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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> Kolmogorov topological space </h1> <div class="navigation"> <span class="skipNav"><a href='#navEnd'>Skip the Navigation Links</a> | </span> <span style="display:inline-block; width: 0.3em;"></span> <a href="/nlab/show/HomePage" accesskey="H" title="Home page">Home Page</a> | <a href="/nlab/all_pages" accesskey="A" title="List of all pages">All Pages</a> | <a href="/nlab/latest_revisions" accesskey="U" title="Latest edits and page creations">Latest Revisions</a> | <a href="https://nforum.ncatlab.org/discussion/14939/#Item_16" title="Discuss this page in its dedicated thread on the nForum" style="color: black">Discuss this page</a> | <form accept-charset="utf-8" action="/nlab/search" id="navigationSearchForm" method="get"> <fieldset class="search"><input type="text" id="searchField" name="query" value="Search" style="display:inline-block; float: left;" onfocus="this.value == 'Search' ? this.value = '' : true" onblur="this.value == '' ? this.value = 'Search' : true" /></fieldset> </form> <span id='navEnd'></span> </div> <div id="revision"> <html xmlns="http://www.w3.org/1999/xhtml" xmlns:svg="http://www.w3.org/2000/svg" xml:lang="en" lang="en"> <head><meta http-equiv="Content-type" content="application/xhtml+xml;charset=utf-8" /><title>Contents</title></head> <body> <div class="rightHandSide"> <div class="toc clickDown" tabindex="0"> <h3 id="context">Context</h3> <h4 id="topology">Topology</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/topology">topology</a></strong> (<a class="existingWikiWord" href="/nlab/show/point-set+topology">point-set topology</a>, <a class="existingWikiWord" href="/nlab/show/point-free+topology">point-free topology</a>)</p> <p>see also <em><a class="existingWikiWord" href="/nlab/show/differential+topology">differential topology</a></em>, <em><a class="existingWikiWord" href="/nlab/show/algebraic+topology">algebraic topology</a></em>, <em><a class="existingWikiWord" href="/nlab/show/functional+analysis">functional analysis</a></em> and <em><a class="existingWikiWord" href="/nlab/show/topological+homotopy+theory">topological</a> <a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopy theory</a></em></p> <p><a class="existingWikiWord" href="/nlab/show/Introduction+to+Topology">Introduction</a></p> <p><strong>Basic concepts</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/open+subset">open subset</a>, <a class="existingWikiWord" href="/nlab/show/closed+subset">closed subset</a>, <a class="existingWikiWord" href="/nlab/show/neighbourhood">neighbourhood</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a>, <a class="existingWikiWord" href="/nlab/show/locale">locale</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/base+for+the+topology">base for the topology</a>, <a class="existingWikiWord" href="/nlab/show/neighbourhood+base">neighbourhood base</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/finer+topology">finer/coarser topology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+closure">closure</a>, <a class="existingWikiWord" href="/nlab/show/topological+interior">interior</a>, <a class="existingWikiWord" href="/nlab/show/topological+boundary">boundary</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/separation+axiom">separation</a>, <a class="existingWikiWord" href="/nlab/show/sober+topological+space">sobriety</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/continuous+function">continuous function</a>, <a class="existingWikiWord" href="/nlab/show/homeomorphism">homeomorphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/uniformly+continuous+function">uniformly continuous function</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+embedding">embedding</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/open+map">open map</a>, <a class="existingWikiWord" href="/nlab/show/closed+map">closed map</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/sequence">sequence</a>, <a class="existingWikiWord" href="/nlab/show/net">net</a>, <a class="existingWikiWord" href="/nlab/show/sub-net">sub-net</a>, <a class="existingWikiWord" href="/nlab/show/filter">filter</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/convergence">convergence</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/category">category</a><a class="existingWikiWord" href="/nlab/show/Top">Top</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/convenient+category+of+topological+spaces">convenient category of topological spaces</a></li> </ul> </li> </ul> <p><strong><a href="Top#UniversalConstructions">Universal constructions</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/initial+topology">initial topology</a>, <a class="existingWikiWord" href="/nlab/show/final+topology">final topology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/subspace">subspace</a>, <a class="existingWikiWord" href="/nlab/show/quotient+space">quotient space</a>,</p> </li> <li> <p>fiber space, <a class="existingWikiWord" href="/nlab/show/space+attachment">space attachment</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/product+space">product space</a>, <a class="existingWikiWord" href="/nlab/show/disjoint+union+space">disjoint union space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/mapping+cylinder">mapping cylinder</a>, <a class="existingWikiWord" href="/nlab/show/mapping+cocylinder">mapping cocylinder</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/mapping+cone">mapping cone</a>, <a class="existingWikiWord" href="/nlab/show/mapping+cocone">mapping cocone</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/mapping+telescope">mapping telescope</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/colimits+of+normal+spaces">colimits of normal spaces</a></p> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/stuff%2C+structure%2C+property">Extra stuff, structure, properties</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/nice+topological+space">nice topological space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/metric+space">metric space</a>, <a class="existingWikiWord" href="/nlab/show/metric+topology">metric topology</a>, <a class="existingWikiWord" href="/nlab/show/metrisable+space">metrisable space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Kolmogorov+space">Kolmogorov space</a>, <a class="existingWikiWord" href="/nlab/show/Hausdorff+space">Hausdorff space</a>, <a class="existingWikiWord" href="/nlab/show/regular+space">regular space</a>, <a class="existingWikiWord" href="/nlab/show/normal+space">normal space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/sober+space">sober space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/compact+space">compact space</a>, <a class="existingWikiWord" href="/nlab/show/proper+map">proper map</a></p> <p><a class="existingWikiWord" href="/nlab/show/sequentially+compact+topological+space">sequentially compact</a>, <a class="existingWikiWord" href="/nlab/show/countably+compact+topological+space">countably compact</a>, <a class="existingWikiWord" href="/nlab/show/locally+compact+topological+space">locally compact</a>, <a class="existingWikiWord" href="/nlab/show/sigma-compact+topological+space">sigma-compact</a>, <a class="existingWikiWord" href="/nlab/show/paracompact+space">paracompact</a>, <a class="existingWikiWord" href="/nlab/show/countably+paracompact+topological+space">countably paracompact</a>, <a class="existingWikiWord" href="/nlab/show/strongly+compact+topological+space">strongly compact</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/compactly+generated+space">compactly generated space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/second-countable+space">second-countable space</a>, <a class="existingWikiWord" href="/nlab/show/first-countable+space">first-countable space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/contractible+space">contractible space</a>, <a class="existingWikiWord" href="/nlab/show/locally+contractible+space">locally contractible space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/connected+space">connected space</a>, <a class="existingWikiWord" href="/nlab/show/locally+connected+space">locally connected space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/simply-connected+space">simply-connected space</a>, <a class="existingWikiWord" href="/nlab/show/locally+simply-connected+space">locally simply-connected space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cell+complex">cell complex</a>, <a class="existingWikiWord" href="/nlab/show/CW-complex">CW-complex</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/pointed+topological+space">pointed space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+vector+space">topological vector space</a>, <a class="existingWikiWord" href="/nlab/show/Banach+space">Banach space</a>, <a class="existingWikiWord" href="/nlab/show/Hilbert+space">Hilbert space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+group">topological group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+vector+bundle">topological vector bundle</a>, <a class="existingWikiWord" href="/nlab/show/topological+K-theory">topological K-theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+manifold">topological manifold</a></p> </li> </ul> <p><strong>Examples</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/empty+space">empty space</a>, <a class="existingWikiWord" href="/nlab/show/point+space">point space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/discrete+space">discrete space</a>, <a class="existingWikiWord" href="/nlab/show/codiscrete+space">codiscrete space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Sierpinski+space">Sierpinski space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/order+topology">order topology</a>, <a class="existingWikiWord" href="/nlab/show/specialization+topology">specialization topology</a>, <a class="existingWikiWord" href="/nlab/show/Scott+topology">Scott topology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Euclidean+space">Euclidean space</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/real+line">real line</a>, <a class="existingWikiWord" href="/nlab/show/plane">plane</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cylinder">cylinder</a>, <a class="existingWikiWord" href="/nlab/show/cone">cone</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/sphere">sphere</a>, <a class="existingWikiWord" href="/nlab/show/ball">ball</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/circle">circle</a>, <a class="existingWikiWord" href="/nlab/show/torus">torus</a>, <a class="existingWikiWord" href="/nlab/show/annulus">annulus</a>, <a class="existingWikiWord" href="/nlab/show/Moebius+strip">Moebius strip</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/polytope">polytope</a>, <a class="existingWikiWord" href="/nlab/show/polyhedron">polyhedron</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/projective+space">projective space</a> (<a class="existingWikiWord" href="/nlab/show/real+projective+space">real</a>, <a class="existingWikiWord" href="/nlab/show/complex+projective+space">complex</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/classifying+space">classifying space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/configuration+space+%28mathematics%29">configuration space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/path">path</a>, <a class="existingWikiWord" href="/nlab/show/loop">loop</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/mapping+spaces">mapping spaces</a>: <a class="existingWikiWord" href="/nlab/show/compact-open+topology">compact-open topology</a>, <a class="existingWikiWord" href="/nlab/show/topology+of+uniform+convergence">topology of uniform convergence</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/loop+space">loop space</a>, <a class="existingWikiWord" href="/nlab/show/path+space">path space</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Zariski+topology">Zariski topology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Cantor+space">Cantor space</a>, <a class="existingWikiWord" href="/nlab/show/Mandelbrot+space">Mandelbrot space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Peano+curve">Peano curve</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/line+with+two+origins">line with two origins</a>, <a class="existingWikiWord" href="/nlab/show/long+line">long line</a>, <a class="existingWikiWord" href="/nlab/show/Sorgenfrey+line">Sorgenfrey line</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/K-topology">K-topology</a>, <a class="existingWikiWord" href="/nlab/show/Dowker+space">Dowker space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Warsaw+circle">Warsaw circle</a>, <a class="existingWikiWord" href="/nlab/show/Hawaiian+earring+space">Hawaiian earring space</a></p> </li> </ul> <p><strong>Basic statements</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Hausdorff+spaces+are+sober">Hausdorff spaces are sober</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/schemes+are+sober">schemes are sober</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/continuous+images+of+compact+spaces+are+compact">continuous images of compact spaces are compact</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/closed+subspaces+of+compact+Hausdorff+spaces+are+equivalently+compact+subspaces">closed subspaces of compact Hausdorff spaces are equivalently compact subspaces</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/open+subspaces+of+compact+Hausdorff+spaces+are+locally+compact">open subspaces of compact Hausdorff spaces are locally compact</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quotient+projections+out+of+compact+Hausdorff+spaces+are+closed+precisely+if+the+codomain+is+Hausdorff">quotient projections out of compact Hausdorff spaces are closed precisely if the codomain is Hausdorff</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/compact+spaces+equivalently+have+converging+subnet+of+every+net">compact spaces equivalently have converging subnet of every net</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Lebesgue+number+lemma">Lebesgue number lemma</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/sequentially+compact+metric+spaces+are+equivalently+compact+metric+spaces">sequentially compact metric spaces are equivalently compact metric spaces</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/compact+spaces+equivalently+have+converging+subnet+of+every+net">compact spaces equivalently have converging subnet of every net</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/sequentially+compact+metric+spaces+are+totally+bounded">sequentially compact metric spaces are totally bounded</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/continuous+metric+space+valued+function+on+compact+metric+space+is+uniformly+continuous">continuous metric space valued function on compact metric space is uniformly continuous</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/paracompact+Hausdorff+spaces+are+normal">paracompact Hausdorff spaces are normal</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/paracompact+Hausdorff+spaces+equivalently+admit+subordinate+partitions+of+unity">paracompact Hausdorff spaces equivalently admit subordinate partitions of unity</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/closed+injections+are+embeddings">closed injections are embeddings</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/proper+maps+to+locally+compact+spaces+are+closed">proper maps to locally compact spaces are closed</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/injective+proper+maps+to+locally+compact+spaces+are+equivalently+the+closed+embeddings">injective proper maps to locally compact spaces are equivalently the closed embeddings</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/locally+compact+and+sigma-compact+spaces+are+paracompact">locally compact and sigma-compact spaces are paracompact</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/locally+compact+and+second-countable+spaces+are+sigma-compact">locally compact and second-countable spaces are sigma-compact</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/second-countable+regular+spaces+are+paracompact">second-countable regular spaces are paracompact</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/CW-complexes+are+paracompact+Hausdorff+spaces">CW-complexes are paracompact Hausdorff spaces</a></p> </li> </ul> <p><strong>Theorems</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Urysohn%27s+lemma">Urysohn's lemma</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Tietze+extension+theorem">Tietze extension theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Tychonoff+theorem">Tychonoff theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/tube+lemma">tube lemma</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Michael%27s+theorem">Michael's theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Brouwer%27s+fixed+point+theorem">Brouwer's fixed point theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+invariance+of+dimension">topological invariance of dimension</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Jordan+curve+theorem">Jordan curve theorem</a></p> </li> </ul> <p><strong>Analysis Theorems</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Heine-Borel+theorem">Heine-Borel theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/intermediate+value+theorem">intermediate value theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/extreme+value+theorem">extreme value theorem</a></p> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/topological+homotopy+theory">topological homotopy theory</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/left+homotopy">left homotopy</a>, <a class="existingWikiWord" href="/nlab/show/right+homotopy">right homotopy</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+equivalence">homotopy equivalence</a>, <a class="existingWikiWord" href="/nlab/show/deformation+retract">deformation retract</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+group">fundamental group</a>, <a class="existingWikiWord" href="/nlab/show/covering+space">covering space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+theorem+of+covering+spaces">fundamental theorem of covering spaces</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+group">homotopy group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/weak+homotopy+equivalence">weak homotopy equivalence</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Whitehead%27s+theorem">Whitehead's theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Freudenthal+suspension+theorem">Freudenthal suspension theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/nerve+theorem">nerve theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+extension+property">homotopy extension property</a>, <a class="existingWikiWord" href="/nlab/show/Hurewicz+cofibration">Hurewicz cofibration</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+cofiber+sequence">cofiber sequence</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Str%C3%B8m+model+category">Strøm model category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/classical+model+structure+on+topological+spaces">classical model structure on topological spaces</a></p> </li> </ul> </div></div> <h4 id="category_theory"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(0,1)</annotation></semantics></math>-Category theory</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/%280%2C1%29-category+theory">(0,1)-category theory</a></strong>: <a class="existingWikiWord" href="/nlab/show/logic">logic</a>, <a class="existingWikiWord" href="/nlab/show/order+theory">order theory</a></p> <p><strong><a class="existingWikiWord" href="/nlab/show/%280%2C1%29-category">(0,1)-category</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/relation+between+preorders+and+%280%2C1%29-categories">relation between preorders and (0,1)-categories</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/proset">proset</a>, <a class="existingWikiWord" href="/nlab/show/partially+ordered+set">partially ordered set</a> (<a class="existingWikiWord" href="/nlab/show/directed+set">directed set</a>, <a class="existingWikiWord" href="/nlab/show/total+order">total order</a>, <a class="existingWikiWord" href="/nlab/show/linear+order">linear order</a>)</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/top">top</a>, <a class="existingWikiWord" href="/nlab/show/true">true</a>,</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/bottom">bottom</a>, <a class="existingWikiWord" href="/nlab/show/false">false</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monotone+function">monotone function</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/implication">implication</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/filter">filter</a>, <a class="existingWikiWord" href="/nlab/show/interval">interval</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/lattice">lattice</a>, <a class="existingWikiWord" href="/nlab/show/semilattice">semilattice</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/meet">meet</a>, <a class="existingWikiWord" href="/nlab/show/logical+conjunction">logical conjunction</a>, <a class="existingWikiWord" href="/nlab/show/and">and</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/join">join</a>, <a class="existingWikiWord" href="/nlab/show/logical+disjunction">logical disjunction</a>, <a class="existingWikiWord" href="/nlab/show/or">or</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/compact+element">compact element</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/lattice+of+subobjects">lattice of subobjects</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/complete+lattice">complete lattice</a>, <a class="existingWikiWord" href="/nlab/show/algebraic+lattice">algebraic lattice</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/distributive+lattice">distributive lattice</a>, <a class="existingWikiWord" href="/nlab/show/completely+distributive+lattice">completely distributive lattice</a>, <a class="existingWikiWord" href="/nlab/show/canonical+extension">canonical extension</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/hyperdoctrine">hyperdoctrine</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/first-order+hyperdoctrine">first-order</a>, <a class="existingWikiWord" href="/nlab/show/Boolean+hyperdoctrine">Boolean</a>, <a class="existingWikiWord" href="/nlab/show/coherent+hyperdoctrine">coherent</a>, <a class="existingWikiWord" href="/nlab/show/tripos">tripos</a></li> </ul> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/%280%2C1%29-topos">(0,1)-topos</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Heyting+algebra">Heyting algebra</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/regular+element">regular element</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Boolean+algebra">Boolean algebra</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/frame">frame</a>, <a class="existingWikiWord" href="/nlab/show/locale">locale</a></p> </li> </ul> <h2 id="theorems">Theorems</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/Stone+duality">Stone duality</a></li> </ul> </div></div> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#definition'>Definition</a></li> <li><a href='#Properties'>Properties</a></li> <ul> <li><a href='#specialization_order'>Specialization order</a></li> <li><a href='#alternative_characterizations'>Alternative Characterizations</a></li> <li><a href='#SeparationAxiomInTermsOfLiftingProperties'>In terms of lifting properties</a></li> <li><a href='#reflection'>Reflection</a></li> </ul> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> </ul> </div> <h2 id="definition">Definition</h2> <p>A <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><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 called a <em>Kolmogorov space</em> if it satisfies the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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/separation+axiom">separation axiom</a>, hence if for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>x</mi> <mn>1</mn></msub><mo>≠</mo><msub><mi>x</mi> <mn>2</mn></msub><mo>∈</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">x_1 \neq x_2 \in X</annotation></semantics></math> any two distinct points, then at least one of them has an <a class="existingWikiWord" href="/nlab/show/open+neighbourhood">open neighbourhood</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>U</mi> <mrow><msub><mi>x</mi> <mi>i</mi></msub></mrow></msub><mo>∈</mo><mi>τ</mi></mrow><annotation encoding="application/x-tex">U_{x_i} \in \tau</annotation></semantics></math> which does not contain the other point. This is equivalent to its <a class="existingWikiWord" href="/nlab/show/contrapositive">contrapositive</a>: for all points <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>x</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>x</mi> <mn>2</mn></msub><mo>∈</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">x_1, x_2 \in X</annotation></semantics></math>, if every <a class="existingWikiWord" href="/nlab/show/open+neighbourhood">open neighbourhood</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>U</mi> <mrow><msub><mi>x</mi> <mi>i</mi></msub></mrow></msub><mo>∈</mo><mi>τ</mi></mrow><annotation encoding="application/x-tex">U_{x_i} \in \tau</annotation></semantics></math> which contains one of the points also contains the other point, then the two points are equal <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>x</mi> <mn>1</mn></msub><mo>=</mo><msub><mi>x</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">x_1 = x_2</annotation></semantics></math>.</p> <div> <p>the <strong>main <a class="existingWikiWord" href="/nlab/show/separation+axioms">separation axioms</a></strong></p> <table><thead><tr><th>number</th><th>name</th><th>statement</th><th>reformulation</th></tr></thead><tbody><tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>T</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">T_0</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Kolmogorov+space">Kolmogorov</a></td><td style="text-align: left;">given two distinct points, at least one of them has an <a class="existingWikiWord" href="/nlab/show/open+neighbourhood">open neighbourhood</a> not containing the other point</td><td style="text-align: left;">every <a class="existingWikiWord" href="/nlab/show/irreducible+closed+subset">irreducible closed subset</a> is the <a class="existingWikiWord" href="/nlab/show/topological+closure">closure</a> of at most one point</td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>T</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">T_1</annotation></semantics></math></td><td style="text-align: left;"></td><td style="text-align: left;">given two distinct points, both have an <a class="existingWikiWord" href="/nlab/show/open+neighbourhood">open neighbourhood</a> not containing the other point</td><td style="text-align: left;">all points are <a class="existingWikiWord" href="/nlab/show/closed+point">closed</a></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>T</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">T_2</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Hausdorff+topological+space">Hausdorff</a></td><td style="text-align: left;">given two distinct points, they have <a class="existingWikiWord" href="/nlab/show/disjoint+subset">disjoint</a> <a class="existingWikiWord" href="/nlab/show/open+neighbourhoods">open neighbourhoods</a></td><td style="text-align: left;">the <a class="existingWikiWord" href="/nlab/show/diagonal">diagonal</a> is a <a class="existingWikiWord" href="/nlab/show/closed+map">closed map</a></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>T</mi> <mrow><mo>></mo><mn>2</mn></mrow></msub></mrow><annotation encoding="application/x-tex">T_{\gt 2}</annotation></semantics></math></td><td style="text-align: left;"></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>T</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">T_1</annotation></semantics></math> and…</td><td style="text-align: left;">all points are <a class="existingWikiWord" href="/nlab/show/closed+point">closed</a> and…</td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>T</mi> <mn>3</mn></msub></mrow><annotation encoding="application/x-tex">T_3</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/regular+Hausdorff+topological+space">regular Hausdorff</a></td><td style="text-align: left;">…given a point and a <a class="existingWikiWord" href="/nlab/show/closed+subset">closed subset</a> not containing it, they have <a class="existingWikiWord" href="/nlab/show/disjoint+subset">disjoint</a> <a class="existingWikiWord" href="/nlab/show/open+neighbourhoods">open neighbourhoods</a></td><td style="text-align: left;">…every <a class="existingWikiWord" href="/nlab/show/neighbourhood">neighbourhood</a> of a point contains the <a class="existingWikiWord" href="/nlab/show/topological+closure">closure</a> of an <a class="existingWikiWord" href="/nlab/show/open+neighbourhood">open neighbourhood</a></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>T</mi> <mn>4</mn></msub></mrow><annotation encoding="application/x-tex">T_4</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/normal+Hausdorff+topological+space">normal Hausdorff</a></td><td style="text-align: left;">…given two <a class="existingWikiWord" href="/nlab/show/disjoint+subset">disjoint</a> <a class="existingWikiWord" href="/nlab/show/closed+subsets">closed subsets</a>, they have <a class="existingWikiWord" href="/nlab/show/disjoint+subset">disjoint</a> <a class="existingWikiWord" href="/nlab/show/open+neighbourhoods">open neighbourhoods</a></td><td style="text-align: left;">…every <a class="existingWikiWord" href="/nlab/show/neighbourhood">neighbourhood</a> of a <a class="existingWikiWord" href="/nlab/show/closed+set">closed set</a> also contains the <a class="existingWikiWord" href="/nlab/show/topological+closure">closure</a> of an <a class="existingWikiWord" href="/nlab/show/open+neighbourhood">open neighbourhood</a> <br /> … every pair of <a class="existingWikiWord" href="/nlab/show/disjoint+subset">disjoint</a> <a class="existingWikiWord" href="/nlab/show/closed+subsets">closed subsets</a> is separated by an <a class="existingWikiWord" href="/nlab/show/Urysohn+function">Urysohn function</a></td></tr> </tbody></table> </div> <h2 id="Properties">Properties</h2> <h3 id="specialization_order">Specialization order</h3> <p>Every topological space <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>O</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(X, O(X))</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/preorder">preorder</a> with respect to the <a class="existingWikiWord" href="/nlab/show/specialization+order">specialization order</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mo>∀</mo> <mrow><mi>U</mi><mo>:</mo><mi>O</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo>∈</mo><mi>U</mi><mo stretchy="false">)</mo><mo>⇒</mo><mo stretchy="false">(</mo><mi>y</mi><mo>∈</mo><mi>U</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\forall_{U:O(X)} (x \in U) \implies (y \in U)</annotation></semantics></math>.</p> <p> <div class='num_prop'> <h6>Proposition</h6> <p>A topological space is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>T</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">T_0</annotation></semantics></math> precisely when it is a <a class="existingWikiWord" href="/nlab/show/partial+order">partial order</a> with respect to the <a class="existingWikiWord" href="/nlab/show/specialization+order">specialization order</a> of the topological space.</p> </div> </p> <p>The proof follows from the <a class="existingWikiWord" href="/nlab/show/contrapositive">contrapositive</a> of the definition of a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>T</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">T_0</annotation></semantics></math>-space.</p> <h3 id="alternative_characterizations">Alternative Characterizations</h3> <p>Regard <a class="existingWikiWord" href="/nlab/show/Sierpinski+space">Sierpinski space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Σ</mi></mrow><annotation encoding="application/x-tex">\Sigma</annotation></semantics></math> as a <a class="existingWikiWord" href="/nlab/show/frame">frame</a> object in the category of topological spaces (meaning: the representable functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Top</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mi>Σ</mi><mo stretchy="false">)</mo><mo>:</mo><msup><mi>Top</mi> <mi>op</mi></msup><mo>→</mo><mi>Set</mi></mrow><annotation encoding="application/x-tex">Top(-, \Sigma): Top^{op} \to Set</annotation></semantics></math> lifts through the monadic forgetful functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Frame</mi><mo>→</mo><mi>Set</mi></mrow><annotation encoding="application/x-tex">Frame \to Set</annotation></semantics></math>), hence as a <a class="existingWikiWord" href="/nlab/show/dualizing+object">dualizing object</a> that induces a contravariant adjunction between frames and topological spaces. Thus for a space <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>, the frame <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Top</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>Σ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Top(X, \Sigma)</annotation></semantics></math> is the frame of <a class="existingWikiWord" href="/nlab/show/open+sets">open sets</a>; for a frame <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>, there is an accompanying topological space of points <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Frame</mi><mo stretchy="false">(</mo><mi>A</mi><mo>,</mo><mi>Σ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Frame(A, \Sigma)</annotation></semantics></math>, a subspace of the product space <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Σ</mi> <mrow><mo stretchy="false">|</mo><mi>A</mi><mo stretchy="false">|</mo></mrow></msup></mrow><annotation encoding="application/x-tex">\Sigma^{{|A|}}</annotation></semantics></math>. The unit of the adjunction is called the <em>double dual embedding</em>.</p> <p> <div class='num_prop'> <h6>Proposition</h6> <p>A <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>T</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">T_0</annotation></semantics></math> precisely when the double dual embedding <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>→</mo><mi>Frame</mi><mo stretchy="false">(</mo><mi>Top</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>Σ</mi><mo stretchy="false">)</mo><mo>,</mo><mi>Σ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">X \to Frame(Top(X, \Sigma), \Sigma)</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/monomorphism">monomorphism</a>.</p> </div> </p> <p>The proof is trivial: the monomorphism condition translates to saying that for any points <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>,</mo><mi>y</mi><mo>∈</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">x, y \in X</annotation></semantics></math>, if the truth values of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>∈</mo><mi>U</mi></mrow><annotation encoding="application/x-tex">x \in U</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>y</mi><mo>∈</mo><mi>U</mi></mrow><annotation encoding="application/x-tex">y \in U</annotation></semantics></math> agree for every open set <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi></mrow><annotation encoding="application/x-tex">U</annotation></semantics></math>, then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>=</mo><mi>y</mi></mrow><annotation encoding="application/x-tex">x = y</annotation></semantics></math>.</p> <div> <h3 id="SeparationAxiomInTermsOfLiftingProperties">In terms of lifting properties</h3> <p>The <a class="existingWikiWord" href="/nlab/show/separation+conditions">separation conditions</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>T</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">T_0</annotation></semantics></math> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>T</mi> <mn>4</mn></msub></mrow><annotation encoding="application/x-tex">T_4</annotation></semantics></math> may equivalently be understood as <a class="existingWikiWord" href="/nlab/show/lifting+properties">lifting properties</a> against certain maps of <a class="existingWikiWord" href="/nlab/show/finite+topological+spaces">finite topological spaces</a>, among others.</p> <p>This is discussed at <em><a class="existingWikiWord" href="/nlab/show/separation+axioms+in+terms+of+lifting+properties">separation axioms in terms of lifting properties</a></em>, to which we refer for further details. Here we just briefly indicate the corresponding lifting diagrams.</p> <p>In the following diagrams, the relevant <a class="existingWikiWord" href="/nlab/show/finite+topological+spaces">finite topological spaces</a> are indicated explicitly by illustration of their <a class="existingWikiWord" href="/nlab/show/underlying">underlying</a> point set and their <a class="existingWikiWord" href="/nlab/show/open+subsets">open subsets</a>:</p> <ul> <li> <p>points (elements) are denoted by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>•</mo></mrow><annotation encoding="application/x-tex">\bullet</annotation></semantics></math> with subscripts indicating where the points map to;</p> </li> <li> <p>boxes are put around <a class="existingWikiWord" href="/nlab/show/open+subsets">open subsets</a>,</p> </li> <li> <p>an arrow <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mo>•</mo> <mi>u</mi></msub><mo>→</mo><msub><mo>•</mo> <mi>c</mi></msub></mrow><annotation encoding="application/x-tex">\bullet_u \to \bullet_c</annotation></semantics></math> means that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mo>•</mo> <mi>c</mi></msub></mrow><annotation encoding="application/x-tex">\bullet_c</annotation></semantics></math> is in the <a class="existingWikiWord" href="/nlab/show/topological+closure">topological closure</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mo>•</mo> <mi>u</mi></msub></mrow><annotation encoding="application/x-tex">\bullet_u</annotation></semantics></math>.</p> </li> </ul> <p>In the lifting diagrams for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>T</mi> <mn>2</mn></msub><mo>−</mo><msub><mi>T</mi> <mn>4</mn></msub></mrow><annotation encoding="application/x-tex">T_2-T_4</annotation></semantics></math> below, an arrow out of the given <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/map">map</a> that determines (classifies) a decomposition of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> into a <a class="existingWikiWord" href="/nlab/show/union">union</a> of <a class="existingWikiWord" href="/nlab/show/subsets">subsets</a> with properties indicated by the picture of the finite space.</p> <p>Notice that the diagrams for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>T</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">T_2</annotation></semantics></math>-<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>T</mi> <mn>4</mn></msub></mrow><annotation encoding="application/x-tex">T_4</annotation></semantics></math> below do not in themselves imply <a class="existingWikiWord" href="/nlab/show/T1"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"> <semantics> <mrow> <msub><mi>T</mi> <mn>1</mn></msub> </mrow> <annotation encoding="application/x-tex">T_1</annotation> </semantics> </math></a>.</p> <p> <div class="num_prop"> <h6>Proposition</h6> <p><strong>(Lifting property encoding <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>T</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">T_0</annotation></semantics></math>)</strong> <br /> The following <a class="existingWikiWord" href="/nlab/show/lifting+property">lifting property</a> in <a class="existingWikiWord" href="/nlab/show/Top">Top</a> equivalently encodes the <a class="existingWikiWord" href="/nlab/show/separation+axiom">separation axiom</a> <a class="existingWikiWord" href="/nlab/show/T0"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"> <semantics> <mrow> <msub><mi>T</mi> <mn>0</mn></msub> </mrow> <annotation encoding="application/x-tex">T_0</annotation> </semantics> </math></a>:</p> <svg xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" width="135.276" height="99.78" viewBox="0 0 135.276 99.78"> <defs> <g> <g id="1d_XmH7ZvqhH3-pcTRrAnIWvYFk=-glyph-0-0"> </g> <g id="1d_XmH7ZvqhH3-pcTRrAnIWvYFk=-glyph-0-1"> <path d="M 3.375 -7.375 C 3.375 -7.859375 3.6875 -8.625 5 -8.703125 C 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transform="matrix(0.7071, -0.70708, -0.70708, -0.7071, 106.49142, 18.65319)"></path> <g fill="rgb(0%, 0%, 0%)" fill-opacity="1"> <use xlink:href="#1d_XmH7ZvqhH3-pcTRrAnIWvYFk=-glyph-3-2" x="66.522" y="48.291"></use> </g> </svg> <p></p> </div> </p> <p> <div class="num_prop"> <h6>Proposition</h6> <p><strong>(Lifting property encoding <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>T</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">T_1</annotation></semantics></math>)</strong> <br /> The following <a class="existingWikiWord" href="/nlab/show/lifting+property">lifting property</a> in <a class="existingWikiWord" href="/nlab/show/Top">Top</a> equivalently encodes the <a class="existingWikiWord" href="/nlab/show/separation+axiom">separation axiom</a> <a class="existingWikiWord" href="/nlab/show/T1"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"> <semantics> <mrow> 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encoding="application/x-tex">T_2</annotation></semantics></math>)</strong> <br /> The following <a class="existingWikiWord" href="/nlab/show/lifting+property">lifting property</a> in <a class="existingWikiWord" href="/nlab/show/Top">Top</a> equivalently encodes the <a class="existingWikiWord" href="/nlab/show/separation+axiom">separation axiom</a> <a class="existingWikiWord" href="/nlab/show/T2"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"> <semantics> <mrow> <msub><mi>T</mi> <mn>2</mn></msub> </mrow> <annotation encoding="application/x-tex">T_2</annotation> </semantics> </math></a>:</p> <svg xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" width="231.836" height="131.601" viewBox="0 0 231.836 131.601"> <defs> <g> <g id="J93l4iDU14g_p7R1ul1qa9DDid8=-glyph-0-0"> </g> <g id="J93l4iDU14g_p7R1ul1qa9DDid8=-glyph-0-1"> <path d="M 3.375 -7.375 C 3.375 -7.859375 3.6875 -8.625 5 -8.703125 C 5.0625 -8.71875 5.109375 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<h6>Proposition</h6> <p><strong>(Lifting property encoding <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>T</mi> <mn>3</mn></msub></mrow><annotation encoding="application/x-tex">T_3</annotation></semantics></math>)</strong> <br /> The following <a class="existingWikiWord" href="/nlab/show/lifting+property">lifting property</a> in <a class="existingWikiWord" href="/nlab/show/Top">Top</a> equivalently encodes the <a class="existingWikiWord" href="/nlab/show/separation+axiom">separation axiom</a> <a class="existingWikiWord" href="/nlab/show/T3"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"> <semantics> <mrow> <msub><mi>T</mi> <mn>3</mn></msub> </mrow> <annotation encoding="application/x-tex">T_3</annotation> </semantics> </math></a>:</p> <svg xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" width="212.503" height="124.727" viewBox="0 0 212.503 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fill-opacity="1"> <use xlink:href="#A_LG5PqGwMF9qFH4yZlgCmYaimQ=-glyph-2-3" x="59.7" y="70.07"></use> </g> </svg> <p></p> </div> </p> <p> <div class="num_prop"> <h6>Proposition</h6> <p><strong>(Lifting property encoding <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>T</mi> <mn>4</mn></msub></mrow><annotation encoding="application/x-tex">T_4</annotation></semantics></math>)</strong> <br /> The following <a class="existingWikiWord" href="/nlab/show/lifting+property">lifting property</a> in <a class="existingWikiWord" href="/nlab/show/Top">Top</a> equivalently encodes the <a class="existingWikiWord" href="/nlab/show/separation+axiom">separation axiom</a> <a class="existingWikiWord" href="/nlab/show/T4"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"> <semantics> <mrow> <msub><mi>T</mi> <mn>4</mn></msub> </mrow> <annotation encoding="application/x-tex">T_4</annotation> </semantics> 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stroke-opacity="1" stroke-miterlimit="10" d="M -80.528562 -43.19025 L -34.700437 -43.19025 " transform="matrix(1, 0, 0, -1, 101.677, 64.466)"></path> <path fill="none" stroke-width="0.47818" stroke-linecap="round" stroke-linejoin="round" stroke="rgb(0%, 0%, 0%)" stroke-opacity="1" stroke-miterlimit="10" d="M -2.488077 2.869424 C -2.031046 1.146767 -1.019327 0.334267 0.00020375 -0.00167 C -1.019327 -0.333701 -2.031046 -1.146201 -2.488077 -2.868858 " transform="matrix(1, 0, 0, -1, 67.21464, 107.65458)"></path> <path fill="none" stroke-width="0.47818" stroke-linecap="butt" stroke-linejoin="miter" stroke="rgb(0%, 0%, 0%)" stroke-opacity="1" stroke-dasharray="3.34735 1.91277" stroke-miterlimit="10" d="M -80.528562 -36.080875 L -14.415281 7.99725 " transform="matrix(1, 0, 0, -1, 101.677, 64.466)"></path> <path fill="none" stroke-width="0.47818" stroke-linecap="round" stroke-linejoin="round" stroke="rgb(0%, 0%, 0%)" stroke-opacity="1" stroke-miterlimit="10" d="M -2.484075 2.869708 C -2.03231 1.145929 -1.0193 0.334405 0.00238742 0.000682961 C -1.020396 -0.33625 -2.032355 -1.149902 -2.488521 -2.869328 " transform="matrix(0.832, -0.55469, -0.55469, -0.832, 87.45933, 56.33783)"></path> <g fill="rgb(0%, 0%, 0%)" fill-opacity="1"> <use xlink:href="#ciWwWK3VtzzU8FkNssdLWNRGT8A=-glyph-1-2" x="46.886" y="75.564"></use> </g> </svg> <p></p> </div> </p> </div> <h3 id="reflection">Reflection</h3> <div class="num_prop" id="KolmogorovQuotient"> <h6 id="proposition">Proposition</h6> <p><strong>(Kolmogorov quotient)</strong></p> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>τ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(X,\tau)</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a>. Consider the <a class="existingWikiWord" href="/nlab/show/relation">relation</a> on the underlying set by which <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>x</mi> <mn>1</mn></msub><mo>∼</mo><msub><mi>x</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">x_1 \sim x_1</annotation></semantics></math> precisely if neither <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>x</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">x_i</annotation></semantics></math> has an <a class="existingWikiWord" href="/nlab/show/open+neighbourhood">open neighbourhood</a> not containing the other. This is an <a class="existingWikiWord" href="/nlab/show/equivalence+relation">equivalence relation</a>. The <a class="existingWikiWord" href="/nlab/show/quotient+topological+space">quotient topological space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>→</mo><mi>X</mi><mo stretchy="false">/</mo><mo>∼</mo></mrow><annotation encoding="application/x-tex">X \to X/\sim</annotation></semantics></math> by this equivalence relation is a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>T</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">T_0</annotation></semantics></math>-space.</p> <p>This construction is the reflector exhibiting Kolmogorov spaces as a <a class="existingWikiWord" href="/nlab/show/reflective+subcategory">reflective subcategory</a> of the <a class="existingWikiWord" href="/nlab/show/category">category</a> <a class="existingWikiWord" href="/nlab/show/Top">Top</a> of all topological spaces.</p> </div> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Hausdorff+topological+space">Hausdorff topological space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/nice+topological+space">nice topological space</a></p> </li> </ul> <table><thead><tr><th><a class="existingWikiWord" href="/nlab/show/preorder">preorder</a></th><th><a class="existingWikiWord" href="/nlab/show/partial+order">partial order</a></th><th><a class="existingWikiWord" href="/nlab/show/equivalence+relation">equivalence relation</a></th><th><a class="existingWikiWord" href="/nlab/show/equality">equality</a></th></tr></thead><tbody><tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Kolmogorov+topological+space">Kolmogorov topological space</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/symmetric+topological+space">symmetric topological space</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/accessible+topological+space">accessible topological space</a></td></tr> </tbody></table> <h2 id="references">References</h2> <ul> <li>Wikipedia, <em><a href="https://en.wikipedia.org/wiki/Kolmogorov_space">Kolmogorov space</a></em></li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on June 6, 2023 at 11:08:40. 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