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> first-countable 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> first-countable 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/18066/#Item_2" 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 #3 to #4: <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='firstcountable_spaces'>First-countable spaces</h1> <div class='maruku_toc'><ul><li><a href='#idea'>Idea</a></li><li><a href='#definitions'>Definitions</a></li><li><a href='#generalizations'>Generalizations</a></li><li><a href='#related_countability_properties'>Related countability properties</a><ul><li><a href='#properties'>Properties</a></li><li><a href='#implications'>Implications</a></li></ul></li><li><a href='#related_concepts'>Related concepts</a></li></ul></div> <h2 id='idea'>Idea</h2> <p>A <a class='existingWikiWord' href='/nlab/show/diff/space'>space</a> (such as a <a class='existingWikiWord' href='/nlab/show/diff/topological+space'>topological space</a>) is <em>first-countable</em> if, in a certain sense, there is only a <a class='existingWikiWord' href='/nlab/show/diff/countable+set'>countable</a> amount of information locally in its topology. (Change ‘locally’ to ‘globally’ to get a <a class='existingWikiWord' href='/nlab/show/diff/second-countable+space'>second-countable space</a>.)</p> <h2 id='definitions'>Definitions</h2> <div class='num_defn'> <h6 id='definition'>Definition</h6> <p>A <a class='existingWikiWord' href='/nlab/show/diff/topological+space'>topological space</a> is <strong>first-countable</strong> if every <a class='existingWikiWord' href='/nlab/show/diff/point'>point</a> <math class='maruku-mathml' display='inline' id='mathml_b320b0e47674d3754e71fa7b1487bf4be90e9340_1' 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/countable+set'>countable</a> <a class='existingWikiWord' href='/nlab/show/diff/neighborhood+base'>local basis</a> <math class='maruku-mathml' display='inline' id='mathml_b320b0e47674d3754e71fa7b1487bf4be90e9340_2' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>B</mi> <mi>x</mi></msub></mrow><annotation encoding='application/x-tex'>B_x</annotation></semantics></math>.</p> </div> <h2 id='generalizations'>Generalizations</h2> <p>The <strong>character</strong> of a space at a point <math class='maruku-mathml' display='inline' id='mathml_b320b0e47674d3754e71fa7b1487bf4be90e9340_3' 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 <a class='existingWikiWord' href='/nlab/show/diff/extremum'>minimum</a> of the <a class='existingWikiWord' href='/nlab/show/diff/cardinal+number'>cardinalities</a> of the possible bases <math class='maruku-mathml' display='inline' id='mathml_b320b0e47674d3754e71fa7b1487bf4be90e9340_4' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>B</mi> <mi>x</mi></msub></mrow><annotation encoding='application/x-tex'>B_x</annotation></semantics></math>. We are implicitly using the <a class='existingWikiWord' href='/nlab/show/diff/axiom+of+choice'>axiom of choice</a> here, to suppose that this set of cardinalities (which really is a <a class='existingWikiWord' href='/nlab/show/diff/small+set'>small set</a> because bounded above by the number of <a class='existingWikiWord' href='/nlab/show/diff/neighborhood'>neighbourhoods</a> of <math class='maruku-mathml' display='inline' id='mathml_b320b0e47674d3754e71fa7b1487bf4be90e9340_5' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>x</mi></mrow><annotation encoding='application/x-tex'>x</annotation></semantics></math>, and <a class='existingWikiWord' href='/nlab/show/diff/inhabited+set'>inhabited</a> by this number as well) has a minimum. But without Choice, we can still consider this collection of cardinalities.</p> <p>Then a first-countable space is simply one whose characters are all countable.</p> <p>The <strong>character</strong>, tout court, of a space is the <a class='existingWikiWord' href='/nlab/show/diff/join'>supremum</a> of the characters of its points; then a first-countable space is simply one with a countable character.</p> <h2 id='related_countability_properties'>Related countability properties</h2> <p>\begin{tikzcd}[column sep=normal, row sep=large] & & & \scriptsize \text{metrisable} & \substack{ \sigma\text{-locally finite } \ \text{base}\wedge \text{regular} \wedge \mathrm{T}_2 } & \ & \text{second countable} & & \sigma\text{-locally discrete base} & \sigma\text{-locally finite base} & \text{first countable} \ \text{separable} & & \text{Lindel"of} & \substack{\text{weakly Lindel"of }\wedge \ \sigma\text{-locally finite base}} & & \text{Fr'echet-Urysohn} \ & & & & & \text{sequential} \ \text{countable chain condition}<br /> & & \text{weakly Lindel"of} & & & \text{countably tight} \arrow[Rightarrow, from=3-4, to=2-2] \arrow[Rightarrow, from=3-4, to=5-3] \arrow[Rightarrow, from=3-4, to=2-5] \arrow[Rightarrow, from=2-2, to=3-1, \text{CC} description] \arrow[Rightarrow, from=2-2, to=3-3, \text{CC} description] \arrow[Rightarrow, from=3-1, to=3-3, \text{if metacompact}] \arrow[Rightarrow, from=3-3, to=5-3] \arrow[Rightarrow, from=3-1, to=5-3, \text{AC} {description, near end}] \arrow[Rightarrow, from=2-2, to=2-4] \arrow[Rightarrow, from=3-1, to=5-1] \arrow[Rightarrow, from=2-4, to=2-5] \arrow[Rightarrow, from=2-5, to=2-6] \arrow[Rightarrow, from=2-6, to=3-6] \arrow[Rightarrow, from=3-6, to=4-6] \arrow[Rightarrow, from=4-6, to=5-6] \arrow[Rightarrow, from=1-4, to=2-4] \arrow[Leftrightarrow, from=1-5, to=1-4, \substack{\text{Nagata-Smirnov} \ \text{metrization thm.} } above] \arrow[Rightarrow, from=1-5, to=2-5] \arrow[Rightarrow, from=5-1, to=3-3, \text{AC} {description, near end}, \text{if paracompact} {near start, below right}] \end{tikzcd}</p> <p>Axioms: <a class='existingWikiWord' href='/nlab/show/diff/axiom+of+choice'>axiom of choice</a> (AC), <a class='existingWikiWord' href='/nlab/show/diff/countable+choice'>countable choice</a> (CC).</p> <h3 id='properties'>Properties</h3> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/second-countable+space'>second-countable</a>: there is a <a class='existingWikiWord' href='/nlab/show/diff/countable+set'>countable</a> <a class='existingWikiWord' href='/nlab/show/diff/topological+base'>base</a> of the topology.</p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/metrisable+topological+space'>metrisable</a>: the topology is induced by a metric.</p> </li> <li> <p><math class='maruku-mathml' display='inline' id='mathml_b320b0e47674d3754e71fa7b1487bf4be90e9340_8' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>σ</mi></mrow><annotation encoding='application/x-tex'>\sigma</annotation></semantics></math>-locally discrete base: the topology of <math class='maruku-mathml' display='inline' id='mathml_b320b0e47674d3754e71fa7b1487bf4be90e9340_9' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> is generated by a <a class='existingWikiWord' href='/nlab/show/diff/countably+locally+discrete+set+of+subsets'>$\sigma$-locally discrete</a> <a class='existingWikiWord' href='/nlab/show/diff/topological+base'>base</a>.</p> </li> <li> <p><math class='maruku-mathml' display='inline' id='mathml_b320b0e47674d3754e71fa7b1487bf4be90e9340_10' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>σ</mi></mrow><annotation encoding='application/x-tex'>\sigma</annotation></semantics></math>-locally finite base: the topology of <math class='maruku-mathml' display='inline' id='mathml_b320b0e47674d3754e71fa7b1487bf4be90e9340_11' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> is generated by a <a class='existingWikiWord' href='/nlab/show/diff/countably+locally+finite+set+of+subsets'>countably locally finite</a> <a class='existingWikiWord' href='/nlab/show/diff/topological+base'>base</a>.</p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/separable+space'>separable</a>: there is a countable <a class='existingWikiWord' href='/nlab/show/diff/dense+subspace'>dense</a> subset.</p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Lindel%C3%B6f+topological+space'>Lindelöf</a>: every <a class='existingWikiWord' href='/nlab/show/diff/open+cover'>open cover</a> has a <a class='existingWikiWord' href='/nlab/show/diff/countable+cover'>countable</a> sub-cover.</p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/weakly+Lindel%C3%B6f+topological+space'>weakly Lindelöf</a>: every <a class='existingWikiWord' href='/nlab/show/diff/open+cover'>open cover</a> has a <a class='existingWikiWord' href='/nlab/show/diff/countable+set'>countable</a> subcollection the union of which is dense.</p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/metacompact+space'>metacompact</a>: every open cover has a point-finite open refinement.</p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/countable+chain+condition'>countable chain condition</a>: A family of pairwise disjoint open subsets is at most countable.</p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/first-countable+space'>first-countable</a>: every point has a countable <a class='existingWikiWord' href='/nlab/show/diff/neighborhood+base'>neighborhood base</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Frechet-Uryson+space'>Frechet-Uryson space</a>: the closure of a set <math class='maruku-mathml' display='inline' id='mathml_b320b0e47674d3754e71fa7b1487bf4be90e9340_12' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>A</mi></mrow><annotation encoding='application/x-tex'>A</annotation></semantics></math> consists precisely of all limit points of sequences in <math class='maruku-mathml' display='inline' id='mathml_b320b0e47674d3754e71fa7b1487bf4be90e9340_13' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>A</mi></mrow><annotation encoding='application/x-tex'>A</annotation></semantics></math></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/sequential+topological+space'>sequential topological space</a>: a set <math class='maruku-mathml' display='inline' id='mathml_b320b0e47674d3754e71fa7b1487bf4be90e9340_14' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>A</mi></mrow><annotation encoding='application/x-tex'>A</annotation></semantics></math> is closed if it contains all limit points of sequences in <math class='maruku-mathml' display='inline' id='mathml_b320b0e47674d3754e71fa7b1487bf4be90e9340_15' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>A</mi></mrow><annotation encoding='application/x-tex'>A</annotation></semantics></math></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/countably+tight+space'>countably tight</a>: for each subset <math class='maruku-mathml' display='inline' id='mathml_b320b0e47674d3754e71fa7b1487bf4be90e9340_16' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>A</mi></mrow><annotation encoding='application/x-tex'>A</annotation></semantics></math> and each point <math class='maruku-mathml' display='inline' id='mathml_b320b0e47674d3754e71fa7b1487bf4be90e9340_17' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>x</mi><mo>∈</mo><mover><mi>A</mi><mo>¯</mo></mover></mrow><annotation encoding='application/x-tex'>x\in \overline A</annotation></semantics></math> there is a countable subset <math class='maruku-mathml' display='inline' id='mathml_b320b0e47674d3754e71fa7b1487bf4be90e9340_18' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>D</mi><mo>⊆</mo><mi>A</mi></mrow><annotation encoding='application/x-tex'>D\subseteq A</annotation></semantics></math> such that <math class='maruku-mathml' display='inline' id='mathml_b320b0e47674d3754e71fa7b1487bf4be90e9340_19' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>x</mi><mo>∈</mo><mover><mi>D</mi><mo>¯</mo></mover></mrow><annotation encoding='application/x-tex'>x\in \overline D</annotation></semantics></math>.</p> </li> </ul> <h3 id='implications'>Implications</h3> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/open+covers+of+metric+spaces+have+open+countably+locally+discrete+refinements'>a metric space has a $\sigma$-locally discrete base</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Nagata-Smirnov+metrization+theorem'>Nagata-Smirnov metrization theorem</a></p> </li> <li> <p>a second-countable space has a <a class='existingWikiWord' href='/nlab/show/diff/countably+locally+finite+set+of+subsets'>$\sigma$-locally finite</a> <a class='existingWikiWord' href='/nlab/show/diff/topological+base'>base</a>: take the the collection of singeltons of all elements of a countable cover of <math class='maruku-mathml' display='inline' id='mathml_b320b0e47674d3754e71fa7b1487bf4be90e9340_20' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math>.</p> </li> <li> <p>second-countable spaces are separable: use the axiom of <a class='existingWikiWord' href='/nlab/show/diff/countable+choice'>countable choice</a> to choose a point in each set of a countable cover.</p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/second-countable+spaces+are+Lindel%C3%B6f'>second-countable spaces are Lindelöf</a>.</p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/weakly+Lindel%C3%B6f+spaces+with+countably+locally+finite+base+are+second+countable'>weakly Lindelöf spaces with countably locally finite base are second countable</a>.</p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/separable+metacompact+spaces+are+Lindel%C3%B6f'>separable metacompact spaces are Lindelöf</a>.</p> </li> <li> <p>separable spaces satisfy the countable chain condition: given a dense set <math class='maruku-mathml' display='inline' id='mathml_b320b0e47674d3754e71fa7b1487bf4be90e9340_21' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>D</mi></mrow><annotation encoding='application/x-tex'>D</annotation></semantics></math> and a family <math class='maruku-mathml' display='inline' id='mathml_b320b0e47674d3754e71fa7b1487bf4be90e9340_22' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>{</mo><msub><mi>U</mi> <mi>α</mi></msub><mo>:</mo><mi>α</mi><mo>∈</mo><mi>A</mi><mo stretchy='false'>}</mo></mrow><annotation encoding='application/x-tex'>\{U_\alpha : \alpha \in A\}</annotation></semantics></math>, the map <math class='maruku-mathml' display='inline' id='mathml_b320b0e47674d3754e71fa7b1487bf4be90e9340_23' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>D</mi><mo>∩</mo><msub><mo lspace='thinmathspace' rspace='thinmathspace'>⋃</mo> <mrow><mi>α</mi><mo>∈</mo><mi>A</mi></mrow></msub><msub><mi>U</mi> <mi>α</mi></msub><mo>→</mo><mi>A</mi></mrow><annotation encoding='application/x-tex'>D \cap \bigcup_{\alpha \in A} U_\alpha \to A</annotation></semantics></math> assigning <math class='maruku-mathml' display='inline' id='mathml_b320b0e47674d3754e71fa7b1487bf4be90e9340_24' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>d</mi></mrow><annotation encoding='application/x-tex'>d</annotation></semantics></math> to the unique <math class='maruku-mathml' display='inline' id='mathml_b320b0e47674d3754e71fa7b1487bf4be90e9340_25' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>α</mi><mo>∈</mo><mi>A</mi></mrow><annotation encoding='application/x-tex'>\alpha \in A</annotation></semantics></math> with <math class='maruku-mathml' display='inline' id='mathml_b320b0e47674d3754e71fa7b1487bf4be90e9340_26' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>d</mi><mo>∈</mo><msub><mi>U</mi> <mi>α</mi></msub></mrow><annotation encoding='application/x-tex'>d \in U_\alpha</annotation></semantics></math> is surjective.</p> </li> <li> <p>separable spaces are weakly Lindelöf: given a countable dense subset and an open cover <a class='existingWikiWord' href='/nlab/show/diff/axiom+of+choice'>choose</a> for each point of the subset an open from the cover.</p> </li> <li> <p>Lindelöf spaces are trivially also weakly Lindelöf.</p> </li> <li> <p>a space with a <math class='maruku-mathml' display='inline' id='mathml_b320b0e47674d3754e71fa7b1487bf4be90e9340_27' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>σ</mi></mrow><annotation encoding='application/x-tex'>\sigma</annotation></semantics></math>-locally finite base is first countable: obviously, every point is contained in at most countably many sets of a <math class='maruku-mathml' display='inline' id='mathml_b320b0e47674d3754e71fa7b1487bf4be90e9340_28' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>σ</mi></mrow><annotation encoding='application/x-tex'>\sigma</annotation></semantics></math>-locally finite base.</p> </li> <li> <p>a first-countable space is obviously Fréchet-Urysohn.</p> </li> <li> <p>a Fréchet-Uryson space is obviously sequential.</p> </li> <li> <p>a sequential space is obviously countably tight.</p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/paracompact+spaces+satisfying+the+countable+chain+condition+are+Lindel%C3%B6f'>paracompact spaces satisfying the countable chain condition are Lindelöf</a>.</p> </li> </ul> <h2 id='related_concepts'>Related concepts</h2> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/sigma-topological+space'>sigma-topological space</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/sigma-frame'>sigma-frame</a></p> </li> </ul><ins class='diffins'> </ins><ins class='diffins'><p> </p></ins> <p> </p> <p> </p> </div> <div class="revisedby"> <p> Last revised on June 1, 2024 at 01:02:25. See the <a href="/nlab/history/first-countable+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/first-countable+space" accesskey="E" class="navlink" id="edit" rel="nofollow">Edit</a><a href="https://nforum.ncatlab.org/discussion/18066/#Item_2">Discuss</a><span class="backintime"><a href="/nlab/revision/diff/first-countable+space/3" accesskey="B" class="navlinkbackintime" id="to_previous_revision" rel="nofollow">Previous revision</a></span><a href="/nlab/show/first-countable+space" accesskey="C" class="navlink" id="see_changes" rel="nofollow">Hide changes</a><a href="/nlab/history/first-countable+space" accesskey="S" class="navlink" id="history" rel="nofollow">History (3 revisions)</a> <a href="/nlab/show/first-countable+space/cite" style="color: black">Cite</a> <a href="/nlab/print/first-countable+space" accesskey="p" id="view_print" rel="nofollow">Print</a> <a href="/nlab/source/first-countable+space" id="view_source" rel="nofollow">Source</a> </div> </div>  </div>  </body> </html>