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>Normal space - 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":"3731926b-b1c4-4b6b-a3fd-7ecb5a11037d","wgCanonicalNamespace":"","wgCanonicalSpecialPageName":false,"wgNamespaceNumber":0,"wgPageName":"Normal_space","wgTitle":"Normal space","wgCurRevisionId":1247621055,"wgRevisionId":1247621055,"wgArticleId":48629,"wgIsArticle":true,"wgIsRedirect":false,"wgAction":"view","wgUserName":null,"wgUserGroups":["*"],"wgCategories":["Articles with short description","Short description is different from Wikidata","Properties of topological spaces","Separation axioms"],"wgPageViewLanguage":"en","wgPageContentLanguage":"en","wgPageContentModel":"wikitext","wgRelevantPageName":"Normal_space","wgRelevantArticleId":48629,"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":10000,"wgRelatedArticlesCompat":[],"wgCentralAuthMobileDomain":false,"wgEditSubmitButtonLabelPublish":true,"wgULSPosition":"interlanguage","wgULSisCompactLinksEnabled":false,"wgVector2022LanguageInHeader":true,"wgULSisLanguageSelectorEmpty":false,"wgWikibaseItemId":"Q1071795","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.math.styles":"ready","ext.cite.styles":"ready","skins.vector.search.codex.styles":"ready","skins.vector.styles":"ready","skins.vector.icons":"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","ext.scribunto.logs","site","mediawiki.page.ready","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.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%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.4"> <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="Normal space - 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/Normal_space"> <link rel="alternate" type="application/x-wiki" title="Edit this page" href="/w/index.php?title=Normal_space&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/Normal_space"> <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-Normal_space rootpage-Normal_space 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=Normal+space" 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=Normal+space" 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=Normal+space" 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=Normal+space" 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-Definitions" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Definitions"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>Definitions</span> </div> </a> <ul id="toc-Definitions-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Examples_of_normal_spaces" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Examples_of_normal_spaces"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Examples of normal spaces</span> </div> </a> <ul id="toc-Examples_of_normal_spaces-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Examples_of_non-normal_spaces" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Examples_of_non-normal_spaces"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Examples of non-normal spaces</span> </div> </a> <ul id="toc-Examples_of_non-normal_spaces-sublist" class="vector-toc-list"> </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">4</span> <span>Properties</span> </div> </a> <ul id="toc-Properties-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Relationships_to_other_separation_axioms" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Relationships_to_other_separation_axioms"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Relationships to other separation axioms</span> </div> </a> <ul id="toc-Relationships_to_other_separation_axioms-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Citations" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Citations"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Citations</span> </div> </a> <ul id="toc-Citations-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 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> </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">Normal space</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Go to an article in another language. Available in 16 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-16" 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">16 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Normaler_Raum" title="Normaler Raum – German" lang="de" hreflang="de" data-title="Normaler Raum" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Espacio_normal" title="Espacio normal – Spanish" lang="es" hreflang="es" data-title="Espacio normal" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Normala_spaco" title="Normala spaco – Esperanto" lang="eo" hreflang="eo" data-title="Normala spaco" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D9%81%D8%B6%D8%A7%DB%8C_%D9%86%D8%B1%D9%85%D8%A7%D9%84" title="فضای نرمال – Persian" lang="fa" hreflang="fa" data-title="فضای نرمال" 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/Espace_normal" title="Espace normal – French" lang="fr" hreflang="fr" data-title="Espace normal" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%A0%95%EA%B7%9C_%EA%B3%B5%EA%B0%84" title="정규 공간 – Korean" lang="ko" hreflang="ko" data-title="정규 공간" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Spazio_normale" title="Spazio normale – Italian" lang="it" hreflang="it" data-title="Spazio normale" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%9E%D7%A8%D7%97%D7%91_%D7%A0%D7%95%D7%A8%D7%9E%D7%9C%D7%99_%D7%91%D7%90%D7%95%D7%A4%D7%9F_%D7%9E%D7%95%D7%A9%D7%9C%D7%9D" title="מרחב נורמלי באופן מושלם – Hebrew" lang="he" hreflang="he" data-title="מרחב נורמלי באופן מושלם" 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/Normale_ruimte" title="Normale ruimte – Dutch" lang="nl" hreflang="nl" data-title="Normale ruimte" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Przestrze%C5%84_T4" title="Przestrzeń T4 – Polish" lang="pl" hreflang="pl" data-title="Przestrzeń T4" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Espa%C3%A7o_normal" title="Espaço normal – Portuguese" lang="pt" hreflang="pt" data-title="Espaço normal" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%9D%D0%BE%D1%80%D0%BC%D0%B0%D0%BB%D1%8C%D0%BD%D0%BE%D0%B5_%D0%BF%D1%80%D0%BE%D1%81%D1%82%D1%80%D0%B0%D0%BD%D1%81%D1%82%D0%B2%D0%BE" title="Нормальное пространство – Russian" lang="ru" hreflang="ru" data-title="Нормальное пространство" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Normaali_avaruus" title="Normaali avaruus – Finnish" lang="fi" hreflang="fi" data-title="Normaali avaruus" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Normalt_rum" title="Normalt rum – Swedish" lang="sv" hreflang="sv" data-title="Normalt rum" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%9D%D0%BE%D1%80%D0%BC%D0%B0%D0%BB%D1%8C%D0%BD%D0%B8%D0%B9_%D0%BF%D1%80%D0%BE%D1%81%D1%82%D1%96%D1%80" title="Нормальний простір – Ukrainian" lang="uk" hreflang="uk" data-title="Нормальний простір" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E6%AD%A3%E8%A7%84%E7%A9%BA%E9%97%B4" title="正规空间 – Chinese" lang="zh" hreflang="zh" data-title="正规空间" 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/Q1071795#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/Normal_space" 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:Normal_space" 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/Normal_space"><span>Read</span></a></li><li id="ca-edit" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Normal_space&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=Normal_space&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/Normal_space"><span>Read</span></a></li><li id="ca-more-edit" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Normal_space&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=Normal_space&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/Normal_space" 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/Normal_space" 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=Normal_space&oldid=1247621055" 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=Normal_space&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=Normal_space&id=1247621055&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%2FNormal_space"><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%2FNormal_space"><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=Normal_space&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=Normal_space&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 id="t-wikibase" class="wb-otherproject-link wb-otherproject-wikibase-dataitem mw-list-item"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q1071795" 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> <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">For normal vector space, see <a href="/wiki/Normal_(geometry)" title="Normal (geometry)">normal (geometry)</a>.</div> <style data-mw-deduplicate="TemplateStyles:r1257001546">.mw-parser-output .infobox-subbox{padding:0;border:none;margin:-3px;width:auto;min-width:100%;font-size:100%;clear:none;float:none;background-color:transparent}.mw-parser-output .infobox-3cols-child{margin:auto}.mw-parser-output .infobox .navbar{font-size:100%}@media screen{html.skin-theme-clientpref-night .mw-parser-output .infobox-full-data:not(.notheme)>div:not(.notheme)[style]{background:#1f1f23!important;color:#f8f9fa}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .infobox-full-data:not(.notheme) div:not(.notheme){background:#1f1f23!important;color:#f8f9fa}}@media(min-width:640px){body.skin--responsive .mw-parser-output .infobox-table{display:table!important}body.skin--responsive .mw-parser-output .infobox-table>caption{display:table-caption!important}body.skin--responsive .mw-parser-output .infobox-table>tbody{display:table-row-group}body.skin--responsive .mw-parser-output .infobox-table tr{display:table-row!important}body.skin--responsive .mw-parser-output .infobox-table th,body.skin--responsive .mw-parser-output .infobox-table td{padding-left:inherit;padding-right:inherit}}</style><table class="infobox" style="width:auto;"><tbody><tr><th colspan="2" class="infobox-above" style="background:#ccc;font-size:120%;padding:0.3em 0.3em 0.4em;"><a href="/wiki/Separation_axiom" title="Separation axiom">Separation axioms</a><br />in <a href="/wiki/Topological_spaces" class="mw-redirect" title="Topological spaces">topological spaces</a></th></tr><tr><th colspan="2" class="infobox-header" style="background:#ddd;"><a href="/wiki/Andrey_Kolmogorov" title="Andrey Kolmogorov">Kolmogorov</a> classification</th></tr><tr><th scope="row" class="infobox-label" style="text-align:right;padding-right:0.5em;"><a href="/wiki/Kolmogorov_space" title="Kolmogorov space">T<sub><span style="position:relative; bottom:0.25em;">0</span></sub></a><span style="padding-left:0.5em;"> </span></th><td class="infobox-data" style="font-size:90%; padding-top:0.4em;line-height:1.2em;">(Kolmogorov)</td></tr><tr><th scope="row" class="infobox-label" style="text-align:right;padding-right:0.5em;"><a href="/wiki/T1_space" title="T1 space">T<sub><span style="position:relative; bottom:0.25em;">1</span></sub></a><span style="padding-left:0.5em;"> </span></th><td class="infobox-data" style="font-size:90%; padding-top:0.4em;line-height:1.2em;">(Fréchet)</td></tr><tr><th scope="row" class="infobox-label" style="text-align:right;padding-right:0.5em;"><a href="/wiki/Hausdorff_space" title="Hausdorff space">T<sub><span style="position:relative; bottom:0.25em;">2</span></sub></a><span style="padding-left:0.5em;"> </span></th><td class="infobox-data" style="font-size:90%; padding-top:0.4em;line-height:1.2em;">(Hausdorff)</td></tr><tr><th scope="row" class="infobox-label" style="text-align:right;padding-right:0.5em;"><a href="/wiki/Urysohn_and_completely_Hausdorff_spaces" title="Urysohn and completely Hausdorff spaces">T<sub><span style="position:relative; bottom:0.25em;">2<span style="font-size:120%">½</span></span></sub></a></th><td class="infobox-data" style="font-size:90%; padding-top:0.4em;line-height:1.2em;">(Urysohn)</td></tr><tr><th scope="row" class="infobox-label" style="text-align:right;padding-right:0.5em;"><a href="/wiki/Urysohn_and_completely_Hausdorff_spaces" title="Urysohn and completely Hausdorff spaces">completely T<sub><span style="position:relative; bottom:0.25em;">2</span></sub></a><span style="padding-left:0.5em;"> </span></th><td class="infobox-data" style="font-size:90%; padding-top:0.4em;line-height:1.2em;">(completely Hausdorff)</td></tr><tr><th scope="row" class="infobox-label" style="text-align:right;padding-right:0.5em;"><a href="/wiki/Regular_space" title="Regular space">T<sub><span style="position:relative; bottom:0.25em;">3</span></sub></a><span style="padding-left:0.5em;"> </span></th><td class="infobox-data" style="font-size:90%; padding-top:0.4em;line-height:1.2em;">(regular Hausdorff)</td></tr><tr><th scope="row" class="infobox-label" style="text-align:right;padding-right:0.5em;"><a href="/wiki/Tychonoff_space" title="Tychonoff space">T<sub><span style="position:relative; bottom:0.25em;"><span style="font-size:120%">3½</span></span></sub></a></th><td class="infobox-data" style="font-size:90%; padding-top:0.4em;line-height:1.2em;">(Tychonoff)</td></tr><tr><th scope="row" class="infobox-label" style="text-align:right;padding-right:0.5em;"><a class="mw-selflink selflink">T<sub><span style="position:relative; bottom:0.25em;">4</span></sub></a><span style="padding-left:0.5em;"> </span></th><td class="infobox-data" style="font-size:90%; padding-top:0.4em;line-height:1.2em;">(normal Hausdorff)</td></tr><tr><th scope="row" class="infobox-label" style="text-align:right;padding-right:0.5em;"><a class="mw-selflink selflink">T<sub><span style="position:relative; bottom:0.25em;">5</span></sub></a><span style="padding-left:0.5em;"> </span></th><td class="infobox-data" style="font-size:90%; padding-top:0.4em;line-height:1.2em;">(completely normal<br /> Hausdorff)</td></tr><tr><th scope="row" class="infobox-label" style="text-align:right;padding-right:0.5em;"><a class="mw-selflink selflink">T<sub><span style="position:relative; bottom:0.25em;">6</span></sub></a><span style="padding-left:0.5em;"> </span></th><td class="infobox-data" style="font-size:90%; padding-top:0.4em;line-height:1.2em;">(perfectly normal<br /> Hausdorff)</td></tr><tr><td colspan="2" class="infobox-below" style="background:#ddd;"><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><div class="hlist"><ul><li><a href="/wiki/History_of_the_separation_axioms" title="History of the separation axioms">History</a></li></ul></div></td></tr></tbody></table> <p>In <a href="/wiki/Topology" title="Topology">topology</a> and related branches of <a href="/wiki/Mathematics" title="Mathematics">mathematics</a>, a <b>normal space</b> is a <a href="/wiki/Topological_space" title="Topological space">topological space</a> <i>X</i> that satisfies <b>Axiom T<sub>4</sub></b>: every two disjoint <a href="/wiki/Closed_set" title="Closed set">closed sets</a> of <i>X</i> have disjoint <a href="/wiki/Open_neighborhood" class="mw-redirect" title="Open neighborhood">open neighborhoods</a>. A normal <a href="/wiki/Hausdorff_space" title="Hausdorff space">Hausdorff space</a> is also called a <b>T<sub>4</sub> space</b>. These conditions are examples of <a href="/wiki/Separation_axiom" title="Separation axiom">separation axioms</a> and their further strengthenings define <b>completely normal Hausdorff spaces</b>, or <b>T<sub>5</sub> spaces</b>, and <b>perfectly normal Hausdorff spaces</b>, or <b>T<sub>6</sub> spaces</b>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Definitions">Definitions</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Normal_space&action=edit&section=1" title="Edit section: Definitions"><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 a <b>normal space</b> if, given any <a href="/wiki/Disjoint_sets" title="Disjoint sets">disjoint</a> <a href="/wiki/Closed_set" title="Closed set">closed sets</a> <i>E</i> and <i>F</i>, there are <a href="/wiki/Neighbourhood_(topology)" class="mw-redirect" title="Neighbourhood (topology)">neighbourhoods</a> <i>U</i> of <i>E</i> and <i>V</i> of <i>F</i> that are also disjoint. More intuitively, this condition says that <i>E</i> and <i>F</i> can be <a href="/wiki/Separated_set" class="mw-redirect" title="Separated set">separated by neighbourhoods</a>. </p> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Normal_space.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/06/Normal_space.svg/203px-Normal_space.svg.png" decoding="async" width="203" height="102" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/06/Normal_space.svg/305px-Normal_space.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/06/Normal_space.svg/406px-Normal_space.svg.png 2x" data-file-width="1200" data-file-height="600" /></a><figcaption>The closed sets <i>E</i> and <i>F</i>, here represented by closed disks on opposite sides of the picture, are separated by their respective neighbourhoods <i>U</i> and <i>V</i>, here represented by larger, but still disjoint, open disks.</figcaption></figure> <p>A <b>T<sub>4</sub> space</b> is a <a href="/wiki/T1_space" title="T1 space">T<sub>1</sub> space</a> <i>X</i> that is normal; this is equivalent to <i>X</i> being normal and <a href="/wiki/Hausdorff_space" title="Hausdorff space">Hausdorff</a>. </p><p>A <b>completely normal space</b>, or <b><style data-mw-deduplicate="TemplateStyles:r1238216509">.mw-parser-output .vanchor>:target~.vanchor-text{background-color:#b1d2ff}@media screen{html.skin-theme-clientpref-night .mw-parser-output .vanchor>:target~.vanchor-text{background-color:#0f4dc9}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .vanchor>:target~.vanchor-text{background-color:#0f4dc9}}</style><span class="vanchor"><span id="hereditarily_normal_space"></span><span class="vanchor-text">hereditarily normal space</span></span></b>, is a topological space <i>X</i> such that every <a href="/wiki/Subspace_(topology)" class="mw-redirect" title="Subspace (topology)">subspace</a> of <i>X</i> is a normal space. It turns out that <i>X</i> is completely normal if and only if every two <a href="/wiki/Separated_set" class="mw-redirect" title="Separated set">separated sets</a> can be separated by neighbourhoods. Also, <i>X</i> is completely normal if and only if every open subset of <i>X</i> is normal with the subspace topology. </p><p>A <b>T<sub>5</sub> space</b>, or <b>completely T<sub>4</sub> space</b>, is a completely normal T<sub>1</sub> space <i>X</i>, which implies that <i>X</i> is Hausdorff; equivalently, every subspace of <i>X</i> must be a T<sub>4</sub> space. </p><p>A <b>perfectly normal space</b> is a topological space <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"></span> in which every two disjoint closed sets <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 E}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle E}"></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 F}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/545fd099af8541605f7ee55f08225526be88ce57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}"></span> can be <a href="/wiki/Precisely_separated_by_a_function" class="mw-redirect" title="Precisely separated by a function">precisely separated by a function</a>, in the sense that there is a continuous function <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"></span> from <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"></span> to the interval <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 [0,1]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [0,1]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/738f7d23bb2d9642bab520020873cccbef49768d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.653ex; height:2.843ex;" alt="{\displaystyle [0,1]}"></span> such that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f^{-1}(0)=E}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f^{-1}(0)=E}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f05f669bbcf6d4c18a2ab5227b7a52f8e3814c41" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.499ex; height:3.176ex;" alt="{\displaystyle f^{-1}(0)=E}"></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 f^{-1}(1)=F}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mi>F</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f^{-1}(1)=F}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/210aba5e27378d9856eccd1c921b0c8cb937b759" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.464ex; height:3.176ex;" alt="{\displaystyle f^{-1}(1)=F}"></span>.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> This is a stronger separation property than normality, as by <a href="/wiki/Urysohn%27s_lemma" title="Urysohn's lemma">Urysohn's lemma</a> disjoint closed sets in a normal space can be <a href="/wiki/Separated_by_a_function" class="mw-redirect" title="Separated by a function">separated by a function</a>, in the sense 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 E\subseteq f^{-1}(0)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo>⊆</mo> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E\subseteq f^{-1}(0)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/518e9a368414fdeaa496a174474aff29acfc7b74" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.499ex; height:3.176ex;" alt="{\displaystyle E\subseteq f^{-1}(0)}"></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 F\subseteq f^{-1}(1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo>⊆</mo> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F\subseteq f^{-1}(1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7e0137fef2864ed522f7d273f059e28c5441408f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.464ex; height:3.176ex;" alt="{\displaystyle F\subseteq f^{-1}(1)}"></span>, but not precisely separated in general. It turns out that <i>X</i> is perfectly normal if and only if <i>X</i> is normal and every closed set is a <a href="/wiki/G-delta_set" class="mw-redirect" title="G-delta set">G<sub>δ</sub> set</a>. Equivalently, <i>X</i> is perfectly normal if and only if every closed set is the <a href="/wiki/Zero_set" class="mw-redirect" title="Zero set">zero set</a> of a <a href="/wiki/Continuous_function" title="Continuous function">continuous function</a>. The equivalence between these three characterizations is called <b>Vedenissoff's theorem</b>.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> Every perfectly normal space is completely normal, because perfect normality is a <a href="/wiki/Hereditary_property" title="Hereditary property">hereditary property</a>.<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Munkres_p213_5-0" class="reference"><a href="#cite_note-Munkres_p213-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </p><p>A <b>T<sub>6</sub> space</b>, or <b>perfectly T<sub>4</sub> space</b>, is a perfectly normal Hausdorff space. </p><p>Note that the terms "normal space" and "T<sub>4</sub>" and derived concepts occasionally have a different meaning. (Nonetheless, "T<sub>5</sub>" always means the same as "completely T<sub>4</sub>", whatever the meaning of T<sub>4</sub> may be.) The definitions given here are the ones usually used today. For more on this issue, see <a href="/wiki/History_of_the_separation_axioms" title="History of the separation axioms">History of the separation axioms</a>. </p><p>Terms like "normal <a href="/wiki/Regular_space" title="Regular space">regular space</a>" and "normal Hausdorff space" also turn up in the literature—they simply mean that the space both is normal and satisfies the other condition mentioned. In particular, a normal Hausdorff space is the same thing as a T<sub>4</sub> space. Given the historical confusion of the meaning of the terms, verbal descriptions when applicable are helpful, that is, "normal Hausdorff" instead of "T<sub>4</sub>", or "completely normal Hausdorff" instead of "T<sub>5</sub>". </p><p><a href="/wiki/Fully_normal_space" class="mw-redirect" title="Fully normal space">Fully normal spaces</a> and <a href="/wiki/Paracompact_Hausdorff_space" class="mw-redirect" title="Paracompact Hausdorff space">fully T<sub>4</sub> spaces</a> are discussed elsewhere; they are related to <a href="/wiki/Paracompactness" class="mw-redirect" title="Paracompactness">paracompactness</a>. </p><p>A <a href="/wiki/Locally_normal_space" title="Locally normal space">locally normal space</a> is a topological space where every point has an open neighbourhood that is normal. Every normal space is locally normal, but the converse is not true. A classical example of a completely regular locally normal space that is not normal is the <a href="/wiki/Nemytskii_plane" class="mw-redirect" title="Nemytskii plane">Nemytskii plane</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Examples_of_normal_spaces">Examples of normal spaces</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Normal_space&action=edit&section=2" title="Edit section: Examples of normal spaces"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Most spaces encountered in <a href="/wiki/Mathematical_analysis" title="Mathematical analysis">mathematical analysis</a> are normal Hausdorff spaces, or at least normal regular spaces: </p> <ul><li>All <a href="/wiki/Metric_spaces" class="mw-redirect" title="Metric spaces">metric spaces</a> (and hence all <a href="/wiki/Metrizable_space" title="Metrizable space">metrizable spaces</a>) are perfectly normal Hausdorff;</li> <li>All <a href="/wiki/Pseudometric_space" title="Pseudometric space">pseudometric spaces</a> (and hence all <a href="/wiki/Pseudometrisable_space" class="mw-redirect" title="Pseudometrisable space">pseudometrisable spaces</a>) are perfectly normal regular, although not in general Hausdorff;</li> <li>All <a href="/wiki/Compact_space" title="Compact space">compact</a> Hausdorff spaces are normal;</li> <li>In particular, the <a href="/wiki/Stone%E2%80%93%C4%8Cech_compactification" title="Stone–Čech compactification">Stone–Čech compactification</a> of a <a href="/wiki/Tychonoff_space" title="Tychonoff space">Tychonoff space</a> is normal Hausdorff;</li> <li>Generalizing the above examples, all <a href="/wiki/Paracompact" class="mw-redirect" title="Paracompact">paracompact</a> Hausdorff spaces are normal, and all paracompact regular spaces are normal;</li> <li>All paracompact <a href="/wiki/Topological_manifold" title="Topological manifold">topological manifolds</a> are perfectly normal Hausdorff. However, there exist non-paracompact manifolds that are not even normal.</li> <li>All <a href="/wiki/Order_topology" title="Order topology">order topologies</a> on <a href="/wiki/Totally_ordered_set" class="mw-redirect" title="Totally ordered set">totally ordered sets</a> are hereditarily normal and Hausdorff.</li> <li>Every regular <a href="/wiki/Second-countable_space" title="Second-countable space">second-countable space</a> is completely normal, and every regular <a href="/wiki/Lindel%C3%B6f_space" title="Lindelöf space">Lindelöf space</a> is normal.</li></ul> <p>Also, all <a href="/wiki/Fully_normal_space" class="mw-redirect" title="Fully normal space">fully normal spaces</a> are normal (even if not regular). <a href="/wiki/Sierpi%C5%84ski_space" title="Sierpiński space">Sierpiński space</a> is an example of a normal space that is not regular. </p> <div class="mw-heading mw-heading2"><h2 id="Examples_of_non-normal_spaces">Examples of non-normal spaces</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Normal_space&action=edit&section=3" title="Edit section: Examples of non-normal spaces"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>An important example of a non-normal topology is given by the <a href="/wiki/Zariski_topology" title="Zariski topology">Zariski topology</a> on an <a href="/wiki/Algebraic_variety" title="Algebraic variety">algebraic variety</a> or on the <a href="/wiki/Spectrum_of_a_ring" title="Spectrum of a ring">spectrum of a ring</a>, which is used in <a href="/wiki/Algebraic_geometry" title="Algebraic geometry">algebraic geometry</a>. </p><p>A non-normal space of some relevance to analysis is the <a href="/wiki/Topological_vector_space" title="Topological vector space">topological vector space</a> of all <a href="/wiki/Function_(mathematics)" title="Function (mathematics)">functions</a> from the <a href="/wiki/Real_line" class="mw-redirect" title="Real line">real line</a> <b>R</b> to itself, with the <a href="/wiki/Topology_of_pointwise_convergence" class="mw-redirect" title="Topology of pointwise convergence">topology of pointwise convergence</a>. More generally, a theorem of <a href="/wiki/Arthur_Harold_Stone" title="Arthur Harold Stone">Arthur Harold Stone</a> states that the <a href="/wiki/Product_topology" title="Product topology">product</a> of <a href="/wiki/Uncountable" class="mw-redirect" title="Uncountable">uncountably many</a> non-<a href="/wiki/Compact_space" title="Compact space">compact</a> metric spaces is never normal. </p> <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=Normal_space&action=edit&section=4" title="Edit section: Properties"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Every closed subset of a normal space is normal. The continuous and closed image of a normal space is normal.<sup id="cite_ref-FOOTNOTEWillard1970[httpsarchiveorgdetailsgeneraltopology00will_0page100_100–101]_6-0" class="reference"><a href="#cite_note-FOOTNOTEWillard1970[httpsarchiveorgdetailsgeneraltopology00will_0page100_100–101]-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </p><p>The main significance of normal spaces lies in the fact that they admit "enough" <a href="/wiki/Continuous_function_(topology)" class="mw-redirect" title="Continuous function (topology)">continuous</a> <a href="/wiki/Real_number" title="Real number">real</a>-valued <a href="/wiki/Function_(mathematics)" title="Function (mathematics)">functions</a>, as expressed by the following theorems valid for any normal space <i>X</i>. </p><p><a href="/wiki/Urysohn%27s_lemma" title="Urysohn's lemma">Urysohn's lemma</a>: If <i>A</i> and <i>B</i> are two <a href="/wiki/Disjoint_sets" title="Disjoint sets">disjoint</a> closed subsets of <i>X</i>, then there exists a continuous function <i>f</i> from <i>X</i> to the real line <b>R</b> such that <i>f</i>(<i>x</i>) = 0 for all <i>x</i> in <i>A</i> and <i>f</i>(<i>x</i>) = 1 for all <i>x</i> in <i>B</i>. In fact, we can take the values of <i>f</i> to be entirely within the <a href="/wiki/Unit_interval" title="Unit interval">unit interval</a> [0,1]. In fancier terms, disjoint closed sets are not only separated by neighbourhoods, but also <a href="/wiki/Separated_by_a_function" class="mw-redirect" title="Separated by a function">separated by a function</a>. </p><p>More generally, the <a href="/wiki/Tietze_extension_theorem" title="Tietze extension theorem">Tietze extension theorem</a>: If <i>A</i> is a closed subset of <i>X</i> and <i>f</i> is a continuous function from <i>A</i> to <b>R</b>, then there exists a continuous function <i>F</i>: <i>X</i> → <b>R</b> that extends <i>f</i> in the sense that <i>F</i>(<i>x</i>) = <i>f</i>(<i>x</i>) for all <i>x</i> in <i>A</i>. </p><p>The map <i><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \emptyset \rightarrow X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∅</mi> <mo stretchy="false">→</mo> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \emptyset \rightarrow X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0824e51cce520be5752b00c0d65587ba3bf164a0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.757ex; height:2.509ex;" alt="{\displaystyle \emptyset \rightarrow X}"></span></i> has the <a href="/wiki/Lifting_property" title="Lifting property">lifting property</a> with respect to a map from a certain finite topological space with five points (two open and three closed) to the space with one open and two closed points.<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> </p><p>If <b>U</b> is a locally finite <a href="/wiki/Open_cover" class="mw-redirect" title="Open cover">open cover</a> of a normal space <i>X</i>, then there is a <a href="/wiki/Partition_of_unity" title="Partition of unity">partition of unity</a> precisely subordinate to <b>U</b>. This shows the relationship of normal spaces to <a href="/wiki/Paracompactness" class="mw-redirect" title="Paracompactness">paracompactness</a>. </p><p>In fact, any space that satisfies any one of these three conditions must be normal. </p><p>A <a href="/wiki/Product_space" class="mw-redirect" title="Product space">product</a> of normal spaces is not necessarily normal. This fact was first proved by <a href="/wiki/Robert_Sorgenfrey" title="Robert Sorgenfrey">Robert Sorgenfrey</a>. An example of this phenomenon is the <a href="/wiki/Sorgenfrey_plane" title="Sorgenfrey plane">Sorgenfrey plane</a>. In fact, since there exist spaces which are <a href="/wiki/Dowker_space" title="Dowker space">Dowker</a>, a product of a normal space and [0, 1] need not to be normal. Also, a subset of a normal space need not be normal (i.e. not every normal Hausdorff space is a completely normal Hausdorff space), since every Tychonoff space is a subset of its Stone–Čech compactification (which is normal Hausdorff). A more explicit example is the <a href="/wiki/Tychonoff_plank" title="Tychonoff plank">Tychonoff plank</a>. The only large class of product spaces of normal spaces known to be normal are the products of compact Hausdorff spaces, since both compactness (<a href="/wiki/Tychonoff%27s_theorem" title="Tychonoff's theorem">Tychonoff's theorem</a>) and the T<sub>2</sub> axiom are preserved under arbitrary products.<sup id="cite_ref-FOOTNOTEWillard1970Section_17_8-0" class="reference"><a href="#cite_note-FOOTNOTEWillard1970Section_17-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Relationships_to_other_separation_axioms">Relationships to other separation axioms</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Normal_space&action=edit&section=5" title="Edit section: Relationships to other separation axioms"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>If a normal space is <a href="/wiki/R0_space" class="mw-redirect" title="R0 space">R<sub>0</sub></a>, then it is in fact <a href="/wiki/Completely_regular" class="mw-redirect" title="Completely regular">completely regular</a>. Thus, anything from "normal R<sub>0</sub>" to "normal completely regular" is the same as what we usually call <i>normal regular</i>. Taking <a href="/wiki/Kolmogorov_quotient" class="mw-redirect" title="Kolmogorov quotient">Kolmogorov quotients</a>, we see that all normal <a href="/wiki/T1_space" title="T1 space">T<sub>1</sub> spaces</a> are <a href="/wiki/Tychonoff_space" title="Tychonoff space">Tychonoff</a>. These are what we usually call <i>normal Hausdorff</i> spaces. </p><p>A topological space is said to be <a href="/wiki/Pseudonormal_space" title="Pseudonormal space">pseudonormal</a> if given two disjoint closed sets in it, one of which is countable, there are disjoint open sets containing them. Every normal space is pseudonormal, but not vice versa. </p><p>Counterexamples to some variations on these statements can be found in the lists above. Specifically, <a href="/wiki/Sierpi%C5%84ski_space" title="Sierpiński space">Sierpiński space</a> is normal but not regular, while the space of functions from <b>R</b> to itself is Tychonoff but not normal. </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Normal_space&action=edit&section=6" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Collectionwise_normal_space" title="Collectionwise normal space">Collectionwise normal space</a> – Property of topological spaces stronger than normality</li> <li><a href="/wiki/Monotonically_normal_space" title="Monotonically normal space">Monotonically normal space</a> – Property of topological spaces stronger than normality</li></ul> <div class="mw-heading mw-heading2"><h2 id="Citations">Citations</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Normal_space&action=edit&section=7" title="Edit section: Citations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text">Willard, Exercise 15C</span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text">Engelking, Theorem 1.5.19. This is stated under the assumption of a T<sub>1</sub> space, but the proof does not make use of that assumption.</span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://math.stackexchange.com/questions/72138">"Why are these two definitions of a perfectly normal space equivalent?"</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Why+are+these+two+definitions+of+a+perfectly+normal+space+equivalent%3F&rft_id=https%3A%2F%2Fmath.stackexchange.com%2Fquestions%2F72138&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANormal+space" class="Z3988"></span></span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text">Engelking, Theorem 2.1.6, p. 68</span> </li> <li id="cite_note-Munkres_p213-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-Munkres_p213_5-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFMunkres2000">Munkres 2000</a>, p. 213</span> </li> <li id="cite_note-FOOTNOTEWillard1970[httpsarchiveorgdetailsgeneraltopology00will_0page100_100–101]-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEWillard1970[httpsarchiveorgdetailsgeneraltopology00will_0page100_100–101]_6-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFWillard1970">Willard 1970</a>, pp. <a rel="nofollow" class="external text" href="https://archive.org/details/generaltopology00will_0/page/100">100–101</a>.</span> </li> <li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://ncatlab.org/nlab/show/separation+axioms##TableOfMainSeparationAxiomsAsLiftingProperties">"separation axioms in nLab"</a>. <i>ncatlab.org</i><span class="reference-accessdate">. Retrieved <span class="nowrap">2021-10-12</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=ncatlab.org&rft.atitle=separation+axioms+in+nLab&rft_id=https%3A%2F%2Fncatlab.org%2Fnlab%2Fshow%2Fseparation%2Baxioms%23%23TableOfMainSeparationAxiomsAsLiftingProperties&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANormal+space" class="Z3988"></span></span> </li> <li id="cite_note-FOOTNOTEWillard1970Section_17-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEWillard1970Section_17_8-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFWillard1970">Willard 1970</a>, Section 17.</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=Normal_space&action=edit&section=8" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Ryszard_Engelking" title="Ryszard Engelking">Engelking, Ryszard</a>, <i>General Topology</i>, Heldermann Verlag Berlin, 1989. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/3-88538-006-4" title="Special:BookSources/3-88538-006-4">3-88538-006-4</a></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKemoto2004" class="citation encyclopaedia cs1">Kemoto, Nobuyuki (2004). "Higher Separation Axioms". In K.P. Hart; J. Nagata; J.E. Vaughan (eds.). <i>Encyclopedia of General Topology</i>. Amsterdam: <a href="/wiki/Elsevier_Science" class="mw-redirect" title="Elsevier Science">Elsevier Science</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-444-50355-8" title="Special:BookSources/978-0-444-50355-8"><bdi>978-0-444-50355-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Higher+Separation+Axioms&rft.btitle=Encyclopedia+of+General+Topology&rft.place=Amsterdam&rft.pub=Elsevier+Science&rft.date=2004&rft.isbn=978-0-444-50355-8&rft.aulast=Kemoto&rft.aufirst=Nobuyuki&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANormal+space" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMunkres2000" class="citation book cs1"><a href="/wiki/James_Munkres" title="James Munkres">Munkres, James R.</a> (2000). <i>Topology</i> (2nd ed.). <a href="/wiki/Prentice-Hall" class="mw-redirect" title="Prentice-Hall">Prentice-Hall</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-13-181629-9" title="Special:BookSources/978-0-13-181629-9"><bdi>978-0-13-181629-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Topology&rft.edition=2nd&rft.pub=Prentice-Hall&rft.date=2000&rft.isbn=978-0-13-181629-9&rft.aulast=Munkres&rft.aufirst=James+R.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANormal+space" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSorgenfrey1947" class="citation journal cs1">Sorgenfrey, R.H. (1947). <a rel="nofollow" class="external text" href="https://doi.org/10.1090%2FS0002-9904-1947-08858-3">"On the topological product of paracompact spaces"</a>. <i>Bull. Amer. Math. Soc</i>. <b>53</b> (6): 631–632. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1090%2FS0002-9904-1947-08858-3">10.1090/S0002-9904-1947-08858-3</a></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Bull.+Amer.+Math.+Soc.&rft.atitle=On+the+topological+product+of+paracompact+spaces&rft.volume=53&rft.issue=6&rft.pages=631-632&rft.date=1947&rft_id=info%3Adoi%2F10.1090%2FS0002-9904-1947-08858-3&rft.aulast=Sorgenfrey&rft.aufirst=R.H.&rft_id=https%3A%2F%2Fdoi.org%2F10.1090%252FS0002-9904-1947-08858-3&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANormal+space" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFStone1948" class="citation journal cs1">Stone, A. H. (1948). <a rel="nofollow" class="external text" href="https://doi.org/10.1090%2FS0002-9904-1948-09118-2">"Paracompactness and product spaces"</a>. <i>Bull. Amer. Math. Soc</i>. <b>54</b> (10): 977–982. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1090%2FS0002-9904-1948-09118-2">10.1090/S0002-9904-1948-09118-2</a></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Bull.+Amer.+Math.+Soc.&rft.atitle=Paracompactness+and+product+spaces&rft.volume=54&rft.issue=10&rft.pages=977-982&rft.date=1948&rft_id=info%3Adoi%2F10.1090%2FS0002-9904-1948-09118-2&rft.aulast=Stone&rft.aufirst=A.+H.&rft_id=https%3A%2F%2Fdoi.org%2F10.1090%252FS0002-9904-1948-09118-2&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANormal+space" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWillard1970" class="citation book cs1">Willard, Stephen (1970). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/generaltopology00will_0"><i>General Topology</i></a></span>. Reading, MA: Addison-Wesley. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-486-43479-7" title="Special:BookSources/978-0-486-43479-7"><bdi>978-0-486-43479-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=General+Topology&rft.place=Reading%2C+MA&rft.pub=Addison-Wesley&rft.date=1970&rft.isbn=978-0-486-43479-7&rft.aulast=Willard&rft.aufirst=Stephen&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fgeneraltopology00will_0&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANormal+space" class="Z3988"></span></li></ul>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" href="https://en.wikipedia.org/w/index.php?title=Normal_space&oldid=1247621055">https://en.wikipedia.org/w/index.php?title=Normal_space&oldid=1247621055</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:Properties_of_topological_spaces" title="Category:Properties of topological spaces">Properties of topological spaces</a></li><li><a href="/wiki/Category:Separation_axioms" title="Category:Separation axioms">Separation axioms</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></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 25 September 2024, at 03:32<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=Normal_space&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-f69cdc8f6-7jdc6","wgBackendResponseTime":153,"wgPageParseReport":{"limitreport":{"cputime":"0.605","walltime":"0.850","ppvisitednodes":{"value":1638,"limit":1000000},"postexpandincludesize":{"value":28445,"limit":2097152},"templateargumentsize":{"value":1783,"limit":2097152},"expansiondepth":{"value":14,"limit":100},"expensivefunctioncount":{"value":2,"limit":500},"unstrip-depth":{"value":1,"limit":20},"unstrip-size":{"value":28243,"limit":5000000},"entityaccesscount":{"value":0,"limit":400},"timingprofile":["100.00% 717.297 1 -total"," 24.75% 177.497 2 Template:Annotated_link"," 18.18% 130.419 1 Template:Short_description"," 17.43% 125.036 1 Template:Separation_axioms"," 17.43% 125.015 1 Template:Reflist"," 16.72% 119.922 1 Template:Infobox"," 13.94% 99.972 2 Template:Cite_web"," 9.57% 68.649 6 Template:Main_other"," 9.09% 65.204 1 Template:SDcat"," 8.72% 62.582 1 Template:Hlist"]},"scribunto":{"limitreport-timeusage":{"value":"0.402","limit":"10.000"},"limitreport-memusage":{"value":20606510,"limit":52428800},"limitreport-logs":"anchor_id_list = table#1 {\n [\"CITEREFKemoto2004\"] = 1,\n [\"CITEREFMunkres2000\"] = 1,\n [\"CITEREFSorgenfrey1947\"] = 1,\n [\"CITEREFStone1948\"] = 1,\n [\"CITEREFWillard1970\"] = 1,\n}\ntemplate_list = table#1 {\n [\"Annotated link\"] = 2,\n [\"Cite book\"] = 2,\n [\"Cite encyclopedia\"] = 1,\n [\"Cite journal\"] = 2,\n [\"Cite web\"] = 2,\n [\"For\"] = 1,\n [\"Harvnb\"] = 1,\n [\"ISBN\"] = 1,\n [\"Reflist\"] = 1,\n [\"Separation axioms\"] = 1,\n [\"Sfn\"] = 2,\n [\"Short description\"] = 1,\n [\"Visible anchor\"] = 1,\n}\narticle_whitelist = table#1 {\n}\n"},"cachereport":{"origin":"mw-web.codfw.main-f69cdc8f6-9j6r5","timestamp":"20241122145936","ttl":2592000,"transientcontent":false}}});});</script> <script type="application/ld+json">{"@context":"https:\/\/schema.org","@type":"Article","name":"Normal space","url":"https:\/\/en.wikipedia.org\/wiki\/Normal_space","sameAs":"http:\/\/www.wikidata.org\/entity\/Q1071795","mainEntity":"http:\/\/www.wikidata.org\/entity\/Q1071795","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":"2002-04-10T21:11:39Z","dateModified":"2024-09-25T03:32:56Z","headline":"topological space in which every pair of disjoint closed sets has disjoint open neighborhoods"}</script> </body> </html>