CINXE.COM

<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.1 plus MathML 2.0 plus SVG 1.1//EN" "http://www.w3.org/2002/04/xhtml-math-svg/xhtml-math-svg-flat.dtd" > <html xmlns="http://www.w3.org/1999/xhtml"> <head> <title> topological manifold in nLab </title> <meta http-equiv="Content-Type" content="text/html; charset=UTF-8" /> <meta name="robots" content="index,follow" /> <meta name="viewport" content="width=device-width, initial-scale=1" /> <link href="/stylesheets/instiki.css?1676280126" media="all" rel="stylesheet" type="text/css" /> <link href="/stylesheets/mathematics.css?1660229990" media="all" rel="stylesheet" type="text/css" /> <link href="/stylesheets/syntax.css?1660229990" media="all" rel="stylesheet" type="text/css" /> <link href="/stylesheets/nlab.css?1676280126" media="all" rel="stylesheet" type="text/css" /> <link rel="stylesheet" type="text/css" href="https://cdn.jsdelivr.net/gh/dreampulse/computer-modern-web-font@master/fonts.css"/> <style type="text/css"> h1#pageName, div.info, .newWikiWord a, a.existingWikiWord, .newWikiWord a:hover, [actiontype="toggle"]:hover, #TextileHelp h3 { color: #226622; } a:visited.existingWikiWord { color: #164416; } </style> <style type="text/css"><![CDATA[/*></style> <script src="/javascripts/prototype.js?1660229990" type="text/javascript"></script> <script src="/javascripts/effects.js?1660229990" type="text/javascript"></script> <script src="/javascripts/dragdrop.js?1660229990" type="text/javascript"></script> <script src="/javascripts/controls.js?1660229990" type="text/javascript"></script> <script src="/javascripts/application.js?1660229990" type="text/javascript"></script> <script src="/javascripts/page_helper.js?1660229990" type="text/javascript"></script> <script src="/javascripts/thm_numbering.js?1660229990" type="text/javascript"></script> <script type="text/x-mathjax-config"> <![CDATA[//><!]]> </script> <script type="text/javascript"> <![CDATA[//><!]]> </script> <link href="https://ncatlab.org/nlab/atom_with_headlines" rel="alternate" title="Atom with headlines" type="application/atom+xml" /> <link href="https://ncatlab.org/nlab/atom_with_content" rel="alternate" title="Atom with full content" type="application/atom+xml" /> <script type="text/javascript"> document.observe("dom:loaded", function() { generateThmNumbers(); }); </script> </head> <body> <div id="Container"> <div id="Content"> <h1 id="pageName"> <span style="float: left; margin: 0.5em 0.25em -0.25em 0"> <svg xmlns="http://www.w3.org/2000/svg" width="1.872em" height="1.8em" viewBox="0 0 190 181"> <path fill="#226622" d="M72.8 145c-1.6 17.3-15.7 10-23.6 20.2-5.6 7.3 4.8 15 11.4 15 11.5-.2 19-13.4 26.4-20.3 3.3-3 8.2-4 11.2-7.2a14 14 0 0 0 2.9-11.1c-1.4-9.6-12.4-18.6-16.9-27.2-5-9.6-10.7-27.4-24.1-27.7-17.4-.3-.4 26 4.7 30.7 2.4 2.3 5.4 4.1 7.3 6.9 1.6 2.3 2.1 5.8-1 7.2-5.9 2.6-12.4-6.3-15.5-10-8.8-10.6-15.5-23-26.2-31.8-5.2-4.3-11.8-8-18-3.7-7.3 4.9-4.2 12.9.2 18.5a81 81 0 0 0 30.7 23c3.3 1.5 12.8 5.6 10 10.7-2.5 5.2-11.7 3-15.6 1.1-8.4-3.8-24.3-21.3-34.4-13.7-3.5 2.6-2.3 7.6-1.2 11.1 2.8 9 12.2 17.2 20.9 20.5 17.3 6.7 34.3-8 50.8-12.1z"/> <path fill="#a41e32" d="M145.9 121.3c-.2-7.5 0-19.6-4.5-26-5.4-7.5-12.9-1-14.1 5.8-1.4 7.8 2.7 14.1 4.8 21.3 3.4 12 5.8 29-.8 40.1-3.6-6.7-5.2-13-7-20.4-2.1-8.2-12.8-13.2-15.1-1.9-2 9.7 9 21.2 12 30.1 1.2 4 2 8.8 6.4 10.3 6.9 2.3 13.3-4.7 17.7-8.8 12.2-11.5 36.6-20.7 43.4-36.4 6.7-15.7-13.7-14-21.3-7.2-9.1 8-11.9 20.5-23.6 25.1 7.5-23.7 31.8-37.6 38.4-61.4 2-7.3-.8-29.6-13-19.8-14.5 11.6-6.6 37.6-23.3 49.2z"/> <path fill="#193c78" d="M86.3 47.5c0-13-10.2-27.6-5.8-40.4 2.8-8.4 14.1-10.1 17-1 3.8 11.6-.3 26.3-1.8 38 11.7-.7 10.5-16 14.8-24.3 2.1-4.2 5.7-9.1 11-6.7 6 2.7 7.4 9.2 6.6 15.1-2.2 14-12.2 18.8-22.4 27-3.4 2.7-8 6.6-5.9 11.6 2 4.4 7 4.5 10.7 2.8 7.4-3.3 13.4-16.5 21.7-16 14.6.7 12 21.9.9 26.2-5 1.9-10.2 2.3-15.2 3.9-5.8 1.8-9.4 8.7-15.7 8.9-6.1.1-9-6.9-14.3-9-14.4-6-33.3-2-44.7-14.7-3.7-4.2-9.6-12-4.9-17.4 9.3-10.7 28 7.2 35.7 12 2 1.1 11 6.9 11.4 1.1.4-5.2-10-8.2-13.5-10-11.1-5.2-30-15.3-35-27.3-2.5-6 2.8-13.8 9.4-13.6 6.9.2 13.4 7 17.5 12C70.9 34 75 43.8 86.3 47.4z"/> </svg> </span> <span class="webName">nLab</span> topological manifold </h1> <div class="navigation"> <span class="skipNav"><a href='#navEnd'>Skip the Navigation Links</a> | </span> <span style="display:inline-block; width: 0.3em;"></span> <a href="/nlab/show/HomePage" accesskey="H" title="Home page">Home Page</a> | <a href="/nlab/all_pages" accesskey="A" title="List of all pages">All Pages</a> | <a href="/nlab/latest_revisions" accesskey="U" title="Latest edits and page creations">Latest Revisions</a> | <a href="https://nforum.ncatlab.org/discussion/7772/#Item_6" title="Discuss this page in its dedicated thread on the nForum" style="color: black">Discuss this page</a> | <form accept-charset="utf-8" action="/nlab/search" id="navigationSearchForm" method="get"> <fieldset class="search"><input type="text" id="searchField" name="query" value="Search" style="display:inline-block; float: left;" onfocus="this.value == 'Search' ? this.value = '' : true" onblur="this.value == '' ? this.value = 'Search' : true" /></fieldset> </form> <span id='navEnd'></span> </div> <div id="revision"> <html xmlns="http://www.w3.org/1999/xhtml" xmlns:svg="http://www.w3.org/2000/svg" xml:lang="en" lang="en"> <head><meta http-equiv="Content-type" content="application/xhtml+xml;charset=utf-8" /><title>Contents</title></head> <body> <div class="rightHandSide"> <div class="toc clickDown" tabindex="0"> <h3 id="context">Context</h3> <h4 id="topology">Topology</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/topology">topology</a></strong> (<a class="existingWikiWord" href="/nlab/show/point-set+topology">point-set topology</a>, <a class="existingWikiWord" href="/nlab/show/point-free+topology">point-free topology</a>)</p> <p>see also <em><a class="existingWikiWord" href="/nlab/show/differential+topology">differential topology</a></em>, <em><a class="existingWikiWord" href="/nlab/show/algebraic+topology">algebraic topology</a></em>, <em><a class="existingWikiWord" href="/nlab/show/functional+analysis">functional analysis</a></em> and <em><a class="existingWikiWord" href="/nlab/show/topological+homotopy+theory">topological</a> <a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopy theory</a></em></p> <p><a class="existingWikiWord" href="/nlab/show/Introduction+to+Topology">Introduction</a></p> <p><strong>Basic concepts</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/open+subset">open subset</a>, <a class="existingWikiWord" href="/nlab/show/closed+subset">closed subset</a>, <a class="existingWikiWord" href="/nlab/show/neighbourhood">neighbourhood</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a>, <a class="existingWikiWord" href="/nlab/show/locale">locale</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/base+for+the+topology">base for the topology</a>, <a class="existingWikiWord" href="/nlab/show/neighbourhood+base">neighbourhood base</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/finer+topology">finer/coarser topology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+closure">closure</a>, <a class="existingWikiWord" href="/nlab/show/topological+interior">interior</a>, <a class="existingWikiWord" href="/nlab/show/topological+boundary">boundary</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/separation+axiom">separation</a>, <a class="existingWikiWord" href="/nlab/show/sober+topological+space">sobriety</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/continuous+function">continuous function</a>, <a class="existingWikiWord" href="/nlab/show/homeomorphism">homeomorphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/uniformly+continuous+function">uniformly continuous function</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+embedding">embedding</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/open+map">open map</a>, <a class="existingWikiWord" href="/nlab/show/closed+map">closed map</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/sequence">sequence</a>, <a class="existingWikiWord" href="/nlab/show/net">net</a>, <a class="existingWikiWord" href="/nlab/show/sub-net">sub-net</a>, <a class="existingWikiWord" href="/nlab/show/filter">filter</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/convergence">convergence</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/category">category</a><a class="existingWikiWord" href="/nlab/show/Top">Top</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/convenient+category+of+topological+spaces">convenient category of topological spaces</a></li> </ul> </li> </ul> <p><strong><a href="Top#UniversalConstructions">Universal constructions</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/initial+topology">initial topology</a>, <a class="existingWikiWord" href="/nlab/show/final+topology">final topology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/subspace">subspace</a>, <a class="existingWikiWord" href="/nlab/show/quotient+space">quotient space</a>,</p> </li> <li> <p>fiber space, <a class="existingWikiWord" href="/nlab/show/space+attachment">space attachment</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/product+space">product space</a>, <a class="existingWikiWord" href="/nlab/show/disjoint+union+space">disjoint union space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/mapping+cylinder">mapping cylinder</a>, <a class="existingWikiWord" href="/nlab/show/mapping+cocylinder">mapping cocylinder</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/mapping+cone">mapping cone</a>, <a class="existingWikiWord" href="/nlab/show/mapping+cocone">mapping cocone</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/mapping+telescope">mapping telescope</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/colimits+of+normal+spaces">colimits of normal spaces</a></p> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/stuff%2C+structure%2C+property">Extra stuff, structure, properties</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/nice+topological+space">nice topological space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/metric+space">metric space</a>, <a class="existingWikiWord" href="/nlab/show/metric+topology">metric topology</a>, <a class="existingWikiWord" href="/nlab/show/metrisable+space">metrisable space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Kolmogorov+space">Kolmogorov space</a>, <a class="existingWikiWord" href="/nlab/show/Hausdorff+space">Hausdorff space</a>, <a class="existingWikiWord" href="/nlab/show/regular+space">regular space</a>, <a class="existingWikiWord" href="/nlab/show/normal+space">normal space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/sober+space">sober space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/compact+space">compact space</a>, <a class="existingWikiWord" href="/nlab/show/proper+map">proper map</a></p> <p><a class="existingWikiWord" href="/nlab/show/sequentially+compact+topological+space">sequentially compact</a>, <a class="existingWikiWord" href="/nlab/show/countably+compact+topological+space">countably compact</a>, <a class="existingWikiWord" href="/nlab/show/locally+compact+topological+space">locally compact</a>, <a class="existingWikiWord" href="/nlab/show/sigma-compact+topological+space">sigma-compact</a>, <a class="existingWikiWord" href="/nlab/show/paracompact+space">paracompact</a>, <a class="existingWikiWord" href="/nlab/show/countably+paracompact+topological+space">countably paracompact</a>, <a class="existingWikiWord" href="/nlab/show/strongly+compact+topological+space">strongly compact</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/compactly+generated+space">compactly generated space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/second-countable+space">second-countable space</a>, <a class="existingWikiWord" href="/nlab/show/first-countable+space">first-countable space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/contractible+space">contractible space</a>, <a class="existingWikiWord" href="/nlab/show/locally+contractible+space">locally contractible space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/connected+space">connected space</a>, <a class="existingWikiWord" href="/nlab/show/locally+connected+space">locally connected space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/simply-connected+space">simply-connected space</a>, <a class="existingWikiWord" href="/nlab/show/locally+simply-connected+space">locally simply-connected space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cell+complex">cell complex</a>, <a class="existingWikiWord" href="/nlab/show/CW-complex">CW-complex</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/pointed+topological+space">pointed space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+vector+space">topological vector space</a>, <a class="existingWikiWord" href="/nlab/show/Banach+space">Banach space</a>, <a class="existingWikiWord" href="/nlab/show/Hilbert+space">Hilbert space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+group">topological group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+vector+bundle">topological vector bundle</a>, <a class="existingWikiWord" href="/nlab/show/topological+K-theory">topological K-theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+manifold">topological manifold</a></p> </li> </ul> <p><strong>Examples</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/empty+space">empty space</a>, <a class="existingWikiWord" href="/nlab/show/point+space">point space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/discrete+space">discrete space</a>, <a class="existingWikiWord" href="/nlab/show/codiscrete+space">codiscrete space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Sierpinski+space">Sierpinski space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/order+topology">order topology</a>, <a class="existingWikiWord" href="/nlab/show/specialization+topology">specialization topology</a>, <a class="existingWikiWord" href="/nlab/show/Scott+topology">Scott topology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Euclidean+space">Euclidean space</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/real+line">real line</a>, <a class="existingWikiWord" href="/nlab/show/plane">plane</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cylinder">cylinder</a>, <a class="existingWikiWord" href="/nlab/show/cone">cone</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/sphere">sphere</a>, <a class="existingWikiWord" href="/nlab/show/ball">ball</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/circle">circle</a>, <a class="existingWikiWord" href="/nlab/show/torus">torus</a>, <a class="existingWikiWord" href="/nlab/show/annulus">annulus</a>, <a class="existingWikiWord" href="/nlab/show/Moebius+strip">Moebius strip</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/polytope">polytope</a>, <a class="existingWikiWord" href="/nlab/show/polyhedron">polyhedron</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/projective+space">projective space</a> (<a class="existingWikiWord" href="/nlab/show/real+projective+space">real</a>, <a class="existingWikiWord" href="/nlab/show/complex+projective+space">complex</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/classifying+space">classifying space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/configuration+space+%28mathematics%29">configuration space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/path">path</a>, <a class="existingWikiWord" href="/nlab/show/loop">loop</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/mapping+spaces">mapping spaces</a>: <a class="existingWikiWord" href="/nlab/show/compact-open+topology">compact-open topology</a>, <a class="existingWikiWord" href="/nlab/show/topology+of+uniform+convergence">topology of uniform convergence</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/loop+space">loop space</a>, <a class="existingWikiWord" href="/nlab/show/path+space">path space</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Zariski+topology">Zariski topology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Cantor+space">Cantor space</a>, <a class="existingWikiWord" href="/nlab/show/Mandelbrot+space">Mandelbrot space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Peano+curve">Peano curve</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/line+with+two+origins">line with two origins</a>, <a class="existingWikiWord" href="/nlab/show/long+line">long line</a>, <a class="existingWikiWord" href="/nlab/show/Sorgenfrey+line">Sorgenfrey line</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/K-topology">K-topology</a>, <a class="existingWikiWord" href="/nlab/show/Dowker+space">Dowker space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Warsaw+circle">Warsaw circle</a>, <a class="existingWikiWord" href="/nlab/show/Hawaiian+earring+space">Hawaiian earring space</a></p> </li> </ul> <p><strong>Basic statements</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Hausdorff+spaces+are+sober">Hausdorff spaces are sober</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/schemes+are+sober">schemes are sober</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/continuous+images+of+compact+spaces+are+compact">continuous images of compact spaces are compact</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/closed+subspaces+of+compact+Hausdorff+spaces+are+equivalently+compact+subspaces">closed subspaces of compact Hausdorff spaces are equivalently compact subspaces</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/open+subspaces+of+compact+Hausdorff+spaces+are+locally+compact">open subspaces of compact Hausdorff spaces are locally compact</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quotient+projections+out+of+compact+Hausdorff+spaces+are+closed+precisely+if+the+codomain+is+Hausdorff">quotient projections out of compact Hausdorff spaces are closed precisely if the codomain is Hausdorff</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/compact+spaces+equivalently+have+converging+subnet+of+every+net">compact spaces equivalently have converging subnet of every net</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Lebesgue+number+lemma">Lebesgue number lemma</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/sequentially+compact+metric+spaces+are+equivalently+compact+metric+spaces">sequentially compact metric spaces are equivalently compact metric spaces</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/compact+spaces+equivalently+have+converging+subnet+of+every+net">compact spaces equivalently have converging subnet of every net</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/sequentially+compact+metric+spaces+are+totally+bounded">sequentially compact metric spaces are totally bounded</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/continuous+metric+space+valued+function+on+compact+metric+space+is+uniformly+continuous">continuous metric space valued function on compact metric space is uniformly continuous</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/paracompact+Hausdorff+spaces+are+normal">paracompact Hausdorff spaces are normal</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/paracompact+Hausdorff+spaces+equivalently+admit+subordinate+partitions+of+unity">paracompact Hausdorff spaces equivalently admit subordinate partitions of unity</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/closed+injections+are+embeddings">closed injections are embeddings</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/proper+maps+to+locally+compact+spaces+are+closed">proper maps to locally compact spaces are closed</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/injective+proper+maps+to+locally+compact+spaces+are+equivalently+the+closed+embeddings">injective proper maps to locally compact spaces are equivalently the closed embeddings</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/locally+compact+and+sigma-compact+spaces+are+paracompact">locally compact and sigma-compact spaces are paracompact</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/locally+compact+and+second-countable+spaces+are+sigma-compact">locally compact and second-countable spaces are sigma-compact</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/second-countable+regular+spaces+are+paracompact">second-countable regular spaces are paracompact</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/CW-complexes+are+paracompact+Hausdorff+spaces">CW-complexes are paracompact Hausdorff spaces</a></p> </li> </ul> <p><strong>Theorems</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Urysohn%27s+lemma">Urysohn's lemma</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Tietze+extension+theorem">Tietze extension theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Tychonoff+theorem">Tychonoff theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/tube+lemma">tube lemma</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Michael%27s+theorem">Michael's theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Brouwer%27s+fixed+point+theorem">Brouwer's fixed point theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+invariance+of+dimension">topological invariance of dimension</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Jordan+curve+theorem">Jordan curve theorem</a></p> </li> </ul> <p><strong>Analysis Theorems</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Heine-Borel+theorem">Heine-Borel theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/intermediate+value+theorem">intermediate value theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/extreme+value+theorem">extreme value theorem</a></p> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/topological+homotopy+theory">topological homotopy theory</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/left+homotopy">left homotopy</a>, <a class="existingWikiWord" href="/nlab/show/right+homotopy">right homotopy</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+equivalence">homotopy equivalence</a>, <a class="existingWikiWord" href="/nlab/show/deformation+retract">deformation retract</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+group">fundamental group</a>, <a class="existingWikiWord" href="/nlab/show/covering+space">covering space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+theorem+of+covering+spaces">fundamental theorem of covering spaces</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+group">homotopy group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/weak+homotopy+equivalence">weak homotopy equivalence</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Whitehead%27s+theorem">Whitehead's theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Freudenthal+suspension+theorem">Freudenthal suspension theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/nerve+theorem">nerve theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+extension+property">homotopy extension property</a>, <a class="existingWikiWord" href="/nlab/show/Hurewicz+cofibration">Hurewicz cofibration</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+cofiber+sequence">cofiber sequence</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Str%C3%B8m+model+category">Strøm model category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/classical+model+structure+on+topological+spaces">classical model structure on topological spaces</a></p> </li> </ul> </div></div> <h4 id="manifolds_and_cobordisms">Manifolds and cobordisms</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/manifolds">manifolds</a></strong> and <strong><a class="existingWikiWord" href="/nlab/show/cobordisms">cobordisms</a></strong></p> <p><a class="existingWikiWord" href="/nlab/show/cobordism+theory">cobordism theory</a>, <em><a class="existingWikiWord" href="/nlab/show/Introduction+to+Cobordism+and+Complex+Oriented+Cohomology">Introduction</a></em></p> <p><strong>Definitions</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/locally+Euclidean+space">locally Euclidean space</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/coordinate+chart">coordinate chart</a>, <a class="existingWikiWord" href="/nlab/show/coordinate+transformation">coordinate transformation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/atlas">atlas</a>,</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/smooth+structure">smooth structure</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/manifold">manifold</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+manifold">topological manifold</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/differentiable+manifold">differentiable manifold</a>, ,<a class="existingWikiWord" href="/nlab/show/smooth+manifold">smooth manifold</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/infinite+dimensional+manifold">infinite dimensional manifold</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Banach+manifold">Banach manifold</a>, <a class="existingWikiWord" href="/nlab/show/Hilbert+manifold">Hilbert manifold</a>, <a class="existingWikiWord" href="/nlab/show/ILH+manifold">ILH manifold</a>, <a class="existingWikiWord" href="/nlab/show/Frechet+manifold">Frechet manifold</a>, <a class="existingWikiWord" href="/nlab/show/convenient+manifold">convenient manifold</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/tangent+bundle">tangent bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/normal+bundle">normal bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/G-structure">G-structure</a>, <a class="existingWikiWord" href="/nlab/show/torsion+of+a+G-structure">torsion of a G-structure</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/orientation">orientation</a>, <a class="existingWikiWord" href="/nlab/show/spin+structure">spin structure</a>, <a class="existingWikiWord" href="/nlab/show/string+structure">string structure</a>, <a class="existingWikiWord" href="/nlab/show/fivebrane+structure">fivebrane structure</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Cartan+geometry">Cartan geometry</a>:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Riemannian+manifold">Riemannian manifold</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/complex+manifold">complex manifold</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/symplectic+manifold">symplectic manifold</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cobordism">cobordism</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/B-bordism">B-bordism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/extended+cobordism">extended cobordism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cobordism+category">cobordism category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2Cn%29-category+of+cobordisms">(∞,n)-category of cobordisms</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/FQFT">functorial quantum field theory</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Thom+spectrum">Thom spectrum</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/cobordism+ring">cobordism ring</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/genus">genus</a></p> </li> </ul> </li> </ul> </li> </ul> <p><strong>Genera and invariants</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/signature+genus">signature genus</a>, <a class="existingWikiWord" href="/nlab/show/Kervaire+invariant">Kervaire invariant</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/A-hat+genus">A-hat genus</a>, <a class="existingWikiWord" href="/nlab/show/Witten+genus">Witten genus</a></p> </li> </ul> <p><strong>Classification</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/2-manifolds">2-manifolds</a>/<a class="existingWikiWord" href="/nlab/show/surfaces">surfaces</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/genus+of+a+surface">genus of a surface</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/3-manifolds">3-manifolds</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Kirby+calculus">Kirby calculus</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/4-manifolds">4-manifolds</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Dehn+surgery">Dehn surgery</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/exotic+smooth+structure">exotic smooth structure</a></p> </li> </ul> </li> </ul> <p><strong>Theorems</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Whitney+embedding+theorem">Whitney embedding theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Thom%27s+transversality+theorem">Thom's transversality theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Pontrjagin-Thom+construction">Pontrjagin-Thom construction</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Galatius-Tillmann-Madsen-Weiss+theorem">Galatius-Tillmann-Madsen-Weiss theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/geometrization+conjecture">geometrization conjecture</a>,</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Poincar%C3%A9+conjecture">Poincaré conjecture</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/elliptization+conjecture">elliptization conjecture</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cobordism+hypothesis">cobordism hypothesis</a>-theorem</p> </li> </ul> </div></div> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#definition'>Definition</a></li> <ul> <li><a href='#LocallyEuclideanTopologicalSpace'>Locally Euclidean topological spaces</a></li> <li><a href='#topological_manifold'>Topological manifold</a></li> <li><a href='#differentiable_manifolds'>Differentiable manifolds</a></li> </ul> <li><a href='#properties'>Properties</a></li> <li><a href='#examples'>Examples</a></li> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> </ul> </div> <h2 id="idea">Idea</h2> <p>A <em>topological manifold</em> is a <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a> (usually required to be <a class="existingWikiWord" href="/nlab/show/Hausdorff+topological+space">Hausdorff</a> and <a class="existingWikiWord" href="/nlab/show/paracompact+topological+space">paracompact</a>) which is <em>locally</em> <a class="existingWikiWord" href="/nlab/show/homeomorphism">homeomorphic</a> to a <a class="existingWikiWord" href="/nlab/show/Euclidean+space">Euclidean space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^n</annotation></semantics></math> equipped with its <a class="existingWikiWord" href="/nlab/show/metric+topology">metric topology</a>.</p> <p>Often one is interested in extra structure on topological manifolds, that make them for instance into <a class="existingWikiWord" href="/nlab/show/differentiable+manifolds">differentiable manifolds</a> or <a class="existingWikiWord" href="/nlab/show/smooth+manifolds">smooth manifolds</a> or <a class="existingWikiWord" href="/nlab/show/analytic+manifolds">analytic manifolds</a> or <a class="existingWikiWord" href="/nlab/show/complex+manifolds">complex manifolds</a>, etc. See at <em><a class="existingWikiWord" href="/nlab/show/manifold">manifold</a></em> for more on the general concept.</p> <p>Topological manifolds form a category <a class="existingWikiWord" href="/nlab/show/TopMfd">TopMfd</a>.</p> <h2 id="definition">Definition</h2> <h3 id="LocallyEuclideanTopologicalSpace">Locally Euclidean topological spaces</h3> <div class="num_defn" id="LocallyEuclideanSpace"> <h6 id="definition_2">Definition</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/locally+Euclidean+topological+space">locally Euclidean topological space</a>)</strong></p> <p>A <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> is <em><a class="existingWikiWord" href="/nlab/show/locally+Euclidean+topological+space">locally Euclidean</a></em> if every point <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>∈</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">x \in X</annotation></semantics></math> has an <a class="existingWikiWord" href="/nlab/show/open+neighbourhood">open neighbourhood</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>U</mi> <mi>x</mi></msub><mo>⊃</mo><mo stretchy="false">{</mo><mi>x</mi><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">U_x \supset \{x\}</annotation></semantics></math> which is <a class="existingWikiWord" href="/nlab/show/homeomorphism">homeomorphic</a> to the <a class="existingWikiWord" href="/nlab/show/Euclidean+space">Euclidean space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^n</annotation></semantics></math> with its <a class="existingWikiWord" href="/nlab/show/metric+topology">metric topology</a>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mi>n</mi></msup><mover><mo>⟶</mo><mrow><mphantom><mi>AA</mi></mphantom><mo>≃</mo><mphantom><mi>AA</mi></mphantom></mrow></mover><msub><mi>U</mi> <mi>x</mi></msub><mo>⊂</mo><mi>X</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \mathbb{R}^n \overset{\phantom{AA} \simeq \phantom{AA}}{\longrightarrow} U_x \subset X \,. </annotation></semantics></math></div></div> <p>The “local” <a class="existingWikiWord" href="/nlab/show/topological+properties">topological properties</a> of Euclidean space are inherited by locally Euclidean spaces:</p> <div class="num_prop" id="LocalPropertiesOfLocallyEuclideanSpace"> <h6 id="proposition">Proposition</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/locally+Euclidean+spaces">locally Euclidean spaces</a> are <a class="existingWikiWord" href="/nlab/show/T1"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"> <semantics> <mrow> <msub><mi>T</mi> <mn>1</mn></msub> </mrow> <annotation encoding="application/x-tex">T_1</annotation> </semantics> </math></a>-<a class="existingWikiWord" href="/nlab/show/separation+axiom">separated</a>, <a class="existingWikiWord" href="/nlab/show/sober+topological+space">sober</a>, <a class="existingWikiWord" href="/nlab/show/locally+connected+topological+space">locally connected</a>, <a class="existingWikiWord" href="/nlab/show/locally+compact+topological+space">locally compact</a>)</strong></p> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/locally+Euclidean+space">locally Euclidean space</a> (def. <a class="maruku-ref" href="#LocallyEuclideanSpace"></a>). Then</p> <ol> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> satisfies the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>T</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">T_1</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/separation+axiom">separation axiom</a>;</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/sober+topological+space">sober</a>;</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/locally+connected+topological+space">locally connected</a>;</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/locally+compact+topological+space">locally compact</a> in the sense that every open neighbourhood of a point contains a <a class="existingWikiWord" href="/nlab/show/compact+topological+space">compact</a> neighbourhood.</p> </li> </ol> </div> <div class="proof"> <h6 id="proof">Proof</h6> <p>Regarding the first statement:</p> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>≠</mo><mi>y</mi></mrow><annotation encoding="application/x-tex">x \neq y</annotation></semantics></math> be two distinct points in the locally Euclidean space. We need to show that there is an open neighbourhood <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>U</mi> <mi>x</mi></msub></mrow><annotation encoding="application/x-tex">U_x</annotation></semantics></math> around <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math> that does not contain <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>y</mi></mrow><annotation encoding="application/x-tex">y</annotation></semantics></math>.</p> <p>By definition, there is a Euclidean open neighbourhood <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mi>n</mi></msup><munderover><mo>→</mo><mo>≃</mo><mi>ϕ</mi></munderover><msub><mi>U</mi> <mi>x</mi></msub><mo>⊂</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">\mathbb{R}^n \underoverset{\simeq}{\phi}{\to} U_x \subset X</annotation></semantics></math> around <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math>. If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>U</mi> <mi>x</mi></msub></mrow><annotation encoding="application/x-tex">U_x</annotation></semantics></math> does not contain <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>y</mi></mrow><annotation encoding="application/x-tex">y</annotation></semantics></math>, then it already is an open neighbourhood as required. If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>U</mi> <mi>x</mi></msub></mrow><annotation encoding="application/x-tex">U_x</annotation></semantics></math> does contain <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>y</mi></mrow><annotation encoding="application/x-tex">y</annotation></semantics></math>, then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ϕ</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>≠</mo><msup><mi>ϕ</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\phi^{-1}(x) \neq \phi^{-1}(y)</annotation></semantics></math> are equivalently two distinct points in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^n</annotation></semantics></math>. But Euclidean space, as every <a class="existingWikiWord" href="/nlab/show/metric+space">metric space</a>, is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>T</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">T_1</annotation></semantics></math>, and hence we may find an open neighbourhood <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>V</mi> <mrow><msup><mi>ϕ</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></msub><mo>⊂</mo><msup><mi>ℝ</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">V_{\phi^{-1}(x)} \subset \mathbb{R}^n </annotation></semantics></math> not containing <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ϕ</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\phi^{-1}(y)</annotation></semantics></math>. By the nature of the <a class="existingWikiWord" href="/nlab/show/subspace+topology">subspace topology</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi><mo stretchy="false">(</mo><msub><mi>V</mi> <mrow><msup><mi>ϕ</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">)</mo><mo>⊂</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">\phi(V_{\phi^{-1}(x)}) \subset X</annotation></semantics></math> is an open neighbourhood as required.</p> <p>Regarding the second statement:</p> <p>We need to show that the map</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Cl</mi><mo stretchy="false">(</mo><mo stretchy="false">{</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">}</mo><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>X</mi><mo>→</mo><mi>IrrClSub</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> Cl(\{-\}) \;\colon\; X \to IrrClSub(X) </annotation></semantics></math></div> <p>that sends points to the <a class="existingWikiWord" href="/nlab/show/topological+closure">topological closure</a> of their <a class="existingWikiWord" href="/nlab/show/singleton+sets">singleton sets</a> is a <a class="existingWikiWord" href="/nlab/show/bijection">bijection</a> with the set of <a class="existingWikiWord" href="/nlab/show/irreducible+closed+subsets">irreducible closed subsets</a>. By the first statement above the map is <a class="existingWikiWord" href="/nlab/show/injective+function">injective</a> (via <a href="separation+axioms#T1InTermsOfClosureOfPoints">this lemma</a>). Hence it remains to see that every irreducible closed subset is the topological closure of a singleton. We will show something stronger: every irreducible closed subset is a singleton.</p> <p>So let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi><mo>⊂</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">P \subset X</annotation></semantics></math> be an open proper subset such that if there are two open subsets <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>U</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>U</mi> <mn>2</mn></msub><mo>⊂</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">U_1, U_2 \subset X</annotation></semantics></math> with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>U</mi> <mn>1</mn></msub><mo>∩</mo><msub><mi>U</mi> <mn>2</mn></msub><mo>⊂</mo><mi>P</mi></mrow><annotation encoding="application/x-tex">U_1 \cap U_2 \subset P</annotation></semantics></math> then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>U</mi> <mn>1</mn></msub><mo>⊂</mo><mi>P</mi></mrow><annotation encoding="application/x-tex">U_1 \subset P</annotation></semantics></math> or <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>U</mi> <mn>2</mn></msub><mo>⊂</mo><mi>P</mi></mrow><annotation encoding="application/x-tex">U_2 \subset P</annotation></semantics></math>. By <a href="irreducible+closed+subspace#OpenSubsetVersionOfClosedIrreducible">this prop.</a> we need to show that there exists a point <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>∈</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">x \in X</annotation></semantics></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi><mo>=</mo><mi>X</mi><mo>∖</mo><mo stretchy="false">{</mo><mi>x</mi><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">P = X \setminus \{x\}</annotation></semantics></math> it its <a class="existingWikiWord" href="/nlab/show/complement">complement</a>.</p> <p>Now since <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi><mo>⊂</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">P \subset X</annotation></semantics></math> is a proper subset, and since the locally Euclidean space <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> is covered by Euclidean neighbourhoods, there exists a Euclidean neighbourhood <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mi>n</mi></msup><munderover><mo>→</mo><mo>≃</mo><mi>ϕ</mi></munderover><mi>U</mi><mo>⊂</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">\mathbb{R}^n \underoverset{\simeq}{\phi}{\to} U \subset X</annotation></semantics></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi><mo>∩</mo><mi>U</mi><mo>⊂</mo><mi>U</mi></mrow><annotation encoding="application/x-tex">P \cap U \subset U</annotation></semantics></math> is a proper subset. In fact this still satisfies the condition that for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>U</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>U</mi> <mn>2</mn></msub><munder><mo>⊂</mo><mtext>open</mtext></munder><mi>U</mi></mrow><annotation encoding="application/x-tex">U_1, U_2 \underset{\text{open}}{\subset} U</annotation></semantics></math> then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>U</mi> <mn>1</mn></msub><mo>∩</mo><msub><mi>U</mi> <mn>2</mn></msub><mo>⊂</mo><mi>P</mi><mo>∩</mo><mi>U</mi></mrow><annotation encoding="application/x-tex">U_1 \cap U_2 \subset P \cap U</annotation></semantics></math> implies <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>U</mi> <mn>1</mn></msub><mo>⊂</mo><mi>P</mi><mo>∩</mo><mi>U</mi></mrow><annotation encoding="application/x-tex">U_1 \subset P \cap U</annotation></semantics></math> or <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>U</mi> <mn>2</mn></msub><mo>⊂</mo><mi>P</mi><mo>∩</mo><mi>U</mi></mrow><annotation encoding="application/x-tex">U_2 \subset P \cap U</annotation></semantics></math>. Accordingly, by <a href="irreducible+closed+subspace#OpenSubsetVersionOfClosedIrreducible">that prop.</a> it follows that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mi>n</mi></msup><mo>∖</mo><msup><mi>ϕ</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>P</mi><mo>∩</mo><mi>U</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbb{R}^n \setminus \phi^{-1}(P \cap U)</annotation></semantics></math> is an irreducible closed subset of <a class="existingWikiWord" href="/nlab/show/Euclidean+space">Euclidean space</a>. Sine <a class="existingWikiWord" href="/nlab/show/metric+spaces">metric spaces</a> are <a class="existingWikiWord" href="/nlab/show/sober+topological+space">sober topological space</a> as well as <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>T</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">T_1</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/separation+axiom">separated</a>, this means that there exists <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>∈</mo><msup><mi>ℝ</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">x \in \mathbb{R}^n</annotation></semantics></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ϕ</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>P</mi><mo>∩</mo><mi>U</mi><mo stretchy="false">)</mo><mo>=</mo><msup><mi>ℝ</mi> <mi>n</mi></msup><mo>∖</mo><mo stretchy="false">{</mo><mi>x</mi><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\phi^{-1}(P \cap U) = \mathbb{R}^n \setminus \{x\}</annotation></semantics></math>.</p> <p>In conclusion this means that the restriction of an irreducible closed subset in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> to any Euclidean chart is either empty or a singleton set. This means that the irreducible closed subset must be a disjoint union of singletons that are separated by Euclidean neighbourhoods. But by irreducibiliy, this union has to consist of just one point.</p> <p>Regarding the third statement:</p> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>∈</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">x \in X</annotation></semantics></math> be a point and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>U</mi> <mi>x</mi></msub><mo>⊃</mo><mo stretchy="false">{</mo><mi>x</mi><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">U_x \supset \{x\}</annotation></semantics></math> a neighbourhood. We need to find a <a class="existingWikiWord" href="/nlab/show/connected+topological+space">connected</a> open neighbourhood <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Cn</mi> <mi>x</mi></msub><mo>⊂</mo><msub><mi>U</mi> <mi>x</mi></msub></mrow><annotation encoding="application/x-tex">Cn_x \subset U_x</annotation></semantics></math>.</p> <p>By local Euclideanness, there is also a Euclidean neighboruhood <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mi>n</mi></msup><munderover><mo>→</mo><mo>≃</mo><mi>ϕ</mi></munderover><msub><mi>V</mi> <mi>x</mi></msub><mo>⊂</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">\mathbb{R}^n \underoverset{\simeq}{\phi}{\to} V_x \subset X</annotation></semantics></math>. Since <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi></mrow><annotation encoding="application/x-tex">\phi</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/homeomorphism">homeomorphism</a>, and since <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>U</mi> <mi>x</mi></msub><mo>∩</mo><msub><mi>V</mi> <mi>x</mi></msub></mrow><annotation encoding="application/x-tex">U_x \cap V_x</annotation></semantics></math> is open, also <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ϕ</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><msub><mi>U</mi> <mi>x</mi></msub><mo>∩</mo><msub><mi>V</mi> <mi>x</mi></msub><mo stretchy="false">)</mo><mo>⊂</mo><msup><mi>ℝ</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">\phi^{-1}(U_x \cap V_x) \subset \mathbb{R}^n</annotation></semantics></math> is open. This means that there exists an <a class="existingWikiWord" href="/nlab/show/open+ball">open ball</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>B</mi> <mrow><msup><mi>ϕ</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow> <mo>∘</mo></msubsup><mo stretchy="false">(</mo><mi>ϵ</mi><mo stretchy="false">)</mo><mo>⊂</mo><msup><mi>ϕ</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><msub><mi>U</mi> <mi>x</mi></msub><mo>∩</mo><msub><mi>V</mi> <mi>x</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">B_{\phi^{-1}(x)}^\circ(\epsilon) \subset \phi^{-1}(U_x \cap V_x)</annotation></semantics></math>. This is open and connected, and hence so is its homeomorphic image <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi><mo stretchy="false">(</mo><msubsup><mi>B</mi> <mrow><msup><mi>ϕ</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow> <mo>∘</mo></msubsup><mo stretchy="false">(</mo><mi>ϵ</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>⊂</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">\phi(B^\circ_{\phi^{-1}(x)}(\epsilon)) \subset X</annotation></semantics></math>. This is a connected open neighbourhood of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math> as required.</p> <p>Regarding the fourth statement:</p> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>∈</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">x \in X</annotation></semantics></math> be a point and let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>U</mi> <mi>x</mi></msub><mo>⊃</mo><mo stretchy="false">{</mo><mi>x</mi><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">U_x \supset \{x\}</annotation></semantics></math> be an open neighbourhood. We need to find a compact neighbourhood <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>K</mi> <mi>x</mi></msub><mo>⊂</mo><msub><mi>U</mi> <mi>x</mi></msub></mrow><annotation encoding="application/x-tex">K_x \subset U_x</annotation></semantics></math>.</p> <p>By assumption there exists a Euclidean open neighbourhood <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mi>n</mi></msup><munderover><mo>→</mo><mo>≃</mo><mi>ϕ</mi></munderover><msub><mi>V</mi> <mi>x</mi></msub><mo>⊂</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">\mathbb{R}^n \underoverset{\simeq}{\phi}{\to} V_x \subset X</annotation></semantics></math>. By definition of the <a class="existingWikiWord" href="/nlab/show/subspace+topology">subspace topology</a> the intersection <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>U</mi> <mi>x</mi></msub><mo>∩</mo><msub><mi>V</mi> <mi>x</mi></msub></mrow><annotation encoding="application/x-tex">U_x \cap V_x</annotation></semantics></math> is still open as a subspace of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>V</mi> <mi>x</mi></msub></mrow><annotation encoding="application/x-tex">V_x</annotation></semantics></math> and hence <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ϕ</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><msub><mi>U</mi> <mi>x</mi></msub><mo>∩</mo><msub><mi>V</mi> <mi>x</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\phi^{-1}(U_x \cap V_x)</annotation></semantics></math> is an open neighbourhood of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ϕ</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>∈</mo><msup><mi>ℝ</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">\phi^{-1}(x) \in \mathbb{R}^n</annotation></semantics></math>.</p> <p>Since Euclidean spaces are locally compact, there exists a compact neighbourhood <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>K</mi> <mrow><msup><mi>ϕ</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></msub><mo>⊂</mo><msup><mi>ℝ</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">K_{\phi^{-1}(x)} \subset \mathbb{R}^n</annotation></semantics></math> (for instance a sufficiently small <a class="existingWikiWord" href="/nlab/show/closed+ball">closed ball</a> around <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math>, which is compact by the <a class="existingWikiWord" href="/nlab/show/Heine-Borel+theorem">Heine-Borel theorem</a>). Now since <a class="existingWikiWord" href="/nlab/show/continuous+images+of+compact+spaces+are+compact">continuous images of compact spaces are compact</a>, it follows that also <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi><mo stretchy="false">(</mo><mi>K</mi><mo stretchy="false">)</mo><mo>⊂</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">\phi(K) \subset X</annotation></semantics></math> is a compact neighbourhood.</p> </div> <p>But the “global” topological properties of Euclidean space are not generally inherited by locally Euclidean spaces. This sounds obvious, but notice that also Hausdorff-ness is a “global property”:</p> <div class="num_remark"> <h6 id="remark">Remark</h6> <p><strong>(locally Euclidean spaces are not necessarily <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>T</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">T_2</annotation></semantics></math>)</strong></p> <p>It might superficially seem that every locally Euclidean space (def. <a class="maruku-ref" href="#LocallyEuclideanSpace"></a>) is necessarily a <a class="existingWikiWord" href="/nlab/show/Hausdorff+topological+space">Hausdorff topological space</a>, since <a class="existingWikiWord" href="/nlab/show/Euclidean+space">Euclidean space</a>, like any <a class="existingWikiWord" href="/nlab/show/metric+space">metric space</a>, is Hausdorff, and since by definition the neighbourhood of every point in a locally Euclidean spaces looks like Euclidean space.</p> <p>But this is not so, Hausdorffness is a “non-local condition”.</p> </div> <p> <div class='num_remark' id='NonHausdorffManifolds'> <h6>Example</h6> <p>counter-example: <strong>(<a class="existingWikiWord" href="/nlab/show/non-Hausdorff+locally+Euclidean+spaces">non-Hausdorff locally Euclidean spaces</a>)</strong> <br /> An example of a <a class="existingWikiWord" href="/nlab/show/locally+Euclidean+space">locally Euclidean space</a> (def. <a class="maruku-ref" href="#LocallyEuclideanSpace"></a>) which is a <a class="existingWikiWord" href="/nlab/show/non-Hausdorff+topological+space">non-Hausdorff topological space</a>, is the <a class="existingWikiWord" href="/nlab/show/line+with+two+origins">line with two origins</a>.</p> </div> </p> <div class="num_lemma" id="PathConnectedFromConnectedLocallyEuclideanSpace"> <h6 id="lemma">Lemma</h6> <p><strong>(connected locally Euclidean spaces are path-connected)</strong></p> <p>A locally Euclidean space which is <a class="existingWikiWord" href="/nlab/show/connected+topological+space">connected</a> is also <a class="existingWikiWord" href="/nlab/show/path-connected+topological+space">path-connected</a>.</p> </div> <div class="proof"> <h6 id="proof_2">Proof</h6> <p>Fix any <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>∈</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">x \in X</annotation></semantics></math>. Write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>PConn</mi> <mi>x</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>⊂</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">PConn_x(X) \subset X</annotation></semantics></math> for the subset of all those points of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math> which are connected to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math> by a path, hence</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>PConn</mi> <mi>x</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mrow><mo>{</mo><mi>y</mi><mo>∈</mo><mi>X</mi><mspace width="thickmathspace"></mspace><mo stretchy="false">|</mo><mspace width="thickmathspace"></mspace><munder><mo>∃</mo><mrow><mo stretchy="false">[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy="false">]</mo><munderover><mo>→</mo><mi>cts</mi><mi>γ</mi></munderover><mi>X</mi></mrow></munder><mrow><mo>(</mo><mrow><mo>(</mo><mi>γ</mi><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo><mo>=</mo><mi>x</mi><mo>)</mo></mrow><mphantom><mi>A</mi></mphantom><mtext>and</mtext><mphantom><mi>a</mi></mphantom><mrow><mo>(</mo><mi>γ</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo>=</mo><mi>y</mi><mo>)</mo></mrow><mo>)</mo></mrow><mo>}</mo></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> PConn_x(X) \;\colon\; \left\{ y \in X \;\vert\; \underset{[0,1] \underoverset{cts}{\gamma}{\to} X }{\exists} \left( \left(\gamma(0) = x\right) \phantom{A} \text{and} \phantom{a} \left( \gamma(1) = y \right) \right) \right\} \,. </annotation></semantics></math></div> <p>Observe now that both <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>PConn</mi> <mi>x</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>⊂</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">PConn_x(X) \subset X</annotation></semantics></math> as well as its <a class="existingWikiWord" href="/nlab/show/complement">complement</a> are <a class="existingWikiWord" href="/nlab/show/open+subsets">open subsets</a>:</p> <p>To see this it is sufficient to find for every point <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>y</mi><mo lspace="0em" rspace="thinmathspace">on</mo><msub><mi>PConn</mi> <mi>x</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">y \on PConn_x(X)</annotation></semantics></math> an <a class="existingWikiWord" href="/nlab/show/open+neighbourhood">open neighbourhood</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>U</mi> <mi>y</mi></msub><mo>⊃</mo><mo stretchy="false">{</mo><mi>y</mi><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">U_y \supset \{y\}</annotation></semantics></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>U</mi> <mi>y</mi></msub><mo>⊂</mo><msub><mi>PConn</mi> <mi>x</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">U_y \subset PConn_x(X)</annotation></semantics></math>, and similarly for the complement.</p> <p>Now by assumption every point <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>y</mi><mo>∈</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">y \in X</annotation></semantics></math> has a Euclidean neighbourhood <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mi>n</mi></msup><mover><mo>→</mo><mo>≃</mo></mover><msub><mi>U</mi> <mi>y</mi></msub><mo>⊂</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">\mathbb{R}^n \overset{\simeq}{\to} U_y \subset X</annotation></semantics></math>. Since Euclidean space is path connected, there is for every <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>z</mi><mo>∈</mo><msub><mi>U</mi> <mi>y</mi></msub></mrow><annotation encoding="application/x-tex">z \in U_y</annotation></semantics></math> a path <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>γ</mi><mo stretchy="false">˜</mo></mover><mo lspace="verythinmathspace">:</mo><mo stretchy="false">[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy="false">]</mo><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">\tilde \gamma \colon [0,1] \to X</annotation></semantics></math> connecting <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>y</mi></mrow><annotation encoding="application/x-tex">y</annotation></semantics></math> with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>z</mi></mrow><annotation encoding="application/x-tex">z</annotation></semantics></math>, i.e. with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>γ</mi><mo stretchy="false">˜</mo></mover><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo><mo>=</mo><mi>y</mi></mrow><annotation encoding="application/x-tex">\tilde \gamma(0) = y</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>γ</mi><mo stretchy="false">˜</mo></mover><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo>=</mo><mi>z</mi></mrow><annotation encoding="application/x-tex">\tilde \gamma(1) = z</annotation></semantics></math>. Accordingly the composite path</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mo stretchy="false">[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy="false">]</mo></mtd> <mtd><mover><mo>⟶</mo><mrow><mover><mi>γ</mi><mo stretchy="false">˜</mo></mover><mo>⋅</mo><mi>γ</mi></mrow></mover></mtd> <mtd><mi>X</mi></mtd></mtr> <mtr><mtd><mi>t</mi></mtd> <mtd><mover><mo>↦</mo><mphantom><mi>AAA</mi></mphantom></mover></mtd> <mtd><mrow><mo>{</mo><mrow><mtable><mtr><mtd><mi>γ</mi><mo stretchy="false">(</mo><mn>2</mn><mi>t</mi><mo stretchy="false">)</mo></mtd> <mtd><mo stretchy="false">|</mo></mtd> <mtd><mi>t</mi><mo>≤</mo><mn>1</mn><mo stretchy="false">/</mo><mn>2</mn></mtd></mtr> <mtr><mtd><mover><mo stretchy="false">(</mo><mo stretchy="false">˜</mo></mover><mn>2</mn><mi>t</mi><mo>−</mo><mn>1</mn><mo stretchy="false">/</mo><mn>2</mn><mo stretchy="false">)</mo></mtd> <mtd><mo stretchy="false">|</mo></mtd> <mtd><mi>t</mi><mo>≥</mo><mn>1</mn><mo stretchy="false">/</mo><mn>2</mn></mtd></mtr></mtable></mrow></mrow></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ [0,1] &\overset{\tilde \gamma\cdot\gamma}{\longrightarrow}& X \\ t &\overset{\phantom{AAA}}{\mapsto}& \left\{ \array{ \gamma(2t) &\vert& t \leq 1/2 \\ \tilde(2t-1/2) &\vert& t \geq 1/2 } \right. } </annotation></semantics></math></div> <p>connects <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math> with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>z</mi><mo>∈</mo><msub><mi>U</mi> <mi>y</mi></msub></mrow><annotation encoding="application/x-tex">z \in U_y</annotation></semantics></math>. Hence <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>U</mi> <mi>y</mi></msub><mo>⊂</mo><msub><mi>PConn</mi> <mi>x</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">U_y \subset PConn_x(X)</annotation></semantics></math>.</p> <p>Similarly, if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>y</mi></mrow><annotation encoding="application/x-tex">y</annotation></semantics></math> is not connected to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math> by a path, then also all point in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>U</mi> <mi>y</mi></msub></mrow><annotation encoding="application/x-tex">U_y</annotation></semantics></math> cannot be connected to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math> by a path, for if they were, then the analogous concatenation of paths would give a path from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>y</mi></mrow><annotation encoding="application/x-tex">y</annotation></semantics></math>, contrary to the assumption.</p> <p>It follows that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>=</mo><msub><mi>PConn</mi> <mi>x</mi></msub><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo><mo>⊔</mo><mo stretchy="false">(</mo><mi>X</mi><mo>∖</mo><msub><mi>PConn</mi> <mi>x</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> X = PConn_x(C) \sqcup (X \setminus PConn_x(X)) </annotation></semantics></math></div> <p>is a decomposition of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> as the <a class="existingWikiWord" href="/nlab/show/disjoint+union">disjoint union</a> of two open subsets. By the assumption that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> is connected, exactly one of these open subsets is empty. Since <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>PConn</mi> <mi>x</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">PConn_x(X)</annotation></semantics></math> is not empty, as it contains <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math>, it follows that its compement is empty, hence that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>PConn</mi> <mi>x</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>=</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">PConn_x(X) = X</annotation></semantics></math>, hence that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>τ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(X,\tau)</annotation></semantics></math> is path connected.</p> </div> <div class="num_prop" id="RegularityConditionsForTopologicalManifoldsComparison"> <h6 id="proposition_2">Proposition</h6> <p><strong>(equivalence of regularity conditions for Hausdorff locally Euclidean spaces)</strong></p> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/locally+Euclidean+space">locally Euclidean space</a> (def. <a class="maruku-ref" href="#LocallyEuclideanSpace"></a>) which is <a class="existingWikiWord" href="/nlab/show/Hausdorff+topological+space">Hausdorff</a>.</p> <p>Then the following are equivalent:</p> <ol> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/sigma-compact+topological+space">sigma-compact</a>,</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/second-countable+topological+space">second-countable</a>,</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/paracompact+topological+space">paracompact</a> and has a <a class="existingWikiWord" href="/nlab/show/countable+set">countable set</a> of <a class="existingWikiWord" href="/nlab/show/connected+components">connected components</a>,</p> </li> </ol> </div> <div class="proof"> <h6 id="proof_3">Proof</h6> <p>Generally, observe that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/locally+compact">locally compact</a>: By prop. <a class="maruku-ref" href="#LocalPropertiesOfLocallyEuclideanSpace"></a> every locally Euclidean space is locally compact in the sense that every point has a <a class="existingWikiWord" href="/nlab/show/neighbourhood+base">neighbourhood base</a> of compact neighbourhoods, and since <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> is assumed to be Hausdorff, this implies all the other variants of definition of local compactness, by <a href="locally+compact+topological+space#InHausdorffSpacesDefinitionsOfLocalCompactnessAgree">this prop.</a>.</p> <p><strong>1) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⇒</mo></mrow><annotation encoding="application/x-tex">\Rightarrow</annotation></semantics></math> 2)</strong></p> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> be sigma-compact. We show that then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/second-countable+topological+space">second-countable</a>:</p> <p>By sigma-compactness there exists a <a class="existingWikiWord" href="/nlab/show/countable+set">countable set</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><msub><mi>K</mi> <mi>i</mi></msub><mo>⊂</mo><mi>X</mi><msub><mo stretchy="false">}</mo> <mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\{K_i \subset X\}_{i \in I}</annotation></semantics></math> of compact subspaces. By <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> being locally Euclidean, each admits an <a class="existingWikiWord" href="/nlab/show/open+cover">open cover</a> by restrictions of <a class="existingWikiWord" href="/nlab/show/Euclidean+spaces">Euclidean spaces</a>. By their compactness, each of these has a subcover <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><msup><mi>ℝ</mi> <mi>n</mi></msup><mover><mo>→</mo><mrow><msub><mi>ϕ</mi> <mrow><mi>i</mi><mo>,</mo><mi>j</mi></mrow></msub></mrow></mover><mi>X</mi><msub><mo stretchy="false">}</mo> <mrow><mi>j</mi><mo>∈</mo><msub><mi>J</mi> <mi>i</mi></msub></mrow></msub></mrow><annotation encoding="application/x-tex">\{ \mathbb{R}^n \overset{\phi_{i,j}}{\to} X \}_{j \in J_i}</annotation></semantics></math> with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>J</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">J_i</annotation></semantics></math> a finite set. Since <a class="existingWikiWord" href="/nlab/show/countable+unions+of+countable+sets+are+countable">countable unions of countable sets are countable</a>, we have obtained a countable cover by Euclidean spaces <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><msup><mi>ℝ</mi> <mi>n</mi></msup><mover><mo>→</mo><mrow><msub><mi>ϕ</mi> <mrow><mi>i</mi><mo>,</mo><mi>j</mi></mrow></msub></mrow></mover><mi>X</mi><msub><mo stretchy="false">}</mo> <mrow><mi>i</mi><mo>∈</mo><mi>I</mi><mo>,</mo><mi>j</mi><mo>∈</mo><msub><mi>J</mi> <mi>i</mi></msub></mrow></msub></mrow><annotation encoding="application/x-tex">\{ \mathbb{R}^n \overset{\phi_{i,j}}{\to} X\}_{i \in I, j \in J_i}</annotation></semantics></math>. Now Euclidean space itself is second countable (by <a href="second-countable+space#SecondCountableEuclideanSpace">this example</a>), hence admits a countable set <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>β</mi> <mrow><msup><mi>ℝ</mi> <mi>n</mi></msup></mrow></msub></mrow><annotation encoding="application/x-tex">\beta_{\mathbb{R}^n}</annotation></semantics></math> of base open sets. As a result the union <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><munder><mo>∪</mo><mfrac linethickness="0"><mrow><mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></mrow><mrow><mrow><mi>j</mi><mo>∈</mo><msub><mi>J</mi> <mi>i</mi></msub></mrow></mrow></mfrac></munder><msub><mi>ϕ</mi> <mrow><mi>i</mi><mo>,</mo><mi>j</mi></mrow></msub><mo stretchy="false">(</mo><msub><mi>β</mi> <mrow><msup><mi>ℝ</mi> <mi>n</mi></msup></mrow></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\underset{{i \in I} \atop {j \in J_i}}{\cup} \phi_{i,j}(\beta_{\mathbb{R}^n})</annotation></semantics></math> is a base of opens for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>. But this is a countable union of countable sets, and since <a class="existingWikiWord" href="/nlab/show/countable+unions+of+countable+sets+are+countable">countable unions of countable sets are countable</a> we have obtained a countable base for the topology of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>. This means that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> is second-countable.</p> <p><strong>1) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⇒</mo></mrow><annotation encoding="application/x-tex">\Rightarrow</annotation></semantics></math> 3)</strong></p> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> be sigma-compact. We show that then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> is paracompact with a countable set of connected components:</p> <p>Since <a class="existingWikiWord" href="/nlab/show/locally+compact+and+sigma-compact+spaces+are+paracompact">locally compact and sigma-compact spaces are paracompact</a>, it follows that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> is paracompact. By <a class="existingWikiWord" href="/nlab/show/locally+connected+topological+space">local connectivity</a> (prop. <a class="maruku-ref" href="#LocalPropertiesOfLocallyEuclideanSpace"></a>) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/disjoint+union+space">disjoint union space</a> of its <a class="existingWikiWord" href="/nlab/show/connected+components">connected components</a> (<a href="locally+connected+topological+space#AlternativeCharacterizationsOfLocalConnectivity">this prop.</a>). Since, by the previous statement, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> is also second-countable it cannot have an uncountable set of connected components.</p> <p><strong>2)<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⇒</mo></mrow><annotation encoding="application/x-tex">\Rightarrow</annotation></semantics></math> 1)</strong> Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> be second-countable, we need to show that it is sigma-compact.</p> <p>This follows since <a class="existingWikiWord" href="/nlab/show/locally+compact+and+second-countable+spaces+are+sigma-compact">locally compact and second-countable spaces are sigma-compact</a>.</p> <p><strong>3) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⇒</mo></mrow><annotation encoding="application/x-tex">\Rightarrow</annotation></semantics></math> 1)</strong></p> <p>Now let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> be paracompact with countably many connected components. We show that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> is sigma-compact.</p> <p>Since <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> is locally compact, there exists a cover <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><msub><mi>K</mi> <mi>i</mi></msub><mo>=</mo><mi>Cl</mi><mo stretchy="false">(</mo><msub><mi>U</mi> <mi>i</mi></msub><mo stretchy="false">)</mo><mo>⊂</mo><mi>X</mi><msub><mo stretchy="false">}</mo> <mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\{K_i = Cl(U_i) \subset X\}_{i \in I}</annotation></semantics></math> by <a class="existingWikiWord" href="/nlab/show/compact+topological+space">compact</a> <a class="existingWikiWord" href="/nlab/show/subspaces">subspaces</a>. By paracompactness there is a locally finite refinement of this cover. Since <a class="existingWikiWord" href="/nlab/show/paracompact+Hausdorff+spaces+are+normal">paracompact Hausdorff spaces are normal</a>, the <a class="existingWikiWord" href="/nlab/show/shrinking+lemma">shrinking lemma</a> applies to this refinement and yields a locally finite open cover</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>𝒱</mi><mo>≔</mo><mo stretchy="false">{</mo><msub><mi>V</mi> <mi>j</mi></msub><mo>⊂</mo><mi>X</mi><msub><mo stretchy="false">}</mo> <mrow><mi>j</mi><mo>∈</mo><mi>J</mi></mrow></msub></mrow><annotation encoding="application/x-tex"> \mathcal{V} \coloneqq \{V_j \subset X \}_{j \in J} </annotation></semantics></math></div> <p>as well as a locally finite cover <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><mi>Cl</mi><mo stretchy="false">(</mo><msub><mi>V</mi> <mi>j</mi></msub><mo stretchy="false">)</mo><mo>⊂</mo><mi>X</mi><msub><mo stretchy="false">}</mo> <mrow><mi>j</mi><mo>∈</mo><mi>J</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\{Cl(V_j) \subset X\}_{j \in J}</annotation></semantics></math> by closed subsets. Since this is a refinement of the orignal cover, all the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Cl</mi><mo stretchy="false">(</mo><msub><mi>V</mi> <mi>j</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Cl(V_j)</annotation></semantics></math> are contained in one of the compact subspaces <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>K</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">K_i</annotation></semantics></math>. Since <a class="existingWikiWord" href="/nlab/show/subsets+are+closed+in+a+closed+subspace+precisely+if+they+are+closed+in+the+ambient+space">subsets are closed in a closed subspace precisely if they are closed in the ambient space</a>, the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Cl</mi><mo stretchy="false">(</mo><msub><mi>V</mi> <mi>j</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Cl(V_j)</annotation></semantics></math> are also closed as subsets of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>K</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">K_i</annotation></semantics></math>. Since <a class="existingWikiWord" href="/nlab/show/closed+subsets+of+compact+spaces+are+compact">closed subsets of compact spaces are compact</a> it follows that the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Cl</mi><mo stretchy="false">(</mo><msub><mi>V</mi> <mi>j</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Cl(V_j)</annotation></semantics></math> are themselves compact and hence form a locally finite cover by compact subspaces.</p> <p>Now fix any <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>j</mi> <mn>0</mn></msub><mo>∈</mo><mi>J</mi></mrow><annotation encoding="application/x-tex">j_0 \in J</annotation></semantics></math>.</p> <p>We claim that for every <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>j</mi><mo>∈</mo><mi>J</mi></mrow><annotation encoding="application/x-tex">j \in J</annotation></semantics></math> there is a finite sequence of indices <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>j</mi> <mn>0</mn></msub><mo>,</mo><msub><mi>j</mi> <mn>1</mn></msub><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>j</mi> <mi>n</mi></msub><mo>=</mo><mi>j</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(j_0, j_1, \cdots, j_n = j)</annotation></semantics></math> with the property that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>V</mi> <mrow><msub><mi>j</mi> <mi>k</mi></msub></mrow></msub><mo>∩</mo><msub><mi>V</mi> <mrow><msub><mi>j</mi> <mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub></mrow></msub><mo>≠</mo><mi>∅</mi></mrow><annotation encoding="application/x-tex">V_{j_k} \cap V_{j_{k+1}} \neq \emptyset</annotation></semantics></math>.</p> <p>To see this, first observe that it is sufficient to show sigma-compactness for the case that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/connected+topological+space">connected</a>. From this the general statement follows since <a class="existingWikiWord" href="/nlab/show/countable+unions+of+countable+sets+are+countable">countable unions of countable sets are countable</a>. Hence assume that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> is connected. It follows from lemma <a class="maruku-ref" href="#PathConnectedFromConnectedLocallyEuclideanSpace"></a> that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/path-connected+topological+space">path-connected</a>.</p> <p>Hence for any <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>∈</mo><msub><mi>V</mi> <mrow><msub><mi>j</mi> <mn>0</mn></msub></mrow></msub></mrow><annotation encoding="application/x-tex">x \in V_{j_0}</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>y</mi><mo>∈</mo><msub><mi>V</mi> <mi>j</mi></msub></mrow><annotation encoding="application/x-tex">y \in V_{j}</annotation></semantics></math> there is a path <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>γ</mi><mo lspace="verythinmathspace">:</mo><mo stretchy="false">[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy="false">]</mo><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">\gamma \colon [0,1] \to X</annotation></semantics></math> connecting <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math> with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>y</mi></mrow><annotation encoding="application/x-tex">y</annotation></semantics></math>. Since the <a class="existingWikiWord" href="/nlab/show/closed+interval">closed interval</a> is compact and since <a class="existingWikiWord" href="/nlab/show/continuous+images+of+compact+spaces+are+compact">continuous images of compact spaces are compact</a>, it follows that there is a finite subset of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>V</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">V_i</annotation></semantics></math> that covers the image of this path. This proves the claim.</p> <p>It follows that there is a function</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>f</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>𝒱</mi><mo>⟶</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex"> f \;\colon\; \mathcal{V} \longrightarrow \mathbb{N} </annotation></semantics></math></div> <p>which sends each <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>V</mi> <mi>j</mi></msub></mrow><annotation encoding="application/x-tex">V_j</annotation></semantics></math> to the <a class="existingWikiWord" href="/nlab/show/minimum">minimum</a> natural number as above.</p> <p>We claim now that for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">n \in \mathbb{N}</annotation></semantics></math> the <a class="existingWikiWord" href="/nlab/show/preimage">preimage</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mi>⋯</mi><mo>,</mo><mi>n</mi><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{0,1, \cdots, n\}</annotation></semantics></math> under this function is a <a class="existingWikiWord" href="/nlab/show/finite+set">finite set</a>. Since <a class="existingWikiWord" href="/nlab/show/countable+unions+of+countable+sets+are+countable">countable unions of countable sets are countable</a> this implies that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><mi>Cl</mi><mo stretchy="false">(</mo><msub><mi>V</mi> <mi>j</mi></msub><mo stretchy="false">)</mo><mo>⊂</mo><mi>X</mi><msub><mo stretchy="false">}</mo> <mrow><mi>j</mi><mo>∈</mo><mi>J</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\{ Cl(V_j) \subset X\}_{j \in J}</annotation></semantics></math> is a countable cover of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> by compact subspaces, hence that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> is sigma-compact.</p> <p>We prove this last claim by <a class="existingWikiWord" href="/nlab/show/induction">induction</a>. It is true for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">n = 0</annotation></semantics></math> by construction. Assume it is true for some <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">n \in \mathbb{N}</annotation></semantics></math>, hence that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>f</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mo stretchy="false">{</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mi>⋯</mi><mo>,</mo><mi>n</mi><mo stretchy="false">}</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f^{-1}(\{0,1, \cdots, n\})</annotation></semantics></math> is a finite set. Since finite unions of compact subspaces are again compact (<a href="compact+space#UnionsAndIntersectionOfCompactSubspaces">this prop.</a>) it follows that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>K</mi> <mi>n</mi></msub><mo>≔</mo><munder><mo>∪</mo><mrow><mi>V</mi><mo>∈</mo><msup><mi>f</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mo stretchy="false">{</mo><mn>0</mn><mo>,</mo><mi>⋯</mi><mo>,</mo><mi>n</mi><mo stretchy="false">}</mo><mo stretchy="false">)</mo></mrow></munder><mi>V</mi></mrow><annotation encoding="application/x-tex"> K_n \coloneqq \underset{V \in f^{-1}(\{0,\cdots, n\})}{\cup} V </annotation></semantics></math></div> <p>is compact. By local finiteness of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><msub><mi>V</mi> <mi>j</mi></msub><msub><mo stretchy="false">}</mo> <mrow><mi>j</mi><mo>∈</mo><mi>J</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\{V_j\}_{j \in J}</annotation></semantics></math>, every point <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>∈</mo><msub><mi>K</mi> <mi>n</mi></msub></mrow><annotation encoding="application/x-tex">x \in K_n</annotation></semantics></math> has an open neighbourhood <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>W</mi> <mi>x</mi></msub></mrow><annotation encoding="application/x-tex">W_x</annotation></semantics></math> that intersects only a finite set of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>V</mi> <mi>j</mi></msub></mrow><annotation encoding="application/x-tex">V_j</annotation></semantics></math>. By compactness of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>K</mi> <mi>n</mi></msub></mrow><annotation encoding="application/x-tex">K_n</annotation></semantics></math>, the cover <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><msub><mi>W</mi> <mi>x</mi></msub><mo>⊂</mo><mi>X</mi><msub><mo stretchy="false">}</mo> <mrow><mi>x</mi><mo>∈</mo><msub><mi>K</mi> <mi>n</mi></msub></mrow></msub></mrow><annotation encoding="application/x-tex">\{W_x \subset X\}_{x \in K_n}</annotation></semantics></math> has a finite subcover. In conclusion this implies that only a finite number of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>V</mi> <mi>j</mi></msub></mrow><annotation encoding="application/x-tex">V_j</annotation></semantics></math> intersect <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>K</mi> <mi>n</mi></msub></mrow><annotation encoding="application/x-tex">K_n</annotation></semantics></math>.</p> <p>Now by definition <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>f</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mo stretchy="false">{</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mi>⋯</mi><mo>,</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo stretchy="false">}</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f^{-1}(\{0,1,\cdots, n+1\})</annotation></semantics></math> is a subset of those <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>V</mi> <mi>j</mi></msub></mrow><annotation encoding="application/x-tex">V_j</annotation></semantics></math> which intersect <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>K</mi> <mi>n</mi></msub></mrow><annotation encoding="application/x-tex">K_n</annotation></semantics></math>, and hence itself finite.</p> </div> <h3 id="topological_manifold">Topological manifold</h3> <div class="num_defn" id="TopologicalManifold"> <h6 id="definition_3">Definition</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/topological+manifold">topological manifold</a>)</strong></p> <p>A <em>topological manifold is a <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a> which is</em></p> <ol> <li> <p><a class="existingWikiWord" href="/nlab/show/locally+Euclidean+topological+space">locally Euclidean</a> (def. <a class="maruku-ref" href="#LocallyEuclideanSpace"></a>),</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/paracompact+Hausdorff+topological+space">paracompact Hausdorff</a>.</p> </li> </ol> <p>If the local <a class="existingWikiWord" href="/nlab/show/Euclidean+spaces">Euclidean</a> neighbourhoods <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mi>n</mi></msup><mover><mo>→</mo><mo>≃</mo></mover><mi>U</mi><mo>⊂</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">\mathbb{R}^n \overset{\simeq}{\to} U \subset X</annotation></semantics></math> are all of <a class="existingWikiWord" href="/nlab/show/dimension">dimension</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math> for a fixed <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">n \in \mathbb{N}</annotation></semantics></math>, then the topological manifold is said to be a <em><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-dimensional manifold</em> or <em><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-fold</em>. This is usually assumed to be the case.</p> </div> <div class="num_remark"> <h6 id="remark_2">Remark</h6> <p><strong>(varying terminology)</strong></p> <p>Often a topological manifold (def. <a class="maruku-ref" href="#TopologicalManifold"></a>) is required to be <a class="existingWikiWord" href="/nlab/show/sigma-compact">sigma-compact</a>. But by prop. <a class="maruku-ref" href="#RegularityConditionsForTopologicalManifoldsComparison"></a> this is not an extra condition as long as there is a <a class="existingWikiWord" href="/nlab/show/countable+set">countable set</a> of <a class="existingWikiWord" href="/nlab/show/connected+components">connected components</a>.</p> </div> <h3 id="differentiable_manifolds">Differentiable manifolds</h3> <div class="num_defn" id="Charts"> <h6 id="definition_4">Definition</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/local+chart">local chart</a> and <a class="existingWikiWord" href="/nlab/show/atlas">atlas</a> and <a class="existingWikiWord" href="/nlab/show/gluing+function">gluing function</a>)</strong></p> <p>Given an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-dimensional topological manifold <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> (def. <a class="maruku-ref" href="#TopologicalManifold"></a>), then</p> <div style="float:right;margin:0 10px 10px 0;"> <img src="https://ncatlab.org/nlab/files/ChartsOfAManifold.png" width="400" /> </div> <ol> <li> <p>an <a class="existingWikiWord" href="/nlab/show/open+subset">open subset</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo>⊂</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">U \subset X</annotation></semantics></math> and a <a class="existingWikiWord" href="/nlab/show/homeomorphism">homeomorphism</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi><mo lspace="verythinmathspace">:</mo><msup><mi>ℝ</mi> <mi>n</mi></msup><mover><mo>→</mo><mrow><mphantom><mi>A</mi></mphantom><mo>≃</mo><mphantom><mi>A</mi></mphantom></mrow></mover><mi>U</mi></mrow><annotation encoding="application/x-tex">\phi \colon \mathbb{R}^n \overset{\phantom{A}\simeq\phantom{A}}{\to} U</annotation></semantics></math> is also called a <em><a class="existingWikiWord" href="/nlab/show/local+coordinate+chart">local coordinate chart</a></em> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>.</p> </li> <li> <p>an <a class="existingWikiWord" href="/nlab/show/open+cover">open cover</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> by local charts <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mrow><mo>{</mo><msup><mi>ℝ</mi> <mi>n</mi></msup><mover><mo>→</mo><mrow><msub><mi>ϕ</mi> <mi>i</mi></msub></mrow></mover><mi>U</mi><mo>⊂</mo><mi>X</mi><mo>}</mo></mrow> <mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\left\{ \mathbb{R}^n \overset{\phi_i}{\to} U \subset X \right\}_{i \in I}</annotation></semantics></math> is called an <em><a class="existingWikiWord" href="/nlab/show/atlas">atlas</a></em> of the topological manifold.</p> </li> <li> <p>denoting for each <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi><mo>,</mo><mi>j</mi><mo>∈</mo><mi>I</mi></mrow><annotation encoding="application/x-tex">i,j \in I</annotation></semantics></math> the <a class="existingWikiWord" href="/nlab/show/intersection">intersection</a> of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math>th chart with the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>j</mi></mrow><annotation encoding="application/x-tex">j</annotation></semantics></math>th chart in such an atlas by</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>U</mi> <mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>≔</mo><msub><mi>U</mi> <mi>i</mi></msub><mo>∩</mo><msub><mi>U</mi> <mi>j</mi></msub></mrow><annotation encoding="application/x-tex"> U_{i j} \coloneqq U_i \cap U_j </annotation></semantics></math></div> <p>then the induced homeomorphism</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mi>n</mi></msup><mo>⊃</mo><mphantom><mi>AA</mi></mphantom><msubsup><mi>ϕ</mi> <mi>i</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><msub><mi>U</mi> <mrow><mi>i</mi><mi>j</mi></mrow></msub><mo stretchy="false">)</mo><mover><mo>⟶</mo><mrow><mphantom><mi>A</mi></mphantom><msub><mi>ϕ</mi> <mi>i</mi></msub><mphantom><mi>A</mi></mphantom></mrow></mover><msub><mi>U</mi> <mrow><mi>i</mi><mi>j</mi></mrow></msub><mover><mo>⟶</mo><mrow><mphantom><mi>A</mi></mphantom><msubsup><mi>ϕ</mi> <mi>j</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mphantom><mi>A</mi></mphantom></mrow></mover><msubsup><mi>ϕ</mi> <mi>j</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><msub><mi>U</mi> <mrow><mi>i</mi><mi>j</mi></mrow></msub><mo stretchy="false">)</mo><mphantom><mi>AA</mi></mphantom><mo>⊂</mo><msup><mi>ℝ</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex"> \mathbb{R}^n \supset \phantom{AA} \phi_i^{-1}(U_{i j}) \overset{\phantom{A}\phi_i\phantom{A}}{\longrightarrow} U_{i j} \overset{\phantom{A}\phi_j^{-1}\phantom{A}}{\longrightarrow} \phi_j^{-1}(U_{i j}) \phantom{AA} \subset \mathbb{R}^n </annotation></semantics></math></div> <p>is called the <em><a class="existingWikiWord" href="/nlab/show/gluing+function">gluing function</a></em> from chart <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math> to chart <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>j</mi></mrow><annotation encoding="application/x-tex">j</annotation></semantics></math>.</p> </li> </ol> <blockquote> <p>graphics grabbed from <a class="existingWikiWord" href="/nlab/show/The+Geometry+of+Physics+-+An+Introduction">Frankel</a></p> </blockquote> </div> <div class="num_defn" id="Differentiable"> <h6 id="definition_5">Definition</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/differentiable+manifold">differentiable</a> and <a class="existingWikiWord" href="/nlab/show/smooth+manifolds">smooth manifolds</a>)</strong></p> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo>∈</mo><mi>ℕ</mi><mo>∪</mo><mo stretchy="false">{</mo><mn>∞</mn><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">p \in \mathbb{N} \cup \{\infty\}</annotation></semantics></math> then a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math>-fold <em><a class="existingWikiWord" href="/nlab/show/differentiable+manifold">differentiable manifold</a></em> is</p> <ol> <li> <p>a <a class="existingWikiWord" href="/nlab/show/topological+manifold">topological manifold</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> (def. <a class="maruku-ref" href="#TopologicalManifold"></a>);</p> </li> <li> <p>an <a class="existingWikiWord" href="/nlab/show/atlas">atlas</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><msup><mi>ℝ</mi> <mi>n</mi></msup><mover><mo>→</mo><mrow><msub><mi>ϕ</mi> <mi>i</mi></msub></mrow></mover><mi>X</mi><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{\mathbb{R}^n \overset{\phi_i}{\to} X\}</annotation></semantics></math> (def. <a class="maruku-ref" href="#Charts"></a>) all whose <a class="existingWikiWord" href="/nlab/show/gluing+functions">gluing functions</a> are <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math> times continuously <a class="existingWikiWord" href="/nlab/show/differentiable+function">differentiable</a>.</p> </li> </ol> <p>A <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math>-fold <a class="existingWikiWord" href="/nlab/show/differentiable+function">differentiable function</a> between <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math>-fold differentiable manifolds</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mo stretchy="false">{</mo><msup><mi>ℝ</mi> <mi>n</mi></msup><mover><mo>→</mo><mrow><msub><mi>ϕ</mi> <mi>i</mi></msub></mrow></mover><msub><mi>U</mi> <mi>i</mi></msub><mo>⊂</mo><mi>X</mi><msub><mo stretchy="false">}</mo> <mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></msub><mo stretchy="false">)</mo><mover><mo>⟶</mo><mrow><mphantom><mi>AA</mi></mphantom><mi>f</mi><mphantom><mi>AA</mi></mphantom></mrow></mover><mo stretchy="false">(</mo><mi>Y</mi><mo>,</mo><mo stretchy="false">{</mo><msup><mi>ℝ</mi> <mrow><mi>n</mi><mo>′</mo></mrow></msup><mover><mo>→</mo><mrow><msub><mi>ψ</mi> <mi>j</mi></msub></mrow></mover><msub><mi>V</mi> <mi>j</mi></msub><mo>⊂</mo><mi>Y</mi><msub><mo stretchy="false">}</mo> <mrow><mi>j</mi><mo>∈</mo><mi>J</mi></mrow></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> (X, \{\mathbb{R}^{n} \overset{\phi_i}{\to} U_i \subset X\}_{i \in I}) \overset{\phantom{AA}f\phantom{AA}}{\longrightarrow} (Y, \{\mathbb{R}^{n'} \overset{\psi_j}{\to} V_j \subset Y\}_{j \in J}) </annotation></semantics></math></div> <p>is</p> <ul> <li>a <a class="existingWikiWord" href="/nlab/show/continuous+function">continuous function</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo lspace="verythinmathspace">:</mo><mi>X</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">f \colon X \to Y</annotation></semantics></math></li> </ul> <p>such that</p> <ul> <li> <p>for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow><annotation encoding="application/x-tex">i \in I</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>j</mi><mo>∈</mo><mi>J</mi></mrow><annotation encoding="application/x-tex">j \in J</annotation></semantics></math> then</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mi>n</mi></msup><mo>⊃</mo><mphantom><mi>AA</mi></mphantom><mo stretchy="false">(</mo><mi>f</mi><mo>∘</mo><msub><mi>ϕ</mi> <mi>i</mi></msub><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><msub><mi>V</mi> <mi>j</mi></msub><mo stretchy="false">)</mo><mover><mo>⟶</mo><mrow><msub><mi>ϕ</mi> <mi>i</mi></msub></mrow></mover><msup><mi>f</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><msub><mi>V</mi> <mi>j</mi></msub><mo stretchy="false">)</mo><mover><mo>⟶</mo><mi>f</mi></mover><msub><mi>V</mi> <mi>j</mi></msub><mover><mo>⟶</mo><mrow><msubsup><mi>ψ</mi> <mi>j</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup></mrow></mover><msup><mi>ℝ</mi> <mrow><mi>n</mi><mo>′</mo></mrow></msup></mrow><annotation encoding="application/x-tex"> \mathbb{R}^n \supset \phantom{AA} (f\circ \phi_i)^{-1}(V_j) \overset{\phi_i}{\longrightarrow} f^{-1}(V_j) \overset{f}{\longrightarrow} V_j \overset{\psi_j^{-1}}{\longrightarrow} \mathbb{R}^{n'} </annotation></semantics></math></div> <p>is a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math>-fold <a class="existingWikiWord" href="/nlab/show/differentiable+function">differentiable function</a> between open subsets of <a class="existingWikiWord" href="/nlab/show/Euclidean+space">Euclidean space</a>.</p> </li> </ul> </div> <p>Notice that this in in general a non-trivial condition even if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>=</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">X = Y</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> is the identity function. In this case the above exhibits a passage to a different, but equivalent, differentiable atlas.</p> <h2 id="properties">Properties</h2> <div class="num_prop" id="OpenSubsetsOfDifferentiableManifoldsAreDifferentiableManifolds"> <h6 id="proposition_3">Proposition</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> be a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math>-fold differentiable manifold and let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi><mo>⊂</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">S \subset X</annotation></semantics></math> be an <a class="existingWikiWord" href="/nlab/show/open+subset">open subset</a> of the underlying <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>τ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(X,\tau)</annotation></semantics></math>.</p> <p>Then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math> carries the structure of a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math>-fold differentiable manifold such that the inclusion map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi><mo>↪</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">S \hookrightarrow X</annotation></semantics></math> is an <a class="existingWikiWord" href="/nlab/show/open+embedding">open</a> <a class="existingWikiWord" href="/nlab/show/embedding+of+differentiable+manifolds">embedding of differentiable manifolds</a>.</p> </div> <div class="proof"> <h6 id="proof_4">Proof</h6> <p>Since the underlying <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/locally+connected+topological+space">locally connected</a> (<a href="topological+manifold#LocalPropertiesOfLocallyEuclideanSpace">this prop.</a>) it is the <a class="existingWikiWord" href="/nlab/show/disjoint+union+space">disjoint union space</a> of its <a class="existingWikiWord" href="/nlab/show/connected+components">connected components</a> (<a href="locally+connected+topological+space#AlternativeCharacterizationsOfLocalConnectivity">this prop.</a>).</p> <p>Therefore we are reduced to showing the statement for the case that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> has a single <a class="existingWikiWord" href="/nlab/show/connected+component">connected component</a>. By <a href="topological+manifold#RegularityConditionsForTopologicalManifoldsComparison">this prop</a> this implies that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/second-countable+topological+space">second-countable topological space</a>.</p> <p>Now a <a class="existingWikiWord" href="/nlab/show/subspace">subspace</a> of a second-countable Hausdorff space is clearly itself second countable and Hausdorff.</p> <p>Similarly it is immediate that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math> is still <a class="existingWikiWord" href="/nlab/show/locally+Euclidean+space">locally Euclidean</a>: since <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> is locally Euclidean every point <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>∈</mo><mi>S</mi><mo>⊂</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">x \in S \subset X</annotation></semantics></math> has a Euclidean neighbourhood in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> and since <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math> is open there exists an open ball in that (itself <a class="existingWikiWord" href="/nlab/show/homeomorphism">homeomorphic</a> to Euclidean space) which is a Euclidean neighbourhood of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math> contained in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math>.</p> <p>For the differentiable structure we pick these Euclidean neighbourhoods from the given atlas. Then the <a class="existingWikiWord" href="/nlab/show/gluing+functions">gluing functions</a> for the Euclidean charts on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math> are <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math>-fold differentiable follows since these are restrictions of the gluing functions for the atlas of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>.</p> </div> <h2 id="examples">Examples</h2> <p>See the examples at <em><a class="existingWikiWord" href="/nlab/show/differentiable+manifold">differentiable manifold</a></em>.</p> <h2 id="related_concepts">Related concepts</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/triangulation+theorem">triangulation theorem</a></li> </ul> <h2 id="references">References</h2> <p>Historical articles:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Hellmuth+Kneser">Hellmuth Kneser</a>, <em>Die Topologie der Mannigfaltigkeiten</em>, Jahresbericht der Deutschen Mathematiker-Vereinigung (1926), Volume: 34, page 1-13 (<a href="https://eudml.org/doc/145701">eudml:145701</a>)</p> </li> <li> <p>J. W. Cannon, <em>The recognition problem: What is a topological manifold?</em>, Bull. Amer. Math. Soc. 84 (1978), 832-866 (<a href="https://doi.org/10.1090/S0002-9904-1978-14527-3">doi:10.1090/S0002-9904-1978-14527-3</a>)</p> </li> </ul> <p>Textbook accounts:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/John+M.+Lee">John M. Lee</a>, <em>Introduction to topological manifolds</em>, Graduate Texts in Mathematics <strong>202</strong>, Springer (2000) [ISBN: 0-387-98759-2, 0-387-95026-5] <p>Second edition: Springer (2011) [ISBN:978-1-4419-7939-1, <a href="https://doi.org/10.1007/978-1-4419-7940-7">doi:10.1007/978-1-4419-7940-7</a>, errata <a href="https://sites.math.washington.edu/~lee/Books/ITM/errata.pdf">pdf</a>]</p> </li> </ul> <p>See also:</p> <ul> <li>Wikipedia, <em><a href="https://en.wikipedia.org/wiki/Topological_manifold">Topological manifold</a></em></li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on June 11, 2024 at 17:08:56. See the <a href="/nlab/history/topological+manifold" style="color: #005c19">history</a> of this page for a list of all contributions to it. </p> </div> <div class="navigation navfoot"> <a href="/nlab/edit/topological+manifold" accesskey="E" class="navlink" id="edit" rel="nofollow">Edit</a><a href="https://nforum.ncatlab.org/discussion/7772/#Item_6">Discuss</a><span class="backintime"><a href="/nlab/revision/topological+manifold/28" accesskey="B" class="navlinkbackintime" id="to_previous_revision" rel="nofollow">Previous revision</a></span><a href="/nlab/show/diff/topological+manifold" accesskey="C" class="navlink" id="see_changes" rel="nofollow">Changes from previous revision</a><a href="/nlab/history/topological+manifold" accesskey="S" class="navlink" id="history" rel="nofollow">History (28 revisions)</a> <a href="/nlab/show/topological+manifold/cite" style="color: black">Cite</a> <a href="/nlab/print/topological+manifold" accesskey="p" id="view_print" rel="nofollow">Print</a> <a href="/nlab/source/topological+manifold" id="view_source" rel="nofollow">Source</a> </div> </div>  </div>  </body> </html>