li>(<a href="/wiki/Almost-contact_manifold" title="Almost-contact manifold">Almost</a>) <a href="/wiki/Contact_manifold" class="mw-redirect" title="Contact manifold">Contact</a></li> <li><a href="/wiki/Fibered_manifold" title="Fibered manifold">Fibered</a></li> <li><a href="/wiki/Finsler_manifold" title="Finsler manifold">Finsler</a></li> <li><a href="/wiki/Flat_manifold" title="Flat manifold">Flat</a></li> <li><a href="/wiki/G-structure_on_a_manifold" title="G-structure on a manifold">G-structure</a></li> <li><a href="/wiki/Hadamard_manifold" title="Hadamard manifold">Hadamard</a></li> <li><a href="/wiki/Hermitian_manifold" title="Hermitian manifold">Hermitian</a></li> <li><a href="/wiki/Hyperbolic_manifold" title="Hyperbolic manifold">Hyperbolic</a></li> <li><a href="/wiki/K%C3%A4hler_manifold" title="Kähler manifold">Kähler</a></li> <li><a href="/wiki/Kenmotsu_manifold" title="Kenmotsu manifold">Kenmotsu</a></li> <li><a href="/wiki/Lie_group" title="Lie group">Lie group</a> <ul><li><a href="/wiki/Lie_group%E2%80%93Lie_algebra_correspondence" title="Lie group–Lie algebra correspondence">Lie algebra</a></li></ul></li> <li><a href="/wiki/Manifold_with_boundary" class="mw-redirect" title="Manifold with boundary">Manifold with boundary</a></li> <li><a href="/wiki/Orientability" title="Orientability">Oriented</a></li> <li><a href="/wiki/Parallelizable_manifold" title="Parallelizable manifold">Parallelizable</a></li> <li><a href="/wiki/Poisson_manifold" title="Poisson manifold">Poisson</a></li> <li><a href="/wiki/Prime_manifold" title="Prime manifold">Prime</a></li> <li><a href="/wiki/Quaternionic_manifold" title="Quaternionic manifold">Quaternionic</a></li> <li><a href="/wiki/Hypercomplex_manifold" title="Hypercomplex manifold">Hypercomplex</a></li> <li>(<a href="/wiki/Pseudo-Riemannian_manifold" title="Pseudo-Riemannian manifold">Pseudo−</a>, <a href="/wiki/Sub-Riemannian_manifold" title="Sub-Riemannian manifold">Sub−</a>) <a href="/wiki/Riemannian_manifold" title="Riemannian manifold">Riemannian</a></li> <li><a href="/wiki/Rizza_manifold" title="Rizza manifold">Rizza</a></li> <li>(<a href="/wiki/Almost_symplectic_manifold" title="Almost symplectic manifold">Almost</a>) <a href="/wiki/Symplectic_manifold" title="Symplectic manifold">Symplectic</a></li> <li><a href="/wiki/Tame_manifold" title="Tame manifold">Tame</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Tensor" title="Tensor">Tensors</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%">Vectors</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/Distribution_(differential_geometry)" title="Distribution (differential geometry)">Distribution</a></li> <li><a href="/wiki/Lie_bracket_of_vector_fields" title="Lie bracket of vector fields">Lie bracket</a></li> <li><a href="/wiki/Pushforward_(differential)" title="Pushforward (differential)">Pushforward</a></li> <li><a href="/wiki/Tangent_space" title="Tangent space">Tangent space</a> <ul><li><a href="/wiki/Tangent_bundle" title="Tangent bundle">bundle</a></li></ul></li> <li><a href="/wiki/Torsion_tensor" title="Torsion tensor">Torsion</a></li> <li><a href="/wiki/Vector_field" title="Vector field">Vector field</a></li> <li><a href="/wiki/Vector_flow" title="Vector flow">Vector flow</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Covectors</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/Closed_and_exact_differential_forms" title="Closed and exact differential forms">Closed/Exact</a></li> <li><a href="/wiki/Covariant_derivative" title="Covariant derivative">Covariant derivative</a></li> <li><a href="/wiki/Cotangent_space" title="Cotangent space">Cotangent space</a> <ul><li><a href="/wiki/Cotangent_bundle" title="Cotangent bundle">bundle</a></li></ul></li> <li><a href="/wiki/De_Rham_cohomology" title="De Rham cohomology">De Rham cohomology</a></li> <li><a href="/wiki/Differential_form" title="Differential form">Differential form</a> <ul><li><a href="/wiki/Vector-valued_differential_form" title="Vector-valued differential form">Vector-valued</a></li></ul></li> <li><a href="/wiki/Exterior_derivative" title="Exterior derivative">Exterior derivative</a></li> <li><a href="/wiki/Interior_product" title="Interior product">Interior product</a></li> <li><a href="/wiki/Pullback_(differential_geometry)" title="Pullback (differential geometry)">Pullback</a></li> <li><a href="/wiki/Ricci_curvature" title="Ricci curvature">Ricci curvature</a> <ul><li><a href="/wiki/Ricci_flow" title="Ricci flow">flow</a></li></ul></li> <li><a href="/wiki/Riemann_curvature_tensor" title="Riemann curvature tensor">Riemann curvature tensor</a></li> <li><a href="/wiki/Tensor_field" title="Tensor field">Tensor field</a> <ul><li><a href="/wiki/Tensor_density" title="Tensor density">density</a></li></ul></li> <li><a href="/wiki/Volume_form" title="Volume form">Volume form</a></li> <li><a href="/wiki/Wedge_product" class="mw-redirect" title="Wedge product">Wedge product</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Fiber_bundle" title="Fiber bundle">Bundles</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Adjoint_bundle" title="Adjoint bundle">Adjoint</a></li> <li><a href="/wiki/Affine_bundle" title="Affine bundle">Affine</a></li> <li><a href="/wiki/Associated_bundle" title="Associated bundle">Associated</a></li> <li><a href="/wiki/Cotangent_bundle" title="Cotangent bundle">Cotangent</a></li> <li><a href="/wiki/Dual_bundle" title="Dual bundle">Dual</a></li> <li><a href="/wiki/Fiber_bundle" title="Fiber bundle">Fiber</a></li> <li>(<a href="/wiki/Cofibration" title="Cofibration">Co</a>) <a href="/wiki/Fibration" title="Fibration">Fibration</a></li> <li><a href="/wiki/Jet_bundle" title="Jet bundle">Jet</a></li> <li><a href="/wiki/Lie_algebra_bundle" title="Lie algebra bundle">Lie algebra</a></li> <li>(<a href="/wiki/Stable_normal_bundle" title="Stable normal bundle">Stable</a>) <a href="/wiki/Normal_bundle" title="Normal bundle">Normal</a></li> <li><a href="/wiki/Principal_bundle" title="Principal bundle">Principal</a></li> <li><a href="/wiki/Spinor_bundle" title="Spinor bundle">Spinor</a></li> <li><a href="/wiki/Subbundle" title="Subbundle">Subbundle</a></li> <li><a href="/wiki/Tangent_bundle" title="Tangent bundle">Tangent</a></li> <li><a href="/wiki/Tensor_bundle" title="Tensor bundle">Tensor</a></li> <li><a href="/wiki/Vector_bundle" title="Vector bundle">Vector</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Connection_(mathematics)" title="Connection (mathematics)">Connections</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/Affine_connection" title="Affine connection">Affine</a></li> <li><a href="/wiki/Cartan_connection" title="Cartan connection">Cartan</a></li> <li><a href="/wiki/Ehresmann_connection" title="Ehresmann connection">Ehresmann</a></li> <li><a href="/wiki/Connection_form" title="Connection form">Form</a></li> <li><a href="/wiki/Connection_(fibred_manifold)" title="Connection (fibred manifold)">Generalized</a></li> <li><a href="/wiki/Koszul_connection" class="mw-redirect" title="Koszul connection">Koszul</a></li> <li><a href="/wiki/Levi-Civita_connection" title="Levi-Civita connection">Levi-Civita</a></li> <li><a href="/wiki/Connection_(principal_bundle)" title="Connection (principal bundle)">Principal</a></li> <li><a href="/wiki/Connection_(vector_bundle)" title="Connection (vector bundle)">Vector</a></li> <li><a href="/wiki/Parallel_transport" title="Parallel transport">Parallel transport</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Related</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/Classification_of_manifolds" title="Classification of manifolds">Classification of manifolds</a></li> <li><a href="/wiki/Gauge_theory_(mathematics)" title="Gauge theory (mathematics)">Gauge theory</a></li> <li><a href="/wiki/History_of_manifolds_and_varieties" title="History of manifolds and varieties">History</a></li> <li><a href="/wiki/Morse_theory" title="Morse theory">Morse theory</a></li> <li><a href="/wiki/Moving_frame" title="Moving frame">Moving frame</a></li> <li><a href="/wiki/Singularity_theory" title="Singularity theory">Singularity theory</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Generalizations</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_manifold" title="Banach manifold">Banach manifold</a></li> <li><a href="/wiki/Diffeology" title="Diffeology">Diffeology</a></li> <li><a href="/wiki/Diffiety" title="Diffiety">Diffiety</a></li> <li><a href="/wiki/Fr%C3%A9chet_manifold" title="Fréchet manifold">Fréchet manifold</a></li> <li><a href="/wiki/K-theory" title="K-theory">K-theory</a></li> <li><a href="/wiki/Orbifold" title="Orbifold">Orbifold</a></li> <li><a href="/wiki/Secondary_calculus_and_cohomological_physics" title="Secondary calculus and cohomological physics">Secondary calculus</a> <ul><li><a href="/wiki/Differential_calculus_over_commutative_algebras" title="Differential calculus over commutative algebras">over commutative algebras</a></li></ul></li> <li><a href="/wiki/Sheaf_(mathematics)" title="Sheaf (mathematics)">Sheaf</a></li> <li><a href="/wiki/Stratifold" title="Stratifold">Stratifold</a></li> <li><a href="/wiki/Supermanifold" title="Supermanifold">Supermanifold</a></li> <li><a href="/wiki/Stratified_space" title="Stratified space">Stratified space</a></li></ul> </div></td></tr></tbody></table></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236075235"><style data-mw-deduplicate="TemplateStyles:r1038841319">.mw-parser-output .tooltip-dotted{border-bottom:1px dotted;cursor:help}</style></div><div role="navigation" class="navbox authority-control" aria-label="Navbox" style="padding:3px"><table class="nowraplinks hlist navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Help:Authority_control" title="Help:Authority control">Authority control databases</a>: National <span class="mw-valign-text-top noprint" typeof="mw:File/Frameless"><a href="https://www.wikidata.org/wiki/Q2721559#identifiers" title="Edit this at Wikidata"><img alt="Edit this at Wikidata" src="//upload.wikimedia.org/wikipedia/en/thumb/8/8a/OOjs_UI_icon_edit-ltr-progressive.svg/10px-OOjs_UI_icon_edit-ltr-progressive.svg.png" decoding="async" width="10" height="10" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/8/8a/OOjs_UI_icon_edit-ltr-progressive.svg/15px-OOjs_UI_icon_edit-ltr-progressive.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/8/8a/OOjs_UI_icon_edit-ltr-progressive.svg/20px-OOjs_UI_icon_edit-ltr-progressive.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"><ul><li><span class="uid"><a rel="nofollow" class="external text" href="https://id.loc.gov/authorities/sh85136084">United States</a></span></li><li><span class="uid"><span class="rt-commentedText tooltip tooltip-dotted" title="topologické variety"><a rel="nofollow" class="external text" href="https://aleph.nkp.cz/F/?func=find-c&local_base=aut&ccl_term=ica=ph520056&CON_LNG=ENG">Czech Republic</a></span></span></li><li><span class="uid"><a rel="nofollow" class="external text" href="https://www.nli.org.il/en/authorities/987007541467005171">Israel</a></span></li></ul></div></td></tr></tbody></table></div>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1&useformat=desktop" 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=Topological_manifold&oldid=1251981624">https://en.wikipedia.org/w/index.php?title=Topological_manifold&oldid=1251981624</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:Manifolds" title="Category:Manifolds">Manifolds</a></li><li><a href="/wiki/Category:Properties_of_topological_spaces" title="Category:Properties of topological spaces">Properties of topological spaces</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Hidden categories: <ul><li><a href="/wiki/Category:Articles_with_short_description" title="Category:Articles with short description">Articles with short description</a></li><li><a href="/wiki/Category:Short_description_is_different_from_Wikidata" title="Category:Short description is different from Wikidata">Short description is different from Wikidata</a></li><li><a href="/wiki/Category:Commons_category_link_is_on_Wikidata" title="Category:Commons category link is on Wikidata">Commons category link is on Wikidata</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> This page was last edited on 19 October 2024, at 04:42<span class="anonymous-show"> (UTC)</span>.</li> <li id="footer-info-copyright">Text is available under the <a href="/wiki/Wikipedia:Text_of_the_Creative_Commons_Attribution-ShareAlike_4.0_International_License" title="Wikipedia:Text of the Creative Commons Attribution-ShareAlike 4.0 International License">Creative Commons Attribution-ShareAlike 4.0 License</a>; additional terms may apply. By using this site, you agree to the <a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Terms_of_Use" class="extiw" title="foundation:Special:MyLanguage/Policy:Terms of Use">Terms of Use</a> and <a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Privacy_policy" class="extiw" title="foundation:Special:MyLanguage/Policy:Privacy policy">Privacy Policy</a>. Wikipedia® is a registered trademark of the <a rel="nofollow" class="external text" href="https://wikimediafoundation.org/">Wikimedia Foundation, Inc.</a>, a non-profit organization.</li> </ul> <ul id="footer-places"> <li id="footer-places-privacy"><a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Privacy_policy">Privacy policy</a></li> <li id="footer-places-about"><a href="/wiki/Wikipedia:About">About Wikipedia</a></li> <li id="footer-places-disclaimers"><a href="/wiki/Wikipedia:General_disclaimer">Disclaimers</a></li> <li id="footer-places-contact"><a href="//en.wikipedia.org/wiki/Wikipedia:Contact_us">Contact Wikipedia</a></li> <li id="footer-places-wm-codeofconduct"><a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Universal_Code_of_Conduct">Code of Conduct</a></li> <li id="footer-places-developers"><a href="https://developer.wikimedia.org">Developers</a></li> <li id="footer-places-statslink"><a href="https://stats.wikimedia.org/#/en.wikipedia.org">Statistics</a></li> <li id="footer-places-cookiestatement"><a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Cookie_statement">Cookie statement</a></li> <li id="footer-places-mobileview"><a href="//en.m.wikipedia.org/w/index.php?title=Topological_manifold&mobileaction=toggle_view_mobile" class="noprint stopMobileRedirectToggle">Mobile view</a></li> </ul> <ul id="footer-icons" class="noprint"> <li id="footer-copyrightico"><a href="https://wikimediafoundation.org/" class="cdx-button cdx-button--fake-button cdx-button--size-large cdx-button--fake-button--enabled"><img src="/static/images/footer/wikimedia-button.svg" width="84" height="29" alt="Wikimedia Foundation" loading="lazy"></a></li> <li id="footer-poweredbyico"><a href="https://www.mediawiki.org/" class="cdx-button cdx-button--fake-button cdx-button--size-large cdx-button--fake-button--enabled"><img src="/w/resources/assets/poweredby_mediawiki.svg" alt="Powered by MediaWiki" width="88" height="31" loading="lazy"></a></li> </ul> </footer> </div> </div> </div> <div class="vector-settings" id="p-dock-bottom"> <ul></ul> </div><script>(RLQ=window.RLQ||[]).push(function(){mw.config.set({"wgHostname":"mw-web.codfw.main-5b4c46c98-jx5th","wgBackendResponseTime":143,"wgPageParseReport":{"limitreport":{"cputime":"0.442","walltime":"0.639","ppvisitednodes":{"value":1489,"limit":1000000},"postexpandincludesize":{"value":54669,"limit":2097152},"templateargumentsize":{"value":955,"limit":2097152},"expansiondepth":{"value":12,"limit":100},"expensivefunctioncount":{"value":12,"limit":500},"unstrip-depth":{"value":1,"limit":20},"unstrip-size":{"value":59421,"limit":5000000},"entityaccesscount":{"value":1,"limit":400},"timingprofile":["100.00% 451.953 1 -total"," 35.45% 160.208 1 Template:Reflist"," 32.92% 148.771 11 Template:Cite_book"," 20.13% 90.959 1 Template:Manifolds"," 19.56% 88.387 1 Template:Navbox"," 17.84% 80.645 1 Template:Short_description"," 9.84% 44.479 2 Template:Pagetype"," 8.80% 39.791 9 Template:Main"," 6.46% 29.183 1 Template:Authority_control"," 5.07% 22.909 4 Template:Main_other"]},"scribunto":{"limitreport-timeusage":{"value":"0.272","limit":"10.000"},"limitreport-memusage":{"value":4837788,"limit":52428800}},"cachereport":{"origin":"mw-web.codfw.main-6855777b7b-t42ml","timestamp":"20241204113645","ttl":2592000,"transientcontent":false}}});});</script> <script type="application/ld+json">{"@context":"https:\/\/schema.org","@type":"Article","name":"Topological manifold","url":"https:\/\/en.wikipedia.org\/wiki\/Topological_manifold","sameAs":"http:\/\/www.wikidata.org\/entity\/Q2721559","mainEntity":"http:\/\/www.wikidata.org\/entity\/Q2721559","author":{"@type":"Organization","name":"Contributors to Wikimedia projects"},"publisher":{"@type":"Organization","name":"Wikimedia Foundation, Inc.","logo":{"@type":"ImageObject","url":"https:\/\/www.wikimedia.org\/static\/images\/wmf-hor-googpub.png"}},"datePublished":"2002-05-20T03:50:30Z","dateModified":"2024-10-19T04:42:45Z","headline":"topological space (which may also be a separated space) which locally resembles real n-dimensional space"}</script> </body> </html>