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> compact space (changes) in nLab </title> <meta http-equiv="Content-Type" content="text/html; charset=UTF-8" /> <meta name="robots" content="noindex,nofollow" /> <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> compact space (changes) </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/diff/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/1976/#Item_89" 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"> <p class="show_diff"> Showing changes from revision #97 to #98: <ins class="diffins">Added</ins> | <del class="diffdel">Removed</del> | <del class="diffmod">Chan</del><ins class="diffmod">ged</ins> </p> <div class='rightHandSide'> <div class='toc clickDown' tabindex='0'> <h3 id='context'>Context</h3> <h4 id='topology'>Topology</h4> <div class='hide'> <p><strong><a class='existingWikiWord' href='/nlab/show/diff/topology'>topology</a></strong> (<a class='existingWikiWord' href='/nlab/show/diff/general+topology'>point-set topology</a>, <a class='existingWikiWord' href='/nlab/show/diff/point-free+topology'>point-free topology</a>)</p> <p>see also <em><a class='existingWikiWord' href='/nlab/show/diff/differential+topology'>differential topology</a></em>, <em><a class='existingWikiWord' href='/nlab/show/diff/algebraic+topology'>algebraic topology</a></em>, <em><a class='existingWikiWord' href='/nlab/show/diff/functional+analysis'>functional analysis</a></em> and <em><a class='existingWikiWord' href='/nlab/show/diff/topological+homotopy+theory'>topological</a> <a class='existingWikiWord' href='/nlab/show/diff/homotopy+theory'>homotopy theory</a></em></p> <p><a class='existingWikiWord' href='/nlab/show/diff/Introduction+to+Topology'>Introduction</a></p> <p><strong>Basic concepts</strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/open+subspace'>open subset</a>, <a class='existingWikiWord' href='/nlab/show/diff/closed+subspace'>closed subset</a>, <a class='existingWikiWord' href='/nlab/show/diff/neighborhood'>neighbourhood</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/topological+space'>topological space</a>, <a class='existingWikiWord' href='/nlab/show/diff/locale'>locale</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/topological+base'>base for the topology</a>, <a class='existingWikiWord' href='/nlab/show/diff/neighborhood+base'>neighbourhood base</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/finer+topology'>finer/coarser topology</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/closed+subspace'>closure</a>, <a class='existingWikiWord' href='/nlab/show/diff/interior'>interior</a>, <a class='existingWikiWord' href='/nlab/show/diff/boundary'>boundary</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/separation+axioms'>separation</a>, <a class='existingWikiWord' href='/nlab/show/diff/sober+topological+space'>sobriety</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/continuous+map'>continuous function</a>, <a class='existingWikiWord' href='/nlab/show/diff/homeomorphism'>homeomorphism</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/uniformly+continuous+map'>uniformly continuous function</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/embedding+of+topological+spaces'>embedding</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/open+map'>open map</a>, <a class='existingWikiWord' href='/nlab/show/diff/closed+map'>closed map</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/sequence'>sequence</a>, <a class='existingWikiWord' href='/nlab/show/diff/net'>net</a>, <a class='existingWikiWord' href='/nlab/show/diff/subnet'>sub-net</a>, <a class='existingWikiWord' href='/nlab/show/diff/filter'>filter</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/convergence'>convergence</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/category'>category</a> <a class='existingWikiWord' href='/nlab/show/diff/Top'>Top</a></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/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/diff/weak+topology'>initial topology</a>, <a class='existingWikiWord' href='/nlab/show/diff/weak+topology'>final topology</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/subspace'>subspace</a>, <a class='existingWikiWord' href='/nlab/show/diff/quotient+space'>quotient space</a>,</p> </li> <li> <p>fiber space, <a class='existingWikiWord' href='/nlab/show/diff/space+attachment'>space attachment</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/product+topological+space'>product space</a>, <a class='existingWikiWord' href='/nlab/show/diff/disjoint+union+topological+space'>disjoint union space</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/mapping+cylinder'>mapping cylinder</a>, <a class='existingWikiWord' href='/nlab/show/diff/cocylinder'>mapping cocylinder</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/mapping+cone'>mapping cone</a>, <a class='existingWikiWord' href='/nlab/show/diff/mapping+cocone'>mapping cocone</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/mapping+telescope'>mapping telescope</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/colimits+of+normal+spaces'>colimits of normal spaces</a></p> </li> </ul> <p><strong><a class='existingWikiWord' href='/nlab/show/diff/stuff%2C+structure%2C+property'>Extra stuff, structure, properties</a></strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/nice+topological+space'>nice topological space</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/metric+space'>metric space</a>, <a class='existingWikiWord' href='/nlab/show/diff/metric+topology'>metric topology</a>, <a class='existingWikiWord' href='/nlab/show/diff/metrisable+topological+space'>metrisable space</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Kolmogorov+topological+space'>Kolmogorov space</a>, <a class='existingWikiWord' href='/nlab/show/diff/Hausdorff+space'>Hausdorff space</a>, <a class='existingWikiWord' href='/nlab/show/diff/regular+space'>regular space</a>, <a class='existingWikiWord' href='/nlab/show/diff/normal+space'>normal space</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/sober+topological+space'>sober space</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/compact+space'>compact space</a>, <a class='existingWikiWord' href='/nlab/show/diff/proper+map'>proper map</a></p> <p><a class='existingWikiWord' href='/nlab/show/diff/sequentially+compact+topological+space'>sequentially compact</a>, <a class='existingWikiWord' href='/nlab/show/diff/countably+compact+topological+space'>countably compact</a>, <a class='existingWikiWord' href='/nlab/show/diff/locally+compact+topological+space'>locally compact</a>, <a class='existingWikiWord' href='/nlab/show/diff/sigma-compact+topological+space'>sigma-compact</a>, <a class='existingWikiWord' href='/nlab/show/diff/paracompact+topological+space'>paracompact</a>, <a class='existingWikiWord' href='/nlab/show/diff/countably+paracompact+topological+space'>countably paracompact</a>, <a class='existingWikiWord' href='/nlab/show/diff/strongly+compact+topological+space'>strongly compact</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/compactly+generated+topological+space'>compactly generated space</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/second-countable+space'>second-countable space</a>, <a class='existingWikiWord' href='/nlab/show/diff/first-countable+space'>first-countable space</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/contractible+space'>contractible space</a>, <a class='existingWikiWord' href='/nlab/show/diff/locally+contractible+space'>locally contractible space</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/connected+space'>connected space</a>, <a class='existingWikiWord' href='/nlab/show/diff/locally+connected+topological+space'>locally connected space</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/simply+connected+space'>simply-connected space</a>, <a class='existingWikiWord' href='/nlab/show/diff/semi-locally+simply-connected+topological+space'>locally simply-connected space</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/cell+complex'>cell complex</a>, <a class='existingWikiWord' href='/nlab/show/diff/CW+complex'>CW-complex</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/pointed+topological+space'>pointed space</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/topological+vector+space'>topological vector space</a>, <a class='existingWikiWord' href='/nlab/show/diff/Banach+space'>Banach space</a>, <a class='existingWikiWord' href='/nlab/show/diff/Hilbert+space'>Hilbert space</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/topological+group'>topological group</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/topological+vector+bundle'>topological vector bundle</a>, <a class='existingWikiWord' href='/nlab/show/diff/topological+K-theory'>topological K-theory</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/topological+manifold'>topological manifold</a></p> </li> </ul> <p><strong>Examples</strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/empty+space'>empty space</a>, <a class='existingWikiWord' href='/nlab/show/diff/point+space'>point space</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/discrete+object'>discrete space</a>, <a class='existingWikiWord' href='/nlab/show/diff/codiscrete+space'>codiscrete space</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Sierpinski+space'>Sierpinski space</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/order+topology'>order topology</a>, <a class='existingWikiWord' href='/nlab/show/diff/specialization+topology'>specialization topology</a>, <a class='existingWikiWord' href='/nlab/show/diff/Scott+topology'>Scott topology</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Euclidean+space'>Euclidean space</a></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/real+number'>real line</a>, <a class='existingWikiWord' href='/nlab/show/diff/plane'>plane</a></li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/cylinder+object'>cylinder</a>, <a class='existingWikiWord' href='/nlab/show/diff/cone'>cone</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/sphere'>sphere</a>, <a class='existingWikiWord' href='/nlab/show/diff/ball'>ball</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/circle'>circle</a>, <a class='existingWikiWord' href='/nlab/show/diff/torus'>torus</a>, <a class='existingWikiWord' href='/nlab/show/diff/annulus'>annulus</a>, <a class='existingWikiWord' href='/nlab/show/diff/M%C3%B6bius+strip'>Moebius strip</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/polytope'>polytope</a>, <a class='existingWikiWord' href='/nlab/show/diff/polyhedron'>polyhedron</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/projective+space'>projective space</a> (<a class='existingWikiWord' href='/nlab/show/diff/real+projective+space'>real</a>, <a class='existingWikiWord' href='/nlab/show/diff/complex+projective+space'>complex</a>)</p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/classifying+space'>classifying space</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/configuration+space+of+points'>configuration space</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/path'>path</a>, <a class='existingWikiWord' href='/nlab/show/diff/loop'>loop</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/compact-open+topology'>mapping spaces</a>: <a class='existingWikiWord' href='/nlab/show/diff/compact-open+topology'>compact-open topology</a>, <a class='existingWikiWord' href='/nlab/show/diff/topology+of+uniform+convergence'>topology of uniform convergence</a></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/loop+space'>loop space</a>, <a class='existingWikiWord' href='/nlab/show/diff/path+space'>path space</a></li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Zariski+topology'>Zariski topology</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Cantor+space'>Cantor space</a>, <a class='existingWikiWord' href='/nlab/show/diff/Mandelbrot+set'>Mandelbrot space</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Peano+curve'>Peano curve</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/line+with+two+origins'>line with two origins</a>, <a class='existingWikiWord' href='/nlab/show/diff/long+line'>long line</a>, <a class='existingWikiWord' href='/nlab/show/diff/Sorgenfrey+line'>Sorgenfrey line</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/K-topology'>K-topology</a>, <a class='existingWikiWord' href='/nlab/show/diff/Dowker+space'>Dowker space</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Warsaw+circle'>Warsaw circle</a>, <a class='existingWikiWord' href='/nlab/show/diff/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/diff/Hausdorff+implies+sober'>Hausdorff spaces are sober</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/schemes+are+sober'>schemes are sober</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/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/diff/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/diff/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/diff/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/diff/compact+spaces+equivalently+have+converging+subnets'>compact spaces equivalently have converging subnet of every net</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Lebesgue+number+lemma'>Lebesgue number lemma</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/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/diff/compact+spaces+equivalently+have+converging+subnets'>compact spaces equivalently have converging subnet of every net</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/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/diff/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/diff/paracompact+Hausdorff+spaces+are+normal'>paracompact Hausdorff spaces are normal</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/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/diff/closed+injections+are+embeddings'>closed injections are embeddings</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/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/diff/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/diff/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/diff/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/diff/second-countable+regular+spaces+are+paracompact'>second-countable regular spaces are paracompact</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/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/diff/Urysohn%27s+lemma'>Urysohn's lemma</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Tietze+extension+theorem'>Tietze extension theorem</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Tychonoff+theorem'>Tychonoff theorem</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/tube+lemma'>tube lemma</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Michael%27s+theorem'>Michael's theorem</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Brouwer%27s+fixed+point+theorem'>Brouwer's fixed point theorem</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/topological+invariance+of+dimension'>topological invariance of dimension</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/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/diff/Heine-Borel+theorem'>Heine-Borel theorem</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/intermediate+value+theorem'>intermediate value theorem</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/extreme+value+theorem'>extreme value theorem</a></p> </li> </ul> <p><strong><a class='existingWikiWord' href='/nlab/show/diff/topological+homotopy+theory'>topological homotopy theory</a></strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/homotopy'>left homotopy</a>, <a class='existingWikiWord' href='/nlab/show/diff/homotopy'>right homotopy</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/homotopy+equivalence'>homotopy equivalence</a>, <a class='existingWikiWord' href='/nlab/show/diff/deformation+retract'>deformation retract</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/fundamental+group'>fundamental group</a>, <a class='existingWikiWord' href='/nlab/show/diff/covering+space'>covering space</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/fundamental+theorem+of+covering+spaces'>fundamental theorem of covering spaces</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/homotopy+group'>homotopy group</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/weak+homotopy+equivalence'>weak homotopy equivalence</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Whitehead+theorem'>Whitehead's theorem</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Freudenthal+suspension+theorem'>Freudenthal suspension theorem</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/nerve+theorem'>nerve theorem</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/homotopy+extension+property'>homotopy extension property</a>, <a class='existingWikiWord' href='/nlab/show/diff/Hurewicz+cofibration'>Hurewicz cofibration</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/topological+cofiber+sequence'>cofiber sequence</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Str%C3%B8m+model+structure'>Strøm model category</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/classical+model+structure+on+topological+spaces'>classical model structure on topological spaces</a></p> </li> </ul> </div> </div> </div> <h1 id='compact_spaces'>Compact spaces</h1> <div class='maruku_toc'><ul><li><a href='#Idea'>Idea</a></li><li><a href='#definitionscharacterizations'>Definitions/characterizations</a><ul><li><a href='#elementary_reformulations'>Elementary reformulations</a></li><li><a href='#compactness_for_locales'>Compactness for locales</a></li><li><a href='#compactness_via_convergence'>Compactness via convergence</a></li><li><a href='#compactness_via_completeness'>Compactness via completeness</a></li><li><a href='#compactness_via_stability_properties'>Compactness via stability properties</a></li></ul></li><li><a href='#properties'>Properties</a><ul><li><a href='#various'>Various</a></li><li><a href='#relation_to_compact_objects_in_'>Relation to compact objects in <math class='maruku-mathml' display='inline' id='mathml_ad54570a1a49ee7346ff0ef44ee886942fb6526d_1' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Top</mi></mrow><annotation encoding='application/x-tex'>Top</annotation></semantics></math></a></li></ul></li><li><a href='#examples'>Examples</a><ul><li><a href='#ExamplesGeneral'>General</a></li><li><a href='#CompactSpacesWhichAreNotSequentiallyCompact'>Compact spaces which are not sequentially compact</a></li></ul></li><li><a href='#in_synthetic_topology'>In synthetic topology</a></li><li><a href='#compact_spaces_and_proper_maps'>Compact spaces and proper maps</a></li><li><a href='#history'>History</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 <a class='existingWikiWord' href='/nlab/show/diff/topological+space'>topological space</a> (or more generally: a <a class='existingWikiWord' href='/nlab/show/diff/convergence+space'>convergence space</a>) is <em>compact</em> if all <a class='existingWikiWord' href='/nlab/show/diff/sequence'>sequences</a> and more generally <a class='existingWikiWord' href='/nlab/show/diff/net'>nets</a> inside it <a class='existingWikiWord' href='/nlab/show/diff/convergence'>converge</a> as much as possible.</p> <p>Compactness is a topological notion that was developed to abstract the key property of a <a class='existingWikiWord' href='/nlab/show/diff/subspace'>subspace</a> of a <a class='existingWikiWord' href='/nlab/show/diff/Euclidean+space'>Euclidean space</a> being “<a class='existingWikiWord' href='/nlab/show/diff/closed+subspace'>closed</a> and <a class='existingWikiWord' href='/nlab/show/diff/bounded+set'>bounded</a>”: <em>every <a class='existingWikiWord' href='/nlab/show/diff/net'>net</a> must <a class='existingWikiWord' href='/nlab/show/diff/limit+point'>accumulate</a> somewhere in the subspace</em>. (Roughly, the reason is that boundedness implies the net cannot escape the subspace, and the point to which it accumulates lies in the subspace by closure. This is the statement of the <em><a class='existingWikiWord' href='/nlab/show/diff/Heine-Borel+theorem'>Heine-Borel theorem</a></em>, see there for for more.)</p> <p>Compactness provides an intrinsic way of formulating this property in the context of general topological spaces, without the need to view them as subspaces of an ambient space. Still, it is common to work with <em>compact <a class='existingWikiWord' href='/nlab/show/diff/subset'>subsets</a></em> of a given space. These are those subsets which are compact spaces with the <a class='existingWikiWord' href='/nlab/show/diff/subspace+topology'>subspace topology</a>.</p> <p>One often wishes to study <a class='existingWikiWord' href='/nlab/show/diff/compactum'>compact Hausdorff spaces</a> (“compacta”) since these enjoy particularly useful properties. For instance they all arise as <a class='existingWikiWord' href='/nlab/show/diff/one-point+compactification'>one-point compactifications</a> (of <a class='existingWikiWord' href='/nlab/show/diff/locally+compact+topological+space'>locally compact</a> <a class='existingWikiWord' href='/nlab/show/diff/Hausdorff+space'>Hausdorff spaces</a>, see <a href='one-point+compactification#CompactHausdorffSpaceIsCompactificationOfComplementOfAnyPoint'>this remark</a>).</p> <p>As for most concepts related to <a class='existingWikiWord' href='/nlab/show/diff/topological+space'>topological spaces</a>, there is also a concept of compactness for <em><a class='existingWikiWord' href='/nlab/show/diff/locale'>locales</a></em>. Observe that (using <a class='existingWikiWord' href='/nlab/show/diff/classical+logic'>classical logic</a>) every <a class='existingWikiWord' href='/nlab/show/diff/locally+compact+locale'>locally compact locale</a> is aready <a class='existingWikiWord' href='/nlab/show/diff/spatial+locale'>spatial</a> (<a href='locally+compact+locale#UsingClassicalLogicLocallyCompactLocaleIsSpatial'>this prop.</a>). Instead of compact Hausdorff locales one usually considers compact <a class='existingWikiWord' href='/nlab/show/diff/regular+space'>regular</a> locales, since regularity is easier to formulate and handle than Hausdorffness in locale theory (these are equivalent, since every locale is <math class='maruku-mathml' display='inline' id='mathml_ad54570a1a49ee7346ff0ef44ee886942fb6526d_2' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>T</mi> <mn>0</mn></msub></mrow><annotation encoding='application/x-tex'>T_0</annotation></semantics></math> <a class='existingWikiWord' href='/nlab/show/diff/separation+axioms'>separated</a> and hence <math class='maruku-mathml' display='inline' id='mathml_ad54570a1a49ee7346ff0ef44ee886942fb6526d_3' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>T</mi> <mn>3</mn></msub></mrow><annotation encoding='application/x-tex'>T_3</annotation></semantics></math> if regular, while every Hausdorff space is <math class='maruku-mathml' display='inline' id='mathml_ad54570a1a49ee7346ff0ef44ee886942fb6526d_4' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>T</mi> <mn>3</mn></msub></mrow><annotation encoding='application/x-tex'>T_3</annotation></semantics></math> if compact).</p> <h2 id='definitionscharacterizations'>Definitions/characterizations</h2> <p>There are many ways to say that a space <math class='maruku-mathml' display='inline' id='mathml_ad54570a1a49ee7346ff0ef44ee886942fb6526d_5' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> is compact. The first is perhaps the most common:</p> <div class='num_defn' id='OpenCover'> <h6 id='definition'>Definition</h6> <p><strong>(<a class='existingWikiWord' href='/nlab/show/diff/open+cover'>open cover</a>)</strong></p> <p>Let <math class='maruku-mathml' display='inline' id='mathml_ad54570a1a49ee7346ff0ef44ee886942fb6526d_6' xmlns='http://www.w3.org/1998/Math/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> be a <a class='existingWikiWord' href='/nlab/show/diff/topological+space'>topological space</a>. Then an <a class='existingWikiWord' href='/nlab/show/diff/open+cover'>open cover</a> is a <a class='existingWikiWord' href='/nlab/show/diff/set'>set</a> <math class='maruku-mathml' display='inline' id='mathml_ad54570a1a49ee7346ff0ef44ee886942fb6526d_7' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>{</mo><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></mrow><annotation encoding='application/x-tex'>\{U_i \subset X\}_{i \in I}</annotation></semantics></math> of <a class='existingWikiWord' href='/nlab/show/diff/open+subspace'>open subsets</a> (i.e. <math class='maruku-mathml' display='inline' id='mathml_ad54570a1a49ee7346ff0ef44ee886942fb6526d_8' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><msub><mi>U</mi> <mi>i</mi></msub><mo>⊂</mo><mi>X</mi><mo stretchy='false'>)</mo><mo>∈</mo><mi>τ</mi><mo>⊂</mo><mi>P</mi><mo stretchy='false'>(</mo><mi>X</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(U_i \subset X) \in \tau \subset P(X)</annotation></semantics></math>) such that their <a class='existingWikiWord' href='/nlab/show/diff/union'>union</a> is all of <math class='maruku-mathml' display='inline' id='mathml_ad54570a1a49ee7346ff0ef44ee886942fb6526d_9' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math></p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_ad54570a1a49ee7346ff0ef44ee886942fb6526d_10' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><munder><mo>∪</mo><mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></munder><msub><mi>U</mi> <mi>i</mi></msub><mo>=</mo><mi>X</mi><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> \underset{i \in I}{\cup} U_i = X \,. </annotation></semantics></math></div> <p>This is called a <em><a class='existingWikiWord' href='/nlab/show/diff/finite+cover'>finite open cover</a></em> if <math class='maruku-mathml' display='inline' id='mathml_ad54570a1a49ee7346ff0ef44ee886942fb6526d_11' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>I</mi></mrow><annotation encoding='application/x-tex'>I</annotation></semantics></math> is a (Kuratowski-)<a class='existingWikiWord' href='/nlab/show/diff/finite+set'>finite set</a>.</p> <p>A <em>subcover</em> of an open cover as above is a <a class='existingWikiWord' href='/nlab/show/diff/subset'>subset</a> <math class='maruku-mathml' display='inline' id='mathml_ad54570a1a49ee7346ff0ef44ee886942fb6526d_12' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>J</mi><mo>⊂</mo><mi>I</mi></mrow><annotation encoding='application/x-tex'>J\subset I</annotation></semantics></math> of the given open subsets, such that their union still exhausts <math class='maruku-mathml' display='inline' id='mathml_ad54570a1a49ee7346ff0ef44ee886942fb6526d_13' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math>, i.e. <math class='maruku-mathml' display='inline' id='mathml_ad54570a1a49ee7346ff0ef44ee886942fb6526d_14' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><munder><mo>∪</mo><mrow><mi>i</mi><mo>∈</mo><mi>J</mi><mo>⊂</mo><mi>I</mi></mrow></munder><msub><mi>U</mi> <mi>i</mi></msub><mo>=</mo><mi>X</mi></mrow><annotation encoding='application/x-tex'>\underset{i \in J \subset I}{\cup} U_i = X</annotation></semantics></math>.</p> </div> <div class='num_defn' id='hb'> <h6 id='definition_2'>Definition</h6> <p><strong>(compact space)</strong></p> <p>A <a class='existingWikiWord' href='/nlab/show/diff/topological+space'>topological space</a> is called <em>compact</em> if every <a class='existingWikiWord' href='/nlab/show/diff/open+cover'>open cover</a> has a finite subcover (def. <a class='maruku-ref' href='#OpenCover'>1</a>).</p> </div> <p>For the purposes of exposition, this definition will be taken as the baseline definition. Throughout the remainder of this section we state a number of propositions of type “A space is compact (in the sense of Definition <a class='maruku-ref' href='#hb'>2</a>) iff it satisfies property P”, which can be read as saying that property P can be taken as an alternative definition of compactness (and may in fact be considered a more convenient or preferable definition over <a class='maruku-ref' href='#hb'>2</a>, depending on author and context).</p> <div class='num_remark' id='DifferingTerminology'> <h6 id='remark'>Remark</h6> <p><strong>(differing terminology)</strong></p> <p>Some authors use “compact” to mean “compact and <a class='existingWikiWord' href='/nlab/show/diff/Hausdorff+space'>Hausdorff</a>” (a much <a class='existingWikiWord' href='/nlab/show/diff/nice+topological+space'>nicer sort of space</a>, and forming a much <a class='existingWikiWord' href='/nlab/show/diff/nice+category+of+spaces'>nicer category of spaces</a>, see at <em><a class='existingWikiWord' href='/nlab/show/diff/compactum'>compact Hausdorff space</a></em>), and use the word “<a class='existingWikiWord' href='/nlab/show/diff/quasicompact+morphism'>quasicompact</a>” to refer to just “compact” as we are using it here. This custom seems to be prevalent among <a class='existingWikiWord' href='/nlab/show/diff/algebraic+geometry'>algebraic geometers</a>, for example, and particularly so within Francophone schools.</p> <blockquote> <p>But it is far from clear to me (<a class='existingWikiWord' href='/nlab/show/diff/Todd+Trimble'>Todd Trimble</a>) that “quasicompact” is very well-established outside such circles (despite some arguments in favor of it), and using simply “compact” for the nicer concept therefore carries some risk of creating misunderstanding among mathematicians at large. My own habit at any rate is to say “compact Hausdorff” for the nicer concept, and I will continue using this on the <math class='maruku-mathml' display='inline' id='mathml_ad54570a1a49ee7346ff0ef44ee886942fb6526d_15' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>n</mi></mrow><annotation encoding='application/x-tex'>n</annotation></semantics></math>Lab until consensus is reached (if that happens).</p> </blockquote> <p>Another term in usage is ‘compactum’ to mean a <a class='existingWikiWord' href='/nlab/show/diff/compactum'>compact Hausdorff space</a> (even when ‘compact’ is not used to imply Hausdorffness).</p> </div> <p>The various reformulations of compactness fall into several families. Some are more or less tautological reformulations based on direct logical or set-theoretic manipulations of the open covering definition; we give these first. Others express compactness in terms of notions of convergence (nets, filters, ultrafilters). A third family expresses compactness of a space in terms of “stable” properties of maps sourced is that space; this is perhaps the most intrinsically categorical of the three families.</p> <h3 id='elementary_reformulations'>Elementary reformulations</h3> <p>If <a class='existingWikiWord' href='/nlab/show/diff/excluded+middle'>excluded middle</a> is assumed, then def. <a class='maruku-ref' href='#hb'>2</a> has the following reformulations in terms of <a class='existingWikiWord' href='/nlab/show/diff/closed+subspace'>closed subsets</a>:</p> <div class='num_prop' id='fip'> <h6 id='proposition'>Proposition</h6> <p><strong>(compactness in terms of closed subsets)</strong></p> <p>Let <math class='maruku-mathml' display='inline' id='mathml_ad54570a1a49ee7346ff0ef44ee886942fb6526d_16' xmlns='http://www.w3.org/1998/Math/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> be a <a class='existingWikiWord' href='/nlab/show/diff/topological+space'>topological space</a>. Assuming <a class='existingWikiWord' href='/nlab/show/diff/excluded+middle'>excluded middle</a>, then the following are equivalent:</p> <ol> <li> <p><math class='maruku-mathml' display='inline' id='mathml_ad54570a1a49ee7346ff0ef44ee886942fb6526d_17' xmlns='http://www.w3.org/1998/Math/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 compact in the sense of def. <a class='maruku-ref' href='#hb'>2</a>.</p> </li> <li> <p>Let <math class='maruku-mathml' display='inline' id='mathml_ad54570a1a49ee7346ff0ef44ee886942fb6526d_18' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>{</mo><msub><mi>C</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'>\{C_i \subset X\}_{i \in I}</annotation></semantics></math> be a set of <a class='existingWikiWord' href='/nlab/show/diff/closed+subspace'>closed subsets</a> such that their <a class='existingWikiWord' href='/nlab/show/diff/intersection'>intersection</a> is <a class='existingWikiWord' href='/nlab/show/diff/empty+set'>empty</a> <math class='maruku-mathml' display='inline' id='mathml_ad54570a1a49ee7346ff0ef44ee886942fb6526d_19' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><munder><mo>∩</mo><mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></munder><msub><mi>C</mi> <mi>i</mi></msub><mo>=</mo><mi>∅</mi></mrow><annotation encoding='application/x-tex'>\underset{i \in I}{\cap} C_i = \emptyset</annotation></semantics></math>, then there is a <a class='existingWikiWord' href='/nlab/show/diff/finite+set'>finite</a> <a class='existingWikiWord' href='/nlab/show/diff/subset'>subset</a> <math class='maruku-mathml' display='inline' id='mathml_ad54570a1a49ee7346ff0ef44ee886942fb6526d_20' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>J</mi><mo>⊂</mo><mi>I</mi></mrow><annotation encoding='application/x-tex'>J \subset I</annotation></semantics></math> such that the corresponding finite intersection is still empty: <math class='maruku-mathml' display='inline' id='mathml_ad54570a1a49ee7346ff0ef44ee886942fb6526d_21' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><munder><mo>∩</mo><mrow><mi>i</mi><mo>∈</mo><mi>J</mi><mo>⊂</mo><mi>i</mi></mrow></munder><msub><mi>C</mi> <mi>i</mi></msub><mo>=</mo><mi>∅</mi></mrow><annotation encoding='application/x-tex'>\underset{i \in J \subset i}{\cap} C_i = \emptyset</annotation></semantics></math>.</p> </li> <li> <p>Let <math class='maruku-mathml' display='inline' id='mathml_ad54570a1a49ee7346ff0ef44ee886942fb6526d_22' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>{</mo><msub><mi>C</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'>\{C_i \subset X\}_{i \in I}</annotation></semantics></math> be a set of <a class='existingWikiWord' href='/nlab/show/diff/closed+subspace'>closed subsets</a> such that it enjoys the <em><a class='existingWikiWord' href='/nlab/show/diff/finite+intersection+property'>finite intersection property</a></em>, meaning that for every <a class='existingWikiWord' href='/nlab/show/diff/finite+set'>finite</a> <a class='existingWikiWord' href='/nlab/show/diff/subset'>subset</a> <math class='maruku-mathml' display='inline' id='mathml_ad54570a1a49ee7346ff0ef44ee886942fb6526d_23' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>J</mi><mo>⊂</mo><mi>I</mi></mrow><annotation encoding='application/x-tex'>J \subset I</annotation></semantics></math> then the corresponding finite intersection is <a class='existingWikiWord' href='/nlab/show/diff/inhabited+set'>non-empty</a> <math class='maruku-mathml' display='inline' id='mathml_ad54570a1a49ee7346ff0ef44ee886942fb6526d_24' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><munder><mo>∩</mo><mrow><mi>i</mi><mo>∈</mo><mi>J</mi><mo>⊂</mo><mi>I</mi></mrow></munder><msub><mi>C</mi> <mi>i</mi></msub><mo>≠</mo><mi>∅</mi></mrow><annotation encoding='application/x-tex'>\underset{i \in J \subset I}{\cap} C_i \neq \emptyset</annotation></semantics></math>. Then also the total intersection is <a class='existingWikiWord' href='/nlab/show/diff/inhabited+set'>inhabited</a>, <math class='maruku-mathml' display='inline' id='mathml_ad54570a1a49ee7346ff0ef44ee886942fb6526d_25' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><munder><mo>∩</mo><mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></munder><msub><mi>C</mi> <mi>i</mi></msub><mo>≠</mo><mi>∅</mi></mrow><annotation encoding='application/x-tex'>\underset{i \in I}{\cap} C_i \neq \emptyset</annotation></semantics></math>.</p> </li> </ol> </div> <div class='proof'> <h6 id='proof'>Proof</h6> <p>The equivalence of the first two statements follows by <a class='existingWikiWord' href='/nlab/show/diff/De+Morgan+laws'>de Morgan's law</a> (<a class='existingWikiWord' href='/nlab/show/diff/complement'>complements</a> interchange <a class='existingWikiWord' href='/nlab/show/diff/union'>unions</a> with <a class='existingWikiWord' href='/nlab/show/diff/intersection'>intersections</a>), the definition of <a class='existingWikiWord' href='/nlab/show/diff/closed+subspace'>closed subsets</a> as the complements of <a class='existingWikiWord' href='/nlab/show/diff/open+subspace'>open sets</a>, and (using <a class='existingWikiWord' href='/nlab/show/diff/excluded+middle'>excluded middle</a>) that dually the complements of closed subsets are the open subsets:</p> <p>Let <math class='maruku-mathml' display='inline' id='mathml_ad54570a1a49ee7346ff0ef44ee886942fb6526d_26' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>{</mo><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></mrow><annotation encoding='application/x-tex'>\{U_i \subset X\}_{i \in I}</annotation></semantics></math> be an <a class='existingWikiWord' href='/nlab/show/diff/open+cover'>open cover</a>. Write <math class='maruku-mathml' display='inline' id='mathml_ad54570a1a49ee7346ff0ef44ee886942fb6526d_27' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>C</mi> <mi>i</mi></msub><mo>≔</mo><mi>X</mi><mo>\</mo><msub><mi>U</mi> <mi>i</mi></msub></mrow><annotation encoding='application/x-tex'>C_i \coloneqq X \backslash U_i</annotation></semantics></math> for the corresponding closed complements. By <a class='existingWikiWord' href='/nlab/show/diff/De+Morgan+laws'>de Morgan's law</a> the condition that <math class='maruku-mathml' display='inline' id='mathml_ad54570a1a49ee7346ff0ef44ee886942fb6526d_28' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><munder><mo>∪</mo><mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></munder><msub><mi>U</mi> <mi>i</mi></msub><mo>=</mo><mi>X</mi></mrow><annotation encoding='application/x-tex'>\underset{i \in I}{\cup} U_i = X</annotation></semantics></math> is equivalent to <math class='maruku-mathml' display='inline' id='mathml_ad54570a1a49ee7346ff0ef44ee886942fb6526d_29' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><munder><mo>∩</mo><mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></munder><msub><mi>C</mi> <mi>i</mi></msub><mo>=</mo><mi>∅</mi></mrow><annotation encoding='application/x-tex'>\underset{i \in I}{\cap} C_i = \emptyset</annotation></semantics></math>. The second statement is that there is then a finite subset <math class='maruku-mathml' display='inline' id='mathml_ad54570a1a49ee7346ff0ef44ee886942fb6526d_30' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>J</mi><mo>⊂</mo><mi>I</mi></mrow><annotation encoding='application/x-tex'>J \subset I</annotation></semantics></math> such that also <math class='maruku-mathml' display='inline' id='mathml_ad54570a1a49ee7346ff0ef44ee886942fb6526d_31' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><munder><mo>∩</mo><mrow><mi>i</mi><mo>∈</mo><mi>J</mi><mo>⊂</mo><mi>I</mi></mrow></munder><msub><mi>C</mi> <mi>i</mi></msub><mo>=</mo><mi>∅</mi></mrow><annotation encoding='application/x-tex'>\underset{i \in J \subset I}{\cap} C_i = \emptyset</annotation></semantics></math>, and under forming complements again this is equivalently the first statement.</p> <p>Then statement 3 is the <a class='existingWikiWord' href='/nlab/show/diff/contrapositive'>contraposition</a> of the second, and contrapositives are equivalent under <a class='existingWikiWord' href='/nlab/show/diff/excluded+middle'>excluded middle</a>.</p> </div> <p>The closed-subset formulations of compactness appear frequently and are often more convenient. For example, <a class='existingWikiWord' href='/nlab/show/diff/compactness+theorem'>compactness theorems</a> in <a class='existingWikiWord' href='/nlab/show/diff/model+theory'>model theory</a> draw on a connection between the finite intersection property and finite satisfiability of sets of axioms.</p> <h3 id='compactness_for_locales'>Compactness for locales</h3> <p>In another direction, the definition (<a class='maruku-ref' href='#hb'>2</a>) also works for <a class='existingWikiWord' href='/nlab/show/diff/locale'>locales</a>, since it refers only to the <a class='existingWikiWord' href='/nlab/show/diff/frame'>frame</a> of open sets. Here is an equivalent way to phrase it that is often convenient for locale theory.</p> <div class='num_prop' id='directed'> <h6 id='proposition_2'>Proposition</h6> <p><math class='maruku-mathml' display='inline' id='mathml_ad54570a1a49ee7346ff0ef44ee886942fb6526d_32' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> is compact iff given any <a class='existingWikiWord' href='/nlab/show/diff/direction'>directed</a> collection of opens whose union is <math class='maruku-mathml' display='inline' id='mathml_ad54570a1a49ee7346ff0ef44ee886942fb6526d_33' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> (a directed open cover), <math class='maruku-mathml' display='inline' id='mathml_ad54570a1a49ee7346ff0ef44ee886942fb6526d_34' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> belongs to the collection.</p> </div> <div class='proof'> <h6 id='proof_2'>Proof</h6> <p>For the “only if” direction: if <math class='maruku-mathml' display='inline' id='mathml_ad54570a1a49ee7346ff0ef44ee886942fb6526d_35' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>𝒰</mi><mo>=</mo><mo stretchy='false'>{</mo><msub><mi>U</mi> <mi>i</mi></msub><msub><mo stretchy='false'>}</mo> <mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></msub></mrow><annotation encoding='application/x-tex'>\mathcal{U}= \{U_i\}_{i \in I}</annotation></semantics></math> is a directed open cover, then by the open-cover definition of compactness, <math class='maruku-mathml' display='inline' id='mathml_ad54570a1a49ee7346ff0ef44ee886942fb6526d_36' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> is the union of finitely many <math class='maruku-mathml' display='inline' id='mathml_ad54570a1a49ee7346ff0ef44ee886942fb6526d_37' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>U</mi> <mi>i</mi></msub></mrow><annotation encoding='application/x-tex'>U_i</annotation></semantics></math>. But by definition of directedness, any finite subfamily of <math class='maruku-mathml' display='inline' id='mathml_ad54570a1a49ee7346ff0ef44ee886942fb6526d_38' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>𝒰</mi></mrow><annotation encoding='application/x-tex'>\mathcal{U}</annotation></semantics></math> has an upper bound in <math class='maruku-mathml' display='inline' id='mathml_ad54570a1a49ee7346ff0ef44ee886942fb6526d_39' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>𝒰</mi></mrow><annotation encoding='application/x-tex'>\mathcal{U}</annotation></semantics></math>; since the only upper bound of <math class='maruku-mathml' display='inline' id='mathml_ad54570a1a49ee7346ff0ef44ee886942fb6526d_40' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> is <math class='maruku-mathml' display='inline' id='mathml_ad54570a1a49ee7346ff0ef44ee886942fb6526d_41' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math>, it follows that <math class='maruku-mathml' display='inline' id='mathml_ad54570a1a49ee7346ff0ef44ee886942fb6526d_42' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> belongs to <math class='maruku-mathml' display='inline' id='mathml_ad54570a1a49ee7346ff0ef44ee886942fb6526d_43' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>𝒰</mi></mrow><annotation encoding='application/x-tex'>\mathcal{U}</annotation></semantics></math>.</p> <p>For the “if” direction: given an open cover <math class='maruku-mathml' display='inline' id='mathml_ad54570a1a49ee7346ff0ef44ee886942fb6526d_44' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>𝒰</mi></mrow><annotation encoding='application/x-tex'>\mathcal{U}</annotation></semantics></math> of <math class='maruku-mathml' display='inline' id='mathml_ad54570a1a49ee7346ff0ef44ee886942fb6526d_45' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math>, let <math class='maruku-mathml' display='inline' id='mathml_ad54570a1a49ee7346ff0ef44ee886942fb6526d_46' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>𝒰</mi><mo>′</mo></mrow><annotation encoding='application/x-tex'>\mathcal{U}'</annotation></semantics></math> be the family of open sets that are unions of finite subfamilies of <math class='maruku-mathml' display='inline' id='mathml_ad54570a1a49ee7346ff0ef44ee886942fb6526d_47' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>𝒰</mi></mrow><annotation encoding='application/x-tex'>\mathcal{U}</annotation></semantics></math>. This <math class='maruku-mathml' display='inline' id='mathml_ad54570a1a49ee7346ff0ef44ee886942fb6526d_48' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>𝒰</mi><mo>′</mo></mrow><annotation encoding='application/x-tex'>\mathcal{U}'</annotation></semantics></math> is clearly directed, and an open cover of <math class='maruku-mathml' display='inline' id='mathml_ad54570a1a49ee7346ff0ef44ee886942fb6526d_49' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> since <math class='maruku-mathml' display='inline' id='mathml_ad54570a1a49ee7346ff0ef44ee886942fb6526d_50' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>𝒰</mi></mrow><annotation encoding='application/x-tex'>\mathcal{U}</annotation></semantics></math> is. By hypothesis, <math class='maruku-mathml' display='inline' id='mathml_ad54570a1a49ee7346ff0ef44ee886942fb6526d_51' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> belongs to <math class='maruku-mathml' display='inline' id='mathml_ad54570a1a49ee7346ff0ef44ee886942fb6526d_52' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>𝒰</mi><mo>′</mo></mrow><annotation encoding='application/x-tex'>\mathcal{U}'</annotation></semantics></math>, so <math class='maruku-mathml' display='inline' id='mathml_ad54570a1a49ee7346ff0ef44ee886942fb6526d_53' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> is a union of finitely many elements of <math class='maruku-mathml' display='inline' id='mathml_ad54570a1a49ee7346ff0ef44ee886942fb6526d_54' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>𝒰</mi></mrow><annotation encoding='application/x-tex'>\mathcal{U}</annotation></semantics></math>. This shows <math class='maruku-mathml' display='inline' id='mathml_ad54570a1a49ee7346ff0ef44ee886942fb6526d_55' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> is compact according to Definition <a class='maruku-ref' href='#hb'>2</a>.</p> </div> <p>As the union is a <a class='existingWikiWord' href='/nlab/show/diff/colimit'>colimit</a> in the <a class='existingWikiWord' href='/nlab/show/diff/category+of+open+subsets'>category of open subsets</a> <math class='maruku-mathml' display='inline' id='mathml_ad54570a1a49ee7346ff0ef44ee886942fb6526d_56' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Op</mi><mo stretchy='false'>(</mo><mi>X</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>Op(X)</annotation></semantics></math>, we can also say</p> <div class='num_prop' id='object'> <h6 id='proposition_3'>Proposition</h6> <p><math class='maruku-mathml' display='inline' id='mathml_ad54570a1a49ee7346ff0ef44ee886942fb6526d_57' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> is compact iff it is a <a class='existingWikiWord' href='/nlab/show/diff/compact+object'>compact object</a> in <math class='maruku-mathml' display='inline' id='mathml_ad54570a1a49ee7346ff0ef44ee886942fb6526d_58' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Op</mi><mo stretchy='false'>(</mo><mi>X</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>Op(X)</annotation></semantics></math>.</p> </div> <div class='proof'> <h6 id='proof_3'>Proof</h6> <p>To say <math class='maruku-mathml' display='inline' id='mathml_ad54570a1a49ee7346ff0ef44ee886942fb6526d_59' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> is a compact object means that <math class='maruku-mathml' display='inline' id='mathml_ad54570a1a49ee7346ff0ef44ee886942fb6526d_60' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>hom</mi><mo stretchy='false'>(</mo><mi>X</mi><mo>,</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mo stretchy='false'>)</mo><mo>:</mo><mi>Op</mi><mo stretchy='false'>(</mo><mi>X</mi><mo stretchy='false'>)</mo><mo>→</mo><mi>Set</mi></mrow><annotation encoding='application/x-tex'>\hom(X, -): Op(X) \to Set</annotation></semantics></math> preserves <a class='existingWikiWord' href='/nlab/show/diff/filtered+colimit'>filtered colimits</a>, or colimits of filtered diagrams. Here we may as well replace <math class='maruku-mathml' display='inline' id='mathml_ad54570a1a49ee7346ff0ef44ee886942fb6526d_61' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Set</mi></mrow><annotation encoding='application/x-tex'>Set</annotation></semantics></math> by the full subcategory consisting of <math class='maruku-mathml' display='inline' id='mathml_ad54570a1a49ee7346ff0ef44ee886942fb6526d_62' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>0</mn><mo>,</mo><mn>1</mn></mrow><annotation encoding='application/x-tex'>0, 1</annotation></semantics></math> (a <math class='maruku-mathml' display='inline' id='mathml_ad54570a1a49ee7346ff0ef44ee886942fb6526d_63' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>0</mn></mrow><annotation encoding='application/x-tex'>0</annotation></semantics></math>-element set or a <math class='maruku-mathml' display='inline' id='mathml_ad54570a1a49ee7346ff0ef44ee886942fb6526d_64' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>1</mn></mrow><annotation encoding='application/x-tex'>1</annotation></semantics></math>-element set) since the hom-sets for posets are of size <math class='maruku-mathml' display='inline' id='mathml_ad54570a1a49ee7346ff0ef44ee886942fb6526d_65' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>0</mn></mrow><annotation encoding='application/x-tex'>0</annotation></semantics></math> or <math class='maruku-mathml' display='inline' id='mathml_ad54570a1a49ee7346ff0ef44ee886942fb6526d_66' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>1</mn></mrow><annotation encoding='application/x-tex'>1</annotation></semantics></math>. Further, in a <a class='existingWikiWord' href='/nlab/show/diff/partial+order'>poset</a> like <math class='maruku-mathml' display='inline' id='mathml_ad54570a1a49ee7346ff0ef44ee886942fb6526d_67' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Op</mi><mo stretchy='false'>(</mo><mi>X</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>Op(X)</annotation></semantics></math>, a filtered diagram is the same as a nonempty directed diagram <math class='maruku-mathml' display='inline' id='mathml_ad54570a1a49ee7346ff0ef44ee886942fb6526d_68' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>D</mi></mrow><annotation encoding='application/x-tex'>D</annotation></semantics></math>, and <math class='maruku-mathml' display='inline' id='mathml_ad54570a1a49ee7346ff0ef44ee886942fb6526d_69' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>hom</mi><mo stretchy='false'>(</mo><mi>X</mi><mo>,</mo><msub><mo lspace='thinmathspace' rspace='thinmathspace'>⋃</mo> <mrow><mi>d</mi><mo>∈</mo><mi>D</mi></mrow></msub><mi>d</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>\hom(X, \bigcup_{d \in D} d)</annotation></semantics></math> has size <math class='maruku-mathml' display='inline' id='mathml_ad54570a1a49ee7346ff0ef44ee886942fb6526d_70' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>1</mn></mrow><annotation encoding='application/x-tex'>1</annotation></semantics></math> iff <math class='maruku-mathml' display='inline' id='mathml_ad54570a1a49ee7346ff0ef44ee886942fb6526d_71' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi><mo>=</mo><msub><mo lspace='thinmathspace' rspace='thinmathspace'>⋃</mo> <mrow><mi>d</mi><mo>∈</mo><mi>D</mi></mrow></msub><mi>d</mi></mrow><annotation encoding='application/x-tex'>X = \bigcup_{d \in D} d</annotation></semantics></math>. On the other hand, <math class='maruku-mathml' display='inline' id='mathml_ad54570a1a49ee7346ff0ef44ee886942fb6526d_72' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mo lspace='thinmathspace' rspace='thinmathspace'>⋃</mo> <mrow><mi>d</mi><mo>∈</mo><mi>D</mi></mrow></msub><mi>hom</mi><mo stretchy='false'>(</mo><mi>X</mi><mo>,</mo><mi>d</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>\bigcup_{d \in D} \hom(X, d)</annotation></semantics></math> has size <math class='maruku-mathml' display='inline' id='mathml_ad54570a1a49ee7346ff0ef44ee886942fb6526d_73' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>1</mn></mrow><annotation encoding='application/x-tex'>1</annotation></semantics></math> iff <math class='maruku-mathml' display='inline' id='mathml_ad54570a1a49ee7346ff0ef44ee886942fb6526d_74' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi><mo>=</mo><mi>d</mi></mrow><annotation encoding='application/x-tex'>X = d</annotation></semantics></math> for some <math class='maruku-mathml' display='inline' id='mathml_ad54570a1a49ee7346ff0ef44ee886942fb6526d_75' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>d</mi></mrow><annotation encoding='application/x-tex'>d</annotation></semantics></math> in <math class='maruku-mathml' display='inline' id='mathml_ad54570a1a49ee7346ff0ef44ee886942fb6526d_76' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>D</mi></mrow><annotation encoding='application/x-tex'>D</annotation></semantics></math>. Thus that <math class='maruku-mathml' display='inline' id='mathml_ad54570a1a49ee7346ff0ef44ee886942fb6526d_77' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>hom</mi><mo stretchy='false'>(</mo><mi>X</mi><mo>,</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>\hom(X, -)</annotation></semantics></math> preserves filtered colimits amounts to the same thing as saying <math class='maruku-mathml' display='inline' id='mathml_ad54570a1a49ee7346ff0ef44ee886942fb6526d_78' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> belongs to any directed open cover <math class='maruku-mathml' display='inline' id='mathml_ad54570a1a49ee7346ff0ef44ee886942fb6526d_79' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>D</mi></mrow><annotation encoding='application/x-tex'>D</annotation></semantics></math>, which is the same as compactness by Proposition <a class='maruku-ref' href='#directed'>2</a>.</p> </div> <h3 id='compactness_via_convergence'>Compactness via convergence</h3> <p>If the <a class='existingWikiWord' href='/nlab/show/diff/ultrafilter+theorem'>ultrafilter theorem</a> (a weak form of the <a class='existingWikiWord' href='/nlab/show/diff/axiom+of+choice'>axiom of choice</a>) is assumed, compactness may be characterized in terms of <a class='existingWikiWord' href='/nlab/show/diff/ultrafilter'>ultrafilter</a> (or ultranet) <a class='existingWikiWord' href='/nlab/show/diff/convergence'>convergence</a>:</p> <div class='num_prop' id='ultrafilter'> <h6 id='proposition_4'>Proposition</h6> <p><math class='maruku-mathml' display='inline' id='mathml_ad54570a1a49ee7346ff0ef44ee886942fb6526d_80' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> is compact iff every <a class='existingWikiWord' href='/nlab/show/diff/ultrafilter'>ultrafilter</a> <math class='maruku-mathml' display='inline' id='mathml_ad54570a1a49ee7346ff0ef44ee886942fb6526d_81' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>𝒰</mi></mrow><annotation encoding='application/x-tex'>\mathcal{U}</annotation></semantics></math> (or ultranet <math class='maruku-mathml' display='inline' id='mathml_ad54570a1a49ee7346ff0ef44ee886942fb6526d_82' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>ν</mi></mrow><annotation encoding='application/x-tex'>\nu</annotation></semantics></math>) on <math class='maruku-mathml' display='inline' id='mathml_ad54570a1a49ee7346ff0ef44ee886942fb6526d_83' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> <a class='existingWikiWord' href='/nlab/show/diff/convergence'>converges</a> to some point <math class='maruku-mathml' display='inline' id='mathml_ad54570a1a49ee7346ff0ef44ee886942fb6526d_84' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>x</mi><mo>∈</mo><mi>X</mi></mrow><annotation encoding='application/x-tex'>x \in X</annotation></semantics></math>, meaning that <math class='maruku-mathml' display='inline' id='mathml_ad54570a1a49ee7346ff0ef44ee886942fb6526d_85' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>𝒰</mi></mrow><annotation encoding='application/x-tex'>\mathcal{U}</annotation></semantics></math> contains the filter of <a class='existingWikiWord' href='/nlab/show/diff/neighborhood'>neighborhoods</a> of <math class='maruku-mathml' display='inline' id='mathml_ad54570a1a49ee7346ff0ef44ee886942fb6526d_86' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>x</mi></mrow><annotation encoding='application/x-tex'>x</annotation></semantics></math> (or that <math class='maruku-mathml' display='inline' id='mathml_ad54570a1a49ee7346ff0ef44ee886942fb6526d_87' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>ν</mi></mrow><annotation encoding='application/x-tex'>\nu</annotation></semantics></math> is eventually in any neighbourhood of <math class='maruku-mathml' display='inline' id='mathml_ad54570a1a49ee7346ff0ef44ee886942fb6526d_88' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>x</mi></mrow><annotation encoding='application/x-tex'>x</annotation></semantics></math>).</p> </div> <p>In any case, if <a class='existingWikiWord' href='/nlab/show/diff/excluded+middle'>excluded middle</a> is assumed, compactness can be characterized in terms of <a class='existingWikiWord' href='/nlab/show/diff/filter'>proper filter</a> or equivalently (see at <em><a class='existingWikiWord' href='/nlab/show/diff/eventuality+filter'>eventuality filter</a></em>) of <a class='existingWikiWord' href='/nlab/show/diff/net'>net</a> <a class='existingWikiWord' href='/nlab/show/diff/convergence'>convergence</a> .</p> <div class='num_prop' id='refinement'> <h6 id='proposition_5'>Proposition</h6> <p>Assuming <a class='existingWikiWord' href='/nlab/show/diff/excluded+middle'>excluded middle</a>, <math class='maruku-mathml' display='inline' id='mathml_ad54570a1a49ee7346ff0ef44ee886942fb6526d_89' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> is compact iff every <a class='existingWikiWord' href='/nlab/show/diff/filter'>proper filter</a>/<a class='existingWikiWord' href='/nlab/show/diff/net'>net</a> on <math class='maruku-mathml' display='inline' id='mathml_ad54570a1a49ee7346ff0ef44ee886942fb6526d_90' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> has a <a class='existingWikiWord' href='/nlab/show/diff/convergence'>convergent</a> proper <a class='existingWikiWord' href='/nlab/show/diff/refinement'>refinement</a>/<a class='existingWikiWord' href='/nlab/show/diff/subnet'>subnet</a>.</p> </div> <p>This is equivalent to the characterization given in the <a href='#Idea'>Idea-section</a> above:</p> <div class='num_prop' id='clustering'> <h6 id='proposition_6'>Proposition</h6> <p>Assuming <a class='existingWikiWord' href='/nlab/show/diff/excluded+middle'>excluded middle</a>, <math class='maruku-mathml' display='inline' id='mathml_ad54570a1a49ee7346ff0ef44ee886942fb6526d_91' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> is compact iff every proper filter <math class='maruku-mathml' display='inline' id='mathml_ad54570a1a49ee7346ff0ef44ee886942fb6526d_92' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>𝒰</mi></mrow><annotation encoding='application/x-tex'>\mathcal{U}</annotation></semantics></math> (or net <math class='maruku-mathml' display='inline' id='mathml_ad54570a1a49ee7346ff0ef44ee886942fb6526d_93' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>ν</mi></mrow><annotation encoding='application/x-tex'>\nu</annotation></semantics></math>) on <math class='maruku-mathml' display='inline' id='mathml_ad54570a1a49ee7346ff0ef44ee886942fb6526d_94' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> has a cluster point <math class='maruku-mathml' display='inline' id='mathml_ad54570a1a49ee7346ff0ef44ee886942fb6526d_95' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>x</mi></mrow><annotation encoding='application/x-tex'>x</annotation></semantics></math>, meaning that every element of <math class='maruku-mathml' display='inline' id='mathml_ad54570a1a49ee7346ff0ef44ee886942fb6526d_96' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>𝒰</mi></mrow><annotation encoding='application/x-tex'>\mathcal{U}</annotation></semantics></math> meets (has <a class='existingWikiWord' href='/nlab/show/diff/inhabited+set'>inhabited</a> intersection with) every neighbourhood of <math class='maruku-mathml' display='inline' id='mathml_ad54570a1a49ee7346ff0ef44ee886942fb6526d_97' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>x</mi></mrow><annotation encoding='application/x-tex'>x</annotation></semantics></math> (or <math class='maruku-mathml' display='inline' id='mathml_ad54570a1a49ee7346ff0ef44ee886942fb6526d_98' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>ν</mi></mrow><annotation encoding='application/x-tex'>\nu</annotation></semantics></math> is frequently in every neighbourhood of <math class='maruku-mathml' display='inline' id='mathml_ad54570a1a49ee7346ff0ef44ee886942fb6526d_99' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>x</mi></mrow><annotation encoding='application/x-tex'>x</annotation></semantics></math>).</p> </div> <p>While the usual definitions (<a class='maruku-ref' href='#hb'>2</a>&<a class='maruku-ref' href='#fip'>1</a>) are for <a class='existingWikiWord' href='/nlab/show/diff/topological+space'>topological spaces</a>, the convergence definitions (<a class='maruku-ref' href='#ultrafilter'>4</a>–<a class='maruku-ref' href='#clustering'>6</a>) make sense in any <a class='existingWikiWord' href='/nlab/show/diff/convergence+space'>convergence space</a>. However, in <a class='existingWikiWord' href='/nlab/show/diff/constructive+mathematics'>constructive mathematics</a>, that every <a class='existingWikiWord' href='/nlab/show/diff/topological+space'>topological space</a> satisfying (<a class='maruku-ref' href='#hb'>2</a>&<a class='maruku-ref' href='#fip'>1</a>) also satisfies (<a class='maruku-ref' href='#refinement'>5</a>&<a class='maruku-ref' href='#clustering'>6</a>) as a <a class='existingWikiWord' href='/nlab/show/diff/convergence+space'>convergence space</a> is the same as <a class='existingWikiWord' href='/nlab/show/diff/excluded+middle'>excluded middle</a>, see <a class='existingWikiWord' href='/nlab/show/diff/compact+spaces+equivalently+have+converging+subnets#Failure in constructive mathematics'>compact spaces equivalently have converging subnets</a> for more details.</p> <h3 id='compactness_via_completeness'>Compactness via completeness</h3> <div id='completeness'> <p>A <a class='existingWikiWord' href='/nlab/show/diff/uniform+space'>uniform space</a> <math class='maruku-mathml' display='inline' id='mathml_ad54570a1a49ee7346ff0ef44ee886942fb6526d_100' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> is compact if and only if it is <a class='existingWikiWord' href='/nlab/show/diff/complete+space'>complete</a> and <a class='existingWikiWord' href='/nlab/show/diff/totally+bounded+space'>totally bounded</a>. Moreover, a compact Hausdorff topological space has a unique compatible uniformity, which is complete and totally bounded.</p> </div> <p>In <a class='existingWikiWord' href='/nlab/show/diff/constructive+mathematics'>constructive mathematics</a>, “complete and totally bounded” is sometimes taken as a substitute for open-cover compactness (to which it is no longer equivalent); see <a class='existingWikiWord' href='/nlab/show/diff/Bishop-compact+space'>Bishop-compact space</a>.</p> <h3 id='compactness_via_stability_properties'>Compactness via stability properties</h3> <p>Frequently in <a class='existingWikiWord' href='/nlab/show/diff/category+theory'>category theory</a>, for example when we discuss <a class='existingWikiWord' href='/nlab/show/diff/internal+logic'>internal logic</a> in <a class='existingWikiWord' href='/nlab/show/diff/topos'>toposes</a>, we are interested in properties of maps that are stable under <a class='existingWikiWord' href='/nlab/show/diff/pullback'>pullback</a>, and it turns out that compactness can be reformulated in terms of stability properties.</p> <p>A first example concerns the property of a topological space <math class='maruku-mathml' display='inline' id='mathml_ad54570a1a49ee7346ff0ef44ee886942fb6526d_101' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> that the unique map <math class='maruku-mathml' display='inline' id='mathml_ad54570a1a49ee7346ff0ef44ee886942fb6526d_102' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo>!</mo><mo>:</mo><mi>X</mi><mo>→</mo><mn>1</mn></mrow><annotation encoding='application/x-tex'>!: X \to 1</annotation></semantics></math> to the <a class='existingWikiWord' href='/nlab/show/diff/point+space'>point space</a> (the <a class='existingWikiWord' href='/nlab/show/diff/terminal+object'>terminal object</a> in <a class='existingWikiWord' href='/nlab/show/diff/Top'>Top</a>) is a <a class='existingWikiWord' href='/nlab/show/diff/closed+map'>closed map</a>. As a statement in ordinary <a class='existingWikiWord' href='/nlab/show/diff/general+topology'>point-set topology</a>, this is plainly a <a class='existingWikiWord' href='/nlab/show/diff/tautology'>tautology</a>, trivially true for any space <math class='maruku-mathml' display='inline' id='mathml_ad54570a1a49ee7346ff0ef44ee886942fb6526d_103' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math>. (Side remark: it is not at all a tautology in the more general setting of <a class='existingWikiWord' href='/nlab/show/diff/locale'>internal locales</a> in <a class='existingWikiWord' href='/nlab/show/diff/topos'>toposes</a>; the word “closed” is reserved for locales <math class='maruku-mathml' display='inline' id='mathml_ad54570a1a49ee7346ff0ef44ee886942fb6526d_104' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> having that property. See also <em><a class='existingWikiWord' href='/nlab/show/diff/closed+morphism'>closed morphism</a></em>.) However, even in ordinary point-set topology, <math class='maruku-mathml' display='inline' id='mathml_ad54570a1a49ee7346ff0ef44ee886942fb6526d_105' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo>!</mo><mo>:</mo><mi>X</mi><mo>→</mo><mn>1</mn></mrow><annotation encoding='application/x-tex'>!: X \to 1</annotation></semantics></math> is usually not <em>stably closed</em>. In more detail: the pullback of <math class='maruku-mathml' display='inline' id='mathml_ad54570a1a49ee7346ff0ef44ee886942fb6526d_106' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo>!</mo><mo>:</mo><mi>X</mi><mo>→</mo><mn>1</mn></mrow><annotation encoding='application/x-tex'>!: X \to 1</annotation></semantics></math> along a (or the) map <math class='maruku-mathml' display='inline' id='mathml_ad54570a1a49ee7346ff0ef44ee886942fb6526d_107' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Y</mi><mo>→</mo><mn>1</mn></mrow><annotation encoding='application/x-tex'>Y \to 1</annotation></semantics></math> is the <a class='existingWikiWord' href='/nlab/show/diff/projection'>projection</a> map</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_ad54570a1a49ee7346ff0ef44ee886942fb6526d_108' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi><mo>×</mo><mi>Y</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding='application/x-tex'>X \times Y \to Y</annotation></semantics></math></div> <p>out of the <a class='existingWikiWord' href='/nlab/show/diff/product+topological+space'>product topological space</a> and the issue is whether this map is closed. This leads us to the following proposition.</p> <div class='num_defn' id='projection'> <h6 id='proposition_7'>Proposition</h6> <p><strong>(<a class='existingWikiWord' href='/nlab/show/diff/closed-projection+characterization+of+compactness'>closed-projection characterization of compactness</a>)</strong></p> <p>A topological space <math class='maruku-mathml' display='inline' id='mathml_ad54570a1a49ee7346ff0ef44ee886942fb6526d_109' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> is compact (def. <a class='maruku-ref' href='#hb'>2</a>) precisely if for any topological space <math class='maruku-mathml' display='inline' id='mathml_ad54570a1a49ee7346ff0ef44ee886942fb6526d_110' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Y</mi></mrow><annotation encoding='application/x-tex'>Y</annotation></semantics></math>, the <a class='existingWikiWord' href='/nlab/show/diff/projection'>projection map</a> <math class='maruku-mathml' display='inline' id='mathml_ad54570a1a49ee7346ff0ef44ee886942fb6526d_111' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi><mo>×</mo><mi>Y</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding='application/x-tex'>X \times Y \to Y</annotation></semantics></math> out of their <a class='existingWikiWord' href='/nlab/show/diff/product+topological+space'>product topological space</a> is <a class='existingWikiWord' href='/nlab/show/diff/closed+map'>closed</a>.</p> </div> <p>Thus, compactness of <math class='maruku-mathml' display='inline' id='mathml_ad54570a1a49ee7346ff0ef44ee886942fb6526d_112' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> is equivalent to <math class='maruku-mathml' display='inline' id='mathml_ad54570a1a49ee7346ff0ef44ee886942fb6526d_113' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo>!</mo><mo>:</mo><mi>X</mi><mo>→</mo><mn>1</mn></mrow><annotation encoding='application/x-tex'>!: X \to 1</annotation></semantics></math> being stably closed. For a <strong>proof</strong>, see <em><a class='existingWikiWord' href='/nlab/show/diff/closed-projection+characterization+of+compactness'>closed-projection characterization of compactness</a></em>.</p> <div class='num_remark'> <h6 id='remark_2'>Remark</h6> <p>Contrary to possible appearance, the equivalence in prop. <a class='maruku-ref' href='#projection'>3</a> does not require the <a class='existingWikiWord' href='/nlab/show/diff/axiom+of+choice'>axiom of choice</a>; see <a href='http://mathoverflow.net/questions/42186/does-compact-iff-projections-are-closed-require-some-form-of-choice/42196'>this MO question</a> and answers, as well as <a href='/toddtrimble/published/Characterizations+of+compactness'>this page</a>. See also the page <a class='existingWikiWord' href='/nlab/show/diff/compactness+and+stable+closure'>compactness and stable closure</a> (under construction). This equivalence is also true for <a class='existingWikiWord' href='/nlab/show/diff/locale'>locales</a>, by way of proper maps; see below.</p> <p>Moreover, the characterization of compact spaces via prop. <a class='maruku-ref' href='#projection'>3</a> may be used to give an attractive proof of the <a class='existingWikiWord' href='/nlab/show/diff/Tychonoff+theorem'>Tychonoff theorem</a>, due to Clementino and Tholen. See <a href='https://ncatlab.org/nlab/show/closed-projection%20characterization%20of%20compactness'>here</a> for details.</p> </div> <p>Of course, the notion of being stably closed applies to maps <math class='maruku-mathml' display='inline' id='mathml_ad54570a1a49ee7346ff0ef44ee886942fb6526d_114' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>p</mi><mo>:</mo><mi>X</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding='application/x-tex'>p: X \to Y</annotation></semantics></math> besides <math class='maruku-mathml' display='inline' id='mathml_ad54570a1a49ee7346ff0ef44ee886942fb6526d_115' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo>!</mo><mo>:</mo><mi>X</mi><mo>→</mo><mn>1</mn></mrow><annotation encoding='application/x-tex'>!: X \to 1</annotation></semantics></math>. An analysis of this notion (that the pullback <math class='maruku-mathml' display='inline' id='mathml_ad54570a1a49ee7346ff0ef44ee886942fb6526d_116' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>f</mi> <mo>*</mo></msup><mi>p</mi><mo>:</mo><mi>X</mi><msub><mo>×</mo> <mi>Y</mi></msub><mi>Z</mi><mo>→</mo><mi>Z</mi></mrow><annotation encoding='application/x-tex'>f^\ast p: X \times_Y Z \to Z</annotation></semantics></math> is a closed map for every map <math class='maruku-mathml' display='inline' id='mathml_ad54570a1a49ee7346ff0ef44ee886942fb6526d_117' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>f</mi><mo>:</mo><mi>Z</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding='application/x-tex'>f: Z \to Y</annotation></semantics></math>) leads to the correct notion of <a class='existingWikiWord' href='/nlab/show/diff/proper+map'>proper map</a> in <a class='existingWikiWord' href='/nlab/show/diff/algebraic+geometry'>algebraic geometry</a> and elsewhere.</p> <p>Closely related to <a class='existingWikiWord' href='/nlab/show/diff/closed-projection+characterization+of+compactness'>closed-projection characterization of compactness</a>, a characterisation of compactness in terms of logical <a class='existingWikiWord' href='/nlab/show/diff/quantification'>quantification</a> is featured in <a class='existingWikiWord' href='/nlab/show/diff/Paul+Taylor'>Paul Taylor</a>’s <a class='existingWikiWord' href='/nlab/show/diff/abstract+Stone+duality'>Abstract Stone Duality</a>:</p> <div class='num_prop' id='quantification'> <h6 id='proposition_8'>Proposition</h6> <p><math class='maruku-mathml' display='inline' id='mathml_ad54570a1a49ee7346ff0ef44ee886942fb6526d_118' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> is compact iff for any space <math class='maruku-mathml' display='inline' id='mathml_ad54570a1a49ee7346ff0ef44ee886942fb6526d_119' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Y</mi></mrow><annotation encoding='application/x-tex'>Y</annotation></semantics></math> and any open subset <math class='maruku-mathml' display='inline' id='mathml_ad54570a1a49ee7346ff0ef44ee886942fb6526d_120' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>U</mi></mrow><annotation encoding='application/x-tex'>U</annotation></semantics></math> of <math class='maruku-mathml' display='inline' id='mathml_ad54570a1a49ee7346ff0ef44ee886942fb6526d_121' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi><mo>×</mo><mi>Y</mi></mrow><annotation encoding='application/x-tex'>X \times Y</annotation></semantics></math>, the subset</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_ad54570a1a49ee7346ff0ef44ee886942fb6526d_122' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mo>∀</mo> <mi>X</mi></msub><mi>U</mi><mo>=</mo><mo stretchy='false'>{</mo><mi>b</mi><mo>:</mo><mi>Y</mi><mspace width='thickmathspace' /><mo stretchy='false'>|</mo><mspace width='thickmathspace' /><mo>∀</mo><mspace width='thickmathspace' /><mi>a</mi><mo>:</mo><mi>X</mi><mo>,</mo><mspace width='thickmathspace' /><mo stretchy='false'>(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo stretchy='false'>)</mo><mo>∈</mo><mi>U</mi><mo stretchy='false'>}</mo></mrow><annotation encoding='application/x-tex'> \forall_X U = \{ b : Y \;|\; \forall\; a: X,\; (a, b) \in U \} </annotation></semantics></math></div> <p>is open in <math class='maruku-mathml' display='inline' id='mathml_ad54570a1a49ee7346ff0ef44ee886942fb6526d_123' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Y</mi></mrow><annotation encoding='application/x-tex'>Y</annotation></semantics></math>.</p> </div> <p>To remove it from dependence on points, we can also write the definition like this:</p> <div class='num_defn' id='pointless'> <h6 id='definition_3'>Definition</h6> <p><math class='maruku-mathml' display='inline' id='mathml_ad54570a1a49ee7346ff0ef44ee886942fb6526d_124' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> is compact iff given any space <math class='maruku-mathml' display='inline' id='mathml_ad54570a1a49ee7346ff0ef44ee886942fb6526d_125' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Y</mi></mrow><annotation encoding='application/x-tex'>Y</annotation></semantics></math> and any open <math class='maruku-mathml' display='inline' id='mathml_ad54570a1a49ee7346ff0ef44ee886942fb6526d_126' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>U</mi></mrow><annotation encoding='application/x-tex'>U</annotation></semantics></math> in <math class='maruku-mathml' display='inline' id='mathml_ad54570a1a49ee7346ff0ef44ee886942fb6526d_127' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi><mo>×</mo><mi>Y</mi></mrow><annotation encoding='application/x-tex'>X \times Y</annotation></semantics></math>, there exists an open <math class='maruku-mathml' display='inline' id='mathml_ad54570a1a49ee7346ff0ef44ee886942fb6526d_128' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mo>∀</mo> <mi>X</mi></msub><mi>U</mi></mrow><annotation encoding='application/x-tex'>\forall_X U</annotation></semantics></math> in <math class='maruku-mathml' display='inline' id='mathml_ad54570a1a49ee7346ff0ef44ee886942fb6526d_129' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Y</mi></mrow><annotation encoding='application/x-tex'>Y</annotation></semantics></math> that satisfies the <a class='existingWikiWord' href='/nlab/show/diff/universal+construction'>universal property</a> of universal <a class='existingWikiWord' href='/nlab/show/diff/quantification'>quantification</a><del class='diffdel'>:</del></p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_ad54570a1a49ee7346ff0ef44ee886942fb6526d_130' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>V</mi><mo>⊆</mo><msub><mo>∀</mo> <mi>X</mi></msub><mi>U</mi><mspace width='thickmathspace' /><mo>⇔</mo><mspace width='thickmathspace' /><mi>X</mi><mo>×</mo><mi>V</mi><mo>⊆</mo><mi>U</mi></mrow><annotation encoding='application/x-tex'> V \subseteq \forall_X U \;\Leftrightarrow\; X \times V \subseteq U </annotation></semantics></math></div> <p><span> for every<del class='diffmod'> open</del><ins class='diffmod'> (not</ins><ins class='diffins'> necessarily</ins><ins class='diffins'> open)</ins></span><math class='maruku-mathml' display='inline' id='mathml_ad54570a1a49ee7346ff0ef44ee886942fb6526d_131' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>V</mi><ins class='diffins'><mo>⊆</mo></ins><ins class='diffins'><mi>Y</mi></ins></mrow><annotation encoding='application/x-tex'><span> V<ins class='diffins'> \subseteq</ins><ins class='diffins'> Y</ins></span></annotation></semantics></math><del class='diffdel'> in </del><del class='diffdel'><math class='maruku-mathml' display='inline' id='mathml_ad54570a1a49ee7346ff0ef44ee886942fb6526d_132' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Y</mi></mrow><annotation encoding='application/x-tex'>Y</annotation></semantics></math></del>.</p><ins class='diffins'> </ins><ins class='diffins'><p>(For a completely point-free definition referring only to open subspaces, see the definitions of <a class='existingWikiWord' href='/nlab/show/diff/proper+morphism'>proper morphism</a> and <a class='existingWikiWord' href='/nlab/show/diff/closed+morphism'>closed morphism</a> of locales.)</p></ins> </div> <p>A <a class='existingWikiWord' href='/nlab/show/diff/De+Morgan+duality'>dual</a> condition is satisfied by an <a class='existingWikiWord' href='/nlab/show/diff/overt+space'>overt space</a>.</p> <h2 id='properties'>Properties</h2> <h3 id='various'>Various</h3> <div class='num_prop' id='UnionsAndIntersectionOfCompactSubspaces'> <h6 id='proposition_9'>Proposition</h6> <p><strong>(unions and intersection of compact spaces)</strong></p> <p>Let <math class='maruku-mathml' display='inline' id='mathml_ad54570a1a49ee7346ff0ef44ee886942fb6526d_133' xmlns='http://www.w3.org/1998/Math/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> be a <a class='existingWikiWord' href='/nlab/show/diff/topological+space'>topological space</a> and let</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_ad54570a1a49ee7346ff0ef44ee886942fb6526d_134' xmlns='http://www.w3.org/1998/Math/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></div> <p>be a <a class='existingWikiWord' href='/nlab/show/diff/set'>set</a> of <a class='existingWikiWord' href='/nlab/show/diff/compact+space'>compact</a> <a class='existingWikiWord' href='/nlab/show/diff/subspace'>subspaces</a>.</p> <ol> <li> <p>If <math class='maruku-mathml' display='inline' id='mathml_ad54570a1a49ee7346ff0ef44ee886942fb6526d_135' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>I</mi></mrow><annotation encoding='application/x-tex'>I</annotation></semantics></math> is a <a class='existingWikiWord' href='/nlab/show/diff/finite+set'>finite set</a>, then the <a class='existingWikiWord' href='/nlab/show/diff/union'>union</a> <math class='maruku-mathml' display='inline' id='mathml_ad54570a1a49ee7346ff0ef44ee886942fb6526d_136' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><munder><mo>∪</mo><mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></munder><msub><mi>K</mi> <mi>i</mi></msub><mo>⊂</mo><mi>X</mi></mrow><annotation encoding='application/x-tex'>\underset{i \in I}{\cup} K_i \subset X</annotation></semantics></math> is itself a compact subspace;</p> </li> <li> <p>If all <math class='maruku-mathml' display='inline' id='mathml_ad54570a1a49ee7346ff0ef44ee886942fb6526d_137' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>K</mi> <mi>i</mi></msub><mo>⊂</mo><mi>X</mi></mrow><annotation encoding='application/x-tex'>K_i \subset X</annotation></semantics></math> are also <a class='existingWikiWord' href='/nlab/show/diff/closed+subspace'>closed subsets</a> then their <a class='existingWikiWord' href='/nlab/show/diff/intersection'>intersection</a> <math class='maruku-mathml' display='inline' id='mathml_ad54570a1a49ee7346ff0ef44ee886942fb6526d_138' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><munder><mo>∩</mo><mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></munder><msub><mi>K</mi> <mi>i</mi></msub><mo>⊂</mo><mi>X</mi></mrow><annotation encoding='application/x-tex'>\underset{i \in I}{\cap} K_i \subset X</annotation></semantics></math> is itself a compact subspace.</p> </li> </ol> </div> <div class='proof'> <h6 id='proof_4'>Proof</h6> <p>Regarding the first statement:</p> <p>Let <math class='maruku-mathml' display='inline' id='mathml_ad54570a1a49ee7346ff0ef44ee886942fb6526d_139' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>{</mo><msub><mi>U</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'>\{U_j \subset X\}_{j \in J}</annotation></semantics></math> be an <a class='existingWikiWord' href='/nlab/show/diff/open+cover'>open cover</a> of the union. Then this is also an open cover of each of the <math class='maruku-mathml' display='inline' id='mathml_ad54570a1a49ee7346ff0ef44ee886942fb6526d_140' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>K</mi> <mi>i</mi></msub></mrow><annotation encoding='application/x-tex'>K_i</annotation></semantics></math>, hence by their compactness there are finite subsets <math class='maruku-mathml' display='inline' id='mathml_ad54570a1a49ee7346ff0ef44ee886942fb6526d_141' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>J</mi> <mi>i</mi></msub><mo>⊂</mo><mi>J</mi></mrow><annotation encoding='application/x-tex'>J_i \subset J</annotation></semantics></math> such that <math class='maruku-mathml' display='inline' id='mathml_ad54570a1a49ee7346ff0ef44ee886942fb6526d_142' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>{</mo><msub><mi>U</mi> <mrow><msub><mi>j</mi> <mi>i</mi></msub></mrow></msub><mo>⊂</mo><mi>X</mi><msub><mo stretchy='false'>}</mo> <mrow><msub><mi>j</mi> <mi>i</mi></msub><mo>∈</mo><msub><mi>J</mi> <mi>i</mi></msub></mrow></msub></mrow><annotation encoding='application/x-tex'>\{U_{j_i} \subset X\}_{j_i \in J_i}</annotation></semantics></math> is a finite cover of <math class='maruku-mathml' display='inline' id='mathml_ad54570a1a49ee7346ff0ef44ee886942fb6526d_143' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>K</mi> <mi>i</mi></msub></mrow><annotation encoding='application/x-tex'>K_i</annotation></semantics></math>. Accordingly then <math class='maruku-mathml' display='inline' id='mathml_ad54570a1a49ee7346ff0ef44ee886942fb6526d_144' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><munder><mo>∪</mo><mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></munder><mo stretchy='false'>{</mo><msub><mi>U</mi> <mrow><msub><mi>j</mi> <mi>i</mi></msub></mrow></msub><mo>⊂</mo><mi>X</mi><msub><mo stretchy='false'>}</mo> <mrow><msub><mi>j</mi> <mi>i</mi></msub><mo>∈</mo><msub><mi>J</mi> <mi>i</mi></msub></mrow></msub></mrow><annotation encoding='application/x-tex'>\underset{i \in I}{\cup} \{ U_{j_i} \subset X \}_{j_i \in J_i}</annotation></semantics></math> is a finite open cover of the union.</p> <p>Regarding the second statement:</p> <p>By the axioms on a <a class='existingWikiWord' href='/nlab/show/diff/topological+space'>topology</a>, the intersection of an arbitrary set of <a class='existingWikiWord' href='/nlab/show/diff/closed+subspace'>closed subsets</a> is again closed. Hence the intersection of the closed compact subspaces is closed. But since <a class='existingWikiWord' href='/nlab/show/diff/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> then for each <math class='maruku-mathml' display='inline' id='mathml_ad54570a1a49ee7346ff0ef44ee886942fb6526d_145' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow><annotation encoding='application/x-tex'>i \in I</annotation></semantics></math> the intersection is a closed subspace of the compact space <math class='maruku-mathml' display='inline' id='mathml_ad54570a1a49ee7346ff0ef44ee886942fb6526d_146' xmlns='http://www.w3.org/1998/Math/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/diff/closed+subspaces+of+compact+spaces+are+compact'>closed subspaces of compact spaces are compact</a> it follows that the intersection is actually compact, too.</p> </div> <div class='num_prop' id='IntersectionCompactWithOpen'> <h6 id='proposition_10'>Proposition</h6> <p><strong>(complements of compact with open subspaces is compact)</strong></p> <p>Let <math class='maruku-mathml' display='inline' id='mathml_ad54570a1a49ee7346ff0ef44ee886942fb6526d_147' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> be a <a class='existingWikiWord' href='/nlab/show/diff/topological+space'>topological space</a>. Let</p> <ol> <li> <p><math class='maruku-mathml' display='inline' id='mathml_ad54570a1a49ee7346ff0ef44ee886942fb6526d_148' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>K</mi><mo>⊂</mo><mi>X</mi></mrow><annotation encoding='application/x-tex'>K\subset X</annotation></semantics></math> be a <a class='existingWikiWord' href='/nlab/show/diff/compact+space'>compact</a> <a class='existingWikiWord' href='/nlab/show/diff/subspace'>subspace</a>;</p> </li> <li> <p><math class='maruku-mathml' display='inline' id='mathml_ad54570a1a49ee7346ff0ef44ee886942fb6526d_149' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>U</mi><mo>⊂</mo><mi>X</mi></mrow><annotation encoding='application/x-tex'>U \subset X</annotation></semantics></math> be an <a class='existingWikiWord' href='/nlab/show/diff/open+subspace'>open subset</a>.</p> </li> </ol> <p>Then the <a class='existingWikiWord' href='/nlab/show/diff/complement'>complement</a></p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_ad54570a1a49ee7346ff0ef44ee886942fb6526d_150' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>K</mi><mo>∖</mo><mi>U</mi><mo>⊂</mo><mi>Xcov</mi></mrow><annotation encoding='application/x-tex'> K \setminus U \subset Xcov </annotation></semantics></math></div> <p>is itself a compact subspace.</p> </div> <div class='proof' id='ProofIntersectionCompactWithOpen'> <h6 id='proof_5'>Proof</h6> <p>Let <math class='maruku-mathml' display='inline' id='mathml_ad54570a1a49ee7346ff0ef44ee886942fb6526d_151' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>{</mo><msub><mi>V</mi> <mi>i</mi></msub><mo>⊂</mo><mi>K</mi><mo>∖</mo><mi>U</mi><mo stretchy='false'>}</mo></mrow><annotation encoding='application/x-tex'>\{V_i \subset K \setminus U\}</annotation></semantics></math> be an <a class='existingWikiWord' href='/nlab/show/diff/open+cover'>open cover</a> of the complement subspace. We need to show that this admits a finite subcover.</p> <p>Observe that by definition of the <a class='existingWikiWord' href='/nlab/show/diff/subspace+topology'>subspace topologies</a> on <math class='maruku-mathml' display='inline' id='mathml_ad54570a1a49ee7346ff0ef44ee886942fb6526d_152' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>K</mi><mo>∖</mo><mi>U</mi></mrow><annotation encoding='application/x-tex'>K \setminus U</annotation></semantics></math> and on <math class='maruku-mathml' display='inline' id='mathml_ad54570a1a49ee7346ff0ef44ee886942fb6526d_153' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>K</mi></mrow><annotation encoding='application/x-tex'>K</annotation></semantics></math>:</p> <ol> <li> <p>the <math class='maruku-mathml' display='inline' id='mathml_ad54570a1a49ee7346ff0ef44ee886942fb6526d_154' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>V</mi> <mi>i</mi></msub><mo>⊂</mo><mi>K</mi><mo>∖</mo><mi>U</mi></mrow><annotation encoding='application/x-tex'>V_i \subset K \setminus U</annotation></semantics></math> are still open when regarded as subsets <math class='maruku-mathml' display='inline' id='mathml_ad54570a1a49ee7346ff0ef44ee886942fb6526d_155' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>V</mi> <mi>i</mi></msub><mo>⊂</mo><mi>K</mi></mrow><annotation encoding='application/x-tex'>V_i \subset K</annotation></semantics></math> of <math class='maruku-mathml' display='inline' id='mathml_ad54570a1a49ee7346ff0ef44ee886942fb6526d_156' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>K</mi></mrow><annotation encoding='application/x-tex'>K</annotation></semantics></math>;</p> </li> <li> <p>also the intersection <math class='maruku-mathml' display='inline' id='mathml_ad54570a1a49ee7346ff0ef44ee886942fb6526d_157' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>K</mi><mo>∩</mo><mi>U</mi><mo>⊂</mo><mi>K</mi></mrow><annotation encoding='application/x-tex'>K \cap U \subset K</annotation></semantics></math> is an open subset of the compact subspace <math class='maruku-mathml' display='inline' id='mathml_ad54570a1a49ee7346ff0ef44ee886942fb6526d_158' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>K</mi></mrow><annotation encoding='application/x-tex'>K</annotation></semantics></math>.</p> </li> </ol> <p>This means that</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_ad54570a1a49ee7346ff0ef44ee886942fb6526d_159' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>{</mo><msub><mi>V</mi> <mi>i</mi></msub><mo>⊂</mo><mi>K</mi><msub><mo stretchy='false'>}</mo> <mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></msub><mo>⊔</mo><mo stretchy='false'>(</mo><mi>U</mi><mo>∩</mo><mi>K</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'> \{ V_i \subset K \}_{i \in I} \sqcup (U \cap K) </annotation></semantics></math></div> <p>is an <a class='existingWikiWord' href='/nlab/show/diff/open+cover'>open cover</a> of <math class='maruku-mathml' display='inline' id='mathml_ad54570a1a49ee7346ff0ef44ee886942fb6526d_160' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>K</mi></mrow><annotation encoding='application/x-tex'>K</annotation></semantics></math>. Hence by the compactness of <math class='maruku-mathml' display='inline' id='mathml_ad54570a1a49ee7346ff0ef44ee886942fb6526d_161' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>K</mi></mrow><annotation encoding='application/x-tex'>K</annotation></semantics></math>, this has a finite subcover. If <math class='maruku-mathml' display='inline' id='mathml_ad54570a1a49ee7346ff0ef44ee886942fb6526d_162' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>U</mi><mo>∩</mo><mi>K</mi></mrow><annotation encoding='application/x-tex'>U \cap K</annotation></semantics></math> is retained in this subcover then it is of the form</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_ad54570a1a49ee7346ff0ef44ee886942fb6526d_163' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>{</mo><msub><mi>V</mi> <mi>j</mi></msub><mo>⊂</mo><mi>K</mi><msub><mo stretchy='false'>}</mo> <mrow><mi>j</mi><mo>⊂</mo><mi>J</mi></mrow></msub><mo>⊔</mo><mo stretchy='false'>(</mo><mi>U</mi><mo>∩</mo><mi>K</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'> \{ V_j \subset K\}_{j \subset J} \sqcup (U \cap K) </annotation></semantics></math></div> <p>otherwise it is of the form</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_ad54570a1a49ee7346ff0ef44ee886942fb6526d_164' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>{</mo><msub><mi>V</mi> <mi>j</mi></msub><mo>⊂</mo><mi>K</mi><msub><mo stretchy='false'>}</mo> <mrow><mi>j</mi><mo>⊂</mo><mi>J</mi></mrow></msub></mrow><annotation encoding='application/x-tex'> \{ V_j \subset K\}_{j \subset J} </annotation></semantics></math></div> <p>with <math class='maruku-mathml' display='inline' id='mathml_ad54570a1a49ee7346ff0ef44ee886942fb6526d_165' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>J</mi><mo>⊂</mo><mi>I</mi></mrow><annotation encoding='application/x-tex'>J \subset I</annotation></semantics></math> a <a class='existingWikiWord' href='/nlab/show/diff/finite+set'>finite</a> <a class='existingWikiWord' href='/nlab/show/diff/subset'>subset</a>.</p> <p>Since either of these is still a cover, also their restriction from <math class='maruku-mathml' display='inline' id='mathml_ad54570a1a49ee7346ff0ef44ee886942fb6526d_166' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>K</mi></mrow><annotation encoding='application/x-tex'>K</annotation></semantics></math> to <math class='maruku-mathml' display='inline' id='mathml_ad54570a1a49ee7346ff0ef44ee886942fb6526d_167' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>K</mi><mo>∖</mo><mi>U</mi></mrow><annotation encoding='application/x-tex'>K \setminus U</annotation></semantics></math> is a cover of <math class='maruku-mathml' display='inline' id='mathml_ad54570a1a49ee7346ff0ef44ee886942fb6526d_168' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>K</mi><mo>∖</mo><mi>U</mi></mrow><annotation encoding='application/x-tex'>K \setminus U</annotation></semantics></math>. But by assumption the <math class='maruku-mathml' display='inline' id='mathml_ad54570a1a49ee7346ff0ef44ee886942fb6526d_169' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>V</mi> <mi>j</mi></msub></mrow><annotation encoding='application/x-tex'>V_j</annotation></semantics></math> are already restricted to <math class='maruku-mathml' display='inline' id='mathml_ad54570a1a49ee7346ff0ef44ee886942fb6526d_170' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>K</mi><mo>∖</mo><mi>U</mi></mrow><annotation encoding='application/x-tex'>K \setminus U</annotation></semantics></math>, while the restriction of <math class='maruku-mathml' display='inline' id='mathml_ad54570a1a49ee7346ff0ef44ee886942fb6526d_171' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mi>U</mi><mo>∩</mo><mi>K</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(U \cap K)</annotation></semantics></math> to <math class='maruku-mathml' display='inline' id='mathml_ad54570a1a49ee7346ff0ef44ee886942fb6526d_172' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>K</mi><mo>∖</mo><mi>U</mi></mrow><annotation encoding='application/x-tex'>K \setminus U</annotation></semantics></math> is empty. Therefore in either of the above cases we find that</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_ad54570a1a49ee7346ff0ef44ee886942fb6526d_173' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>{</mo><msub><mi>V</mi> <mi>j</mi></msub><mo>⊂</mo><mi>K</mi><mo>∖</mo><mi>U</mi><msub><mo stretchy='false'>}</mo> <mrow><mi>j</mi><mo>⊂</mo><mi>J</mi></mrow></msub></mrow><annotation encoding='application/x-tex'> \{ V_j \subset K \setminus U\}_{j \subset J} </annotation></semantics></math></div> <p>is a finite subcover of the original open cover of <math class='maruku-mathml' display='inline' id='mathml_ad54570a1a49ee7346ff0ef44ee886942fb6526d_174' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>K</mi><mo>∖</mo><mi>U</mi></mrow><annotation encoding='application/x-tex'>K \setminus U</annotation></semantics></math>. Therefore <math class='maruku-mathml' display='inline' id='mathml_ad54570a1a49ee7346ff0ef44ee886942fb6526d_175' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>K</mi><mo>∖</mo><mi>U</mi></mrow><annotation encoding='application/x-tex'>K \setminus U</annotation></semantics></math> is compact.</p> </div> <div class='num_prop' id='ProductOfCompactSpacesIsCompact'> <h6 id='proposition_11'>Proposition</h6> <p>Assuming the <a class='existingWikiWord' href='/nlab/show/diff/axiom+of+choice'>axiom of choice</a>, the category of compact spaces admits all small <a class='existingWikiWord' href='/nlab/show/diff/cartesian+product'>products</a>. And, even without the axiom of choice, the category of compact locales admits all small products.</p> </div> <p>\begin{proof} The first statement follows from the <a class='existingWikiWord' href='/nlab/show/diff/Tychonoff+theorem'>Tychonoff theorem</a>: every <a class='existingWikiWord' href='/nlab/show/diff/product+topological+space'>product topological space</a> of compact spaces is itself again compact. \end{proof}</p> <div class='num_remark'> <h6 id='remark_3'>Remark</h6> <p>However, the category of compact spaces does not admit <a class='existingWikiWord' href='/nlab/show/diff/equalizer'>equalizers</a>. An explicit example is where one takes two maps <math class='maruku-mathml' display='inline' id='mathml_ad54570a1a49ee7346ff0ef44ee886942fb6526d_176' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy='false'>]</mo><mo>→</mo><mo stretchy='false'>{</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy='false'>}</mo></mrow><annotation encoding='application/x-tex'>[0, 1] \to \{0, 1\}</annotation></semantics></math> where the codomain is <a class='existingWikiWord' href='/nlab/show/diff/Sierpinski+space'>Sierpinski space</a>, where one of the maps <math class='maruku-mathml' display='inline' id='mathml_ad54570a1a49ee7346ff0ef44ee886942fb6526d_177' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>f</mi></mrow><annotation encoding='application/x-tex'>f</annotation></semantics></math> has <math class='maruku-mathml' display='inline' id='mathml_ad54570a1a49ee7346ff0ef44ee886942fb6526d_178' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>f</mi> <mrow><mo lspace='verythinmathspace' rspace='0em'>−</mo><mn>1</mn></mrow></msup><mo stretchy='false'>(</mo><mn>1</mn><mo stretchy='false'>)</mo><mo>=</mo><mo stretchy='false'>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy='false'>/</mo><mn>2</mn><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>f^{-1}(1) = [0, 1/2)</annotation></semantics></math>, and the other is the constant map at <math class='maruku-mathml' display='inline' id='mathml_ad54570a1a49ee7346ff0ef44ee886942fb6526d_179' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>1</mn></mrow><annotation encoding='application/x-tex'>1</annotation></semantics></math>. If an equalizer in <math class='maruku-mathml' display='inline' id='mathml_ad54570a1a49ee7346ff0ef44ee886942fb6526d_180' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Comp</mi></mrow><annotation encoding='application/x-tex'>Comp</annotation></semantics></math> existed, then <math class='maruku-mathml' display='inline' id='mathml_ad54570a1a49ee7346ff0ef44ee886942fb6526d_181' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>hom</mi><mo stretchy='false'>(</mo><mn>1</mn><mo>,</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mo stretchy='false'>)</mo><mo>:</mo><mi>Comp</mi><mo>→</mo><mi>Set</mi></mrow><annotation encoding='application/x-tex'>\hom(1, -): Comp \to Set</annotation></semantics></math> would have to preserve it, so set-theoretically it would have to be <math class='maruku-mathml' display='inline' id='mathml_ad54570a1a49ee7346ff0ef44ee886942fb6526d_182' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy='false'>/</mo><mn>2</mn><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>[0, 1/2)</annotation></semantics></math>. The topology on <math class='maruku-mathml' display='inline' id='mathml_ad54570a1a49ee7346ff0ef44ee886942fb6526d_183' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy='false'>/</mo><mn>2</mn><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>[0, 1/2)</annotation></semantics></math> would then have to be the same as or finer than the subspace topology in order for the equalizer map to be continuous. But if the subspace topology isn’t compact, then no finer topology would make it compact either. (Here we are invoking the contrapositive of the proposition that if <math class='maruku-mathml' display='inline' id='mathml_ad54570a1a49ee7346ff0ef44ee886942fb6526d_184' xmlns='http://www.w3.org/1998/Math/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 compact and <math class='maruku-mathml' display='inline' id='mathml_ad54570a1a49ee7346ff0ef44ee886942fb6526d_185' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>τ</mi><mo>′</mo><mo>⊆</mo><mi>τ</mi></mrow><annotation encoding='application/x-tex'>\tau' \subseteq \tau</annotation></semantics></math> is a coarser topology, then <math class='maruku-mathml' display='inline' id='mathml_ad54570a1a49ee7346ff0ef44ee886942fb6526d_186' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mi>X</mi><mo>,</mo><mi>τ</mi><mo>′</mo><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(X, \tau')</annotation></semantics></math> is also compact.)</p> <p>On the other hand, the category of <a class='existingWikiWord' href='/nlab/show/diff/compactum'>compact Hausdorff spaces</a> does admit all <a class='existingWikiWord' href='/nlab/show/diff/limit'>limits</a>, and this fact does not require the full strength of the axiom of choice, but only a weaker choice principle called the <a class='existingWikiWord' href='/nlab/show/diff/ultrafilter+theorem'>ultrafilter principle</a>.</p> </div> <div class='num_prop'> <h6 id='proposition_12'>Proposition</h6> <p>The direct <a class='existingWikiWord' href='/nlab/show/diff/image'>image</a> of a compact subspace under a <a class='existingWikiWord' href='/nlab/show/diff/continuous+map'>continuous map</a> is compact. A topological space becomes a <a class='existingWikiWord' href='/nlab/show/diff/bornological+set'>bornological set</a> by taking the bounded sets to be subsets contained in some compact subspace, and under this bornology, every continuous function is a bounded map.</p> <p>If the spaces in question are <math class='maruku-mathml' display='inline' id='mathml_ad54570a1a49ee7346ff0ef44ee886942fb6526d_187' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>T</mi> <mn>1</mn></msub></mrow><annotation encoding='application/x-tex'>T_1</annotation></semantics></math>, then the sets with compact closure also constitute a bornology and continuous maps become bounded. In a non-Hausdorff situation these bornologies might differ because the closure of a compact set need not be compact.</p> </div> <div class='num_prop'> <h6 id='proposition_13'>Proposition</h6> <p>A compact <a class='existingWikiWord' href='/nlab/show/diff/Hausdorff+space'>Hausdorff space</a> must be <a class='existingWikiWord' href='/nlab/show/diff/normal+space'>normal</a>. That is, the <a class='existingWikiWord' href='/nlab/show/diff/separation+axioms'>separation axioms</a> <math class='maruku-mathml' display='inline' id='mathml_ad54570a1a49ee7346ff0ef44ee886942fb6526d_188' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>T</mi> <mn>2</mn></msub></mrow><annotation encoding='application/x-tex'>T_2</annotation></semantics></math> through <math class='maruku-mathml' display='inline' id='mathml_ad54570a1a49ee7346ff0ef44ee886942fb6526d_189' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>T</mi> <mn>4</mn></msub></mrow><annotation encoding='application/x-tex'>T_4</annotation></semantics></math> (when interpreted as an increasing sequence) are equivalent in the presence of compactness.</p> </div> <div class='num_prop'> <h6 id='proposition_14'>Proposition</h6> <p>The <a class='existingWikiWord' href='/nlab/show/diff/Heine-Borel+theorem'>Heine-Borel theorem</a> asserts that a subspace <math class='maruku-mathml' display='inline' id='mathml_ad54570a1a49ee7346ff0ef44ee886942fb6526d_190' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>S</mi><mo>⊂</mo><msup><mi>ℝ</mi> <mi>n</mi></msup></mrow><annotation encoding='application/x-tex'>S \subset \mathbb{R}^n</annotation></semantics></math> of a <a class='existingWikiWord' href='/nlab/show/diff/cartesian+space'>Cartesian space</a> is compact precisely if it is <a class='existingWikiWord' href='/nlab/show/diff/closed+subspace'>closed</a> and <a class='existingWikiWord' href='/nlab/show/diff/bounded+set'>bounded</a>.</p> </div> <div class='num_prop'> <h6 id='proposition_15'>Proposition</h6> <p>We have:</p> <ol> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/compact+subspaces+of+Hausdorff+spaces+are+closed'>compact subspaces of Hausdorff spaces are closed</a>.</p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/closed+subspaces+of+compact+spaces+are+compact'>closed subsets of compact spaces are compact</a></p> </li> </ol> <p>Hence:</p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/closed+subspaces+of+compact+Hausdorff+spaces+are+equivalently+compact+subspaces'>closed subspaces of compact Hausdorff spaces are equivalently compact subspaces</a></li> </ul> </div> <div class='num_prop'> <h6 id='proposition_16'>Proposition</h6> <p><a class='existingWikiWord' href='/nlab/show/diff/sequentially+compact+metric+spaces+are+equivalently+compact+metric+spaces'>sequentially compact metric spaces are equivalently compact metric spaces</a></p> <p><a class='existingWikiWord' href='/nlab/show/diff/countably+compact+metric+spaces+are+equivalently+compact+metric+spaces'>countably compact metric spaces are equivalently compact metric spaces</a></p> <p><a class='existingWikiWord' href='/nlab/show/diff/compact+spaces+equivalently+have+converging+subnets'>compact spaces equivalently have converging subnet of every net</a></p> </div> <div class='num_prop'> <h6 id='proposition_17'>Proposition</h6> <p><a class='existingWikiWord' href='/nlab/show/diff/continuous+images+of+compact+spaces+are+compact'>continuous images of compact spaces are compact</a></p> </div> <div class='num_prop'> <h6 id='proposition_18'>Proposition</h6> <p><a class='existingWikiWord' href='/nlab/show/diff/maps+from+compact+spaces+to+Hausdorff+spaces+are+closed+and+proper'>maps from compact spaces to Hausdorff spaces are closed and proper</a></p> </div> <div class='num_prop'> <h6 id='proposition_19'>Proposition</h6> <p><a class='existingWikiWord' href='/nlab/show/diff/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> </div> <h3 id='relation_to_compact_objects_in_'>Relation to compact objects in <math class='maruku-mathml' display='inline' id='mathml_ad54570a1a49ee7346ff0ef44ee886942fb6526d_191' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Top</mi></mrow><annotation encoding='application/x-tex'>Top</annotation></semantics></math></h3> <p>One might expect that compact topological spaces are precisely the <a class='existingWikiWord' href='/nlab/show/diff/compact+object'>compact objects</a> in <a class='existingWikiWord' href='/nlab/show/diff/Top'>Top</a> in the abstract sense of <a class='existingWikiWord' href='/nlab/show/diff/category+theory'>category theory</a>, but this identification requires care, the naive version fails. See at <em><a href='compact+object#CompactObjectsInTop'>compact object – Compact objects in Top</a></em></p> <h2 id='examples'>Examples</h2> <h3 id='ExamplesGeneral'>General</h3> <div class='num_prop'> <h6 id='proposition_20'>Proposition</h6> <p>A <a class='existingWikiWord' href='/nlab/show/diff/discrete+object'>discrete space</a> is compact iff its underlying set is <a class='existingWikiWord' href='/nlab/show/diff/finite+set'>finite</a>.</p> <p>In <a class='existingWikiWord' href='/nlab/show/diff/constructive+mathematics'>constructive mathematics</a>, a discrete space is compact iff its underlying set is <a class='existingWikiWord' href='/nlab/show/diff/finite+set'>Kuratowski-finite</a>.</p> </div> <div class='num_example' id='CompactClosedInterval'> <h6 id='example'>Example</h6> <p><strong>(<a class='existingWikiWord' href='/nlab/show/diff/interval'>closed intervals</a> are <a class='existingWikiWord' href='/nlab/show/diff/compact+space'>compact</a>)</strong></p> <p>For any <math class='maruku-mathml' display='inline' id='mathml_ad54570a1a49ee7346ff0ef44ee886942fb6526d_192' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>a</mi><mo><</mo><mi>b</mi><mo>∈</mo><mi>ℝ</mi></mrow><annotation encoding='application/x-tex'>a \lt b \in \mathbb{R}</annotation></semantics></math> the <a class='existingWikiWord' href='/nlab/show/diff/interval'>closed interval</a></p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_ad54570a1a49ee7346ff0ef44ee886942fb6526d_193' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>[</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo stretchy='false'>]</mo><mo>⊂</mo><mi>ℝ</mi></mrow><annotation encoding='application/x-tex'> [a,b] \subset \mathbb{R} </annotation></semantics></math></div> <p>regarded with its <a class='existingWikiWord' href='/nlab/show/diff/subspace+topology'>subspace topology</a> of <a class='existingWikiWord' href='/nlab/show/diff/Euclidean+space'>Euclidean space</a> with its <a class='existingWikiWord' href='/nlab/show/diff/metric+topology'>metric topology</a> is a compact topological space.</p> </div> <p>(See <a class='existingWikiWord' href='/nlab/show/diff/closed+intervals+are+compact+topological+spaces'>closed intervals are compact topological spaces</a>.)</p> <p>In contrast:</p> <div class='num_example'> <h6 id='nonexample'>Nonexample</h6> <p>For all <math class='maruku-mathml' display='inline' id='mathml_ad54570a1a49ee7346ff0ef44ee886942fb6526d_194' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>n</mi><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding='application/x-tex'>n \in \mathbb{N}</annotation></semantics></math>, <math class='maruku-mathml' display='inline' id='mathml_ad54570a1a49ee7346ff0ef44ee886942fb6526d_195' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>n</mi><mo>></mo><mn>0</mn></mrow><annotation encoding='application/x-tex'>n \gt 0</annotation></semantics></math>, the <a class='existingWikiWord' href='/nlab/show/diff/Euclidean+space'>Euclidean space</a> <math class='maruku-mathml' display='inline' id='mathml_ad54570a1a49ee7346ff0ef44ee886942fb6526d_196' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>ℝ</mi> <mi>n</mi></msup></mrow><annotation encoding='application/x-tex'>\mathbb{R}^n</annotation></semantics></math>, regarded with its <a class='existingWikiWord' href='/nlab/show/diff/metric+topology'>metric topology</a>, is <em>not</em> compact.</p> </div> <p>This is a special case of the <a class='existingWikiWord' href='/nlab/show/diff/Heine-Borel+theorem'>Heine-Borel theorem</a> (example <a class='maruku-ref' href='#HeineBorelTheorem'>4</a> below), but for illustration it is instructive to consider the direct proof:</p> <div class='proof'> <h6 id='proof_6'>Proof</h6> <p>Pick any <math class='maruku-mathml' display='inline' id='mathml_ad54570a1a49ee7346ff0ef44ee886942fb6526d_197' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>ϵ</mi><mo>∈</mo><mo stretchy='false'>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy='false'>/</mo><mn>2</mn><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>\epsilon \in (0,1/2)</annotation></semantics></math>. Consider the open cover of <math class='maruku-mathml' display='inline' id='mathml_ad54570a1a49ee7346ff0ef44ee886942fb6526d_198' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>ℝ</mi> <mi>n</mi></msup></mrow><annotation encoding='application/x-tex'>\mathbb{R}^n</annotation></semantics></math> given by</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_ad54570a1a49ee7346ff0ef44ee886942fb6526d_199' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mrow><mo>{</mo><msub><mi>U</mi> <mi>n</mi></msub><mo>≔</mo><mo stretchy='false'>(</mo><mi>n</mi><mo>−</mo><mi>ϵ</mi><mo>,</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo>+</mo><mi>ϵ</mi><mo stretchy='false'>)</mo><mo>×</mo><msup><mi>ℝ</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup><mspace width='thickmathspace' /><mo>⊂</mo><mspace width='thickmathspace' /><msup><mi>ℝ</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup><mo>}</mo></mrow> <mrow><mi>n</mi><mo>∈</mo><mi>ℤ</mi></mrow></msub><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> \left\{ U_n \coloneqq (n-\epsilon, n+1+\epsilon) \times \mathbb{R}^{n-1} \;\subset\; \mathbb{R}^{n+1} \right\}_{n \in \mathbb{Z}} \,. </annotation></semantics></math></div> <p>This is not a finite cover, and removing any one of its patches <math class='maruku-mathml' display='inline' id='mathml_ad54570a1a49ee7346ff0ef44ee886942fb6526d_200' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>U</mi> <mi>n</mi></msub></mrow><annotation encoding='application/x-tex'>U_n</annotation></semantics></math>, it ceases to be a cover, since the points of the form <math class='maruku-mathml' display='inline' id='mathml_ad54570a1a49ee7346ff0ef44ee886942fb6526d_201' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mi>n</mi><mo>+</mo><mi>ϵ</mi><mo>,</mo><msub><mi>x</mi> <mn>2</mn></msub><mo>,</mo><msub><mi>x</mi> <mn>3</mn></msub><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>x</mi> <mi>n</mi></msub><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(n + \epsilon, x_2, x_3, \cdots, x_n)</annotation></semantics></math> are contained only in <math class='maruku-mathml' display='inline' id='mathml_ad54570a1a49ee7346ff0ef44ee886942fb6526d_202' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>U</mi> <mi>n</mi></msub></mrow><annotation encoding='application/x-tex'>U_n</annotation></semantics></math> and in no other patch.</p> </div> <div class='num_example'> <h6 id='example_2'>Example</h6> <p>By the <a class='existingWikiWord' href='/nlab/show/diff/Tychonoff+theorem'>Tychonoff theorem</a>, every <a class='existingWikiWord' href='/nlab/show/diff/product+topological+space'>product topological spaces</a> of compact spaces is again compact.</p> </div> <p>This implies the <a class='existingWikiWord' href='/nlab/show/diff/Heine-Borel+theorem'>Heine-Borel theorem</a>, saying that:</p> <div class='num_example' id='HeineBorelTheorem'> <h6 id='example_3'>Example</h6> <p><strong>(<a class='existingWikiWord' href='/nlab/show/diff/Heine-Borel+theorem'>Heine-Borel theorem</a>)</strong></p> <p>Every <a class='existingWikiWord' href='/nlab/show/diff/bounded+set'>bounded</a> and <a class='existingWikiWord' href='/nlab/show/diff/closed+subspace'>closed subspace</a> of a <a class='existingWikiWord' href='/nlab/show/diff/Euclidean+space'>Euclidean space</a> is compact.</p> </div> <p>In particular:</p> <div class='num_example'> <h6 id='nonexample_2'>Nonexample</h6> <p>The <a class='existingWikiWord' href='/nlab/show/diff/interval'>open intervals</a> <math class='maruku-mathml' display='inline' id='mathml_ad54570a1a49ee7346ff0ef44ee886942fb6526d_203' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo stretchy='false'>)</mo><mo>⊂</mo><mi>ℝ</mi></mrow><annotation encoding='application/x-tex'>(a,b) \subset \mathbb{R}</annotation></semantics></math> and the <a class='existingWikiWord' href='/nlab/show/diff/interval'>half-open intervals</a> <math class='maruku-mathml' display='inline' id='mathml_ad54570a1a49ee7346ff0ef44ee886942fb6526d_204' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>[</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo stretchy='false'>)</mo><mo>⊂</mo><mi>ℝ</mi></mrow><annotation encoding='application/x-tex'>[a,b) \subset \mathbb{R}</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_ad54570a1a49ee7346ff0ef44ee886942fb6526d_205' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo stretchy='false'>]</mo><mo>⊂</mo><mi>ℝ</mi></mrow><annotation encoding='application/x-tex'>(a,b] \subset \mathbb{R}</annotation></semantics></math> are not compact.</p> </div> <div class='num_example'> <h6 id='example_4'>Example</h6> <p>The <a class='existingWikiWord' href='/nlab/show/diff/Mandelbrot+set'>Mandelbrot set</a>, regarded as a <a class='existingWikiWord' href='/nlab/show/diff/subspace'>subspace</a> of the <a class='existingWikiWord' href='/nlab/show/diff/plane'>plane</a>, is a compact space (see <a href='/nlab/show/Mandelbrot+set#MndlbrtIsCompact'>this prop.</a>).</p> </div> <div class='num_example'> <h6 id='example_5'>Example</h6> <p>Any set when equipped with the <a class='existingWikiWord' href='/nlab/show/diff/cofinite+topology'>cofinite topology</a> forms a compact space. This space is also <math class='maruku-mathml' display='inline' id='mathml_ad54570a1a49ee7346ff0ef44ee886942fb6526d_206' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>T</mi> <mn>1</mn></msub></mrow><annotation encoding='application/x-tex'>T_1</annotation></semantics></math> (i.e., <a class='existingWikiWord' href='/nlab/show/diff/singleton'>singletons</a> are closed), but is not Hausdorff if the underlying set is <a class='existingWikiWord' href='/nlab/show/diff/infinite+set'>infinite</a>.</p> </div> <h3 id='CompactSpacesWhichAreNotSequentiallyCompact'>Compact spaces which are not sequentially compact</h3> <p>A famous example of a space that is compact, but not <a class='existingWikiWord' href='/nlab/show/diff/sequentially+compact+topological+space'>sequentially compact</a> is the product space</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_ad54570a1a49ee7346ff0ef44ee886942fb6526d_207' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>{</mo><mn>0</mn><mo>,</mo><mn>1</mn><msup><mo stretchy='false'>}</mo> <mi>I</mi></msup><mo>≔</mo><mo stretchy='false'>{</mo><mn>0</mn><mo>,</mo><mn>1</mn><msup><mo stretchy='false'>}</mo> <mrow><mo stretchy='false'>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy='false'>]</mo></mrow></msup><mo>≔</mo><munder><mo lspace='thinmathspace' rspace='thinmathspace'>∏</mo><mrow><mo stretchy='false'>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy='false'>]</mo></mrow></munder><mo stretchy='false'>{</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy='false'>}</mo></mrow><annotation encoding='application/x-tex'> \{0,1\}^{I} \coloneqq \{0, 1\}^{[0, 1]} \coloneqq \underset{[0,1]}{\prod} \{0,1\} </annotation></semantics></math></div> <p>with the <a class='existingWikiWord' href='/nlab/show/diff/product+topological+space'>product topology</a>.</p> <p>This space is compact by the <a class='existingWikiWord' href='/nlab/show/diff/Tychonoff+theorem'>Tychonoff theorem</a>.</p> <p>But it is not <a class='existingWikiWord' href='/nlab/show/diff/sequentially+compact+topological+space'>sequentially compact</a>. We now discuss why. (We essentially follow <a href='#SteenSeebach70'>Steen-Seebach 70, item 105</a>).</p> <p>Points of <math class='maruku-mathml' display='inline' id='mathml_ad54570a1a49ee7346ff0ef44ee886942fb6526d_208' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>{</mo><mn>0</mn><mo>,</mo><mn>1</mn><msup><mo stretchy='false'>}</mo> <mi>I</mi></msup></mrow><annotation encoding='application/x-tex'>\{0,1\}^{I}</annotation></semantics></math> are functions <math class='maruku-mathml' display='inline' id='mathml_ad54570a1a49ee7346ff0ef44ee886942fb6526d_209' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>I</mi><mo>→</mo><mo stretchy='false'>{</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy='false'>}</mo></mrow><annotation encoding='application/x-tex'>I \to \{0,1\}</annotation></semantics></math>, and the product topology is the topology of pointwise convergence.</p> <p>Define a sequence <math class='maruku-mathml' display='inline' id='mathml_ad54570a1a49ee7346ff0ef44ee886942fb6526d_210' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><msub><mi>a</mi> <mi>n</mi></msub><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(a_n)</annotation></semantics></math> in <math class='maruku-mathml' display='inline' id='mathml_ad54570a1a49ee7346ff0ef44ee886942fb6526d_211' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>I</mi> <mi>I</mi></msup></mrow><annotation encoding='application/x-tex'>I^{I}</annotation></semantics></math> with <math class='maruku-mathml' display='inline' id='mathml_ad54570a1a49ee7346ff0ef44ee886942fb6526d_212' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>a</mi> <mi>n</mi></msub><mo stretchy='false'>(</mo><mi>x</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>a_n(x)</annotation></semantics></math> being the nth digit in the binary expansion of <math class='maruku-mathml' display='inline' id='mathml_ad54570a1a49ee7346ff0ef44ee886942fb6526d_213' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>x</mi></mrow><annotation encoding='application/x-tex'>x</annotation></semantics></math> (we make the convention that binary expansions do not end in sequences of <math class='maruku-mathml' display='inline' id='mathml_ad54570a1a49ee7346ff0ef44ee886942fb6526d_214' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>1</mn></mrow><annotation encoding='application/x-tex'>1</annotation></semantics></math>s). Let <math class='maruku-mathml' display='inline' id='mathml_ad54570a1a49ee7346ff0ef44ee886942fb6526d_215' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>a</mi><mo>≔</mo><mo stretchy='false'>(</mo><msub><mi>a</mi> <mrow><msub><mi>n</mi> <mi>k</mi></msub></mrow></msub><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>a \coloneqq (a_{n_k})</annotation></semantics></math> be a subsequence and define <math class='maruku-mathml' display='inline' id='mathml_ad54570a1a49ee7346ff0ef44ee886942fb6526d_216' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>p</mi> <mi>a</mi></msub><mo>∈</mo><mi>I</mi></mrow><annotation encoding='application/x-tex'>p_a \in I</annotation></semantics></math> by the binary expansion that has a <math class='maruku-mathml' display='inline' id='mathml_ad54570a1a49ee7346ff0ef44ee886942fb6526d_217' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>0</mn></mrow><annotation encoding='application/x-tex'>0</annotation></semantics></math> in the <math class='maruku-mathml' display='inline' id='mathml_ad54570a1a49ee7346ff0ef44ee886942fb6526d_218' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>n</mi> <mi>k</mi></msub></mrow><annotation encoding='application/x-tex'>n_k</annotation></semantics></math>th position if <math class='maruku-mathml' display='inline' id='mathml_ad54570a1a49ee7346ff0ef44ee886942fb6526d_219' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>k</mi></mrow><annotation encoding='application/x-tex'>k</annotation></semantics></math> is even and a <math class='maruku-mathml' display='inline' id='mathml_ad54570a1a49ee7346ff0ef44ee886942fb6526d_220' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>1</mn></mrow><annotation encoding='application/x-tex'>1</annotation></semantics></math> if <math class='maruku-mathml' display='inline' id='mathml_ad54570a1a49ee7346ff0ef44ee886942fb6526d_221' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>k</mi></mrow><annotation encoding='application/x-tex'>k</annotation></semantics></math> is odd (and, for definiteness and to ensure that this fits with our convention, define it to be <math class='maruku-mathml' display='inline' id='mathml_ad54570a1a49ee7346ff0ef44ee886942fb6526d_222' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>0</mn></mrow><annotation encoding='application/x-tex'>0</annotation></semantics></math> elsewhere). Then the projection of <math class='maruku-mathml' display='inline' id='mathml_ad54570a1a49ee7346ff0ef44ee886942fb6526d_223' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><msub><mi>a</mi> <mrow><msub><mi>n</mi> <mi>k</mi></msub></mrow></msub><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(a_{n_k})</annotation></semantics></math> at the <math class='maruku-mathml' display='inline' id='mathml_ad54570a1a49ee7346ff0ef44ee886942fb6526d_224' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>p</mi> <mi>a</mi></msub></mrow><annotation encoding='application/x-tex'>p_a</annotation></semantics></math>th coordinate is <math class='maruku-mathml' display='inline' id='mathml_ad54570a1a49ee7346ff0ef44ee886942fb6526d_225' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>1</mn><mo>,</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>0</mn><mo>,</mo><mo>.</mo><mo>.</mo><mo>.</mo></mrow><annotation encoding='application/x-tex'>1, 0, 1,0,...</annotation></semantics></math> which is not convergent. Hence <math class='maruku-mathml' display='inline' id='mathml_ad54570a1a49ee7346ff0ef44ee886942fb6526d_226' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><msub><mi>a</mi> <mrow><msub><mi>n</mi> <mi>k</mi></msub></mrow></msub><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(a_{n_k})</annotation></semantics></math> is not convergent.</p> <p>(Basically that’s the diagonal trick of <a class='existingWikiWord' href='/nlab/show/diff/Cantor%27s+theorem'>Cantor's theorem</a>).</p> <p>However, as <math class='maruku-mathml' display='inline' id='mathml_ad54570a1a49ee7346ff0ef44ee886942fb6526d_227' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>{</mo><mn>0</mn><mo>,</mo><mn>1</mn><msup><mo stretchy='false'>}</mo> <mi>I</mi></msup></mrow><annotation encoding='application/x-tex'>\{0,1\}^I</annotation></semantics></math> is compact, <math class='maruku-mathml' display='inline' id='mathml_ad54570a1a49ee7346ff0ef44ee886942fb6526d_228' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>a</mi></mrow><annotation encoding='application/x-tex'>a</annotation></semantics></math> has a <a class='existingWikiWord' href='/nlab/show/diff/convergence'>convergent</a> <a class='existingWikiWord' href='/nlab/show/diff/subnet'>subnet</a>. An explicit construction of a convergent subset, given a <a class='existingWikiWord' href='/nlab/show/diff/limit+point'>cluster point</a> <math class='maruku-mathml' display='inline' id='mathml_ad54570a1a49ee7346ff0ef44ee886942fb6526d_229' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>a</mi></mrow><annotation encoding='application/x-tex'>a</annotation></semantics></math>, is as follows. A function <math class='maruku-mathml' display='inline' id='mathml_ad54570a1a49ee7346ff0ef44ee886942fb6526d_230' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>a</mi><mo lspace='verythinmathspace'>:</mo><mi>I</mi><mo>→</mo><mo stretchy='false'>{</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy='false'>}</mo></mrow><annotation encoding='application/x-tex'>a \colon I \to \{0,1\}</annotation></semantics></math> is a <a class='existingWikiWord' href='/nlab/show/diff/limit+point'>cluster point</a> of <math class='maruku-mathml' display='inline' id='mathml_ad54570a1a49ee7346ff0ef44ee886942fb6526d_231' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><msub><mi>a</mi> <mi>n</mi></msub><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(a_n)</annotation></semantics></math> if, for any <math class='maruku-mathml' display='inline' id='mathml_ad54570a1a49ee7346ff0ef44ee886942fb6526d_232' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>p</mi> <mn>1</mn></msub><mo>,</mo><mi>…</mi><mo>,</mo><msub><mi>p</mi> <mi>n</mi></msub><mo>∈</mo><mi>I</mi></mrow><annotation encoding='application/x-tex'>p_1, \dots, p_n \in I</annotation></semantics></math> the set</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_ad54570a1a49ee7346ff0ef44ee886942fb6526d_233' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>A</mi><mo stretchy='false'>(</mo><msub><mi>p</mi> <mn>1</mn></msub><mo>,</mo><mi>…</mi><mo>,</mo><msub><mi>p</mi> <mi>n</mi></msub><mo stretchy='false'>)</mo><mo>≔</mo><mo stretchy='false'>{</mo><mi>k</mi><mo>∈</mo><mi>ℝ</mi><mo>:</mo><msub><mi>a</mi> <mi>k</mi></msub><mo stretchy='false'>(</mo><msub><mi>p</mi> <mi>i</mi></msub><mo stretchy='false'>)</mo><mo>=</mo><mi>a</mi><mo stretchy='false'>(</mo><msub><mi>p</mi> <mi>i</mi></msub><mo stretchy='false'>)</mo><mo>∀</mo><mi>i</mi><mo stretchy='false'>}</mo></mrow><annotation encoding='application/x-tex'> A(p_1,\dots,p_n) \coloneqq \{k \in \mathbb{R} : a_k(p_i) = a(p_i) \forall i\} </annotation></semantics></math></div> <p>is infinite. We index our subnet by the family of finite subsets of <math class='maruku-mathml' display='inline' id='mathml_ad54570a1a49ee7346ff0ef44ee886942fb6526d_234' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>I</mi></mrow><annotation encoding='application/x-tex'>I</annotation></semantics></math> and define the subnet function <math class='maruku-mathml' display='inline' id='mathml_ad54570a1a49ee7346ff0ef44ee886942fb6526d_235' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>ℱ</mi><mo stretchy='false'>(</mo><mi>I</mi><mo stretchy='false'>)</mo><mo>→</mo><mi>ℕ</mi></mrow><annotation encoding='application/x-tex'>\mathcal{F}(I) \to \mathbb{N}</annotation></semantics></math> to be</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_ad54570a1a49ee7346ff0ef44ee886942fb6526d_236' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>{</mo><msub><mi>p</mi> <mn>1</mn></msub><mo>,</mo><mi>…</mi><mo>,</mo><msub><mi>p</mi> <mi>n</mi></msub><mo stretchy='false'>}</mo><mo>↦</mo><mi>min</mi><mi>A</mi><mo stretchy='false'>(</mo><msub><mi>p</mi> <mn>1</mn></msub><mo>,</mo><mi>…</mi><mo>,</mo><msub><mi>p</mi> <mi>n</mi></msub><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'> \{p_1,\dots,p_n\} \mapsto \min A(p_1,\dots,p_n) </annotation></semantics></math></div> <p>This is a convergent sub-<a class='existingWikiWord' href='/nlab/show/diff/net'>net</a>.</p> <h2 id='in_synthetic_topology'>In synthetic topology</h2> <p>In <a class='existingWikiWord' href='/nlab/show/diff/synthetic+topology'>synthetic topology</a>, where ‘space’ means simply ‘set’ (or <a class='existingWikiWord' href='/nlab/show/diff/type'>type</a>, i.e. the basic objects of our foundational system), one natural notion of “compact space” is a <a class='existingWikiWord' href='/nlab/show/diff/covert+space'>covert set</a>, i.e. a set whose discrete topology is covert. This includes the expected examples in various <a class='existingWikiWord' href='/nlab/show/diff/big+and+little+toposes'>gros toposes</a>.</p> <h2 id='compact_spaces_and_proper_maps'>Compact spaces and proper maps</h2> <p>A space <math class='maruku-mathml' display='inline' id='mathml_ad54570a1a49ee7346ff0ef44ee886942fb6526d_237' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> is compact if and only if the unique map <math class='maruku-mathml' display='inline' id='mathml_ad54570a1a49ee7346ff0ef44ee886942fb6526d_238' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi><mo>→</mo><mn>1</mn></mrow><annotation encoding='application/x-tex'>X\to 1</annotation></semantics></math> is <a class='existingWikiWord' href='/nlab/show/diff/proper+map'>proper</a>. Thus, properness is a “relativized” version of compactness.</p> <p>For topological spaces, this is either a definition of “proper map” (closed with compact fibers) or follows from the above characterization of compactness in terms of projections being closed maps (if proper maps are defined to be those that are universally closed). For locales, it follows from the definition of proper map (a closed map such that <math class='maruku-mathml' display='inline' id='mathml_ad54570a1a49ee7346ff0ef44ee886942fb6526d_239' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>f</mi> <mo>*</mo></msub></mrow><annotation encoding='application/x-tex'>f_*</annotation></semantics></math> preserves directed joins) and the fact that compact locales are automatically <a class='existingWikiWord' href='/nlab/show/diff/covert+space'>covert</a> (see <a class='existingWikiWord' href='/nlab/show/diff/covert+space'>covert space</a> for a proof).</p> <h2 id='history'>History</h2> <p>Compact spaces were introduced (under the name of “bicompact spaces”) by <a class='existingWikiWord' href='/nlab/show/diff/Pavel+Aleksandrov'>Paul Alexandroff</a> and <a class='existingWikiWord' href='/nlab/show/diff/Pavel+Urysohn'>Paul Urysohn</a> around 1924, see the 92nd volume of Mathematische Annalen, especially <a href='#AlexandroffUrysohn1924'>AU</a>.</p> <h2 id='related_concepts'>Related concepts</h2> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Lebesgue+number+lemma'>Lebesgue number lemma</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/sequentially+compact+topological+space'>sequentially compact topological space</a>, <a class='existingWikiWord' href='/nlab/show/diff/countably+compact+topological+space'>countably compact topological space</a>, <a class='existingWikiWord' href='/nlab/show/diff/paracompact+topological+space'>paracompact topological space</a>, <a class='existingWikiWord' href='/nlab/show/diff/locally+compact+topological+space'>locally compact topological space</a>, <a class='existingWikiWord' href='/nlab/show/diff/strongly+compact+topological+space'>strongly compact topological space</a>, <a class='existingWikiWord' href='/nlab/show/diff/compactly+generated+topological+space'>compactly generated topological space</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/compactification'>compactification</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/one-point+compactification'>one-point compactification</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Stone-%C4%8Cech+compactification'>Stone-Cech compactification</a></p> </li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/compact-open+topology'>compact-open topology</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/closed+manifold'>closed manifold</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/proper+geometric+morphism'>compact topos</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/compactly+supported+cohomology'>compactly supported cohomology</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/complete+algebraic+variety'>complete algebraic variety</a></p> </li> </ul> <h2 id='references'>References</h2> <p>Introduced and studied by <a class='existingWikiWord' href='/nlab/show/diff/Pavel+Aleksandrov'>Paul Alexandroff</a> and <a class='existingWikiWord' href='/nlab/show/diff/Pavel+Urysohn'>Paul Urysohn</a> in</p> <ul> <li id='AlexandroffUrysohn1924'><a class='existingWikiWord' href='/nlab/show/diff/Pavel+Aleksandrov'>Paul Alexandroff</a>, <a class='existingWikiWord' href='/nlab/show/diff/Pavel+Urysohn'>Paul Urysohn</a>, <em>Zur Theorie der topologischen Räume</em>, <a href='https://eudml.org/doc/159069'>EuDML</a>.</li> </ul> <p>For general references see the <a href='topology#References'>list</a> at <em><a class='existingWikiWord' href='/nlab/show/diff/topology'>topology</a></em>.</p> <p>Examples of compact spaces that are not sequentially compact are in</p> <ul> <li id='SteenSeebach70'><a class='existingWikiWord' href='/nlab/show/diff/Lynn+Arthur+Steen'>Lynn Arthur Steen</a>, <a class='existingWikiWord' href='/nlab/show/diff/J.+Arthur+Seebach'>J. Arthur Seebach</a>, <em><a class='existingWikiWord' href='/nlab/show/diff/Counterexamples+in+Topology'>Counterexamples in Topology</a></em>, Springer 1970/1978 (<a href='https://link.springer.com/book/10.1007/978-1-4612-6290-9'>doi:10.1007/978-1-4612-6290-9</a>)</li> </ul> <p>On compact <a class='existingWikiWord' href='/nlab/show/diff/metric+space'>metric spaces</a>:</p> <ul> <li> <p>Roland Speicher, <em>Compact metric spaces</em> (<a href='http://www.mast.queensu.ca/~speicher/Chapter10.pdf'>pdf</a>)</p> </li> <li> <p>Alan Sokal, <em>Compactness of metric spaces</em> (<a href='http://www.ucl.ac.uk/~ucahad0/3103_handout_2.pdf'>pdf</a>)</p> </li> </ul> <p>For <a class='existingWikiWord' href='/nlab/show/diff/proper+base+change+theorem'>proper base change theorem</a> e.g.</p> <ul> <li id='Milne'><a class='existingWikiWord' href='/nlab/show/diff/James+Milne'>James Milne</a>, section 17 of <em><a class='existingWikiWord' href='/nlab/show/diff/Lectures+on+%C3%89tale+Cohomology'>Lectures on Étale Cohomology</a></em></li> </ul> <p> </p> <p> </p> <p> </p> <p> </p> <p> </p> </div> <div class="revisedby"> <p> Last revised on September 6, 2024 at 13:12:35. See the <a href="/nlab/history/compact+space" 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/compact+space" accesskey="E" class="navlink" id="edit" rel="nofollow">Edit</a><a href="https://nforum.ncatlab.org/discussion/1976/#Item_89">Discuss</a><span class="backintime"><a href="/nlab/revision/diff/compact+space/97" accesskey="B" class="navlinkbackintime" id="to_previous_revision" rel="nofollow">Previous revision</a></span><a href="/nlab/show/compact+space" accesskey="C" class="navlink" id="see_changes" rel="nofollow">Hide changes</a><a href="/nlab/history/compact+space" accesskey="S" class="navlink" id="history" rel="nofollow">History (97 revisions)</a> <a href="/nlab/show/compact+space/cite" style="color: black">Cite</a> <a href="/nlab/print/compact+space" accesskey="p" id="view_print" rel="nofollow">Print</a> <a href="/nlab/source/compact+space" id="view_source" rel="nofollow">Source</a> </div> </div>  </div>  </body> </html>