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sober topological space in nLab
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<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 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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/9020/#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"> <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> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#definition'>Definition</a></li> <li><a href='#properties'>Properties</a></li> <ul> <li><a href='#separation'>Separation</a></li> <li><a href='#AsLocalesWithEnoughPoints'>As locales with enough points</a></li> <li><a href='#Reflection'>Soberification reflection</a></li> <li><a href='#enough_points'>Enough points</a></li> </ul> <li><a href='#examples_and_nonexamples'>Examples and Non-examples</a></li> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> </ul> </div> <h2 id="idea">Idea</h2> <p>If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒪</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathcal{O}(X)</annotation></semantics></math> is the topology on 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> (i.e. its <a class="existingWikiWord" href="/nlab/show/frame+of+opens">frame of opens</a>), and if a map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒪</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>→</mo><mi>𝒪</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathcal{O}(X) \to \mathcal{O}(1)</annotation></semantics></math> that preserves finite <a class="existingWikiWord" href="/nlab/show/meets">meets</a> and arbitrary <a class="existingWikiWord" href="/nlab/show/joins">joins</a> (a <a class="existingWikiWord" href="/nlab/show/homomorphism">homomorphism</a> of <a class="existingWikiWord" href="/nlab/show/frames">frames</a>) is considered an instance of “seeing a point <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>1</mn><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">1 \to X</annotation></semantics></math>”, then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> is <em>sober</em> precisely if every point we see is really there (i.e., is induced from a <a class="existingWikiWord" href="/nlab/show/continuous+function">continuous function</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>1</mn><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">1 \to X</annotation></semantics></math>), and if we never see double.</p> <p>The condition that a <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a> be <em>sober</em> is an extra condition akin to a <a class="existingWikiWord" href="/nlab/show/separation+axiom">separation axiom</a>. In fact with <a class="existingWikiWord" href="/nlab/show/classical+logic">classical logic</a> it is a condition implied by the <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 class="existingWikiWord" href="/nlab/show/separation+axiom">separation axiom</a> (<a class="existingWikiWord" href="/nlab/show/Hausdorff+implies+sober">Hausdorff implies sober</a>) and implying <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>.</p> <table><thead><tr><th><a class="existingWikiWord" href="/nlab/show/separation+axioms">separation axioms</a></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><mrow><mtable><mtr><mtd></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><msub><mi>T</mi> <mn>2</mn></msub><mo>=</mo><mtext>Hausdorff</mtext></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mo>⇙</mo></mtd> <mtd></mtd> <mtd><mo>⇘</mo></mtd></mtr> <mtr><mtd><mspace width="thinmathspace"></mspace></mtd> <mtd><msub><mi>T</mi> <mn>1</mn></msub></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><mtext>sober</mtext></mtd> <mtd><mspace width="thinmathspace"></mspace></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mo>⇘</mo></mtd> <mtd></mtd> <mtd><mo>⇙</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><msub><mi>T</mi> <mn>0</mn></msub><mo>=</mo><mtext>Kolmogorov</mtext></mtd></mtr> <mtr><mtd></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex">\array{\\ &&& T_2 = \text{Hausdorff} \\ && \swArrow && \seArrow \\ \, & T_1 && && \text{sober} & \, \\ && \seArrow && \swArrow \\ &&& T_0 = \text{Kolmogorov} \\ } </annotation></semantics></math></td></tr> </tbody></table> <p>(Note that this diagram is not a pullback – there are <a class="existingWikiWord" href="/nlab/show/Hausdorff+implies+sober"><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> sober spaces which are not Hausdorff</a>.)</p> <p>But the sobriety condition on a topological space has deeper meaning. It means that <a class="existingWikiWord" href="/nlab/show/continuous+functions">continuous functions</a> between sober topological spaces are entirely determined by their <a class="existingWikiWord" href="/nlab/show/inverse+image">inverse image</a> functions on the <a class="existingWikiWord" href="/nlab/show/frames+of+opens">frames of opens</a>, disregarding the underlying sets of points. Technically this means that the sober topological spaces are precisely the <a class="existingWikiWord" href="/nlab/show/locales">locales</a> among the topological spaces.</p> <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><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> is <strong>sober</strong> if its <a class="existingWikiWord" href="/nlab/show/points">points</a> are exactly determined by its <a class="existingWikiWord" href="/nlab/show/lattice+of+open+subsets">lattice of open subsets</a>. Different equivalent ways to say this are:</p> <ul> <li> <p>The <a class="existingWikiWord" href="/nlab/show/continuous+map">continuous map</a> from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> to the space of points of the <a class="existingWikiWord" href="/nlab/show/locale">locale</a> that it gives rise to (see there for details) is a <a class="existingWikiWord" href="/nlab/show/homeomorphism">homeomorphism</a>.</p> </li> <li> <p>The <a class="existingWikiWord" href="/nlab/show/function">function</a> from points of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> to the <em><a class="existingWikiWord" href="/nlab/show/completely+prime+filter">completely prime filters</a></em> of its open-set lattice is a <a class="existingWikiWord" href="/nlab/show/bijection">bijection</a>.</p> </li> <li> <p>(Assuming classical logic) every <a class="existingWikiWord" href="/nlab/show/irreducible+closed+set">irreducible closed set</a> (non-empty closed set that is not the <a class="existingWikiWord" href="/nlab/show/union">union</a> of any two proper closed subsets) is the <a class="existingWikiWord" href="/nlab/show/topological+closure">closure</a> of a unique point.</p> </li> </ul> <p>In each case, half of the definition is that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/T0">T0</a>, the other half states that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> has <strong>enough points</strong>:</p> <div class="num_defn" id="EnoughPointsOfATopologicalSpace"> <h6 id="definition_2">Definition</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> has <em>enough points</em> if the following equivalent conditions hold:</p> <ul> <li> <p>The <a class="existingWikiWord" href="/nlab/show/continuous+map">continuous map</a> from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> to the space of points of the <a class="existingWikiWord" href="/nlab/show/locale">locale</a> that it gives rise to (see there for details) is a <a class="existingWikiWord" href="/nlab/show/quotient+space">quotient map</a>.</p> </li> <li> <p>The <a class="existingWikiWord" href="/nlab/show/function">function</a> from points of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> to the <em><a class="existingWikiWord" href="/nlab/show/completely+prime+filter">completely prime filters</a> of its <a class="existingWikiWord" href="/nlab/show/lattice+of+open+subsets">open-set lattice</a> is a <a class="existingWikiWord" href="/nlab/show/surjection">surjection</a>.</em></p> </li> <li> <p>(Assuming classical logic) every <a class="existingWikiWord" href="/nlab/show/irreducible+closed+set">irreducible closed set</a> (non-empty closed set that is not the <a class="existingWikiWord" href="/nlab/show/union">union</a> of any two proper closed subsets) is the <a class="existingWikiWord" href="/nlab/show/topological+closure">closure</a> of a point.</p> </li> </ul> </div> <h2 id="properties">Properties</h2> <h3 id="separation">Separation</h3> <ul> <li>Sobriety is a <a class="existingWikiWord" href="/nlab/show/separation+property">separation property</a> that is stronger than <a class="existingWikiWord" href="/nlab/show/T0">T0</a>, but incomparable with <a class="existingWikiWord" href="/nlab/show/T1">T1</a>. With <a class="existingWikiWord" href="/nlab/show/classical+logic">classical logic</a>, every <a class="existingWikiWord" href="/nlab/show/Hausdorff+space">Hausdorff space</a> is sober (see at <em><a class="existingWikiWord" href="/nlab/show/Hausdorff+implies+sober">Hausdorff implies sober</a></em>), but this can fail <a class="existingWikiWord" href="/nlab/show/constructive+mathematics">constructively</a>.</li> </ul> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Hausdorff</mi><mo>=</mo><msub><mi>T</mi> <mn>2</mn></msub><mo>⇒</mo><mi>sober</mi><mo>⇒</mo><msub><mi>T</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex"> Hausdorff = T_2 \Rightarrow sober \Rightarrow T_0 </annotation></semantics></math></div> <h3 id="AsLocalesWithEnoughPoints">As locales with enough points</h3> <p>What makes the concept of sober topological spaces special is that for them the concept of <a class="existingWikiWord" href="/nlab/show/continuous+functions">continuous functions</a> may be expressed entirely in terms of the relations between their <a class="existingWikiWord" href="/nlab/show/open+subsets">open subsets</a>, disregarding the underlying set of points of which these open are in fact subsets. In order to express this property (proposition <a class="maruku-ref" href="#FrameMorphismsBetweenOpensOfSoberSpaces"></a> below), we first introduce the following terminology:</p> <div class="num_defn" id="HomomorphismOfFramesOfOpens"> <h6 id="definition_3">Definition</h6> <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><msub><mi>τ</mi> <mi>X</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(X,\tau_X)</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>Y</mi><mo>,</mo><msub><mi>τ</mi> <mi>Y</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(Y,\tau_Y)</annotation></semantics></math> be <a class="existingWikiWord" href="/nlab/show/topological+spaces">topological spaces</a>. Then a function</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>τ</mi> <mi>X</mi></msub><mo>⟵</mo><msub><mi>τ</mi> <mi>Y</mi></msub><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>ϕ</mi></mrow><annotation encoding="application/x-tex"> \tau_X \longleftarrow \tau_Y \;\colon\; \phi </annotation></semantics></math></div> <p>between their <a class="existingWikiWord" href="/nlab/show/frame+of+opens">sets of open subsets</a> is called a <em><a class="existingWikiWord" href="/nlab/show/frame">frame</a> <a class="existingWikiWord" href="/nlab/show/homomorphism">homomorphism</a></em> if it preserves</p> <ol> <li> <p>arbitrary <a class="existingWikiWord" href="/nlab/show/unions">unions</a>;</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/finite+number">finite</a><a class="existingWikiWord" href="/nlab/show/intersections">intersections</a>.</p> </li> </ol> <p>In other words, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi></mrow><annotation encoding="application/x-tex">\phi</annotation></semantics></math> is a frame homomorphism if</p> <ol> <li> <p>for every <a class="existingWikiWord" href="/nlab/show/set">set</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>I</mi></mrow><annotation encoding="application/x-tex">I</annotation></semantics></math> and every <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>I</mi></mrow><annotation encoding="application/x-tex">I</annotation></semantics></math>-indexed set <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><msub><mi>U</mi> <mi>i</mi></msub><mo>∈</mo><msub><mi>τ</mi> <mi>Y</mi></msub><msub><mo stretchy="false">}</mo> <mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\{U_i \in \tau_Y\}_{i \in I}</annotation></semantics></math> of elements of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>τ</mi> <mi>Y</mi></msub></mrow><annotation encoding="application/x-tex">\tau_Y</annotation></semantics></math>, then</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi><mrow><mo>(</mo><munder><mo>∪</mo><mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></munder><msub><mi>U</mi> <mi>i</mi></msub><mo>)</mo></mrow><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><munder><mo>∪</mo><mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></munder><mi>ϕ</mi><mo stretchy="false">(</mo><msub><mi>U</mi> <mi>i</mi></msub><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo>∈</mo><msub><mi>τ</mi> <mi>X</mi></msub><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \phi\left(\underset{i \in I}{\cup} U_i\right) \;=\; \underset{i \in I}{\cup} \phi(U_i)\;\;\;\;\in \tau_X \,, </annotation></semantics></math></div></li> <li> <p>for every <a class="existingWikiWord" href="/nlab/show/finite+set">finite set</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>J</mi></mrow><annotation encoding="application/x-tex">J</annotation></semantics></math> and every <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>J</mi></mrow><annotation encoding="application/x-tex">J</annotation></semantics></math>-indexed set <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><msub><mi>U</mi> <mi>j</mi></msub><mo>∈</mo><msub><mi>τ</mi> <mi>Y</mi></msub><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{U_j \in \tau_Y\}</annotation></semantics></math> of elements in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>τ</mi> <mi>Y</mi></msub></mrow><annotation encoding="application/x-tex">\tau_Y</annotation></semantics></math>, then</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi><mrow><mo>(</mo><munder><mo>∩</mo><mrow><mi>j</mi><mo>∈</mo><mi>J</mi></mrow></munder><msub><mi>U</mi> <mi>j</mi></msub><mo>)</mo></mrow><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><munder><mo>∩</mo><mrow><mi>j</mi><mo>∈</mo><mi>J</mi></mrow></munder><mi>ϕ</mi><mo stretchy="false">(</mo><msub><mi>U</mi> <mi>j</mi></msub><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo>∈</mo><msub><mi>τ</mi> <mi>X</mi></msub><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \phi\left(\underset{j \in J}{\cap} U_j\right) \;=\; \underset{j \in J}{\cap} \phi(U_j) \;\;\;\;\in \tau_X \,. </annotation></semantics></math></div></li> </ol> </div> <div class="num_remark" id="PreservationOfInclusionsByFrameHomomorphism"> <h6 id="remark">Remark</h6> <p>A <a class="existingWikiWord" href="/nlab/show/frame">frame</a> <a class="existingWikiWord" href="/nlab/show/homomorphism">homomorphism</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi></mrow><annotation encoding="application/x-tex">\phi</annotation></semantics></math> as in def. <a class="maruku-ref" href="#HomomorphismOfFramesOfOpens"></a> necessarily also preserves inclusions in that</p> <ul> <li> <p>for every inclusion <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>U</mi> <mn>1</mn></msub><mo>⊂</mo><msub><mi>U</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">U_1 \subset U_2</annotation></semantics></math> with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>U</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>U</mi> <mn>2</mn></msub><mo>∈</mo><msub><mi>τ</mi> <mi>Y</mi></msub><mo>⊂</mo><mi>P</mi><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">U_1, U_2 \in \tau_Y \subset P(Y)</annotation></semantics></math> then</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi><mo stretchy="false">(</mo><msub><mi>U</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mo>⊂</mo><mi>ϕ</mi><mo stretchy="false">(</mo><msub><mi>U</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo>∈</mo><msub><mi>τ</mi> <mi>X</mi></msub><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \phi(U_1) \subset \phi(U_2) \;\;\;\;\;\;\; \in \tau_X \,. </annotation></semantics></math></div></li> </ul> <p>This is because inclusions are witnessed by unions</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>U</mi> <mn>1</mn></msub><mo>⊂</mo><msub><mi>U</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo>⇔</mo><mspace width="thickmathspace"></mspace><mrow><mo>(</mo><msub><mi>U</mi> <mn>1</mn></msub><mo>∪</mo><msub><mi>U</mi> <mn>2</mn></msub><mo>=</mo><msub><mi>U</mi> <mn>2</mn></msub><mo>)</mo></mrow></mrow><annotation encoding="application/x-tex"> (U_1 \subset U_2) \;\Leftrightarrow\; \left( U_1 \cup U_2 = U_2 \right) </annotation></semantics></math></div> <p>and by finite intersections:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>U</mi> <mn>1</mn></msub><mo>⊂</mo><msub><mi>U</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo>⇔</mo><mspace width="thickmathspace"></mspace><mrow><mo>(</mo><msub><mi>U</mi> <mn>1</mn></msub><mo>∩</mo><msub><mi>U</mi> <mn>2</mn></msub><mo>=</mo><msub><mi>U</mi> <mn>1</mn></msub><mo>)</mo></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> (U_1 \subset U_2) \;\Leftrightarrow\; \left( U_1 \cap U_2 = U_1 \right) \,. </annotation></semantics></math></div></div> <div class="num_example"> <h6 id="example">Example</h6> <p>For</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>f</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><msub><mi>τ</mi> <mi>X</mi></msub><mo stretchy="false">)</mo><mo>⟶</mo><mo stretchy="false">(</mo><mi>Y</mi><mo>,</mo><msub><mi>τ</mi> <mi>Y</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> f \;\colon\; (X,\tau_X) \longrightarrow (Y, \tau_Y) </annotation></semantics></math></div> <p>a <a class="existingWikiWord" href="/nlab/show/continuous+function">continuous function</a>, then its function of <a class="existingWikiWord" href="/nlab/show/pre-images">pre-images</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>τ</mi> <mi>X</mi></msub><mo>⟵</mo><msub><mi>τ</mi> <mi>Y</mi></msub><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><msup><mi>f</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup></mrow><annotation encoding="application/x-tex"> \tau_X \longleftarrow \tau_Y \;\colon\; f^{-1} </annotation></semantics></math></div> <p>is a frame homomorphism according to def. <a class="maruku-ref" href="#HomomorphismOfFramesOfOpens"></a>.</p> </div> <p>For sober topological spaces the converse holds:</p> <div class="num_prop" id="FrameMorphismsBetweenOpensOfSoberSpaces"> <h6 id="proposition">Proposition</h6> <p>If <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><msub><mi>τ</mi> <mi>X</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(X,\tau_X)</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>Y</mi><mo>,</mo><msub><mi>τ</mi> <mi>Y</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(Y,\tau_Y)</annotation></semantics></math> are <a class="existingWikiWord" href="/nlab/show/sober+topological+spaces">sober topological spaces</a>, then for every frame homomorphism (def. <a class="maruku-ref" href="#HomomorphismOfFramesOfOpens"></a>)</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>τ</mi> <mi>X</mi></msub><mo>⟵</mo><msub><mi>τ</mi> <mi>Y</mi></msub><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>ϕ</mi></mrow><annotation encoding="application/x-tex"> \tau_X \longleftarrow \tau_Y \;\colon\; \phi </annotation></semantics></math></div> <p>there is a unique <a class="existingWikiWord" href="/nlab/show/continuous+function">continuous function</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo lspace="verythinmathspace">:</mo><mi>X</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">f \colon X \to Y</annotation></semantics></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi></mrow><annotation encoding="application/x-tex">\phi</annotation></semantics></math> is the function of forming <a class="existingWikiWord" href="/nlab/show/pre-images">pre-images</a> under <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi><mo>=</mo><msup><mi>f</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \phi = f^{-1} \,. </annotation></semantics></math></div></div> <p>We prove this <a href="#ProofOfFrameMorphismsBetweenOpensOfSoberSpaces">below</a>, after the following lemma.</p> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>*</mo><mo>=</mo><mo stretchy="false">(</mo><mo stretchy="false">{</mo><mn>1</mn><mo stretchy="false">}</mo><mo>,</mo><msub><mi>τ</mi> <mo>*</mo></msub><mo>=</mo><mrow><mo>{</mo><mi>∅</mi><mo>,</mo><mo stretchy="false">{</mo><mn>1</mn><mo stretchy="false">}</mo><mo>}</mo></mrow><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\ast = (\{1\}, \tau_\ast = \left\{\emptyset, \{1\}\right\})</annotation></semantics></math> be the <a class="existingWikiWord" href="/nlab/show/point">point</a> <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a>.</p> <div class="num_lemma" id="FrameHomomorphismsToPointAreIrrClSub"> <h6 id="lemma">Lemma</h6> <p>For <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> a <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a>, then there is a <a class="existingWikiWord" href="/nlab/show/bijection">bijection</a> between the <a class="existingWikiWord" href="/nlab/show/irreducible+closed+subspaces">irreducible closed subspaces</a> of <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> and the <a class="existingWikiWord" href="/nlab/show/frame">frame</a> <a class="existingWikiWord" href="/nlab/show/homomorphisms">homomorphisms</a> from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>τ</mi> <mi>X</mi></msub></mrow><annotation encoding="application/x-tex">\tau_X</annotation></semantics></math> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>τ</mi> <mo>*</mo></msub></mrow><annotation encoding="application/x-tex">\tau_\ast</annotation></semantics></math>, given bys</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><msub><mi>Hom</mi> <mi>Frame</mi></msub><mo stretchy="false">(</mo><msub><mi>τ</mi> <mi>X</mi></msub><mo>,</mo><msub><mi>τ</mi> <mo>*</mo></msub><mo stretchy="false">)</mo></mtd> <mtd><munderover><mo>⟶</mo><mo>≃</mo><mrow></mrow></munderover></mtd> <mtd><mi>IrrClSub</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><mi>ϕ</mi></mtd> <mtd><mo>↦</mo></mtd> <mtd><mi>X</mi><mo>\</mo><msub><mi>U</mi> <mi>∅</mi></msub><mo stretchy="false">(</mo><mi>ϕ</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ Hom_{Frame}(\tau_X, \tau_\ast) &\underoverset{\simeq}{}{\longrightarrow}& IrrClSub(X) \\ \phi &\mapsto& X \backslash U_\emptyset(\phi) } </annotation></semantics></math></div> <p>where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>U</mi> <mi>∅</mi></msub><mo stretchy="false">(</mo><mi>ϕ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">U_\emptyset(\phi)</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/union">union</a> of all elements <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo>∈</mo><msub><mi>τ</mi> <mi>x</mi></msub></mrow><annotation encoding="application/x-tex">U \in \tau_x</annotation></semantics></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi><mo stretchy="false">(</mo><mi>U</mi><mo stretchy="false">)</mo><mo>=</mo><mi>∅</mi></mrow><annotation encoding="application/x-tex">\phi(U) = \emptyset</annotation></semantics></math>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>U</mi> <mi>∅</mi></msub><mo stretchy="false">(</mo><mi>ϕ</mi><mo stretchy="false">)</mo><mo>≔</mo><munder><mo>∪</mo><mfrac linethickness="0"><mrow><mrow><mi>U</mi><mo>∈</mo><msub><mi>τ</mi> <mi>X</mi></msub></mrow></mrow><mrow><mrow><mi>ϕ</mi><mo stretchy="false">(</mo><mi>U</mi><mo stretchy="false">)</mo><mo>=</mo><mi>∅</mi></mrow></mrow></mfrac></munder><mi>U</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> U_{\emptyset}(\phi) \coloneqq \underset{{U \in \tau_X} \atop {\phi(U) = \emptyset} }{\cup} U \,. </annotation></semantics></math></div></div> <p>See also (<a class="existingWikiWord" href="/nlab/show/Stone+Spaces">Johnstone 82, II 1.3</a>).</p> <div class="proof"> <h6 id="proof">Proof</h6> <p>First we need to show that the function is well defined in that given a frame homomorphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi><mo lspace="verythinmathspace">:</mo><msub><mi>τ</mi> <mi>X</mi></msub><mo>→</mo><msub><mi>τ</mi> <mo>*</mo></msub></mrow><annotation encoding="application/x-tex">\phi \colon \tau_X \to \tau_\ast</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><msub><mi>U</mi> <mi>∅</mi></msub><mo stretchy="false">(</mo><mi>ϕ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">X \backslash U_\emptyset(\phi)</annotation></semantics></math> is indeed an irreducible closed subspace.</p> <p>To that end observe that:</p> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mo>*</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\ast)</annotation></semantics></math> <em>If there are two elements <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>U</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>U</mi> <mn>2</mn></msub><mo>∈</mo><msub><mi>τ</mi> <mi>X</mi></msub></mrow><annotation encoding="application/x-tex">U_1, U_2 \in \tau_X</annotation></semantics></math> with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>U</mi> <mn>1</mn></msub><mo>∩</mo><msub><mi>U</mi> <mn>2</mn></msub><mo>⊂</mo><msub><mi>U</mi> <mi>∅</mi></msub><mo stretchy="false">(</mo><mi>ϕ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">U_1 \cap U_2 \subset U_{\emptyset}(\phi)</annotation></semantics></math> then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>U</mi> <mn>1</mn></msub><mo>⊂</mo><msub><mi>U</mi> <mi>∅</mi></msub><mo stretchy="false">(</mo><mi>ϕ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">U_1 \subset U_{\emptyset}(\phi)</annotation></semantics></math> or <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>U</mi> <mn>2</mn></msub><mo>⊂</mo><msub><mi>U</mi> <mi>∅</mi></msub><mo stretchy="false">(</mo><mi>ϕ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">U_2 \subset U_{\emptyset}(\phi)</annotation></semantics></math>.</em></p> <p>This is because</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd><mi>ϕ</mi><mo stretchy="false">(</mo><msub><mi>U</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mo>∩</mo><mi>ϕ</mi><mo stretchy="false">(</mo><msub><mi>U</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mtd> <mtd><mo>=</mo><mi>ϕ</mi><mo stretchy="false">(</mo><msub><mi>U</mi> <mn>1</mn></msub><mo>∩</mo><msub><mi>U</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>⊂</mo><mi>ϕ</mi><mo stretchy="false">(</mo><msub><mi>U</mi> <mi>∅</mi></msub><mo stretchy="false">(</mo><mi>ϕ</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mi>∅</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \begin{aligned} \phi(U_1) \cap \phi(U_2) & = \phi(U_1 \cap U_2) \\ & \subset \phi(U_{\emptyset}(\phi)) \\ & = \emptyset \end{aligned} \,, </annotation></semantics></math></div> <p>where the first equality holds because <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi></mrow><annotation encoding="application/x-tex">\phi</annotation></semantics></math> preserves finite intersections by def. <a class="maruku-ref" href="#HomomorphismOfFramesOfOpens"></a>, the inclusion holds because <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi></mrow><annotation encoding="application/x-tex">\phi</annotation></semantics></math> respects inclusions by remark <a class="maruku-ref" href="#PreservationOfInclusionsByFrameHomomorphism"></a>, and the second equality holds because <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi></mrow><annotation encoding="application/x-tex">\phi</annotation></semantics></math> preserves arbitrary unions by def. <a class="maruku-ref" href="#HomomorphismOfFramesOfOpens"></a>. But in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>τ</mi> <mo>*</mo></msub><mo>=</mo><mo stretchy="false">{</mo><mi>∅</mi><mo>,</mo><mo stretchy="false">{</mo><mn>1</mn><mo stretchy="false">}</mo><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\tau_\ast = \{\emptyset, \{1\}\}</annotation></semantics></math> the intersection of two open subsets is empty precisely if at least one of them is empty, hence <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi><mo stretchy="false">(</mo><msub><mi>U</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mo>=</mo><mi>∅</mi></mrow><annotation encoding="application/x-tex">\phi(U_1) = \emptyset</annotation></semantics></math> or <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi><mo stretchy="false">(</mo><msub><mi>U</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo>=</mo><mi>∅</mi></mrow><annotation encoding="application/x-tex">\phi(U_2) = \emptyset</annotation></semantics></math>. But this means that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>U</mi> <mn>1</mn></msub><mo>⊂</mo><msub><mi>U</mi> <mi>∅</mi></msub><mo stretchy="false">(</mo><mi>ϕ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">U_1 \subset U_{\emptyset}(\phi)</annotation></semantics></math> or <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>U</mi> <mn>2</mn></msub><mo>⊂</mo><msub><mi>U</mi> <mi>∅</mi></msub><mo stretchy="false">(</mo><mi>ϕ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">U_2 \subset U_{\emptyset}(\phi)</annotation></semantics></math>, as claimed.</p> <p>Now according to <a href="irreducible+closed+subspace#OpenSubsetVersionOfClosedIrreducible">this prop.</a>, the condition <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mo>*</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\ast)</annotation></semantics></math> identifies the <a class="existingWikiWord" href="/nlab/show/complement">complement</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>\</mo><msub><mi>U</mi> <mi>∅</mi></msub><mo stretchy="false">(</mo><mi>ϕ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">X \backslash U_{\emptyset}(\phi)</annotation></semantics></math> as an <a class="existingWikiWord" href="/nlab/show/irreducible+closed+subspace">irreducible closed subspace</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>τ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(X,\tau)</annotation></semantics></math>.</p> <p>Conversely, given an irreducible closed subset <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>\</mo><msub><mi>U</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">X \backslash U_0</annotation></semantics></math>, define <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi></mrow><annotation encoding="application/x-tex">\phi</annotation></semantics></math> by</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>U</mi><mo>↦</mo><mrow><mo>{</mo><mrow><mtable><mtr><mtd><mi>∅</mi></mtd> <mtd><mo stretchy="false">|</mo><mspace width="thinmathspace"></mspace><mtext>if</mtext><mspace width="thinmathspace"></mspace><mi>U</mi><mo>⊂</mo><msub><mi>U</mi> <mn>0</mn></msub></mtd></mtr> <mtr><mtd><mo stretchy="false">{</mo><mn>1</mn><mo stretchy="false">}</mo></mtd> <mtd><mo stretchy="false">|</mo><mspace width="thinmathspace"></mspace><mtext>otherwise</mtext></mtd></mtr></mtable></mrow></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \phi \;\colon\; U \mapsto \left\{ \array{ \emptyset & \vert \, \text{if} \, U \subset U_0 \\ \{1\} & \vert \, \text{otherwise} } \right. \,. </annotation></semantics></math></div> <p>This does preserve</p> <ol> <li> <p>arbitrary unions</p> <p>because <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi><mo stretchy="false">(</mo><munder><mo>∪</mo><mi>i</mi></munder><msub><mi>U</mi> <mi>i</mi></msub><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">{</mo><mi>∅</mi><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\phi(\underset{i}{\cup} U_i) = \{\emptyset\}</annotation></semantics></math> precisely if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><munder><mo>∪</mo><mi>i</mi></munder><msub><mi>U</mi> <mi>i</mi></msub><mo>⊂</mo><msub><mi>U</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">\underset{i}{\cup}U_i \subset U_0</annotation></semantics></math> which is the case precisely if all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>U</mi> <mi>i</mi></msub><mo>⊂</mo><msub><mi>U</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">U_i \subset U_0</annotation></semantics></math>, which means that all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi><mo stretchy="false">(</mo><msub><mi>U</mi> <mi>i</mi></msub><mo stretchy="false">)</mo><mo>=</mo><mi>∅</mi></mrow><annotation encoding="application/x-tex">\phi(U_i) = \emptyset</annotation></semantics></math> and because <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><munder><mo>∪</mo><mi>i</mi></munder><mi>∅</mi><mo>=</mo><mi>∅</mi></mrow><annotation encoding="application/x-tex">\underset{i}{\cup}\emptyset = \emptyset</annotation></semantics></math>;</p> <p>while <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi><mo stretchy="false">(</mo><munder><mo>∪</mo><mi>i</mi></munder><msub><mi>U</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">{</mo><mn>1</mn><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\phi(\underset{i}{\cup}U_1) = \{1\}</annotation></semantics></math> as soon as one of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>U</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">U_i</annotation></semantics></math> is not contained in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>U</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">U_0</annotation></semantics></math>, which means that one of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi><mo stretchy="false">(</mo><msub><mi>U</mi> <mi>i</mi></msub><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">{</mo><mn>1</mn><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\phi(U_i) = \{1\}</annotation></semantics></math> which means that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><munder><mo>∪</mo><mi>i</mi></munder><mi>ϕ</mi><mo stretchy="false">(</mo><msub><mi>U</mi> <mi>i</mi></msub><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">{</mo><mn>1</mn><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\underset{i}{\cup} \phi(U_i) = \{1\}</annotation></semantics></math>;</p> </li> <li> <p>finite intersections</p> <p>because if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>U</mi> <mn>1</mn></msub><mo>∩</mo><msub><mi>U</mi> <mn>2</mn></msub><mo>⊂</mo><msub><mi>U</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">U_1 \cap U_2 \subset U_0</annotation></semantics></math>, then by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mo>*</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\ast)</annotation></semantics></math> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>U</mi> <mn>1</mn></msub><mo>∈</mo><msub><mi>U</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">U_1 \in U_0</annotation></semantics></math> or <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>U</mi> <mn>2</mn></msub><mo>∈</mo><msub><mi>U</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">U_2 \in U_0</annotation></semantics></math>, whence <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi><mo stretchy="false">(</mo><msub><mi>U</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mo>=</mo><mi>∅</mi></mrow><annotation encoding="application/x-tex">\phi(U_1) = \emptyset</annotation></semantics></math> or <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi><mo stretchy="false">(</mo><msub><mi>U</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo>=</mo><mi>∅</mi></mrow><annotation encoding="application/x-tex">\phi(U_2) = \emptyset</annotation></semantics></math>, whence with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi><mo stretchy="false">(</mo><msub><mi>U</mi> <mn>1</mn></msub><mo>∩</mo><msub><mi>U</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo>=</mo><mi>∅</mi></mrow><annotation encoding="application/x-tex">\phi(U_1 \cap U_2) = \emptyset</annotation></semantics></math> also <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi><mo stretchy="false">(</mo><msub><mi>U</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mo>∩</mo><mi>ϕ</mi><mo stretchy="false">(</mo><msub><mi>U</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo>=</mo><mi>∅</mi></mrow><annotation encoding="application/x-tex">\phi(U_1) \cap \phi(U_2) = \emptyset</annotation></semantics></math>;</p> <p>while if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>U</mi> <mn>1</mn></msub><mo>∩</mo><msub><mi>U</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">U_1 \cap U_2</annotation></semantics></math> is not contained in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>U</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">U_0</annotation></semantics></math> then neither <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>U</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">U_1</annotation></semantics></math> nor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>U</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">U_2</annotation></semantics></math> is contained in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>U</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">U_0</annotation></semantics></math> and hence with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi><mo stretchy="false">(</mo><msub><mi>U</mi> <mn>1</mn></msub><mo>∩</mo><msub><mi>U</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">{</mo><mn>1</mn><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\phi(U_1 \cap U_2) = \{1\}</annotation></semantics></math> also <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi><mo stretchy="false">(</mo><msub><mi>U</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mo>∩</mo><mi>ϕ</mi><mo stretchy="false">(</mo><msub><mi>U</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">{</mo><mn>1</mn><mo stretchy="false">}</mo><mo>∩</mo><mo stretchy="false">{</mo><mn>1</mn><mo stretchy="false">}</mo><mo>=</mo><mo stretchy="false">{</mo><mn>1</mn><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\phi(U_1) \cap \phi(U_2) = \{1\} \cap \{1\} = \{1\}</annotation></semantics></math>.</p> </li> </ol> <p>Hence this is indeed a frame homomorphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>τ</mi> <mi>X</mi></msub><mo>→</mo><msub><mi>τ</mi> <mo>*</mo></msub></mrow><annotation encoding="application/x-tex">\tau_X \to \tau_\ast</annotation></semantics></math>.</p> <p>Finally, it is clear that these two operations are inverse to each other.</p> </div> <div class="proof" id="ProofOfFrameMorphismsBetweenOpensOfSoberSpaces"> <h6 id="proof_2">Proof</h6> <p>of prop. <a class="maruku-ref" href="#FrameMorphismsBetweenOpensOfSoberSpaces"></a></p> <p>We first consider the special case of frame homomorphisms of the form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>τ</mi> <mo>*</mo></msub><mo>⟵</mo><msub><mi>τ</mi> <mi>X</mi></msub><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>ϕ</mi></mrow><annotation encoding="application/x-tex"> \tau_\ast \longleftarrow \tau_X \;\colon\; \phi </annotation></semantics></math></div> <p>and show that these are in bijection to the underlying set <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>, identified with the continuous functions <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>*</mo><mo>→</mo><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>τ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\ast \to (X,\tau)</annotation></semantics></math>.</p> <p>By lemma <a class="maruku-ref" href="#FrameHomomorphismsToPointAreIrrClSub"></a>, the frame homomorphisms <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi><mo lspace="verythinmathspace">:</mo><msub><mi>τ</mi> <mi>X</mi></msub><mo>→</mo><msub><mi>τ</mi> <mo>*</mo></msub></mrow><annotation encoding="application/x-tex">\phi \colon \tau_X \to \tau_\ast</annotation></semantics></math> are identified with the irreducible closed subspaces <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>\</mo><msub><mi>U</mi> <mi>∅</mi></msub><mo stretchy="false">(</mo><mi>ϕ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">X \backslash U_\emptyset(\phi)</annotation></semantics></math> of <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><msub><mi>τ</mi> <mi>X</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(X,\tau_X)</annotation></semantics></math>. Therefore by assumption of <a class="existingWikiWord" href="/nlab/show/sober+topological+space">sobriety</a> of <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> there is a unique point <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>∈</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">x \in X</annotation></semantics></math> with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>\</mo><msub><mi>U</mi> <mi>∅</mi></msub><mo>=</mo><mi>Cl</mi><mo stretchy="false">(</mo><mo stretchy="false">{</mo><mi>x</mi><mo stretchy="false">}</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">X \backslash U_{\emptyset} = Cl(\{x\})</annotation></semantics></math>. In particular this means that for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>U</mi> <mi>x</mi></msub></mrow><annotation encoding="application/x-tex">U_x</annotation></semantics></math> an open neighbourhood of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math>, then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>U</mi> <mi>x</mi></msub></mrow><annotation encoding="application/x-tex">U_x</annotation></semantics></math> is not a subset of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>U</mi> <mi>∅</mi></msub><mo stretchy="false">(</mo><mi>ϕ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">U_\emptyset(\phi)</annotation></semantics></math>, and so it follows that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi><mo stretchy="false">(</mo><msub><mi>U</mi> <mi>x</mi></msub><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">{</mo><mn>1</mn><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\phi(U_x) = \{1\}</annotation></semantics></math>. In conclusion we have found a unique <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>∈</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">x \in X</annotation></semantics></math> such that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>U</mi><mo>↦</mo><mrow><mo>{</mo><mrow><mtable><mtr><mtd><mo stretchy="false">{</mo><mn>1</mn><mo stretchy="false">}</mo></mtd> <mtd><mo stretchy="false">|</mo><mspace width="thinmathspace"></mspace><mtext>if</mtext><mspace width="thinmathspace"></mspace><mi>x</mi><mo>∈</mo><mi>U</mi></mtd></mtr> <mtr><mtd><mi>∅</mi></mtd> <mtd><mo stretchy="false">|</mo><mspace width="thinmathspace"></mspace><mtext>otherwise</mtext></mtd></mtr></mtable></mrow></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \phi \;\colon\; U \mapsto \left\{ \array{ \{1\} & \vert \,\text{if}\, x \in U \\ \emptyset & \vert \, \text{otherwise} } \right. \,. </annotation></semantics></math></div> <p>This is precisely the <a class="existingWikiWord" href="/nlab/show/inverse+image">inverse image</a> function of the continuous function <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>*</mo><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">\ast \to X</annotation></semantics></math> which sends <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>1</mn><mo>↦</mo><mi>x</mi></mrow><annotation encoding="application/x-tex">1 \mapsto x</annotation></semantics></math>.</p> <p>Hence this establishes the bijection between frame homomorphisms of the form <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>τ</mi> <mo>*</mo></msub><mo>⟵</mo><msub><mi>τ</mi> <mi>X</mi></msub></mrow><annotation encoding="application/x-tex">\tau_\ast \longleftarrow \tau_X</annotation></semantics></math> and continuous functions of the form <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>*</mo><mo>→</mo><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>τ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\ast \to (X,\tau)</annotation></semantics></math>.</p> <p>With this it follows that a general frame homomorphism of the form <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>τ</mi> <mi>X</mi></msub><mover><mo>⟵</mo><mi>ϕ</mi></mover><msub><mi>τ</mi> <mi>Y</mi></msub></mrow><annotation encoding="application/x-tex">\tau_X \overset{\phi}{\longleftarrow} \tau_Y</annotation></semantics></math> defines a function of sets <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mover><mo>⟶</mo><mi>f</mi></mover><mi>Y</mi></mrow><annotation encoding="application/x-tex">X \overset{f}{\longrightarrow} Y</annotation></semantics></math> by <a class="existingWikiWord" href="/nlab/show/composition">composition</a>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>X</mi></mtd> <mtd><mover><mo>⟶</mo><mi>f</mi></mover></mtd> <mtd><mi>Y</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">(</mo><msub><mi>τ</mi> <mo>*</mo></msub><mo>←</mo><msub><mi>τ</mi> <mi>X</mi></msub><mo stretchy="false">)</mo></mtd> <mtd><mo>↦</mo></mtd> <mtd><mo stretchy="false">(</mo><msub><mi>τ</mi> <mo>*</mo></msub><mo>←</mo><msub><mi>τ</mi> <mi>X</mi></msub><mover><mo>⟵</mo><mi>ϕ</mi></mover><msub><mi>τ</mi> <mi>Y</mi></msub><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ X &\overset{f}{\longrightarrow}& Y \\ (\tau_\ast \leftarrow \tau_X) &\mapsto& (\tau_\ast \leftarrow \tau_X \overset{\phi}{\longleftarrow} \tau_Y) } \,. </annotation></semantics></math></div> <p>By the previous analysis, an element <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>U</mi> <mi>Y</mi></msub><mo>∈</mo><msub><mi>τ</mi> <mi>Y</mi></msub></mrow><annotation encoding="application/x-tex">U_Y \in \tau_Y</annotation></semantics></math> is sent to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><mn>1</mn><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{1\}</annotation></semantics></math> under this composite precisely if the corresponding point <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>*</mo><mo>→</mo><mi>X</mi><mover><mo>⟶</mo><mi>f</mi></mover><mi>Y</mi></mrow><annotation encoding="application/x-tex">\ast \to X \overset{f}{\longrightarrow} Y</annotation></semantics></math> is in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>U</mi> <mi>Y</mi></msub></mrow><annotation encoding="application/x-tex">U_Y</annotation></semantics></math>, and similarly for an element <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>U</mi> <mi>X</mi></msub><mo>∈</mo><msub><mi>τ</mi> <mi>X</mi></msub></mrow><annotation encoding="application/x-tex">U_X \in \tau_X</annotation></semantics></math>. It follows that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi><mo stretchy="false">(</mo><msub><mi>U</mi> <mi>Y</mi></msub><mo stretchy="false">)</mo><mo>∈</mo><msub><mi>τ</mi> <mi>X</mi></msub></mrow><annotation encoding="application/x-tex">\phi(U_Y) \in \tau_X</annotation></semantics></math> is precisely that subset of points in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> which are sent by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> to elements of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>U</mi> <mi>Y</mi></msub></mrow><annotation encoding="application/x-tex">U_Y</annotation></semantics></math>, hence that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi><mo>=</mo><msup><mi>f</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup></mrow><annotation encoding="application/x-tex">\phi = f^{-1}</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/pre-image">pre-image</a> function of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math>. Since <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi></mrow><annotation encoding="application/x-tex">\phi</annotation></semantics></math> by definition sends open subsets of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math> to open subsets 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>, it follows that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> is indeed a continuous function. This proves the claim in generality.</p> </div> <h3 id="Reflection">Soberification reflection</h3> <p>The <a class="existingWikiWord" href="/nlab/show/category">category</a> of sober spaces is <a class="existingWikiWord" href="/nlab/show/reflective+subcategory">reflective</a> in the category of all topological spaces; the <a class="existingWikiWord" href="/nlab/show/left+adjoint">left adjoint</a> is called the <strong>soberification</strong>.</p> <p>This reflection is also induced by the <a class="existingWikiWord" href="/nlab/show/idempotent+adjunction">idempotent adjunction</a> between spaces and <a class="existingWikiWord" href="/nlab/show/locales">locales</a>; thus sober spaces are precisely those spaces that are the spaces of points of some <a class="existingWikiWord" href="/nlab/show/locale">locale</a>, and the <a class="existingWikiWord" href="/nlab/show/category">category</a> of sober spaces is <a class="existingWikiWord" href="/nlab/show/equivalence+of+categories">equivalent</a> to the category of <a class="existingWikiWord" href="/nlab/show/locales+with+enough+points">locales with enough points</a>.</p> <p>We now say this in detail.</p> <p>Recall again the <a class="existingWikiWord" href="/nlab/show/point">point</a> topological space <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>*</mo><mo>≔</mo><mo stretchy="false">(</mo><mo stretchy="false">{</mo><mn>1</mn><mo stretchy="false">}</mo><mo>,</mo><msub><mi>τ</mi> <mo>*</mo></msub><mo>=</mo><mrow><mo>{</mo><mi>∅</mi><mo>,</mo><mo stretchy="false">{</mo><mn>1</mn><mo stretchy="false">}</mo><mo>}</mo></mrow><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\ast \coloneqq ( \{1\}, \tau_\ast = \left\{ \emptyset, \{1\}\right\} )</annotation></semantics></math>.</p> <div class="num_defn" id="SoberificationConstruction"> <h6 id="definition_4">Definition</h6> <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>.</p> <p>Define <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi><mi>X</mi></mrow><annotation encoding="application/x-tex">S X</annotation></semantics></math> to be the set</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>S</mi><mi>X</mi><mo>≔</mo><msub><mi>Hom</mi> <mi>Frame</mi></msub><mo stretchy="false">(</mo><msub><mi>τ</mi> <mi>X</mi></msub><mo>,</mo><msub><mi>τ</mi> <mo>*</mo></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> S X \coloneqq Hom_{Frame}( \tau_X, \tau_\ast ) </annotation></semantics></math></div> <p>of <a class="existingWikiWord" href="/nlab/show/frame">frame</a> <a class="existingWikiWord" href="/nlab/show/homomorphisms">homomorphisms</a> from the <a class="existingWikiWord" href="/nlab/show/frame+of+opens">frame of opens</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> to that of the point. Define a <a class="existingWikiWord" href="/nlab/show/topological+space">topology</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>τ</mi> <mrow><mi>S</mi><mi>X</mi></mrow></msub><mo>⊂</mo><mi>P</mi><mo stretchy="false">(</mo><mi>S</mi><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\tau_{S X} \subset P(S X)</annotation></semantics></math> on this set by declaring it to have one element <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>U</mi><mo stretchy="false">˜</mo></mover></mrow><annotation encoding="application/x-tex">\tilde U</annotation></semantics></math> for each element <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo>∈</mo><msub><mi>τ</mi> <mi>X</mi></msub></mrow><annotation encoding="application/x-tex">U \in \tau_X</annotation></semantics></math> and given by</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mover><mi>U</mi><mo stretchy="false">˜</mo></mover><mspace width="thickmathspace"></mspace><mo>≔</mo><mspace width="thickmathspace"></mspace><mrow><mo>{</mo><mi>ϕ</mi><mo>∈</mo><mi>S</mi><mi>X</mi><mspace width="thinmathspace"></mspace><mo stretchy="false">|</mo><mspace width="thinmathspace"></mspace><mi>ϕ</mi><mo stretchy="false">(</mo><mi>U</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">{</mo><mn>1</mn><mo stretchy="false">}</mo><mo>}</mo></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \tilde U \;\coloneqq\; \left\{ \phi \in S X \,\vert\, \phi(U) = \{1\} \right\} \,. </annotation></semantics></math></div> <p>Consider the function</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>X</mi></mtd> <mtd><mover><mo>⟶</mo><mrow><msub><mi>s</mi> <mi>X</mi></msub></mrow></mover></mtd> <mtd><mi>S</mi><mi>X</mi></mtd></mtr> <mtr><mtd><mi>x</mi></mtd> <mtd><mo>↦</mo></mtd> <mtd><mo stretchy="false">(</mo><msub><mi>const</mi> <mi>x</mi></msub><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ X &\overset{s_X}{\longrightarrow}& S X \\ x &\mapsto& (const_x)^{-1} } </annotation></semantics></math></div> <p>which sends an element <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>∈</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">x \in X</annotation></semantics></math> to the function which assigns <a class="existingWikiWord" href="/nlab/show/inverse+images">inverse images</a> of the <a class="existingWikiWord" href="/nlab/show/constant+function">constant function</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>const</mi> <mi>x</mi></msub><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mo stretchy="false">{</mo><mn>1</mn><mo stretchy="false">}</mo><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">const_x \;\colon\; \{1\} \to X</annotation></semantics></math> on that element.</p> </div> <div class="num_lemma" id="SoberificationConstructionWellDefined"> <h6 id="lemma_2">Lemma</h6> <p>The construction <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>S</mi><mi>X</mi><mo>,</mo><msub><mi>τ</mi> <mrow><mi>S</mi><mi>X</mi></mrow></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(S X, \tau_{S X})</annotation></semantics></math> in def. <a class="maruku-ref" href="#SoberificationConstruction"></a> is a <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a>, and the function <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>s</mi> <mi>X</mi></msub><mo lspace="verythinmathspace">:</mo><mi>X</mi><mo>→</mo><mi>S</mi><mi>X</mi></mrow><annotation encoding="application/x-tex">s_X \colon X \to S X</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/continuous+function">continuous function</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>s</mi> <mi>X</mi></msub><mo lspace="verythinmathspace">:</mo><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><msub><mi>τ</mi> <mi>X</mi></msub><mo stretchy="false">)</mo><mo>⟶</mo><mo stretchy="false">(</mo><mi>S</mi><mi>X</mi><mo>,</mo><msub><mi>τ</mi> <mrow><mi>S</mi><mi>X</mi></mrow></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> s_X \colon (X, \tau_X) \longrightarrow (S X, \tau_{S X}) </annotation></semantics></math></div></div> <div class="proof" id="ProofOfSoberificationConstructionWellDefined"> <h6 id="proof_3">Proof</h6> <p>To see that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>τ</mi> <mrow><mi>S</mi><mi>X</mi></mrow></msub><mo>⊂</mo><mi>P</mi><mo stretchy="false">(</mo><mi>S</mi><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\tau_{S X} \subset P(S X)</annotation></semantics></math> is closed under arbitrary unions and finite intersections, observe that the function</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><msub><mi>τ</mi> <mi>X</mi></msub></mtd> <mtd><mover><mo>⟶</mo><mover><mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow><mo>˜</mo></mover></mover></mtd> <mtd><msub><mi>τ</mi> <mrow><mi>S</mi><mi>X</mi></mrow></msub></mtd></mtr> <mtr><mtd><mi>U</mi></mtd> <mtd><mo>↦</mo></mtd> <mtd><mover><mi>U</mi><mo stretchy="false">˜</mo></mover></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ \tau_X &\overset{\widetilde{(-)}}{\longrightarrow}& \tau_{S X} \\ U &\mapsto& \tilde U } </annotation></semantics></math></div> <p>in fact preserves arbitrary unions and finite intersections. Whith this the statement follows by the fact that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>τ</mi> <mi>X</mi></msub></mrow><annotation encoding="application/x-tex">\tau_X</annotation></semantics></math> is closed under these operations.</p> <p>To see that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow><mo>˜</mo></mover></mrow><annotation encoding="application/x-tex">\widetilde{(-)}</annotation></semantics></math> indeed preserves unions, observe that (e.g. <a href="#Johnstone82">Johnstone 82, II 1.3 Lemma</a>)</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd><mi>p</mi><mo>∈</mo><munder><mo>∪</mo><mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></munder><mover><mrow><msub><mi>U</mi> <mi>i</mi></msub></mrow><mo>˜</mo></mover><mspace width="thickmathspace"></mspace></mtd> <mtd><mo>⇔</mo><munder><mo>∃</mo><mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></munder><mi>p</mi><mo stretchy="false">(</mo><msub><mi>U</mi> <mi>i</mi></msub><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">{</mo><mn>1</mn><mo stretchy="false">}</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>⇔</mo><munder><mo>∪</mo><mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></munder><mi>p</mi><mo stretchy="false">(</mo><msub><mi>U</mi> <mi>i</mi></msub><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">{</mo><mn>1</mn><mo stretchy="false">}</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>⇔</mo><mi>p</mi><mrow><mo>(</mo><munder><mo>∪</mo><mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></munder><msub><mi>U</mi> <mi>i</mi></msub><mo>)</mo></mrow><mo>=</mo><mo stretchy="false">{</mo><mn>1</mn><mo stretchy="false">}</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>⇔</mo><mi>p</mi><mo>∈</mo><mover><mrow><munder><mo>∪</mo><mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></munder><msub><mi>U</mi> <mi>i</mi></msub></mrow><mo>˜</mo></mover></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \begin{aligned} p \in \underset{i \in I}{\cup} \widetilde{U_i} \; & \Leftrightarrow \underset{i \in I}{\exists} p(U_i) = \{1\} \\ & \Leftrightarrow \underset{i \in I}{\cup} p(U_i) = \{1\} \\ & \Leftrightarrow p\left(\underset{i \in I}{\cup} U_i\right) = \{1\} \\ & \Leftrightarrow p \in \widetilde{ \underset{i \in I}{\cup} U_i } \end{aligned} \,, </annotation></semantics></math></div> <p>where we used that the frame homomorphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo lspace="verythinmathspace">:</mo><msub><mi>τ</mi> <mi>X</mi></msub><mo>→</mo><msub><mi>τ</mi> <mo>*</mo></msub></mrow><annotation encoding="application/x-tex">p \colon \tau_X \to \tau_\ast</annotation></semantics></math> preserves unions. Similarly for intersections, now with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>I</mi></mrow><annotation encoding="application/x-tex">I</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/finite+set">finite set</a>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd><mi>p</mi><mo>∈</mo><munder><mo>∩</mo><mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></munder><mover><mrow><msub><mi>U</mi> <mi>i</mi></msub></mrow><mo>˜</mo></mover><mspace width="thickmathspace"></mspace></mtd> <mtd><mo>⇔</mo><munder><mo>∀</mo><mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></munder><mi>p</mi><mo stretchy="false">(</mo><msub><mi>U</mi> <mi>i</mi></msub><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">{</mo><mn>1</mn><mo stretchy="false">}</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>⇔</mo><munder><mo>∩</mo><mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></munder><mi>p</mi><mo stretchy="false">(</mo><msub><mi>U</mi> <mi>i</mi></msub><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">{</mo><mn>1</mn><mo stretchy="false">}</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>⇔</mo><mi>p</mi><mrow><mo>(</mo><munder><mo>∩</mo><mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></munder><msub><mi>U</mi> <mi>i</mi></msub><mo>)</mo></mrow><mo>=</mo><mo stretchy="false">{</mo><mn>1</mn><mo stretchy="false">}</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>⇔</mo><mi>p</mi><mo>∈</mo><mover><mrow><munder><mo>∩</mo><mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></munder><msub><mi>U</mi> <mi>i</mi></msub></mrow><mo>˜</mo></mover></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \begin{aligned} p \in \underset{i \in I}{\cap} \widetilde{U_i} \; & \Leftrightarrow \underset{i \in I}{\forall} p(U_i) = \{1\} \\ & \Leftrightarrow \underset{i \in I}{\cap} p(U_i) = \{1\} \\ & \Leftrightarrow p\left(\underset{i \in I}{\cap} U_i\right) = \{1\} \\ & \Leftrightarrow p \in \widetilde{ \underset{i \in I}{\cap} U_i } \end{aligned} \,, </annotation></semantics></math></div> <p>where now we used that the frame homomorphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math> preserves finite intersections.</p> <p>To see that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>s</mi> <mi>X</mi></msub></mrow><annotation encoding="application/x-tex">s_X</annotation></semantics></math> is continuous, observe that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>s</mi> <mi>X</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><mover><mi>U</mi><mo stretchy="false">˜</mo></mover><mo stretchy="false">)</mo><mo>=</mo><mi>U</mi></mrow><annotation encoding="application/x-tex">s_X^{-1}(\tilde U) = U</annotation></semantics></math>, by construction.</p> </div> <div class="num_lemma" id="UnitIntoSXDetectsT0AndSoberity"> <h6 id="lemma_3">Lemma</h6> <p>For <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><msub><mi>τ</mi> <mi>X</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(X, \tau_X)</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a>, the function <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>s</mi> <mi>X</mi></msub><mo lspace="verythinmathspace">:</mo><mi>X</mi><mo>→</mo><mi>S</mi><mi>X</mi></mrow><annotation encoding="application/x-tex">s_X \colon X \to S X</annotation></semantics></math> from def. <a class="maruku-ref" href="#SoberificationConstruction"></a> is</p> <ol> <li> <p>an <a class="existingWikiWord" href="/nlab/show/injection">injection</a> precisely if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/separation+axiom">T0</a>;</p> </li> <li> <p>a <a class="existingWikiWord" href="/nlab/show/bijection">bijection</a> precisely if <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 sober.</p> <p>In this case <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>s</mi> <mi>X</mi></msub></mrow><annotation encoding="application/x-tex">s_X</annotation></semantics></math> is in fact a <a class="existingWikiWord" href="/nlab/show/homeomorphism">homeomorphism</a>.</p> </li> </ol> </div> <div class="proof"> <h6 id="proof_4">Proof</h6> <p>By lemma <a class="maruku-ref" href="#FrameHomomorphismsToPointAreIrrClSub"></a> there is an identification <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi><mi>X</mi><mo>≃</mo><mi>IrrClSub</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">S X \simeq IrrClSub(X)</annotation></semantics></math> and via this <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>s</mi> <mi>X</mi></msub></mrow><annotation encoding="application/x-tex">s_X</annotation></semantics></math> is identified with the map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>↦</mo><mi>Cl</mi><mo stretchy="false">(</mo><mo stretchy="false">{</mo><mi>x</mi><mo stretchy="false">}</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">x \mapsto Cl(\{x\})</annotation></semantics></math>.</p> <p>Hence the second statement follows by definition, and the first statement by <a href="separation+axioms#T0InTermsOfClosureOfPoints">this prop.</a>.</p> <p>That in the second case <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>s</mi> <mi>X</mi></msub></mrow><annotation encoding="application/x-tex">s_X</annotation></semantics></math> is in fact a homeomorphism follows from the definition of the opens <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>U</mi><mo stretchy="false">˜</mo></mover></mrow><annotation encoding="application/x-tex">\tilde U</annotation></semantics></math>: they are identified with the opens <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> in this case (…expand…).</p> </div> <div class="num_lemma" id="SoberificationIsIndeedSober"> <h6 id="lemma_4">Lemma</h6> <p>For <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> a <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a>, then the topological space <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>S</mi><mi>X</mi><mo>,</mo><msub><mi>τ</mi> <mrow><mi>S</mi><mi>X</mi></mrow></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(S X, \tau_{S X})</annotation></semantics></math> from def. <a class="maruku-ref" href="#SoberificationConstruction"></a>, lemma <a class="maruku-ref" href="#SoberificationConstructionWellDefined"></a> is sober.</p> </div> <p>(e.g. <a href="#Johnstone82">Johnstone 82, lemma II 1.7</a>)</p> <div class="proof"> <h6 id="proof_5">Proof</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi><mi>X</mi><mo>\</mo><mover><mi>U</mi><mo stretchy="false">˜</mo></mover></mrow><annotation encoding="application/x-tex">S X \backslash \tilde U</annotation></semantics></math> be an <a class="existingWikiWord" href="/nlab/show/irreducible+closed+subspace">irreducible closed subspace</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>S</mi><mi>X</mi><mo>,</mo><msub><mi>τ</mi> <mrow><mi>S</mi><mi>X</mi></mrow></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(S X, \tau_{S X})</annotation></semantics></math>. We need to show that it is the <a class="existingWikiWord" href="/nlab/show/topological+closure">topological closure</a> of a unique element <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi><mo>∈</mo><mi>S</mi><mi>X</mi></mrow><annotation encoding="application/x-tex">\phi \in S X</annotation></semantics></math>.</p> <p>Observe first that also <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 \backslash U</annotation></semantics></math> is irreducible.</p> <p>To see this use <a href="irreducible+closed+subspace#OpenSubsetVersionOfClosedIrreducible">this prop.</a>, saying that irreducibility 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 \backslash U</annotation></semantics></math> is equivalent to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>U</mi> <mn>1</mn></msub><mo>∩</mo><msub><mi>U</mi> <mn>2</mn></msub><mo>⊂</mo><mi>U</mi><mo>⇒</mo><mo stretchy="false">(</mo><msub><mi>U</mi> <mn>1</mn></msub><mo>⊂</mo><mi>U</mi><mo stretchy="false">)</mo><mi>or</mi><mo stretchy="false">(</mo><msub><mi>U</mi> <mn>2</mn></msub><mo>⊂</mo><mi>U</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">U_1 \cap U_2 \subset U \Rightarrow (U_1 \subset U) or (U_2 \subset U)</annotation></semantics></math>. But if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>U</mi> <mn>1</mn></msub><mo>∩</mo><msub><mi>U</mi> <mn>2</mn></msub><mo>⊂</mo><mi>U</mi></mrow><annotation encoding="application/x-tex">U_1 \cap U_2 \subset U</annotation></semantics></math> then also <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mover><mi>U</mi><mo stretchy="false">˜</mo></mover> <mn>1</mn></msub><mo>∩</mo><msub><mover><mi>U</mi><mo stretchy="false">˜</mo></mover> <mn>2</mn></msub><mo>⊂</mo><mover><mi>U</mi><mo stretchy="false">˜</mo></mover></mrow><annotation encoding="application/x-tex">\tilde U_1 \cap \tilde U_2 \subset \tilde U</annotation></semantics></math> (as in the <a href="#ProofOfSoberificationConstructionWellDefined">proof</a> of lemma <a class="maruku-ref" href="#SoberificationConstructionWellDefined"></a>) and hence by assumption on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>U</mi><mo stretchy="false">˜</mo></mover></mrow><annotation encoding="application/x-tex">\tilde U</annotation></semantics></math> it follows that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mover><mi>U</mi><mo stretchy="false">˜</mo></mover> <mn>1</mn></msub><mo>⊂</mo><mover><mi>U</mi><mo stretchy="false">˜</mo></mover></mrow><annotation encoding="application/x-tex">\tilde U_1 \subset \tilde U</annotation></semantics></math> or <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mover><mi>U</mi><mo stretchy="false">˜</mo></mover> <mn>2</mn></msub><mo>⊂</mo><mover><mi>U</mi><mo stretchy="false">˜</mo></mover></mrow><annotation encoding="application/x-tex">\tilde U_2 \subset \tilde U</annotation></semantics></math>. By lemma <a class="maruku-ref" href="#FrameHomomorphismsToPointAreIrrClSub"></a> this in turn implies <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>U</mi> <mn>1</mn></msub><mo>⊂</mo><mi>U</mi></mrow><annotation encoding="application/x-tex">U_1 \subset U</annotation></semantics></math> or <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>U</mi> <mn>2</mn></msub><mo>⊂</mo><mi>U</mi></mrow><annotation encoding="application/x-tex">U_2 \subset U</annotation></semantics></math>. In conclusion, this shows that also <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 \backslash U</annotation></semantics></math> is irreducible .</p> <p>By lemma <a class="maruku-ref" href="#FrameHomomorphismsToPointAreIrrClSub"></a> this irreducible closed subspace corresponds to a point <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo>∈</mo><mi>S</mi><mi>X</mi></mrow><annotation encoding="application/x-tex">p \in S X</annotation></semantics></math>. By that same lemma, this frame homomorphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo lspace="verythinmathspace">:</mo><msub><mi>τ</mi> <mi>X</mi></msub><mo>→</mo><msub><mi>τ</mi> <mo>*</mo></msub></mrow><annotation encoding="application/x-tex">p \colon \tau_X \to \tau_\ast</annotation></semantics></math> takes the value <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>∅</mi></mrow><annotation encoding="application/x-tex">\emptyset</annotation></semantics></math> on all those opens which are inside <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>. This means that the <a class="existingWikiWord" href="/nlab/show/topological+closure">topological closure</a> of this point is just <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi><mi>X</mi><mo>\</mo><mover><mi>U</mi><mo stretchy="false">˜</mo></mover></mrow><annotation encoding="application/x-tex">S X \backslash \tilde U</annotation></semantics></math>.</p> <p>This shows that there exists at least one point of which <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>\</mo><mover><mi>U</mi><mo stretchy="false">˜</mo></mover></mrow><annotation encoding="application/x-tex">X \backslash \tilde U</annotation></semantics></math> is the topological closure. It remains to see that there is no other such point.</p> <p>So let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>p</mi> <mn>1</mn></msub><mo>≠</mo><msub><mi>p</mi> <mn>2</mn></msub><mo>∈</mo><mi>S</mi><mi>X</mi></mrow><annotation encoding="application/x-tex">p_1 \neq p_2 \in S X</annotation></semantics></math> be two distinct points. This means that there exists <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo>∈</mo><msub><mi>τ</mi> <mi>X</mi></msub></mrow><annotation encoding="application/x-tex">U \in \tau_X</annotation></semantics></math> with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>p</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>U</mi><mo stretchy="false">)</mo><mo>≠</mo><msub><mi>p</mi> <mn>2</mn></msub><mo stretchy="false">(</mo><mi>U</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">p_1(U) \neq p_2(U)</annotation></semantics></math>. Equivalently this says that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>U</mi><mo stretchy="false">˜</mo></mover></mrow><annotation encoding="application/x-tex">\tilde U</annotation></semantics></math> contains one of the two points, but not the other. This means that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>S</mi><mi>X</mi><mo>,</mo><msub><mi>τ</mi> <mrow><mi>S</mi><mi>X</mi></mrow></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(S X, \tau_{S X})</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/separation+axiom">T0</a>. By <a href="separation+axioms#T0InTermsOfClosureOfPoints">this prop.</a> this is equivalent to there being no two points with the same topological closure.</p> </div> <div class="num_prop"> <h6 id="proposition_2">Proposition</h6> <p>For <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><msub><mi>τ</mi> <mi>X</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(X, \tau_X)</annotation></semantics></math> any <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a>, for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>Y</mi><mo>,</mo><msubsup><mi>τ</mi> <mi>Y</mi> <mi>sob</mi></msubsup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(Y,\tau_Y^{sob})</annotation></semantics></math> a sober topological space, and for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo lspace="verythinmathspace">:</mo><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><msub><mi>τ</mi> <mi>X</mi></msub><mo stretchy="false">)</mo><mo>⟶</mo><mo stretchy="false">(</mo><mi>Y</mi><mo>,</mo><msub><mi>τ</mi> <mi>Y</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f \colon (X, \tau_X) \longrightarrow (Y, \tau_Y)</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/continuous+function">continuous function</a>, then it factors uniquely through the soberification <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>s</mi> <mi>X</mi></msub><mo lspace="verythinmathspace">:</mo><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><msub><mi>τ</mi> <mi>X</mi></msub><mo stretchy="false">)</mo><mo>⟶</mo><mo stretchy="false">(</mo><mi>S</mi><mi>X</mi><mo>,</mo><msub><mi>τ</mi> <mrow><mi>S</mi><mi>X</mi></mrow></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">s_X \colon (X, \tau_X) \longrightarrow(S X, \tau_{S X})</annotation></semantics></math> from def. <a class="maruku-ref" href="#SoberificationConstruction"></a>, lemma <a class="maruku-ref" href="#SoberificationConstructionWellDefined"></a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><msub><mi>τ</mi> <mi>X</mi></msub><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>⟶</mo><mi>f</mi></mover></mtd> <mtd><mo stretchy="false">(</mo><mi>Y</mi><mo>,</mo><msubsup><mi>τ</mi> <mi>Y</mi> <mi>sob</mi></msubsup><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><msub><mi>s</mi> <mi>X</mi></msub></mrow></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd><msub><mo>↗</mo> <mrow><mo>∃</mo><mo>!</mo></mrow></msub></mtd></mtr> <mtr><mtd><mo stretchy="false">(</mo><mi>S</mi><mi>X</mi><mo>,</mo><msub><mi>τ</mi> <mrow><mi>S</mi><mi>X</mi></mrow></msub><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ (X, \tau_X) &\overset{f}{\longrightarrow}& (Y, \tau_Y^{sob}) \\ {}^{\mathllap{s_X}}\downarrow & \nearrow_{\exists !} \\ (S X , \tau_{S X}) } \,. </annotation></semantics></math></div></div> <div class="proof"> <h6 id="proof_6">Proof</h6> <p>By the construction in def. <a class="maruku-ref" href="#SoberificationConstruction"></a>, we find that the outer part of the following square <a class="existingWikiWord" href="/nlab/show/commuting+square">commutes</a>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><msub><mi>τ</mi> <mi>X</mi></msub><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>⟶</mo><mi>f</mi></mover></mtd> <mtd><mo stretchy="false">(</mo><mi>Y</mi><mo>,</mo><msubsup><mi>τ</mi> <mi>Y</mi> <mi>sob</mi></msubsup><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><msub><mi>s</mi> <mi>X</mi></msub></mrow></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd><mo>↗</mo></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><msub><mi>s</mi> <mrow><mi>S</mi><mi>X</mi></mrow></msub></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><mo stretchy="false">(</mo><mi>S</mi><mi>X</mi><mo>,</mo><msub><mi>τ</mi> <mrow><mi>S</mi><mi>X</mi></mrow></msub><mo stretchy="false">)</mo></mtd> <mtd><munder><mo>⟶</mo><mrow><mi>S</mi><mi>f</mi></mrow></munder></mtd> <mtd><mo stretchy="false">(</mo><mi>S</mi><mi>S</mi><mi>X</mi><mo>,</mo><msub><mi>τ</mi> <mrow><mi>S</mi><mi>S</mi><mi>X</mi></mrow></msub><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ (X, \tau_X) &\overset{f}{\longrightarrow}& (Y, \tau^{sob}_Y) \\ {}^{\mathllap{s_X}}\downarrow & \nearrow& \downarrow^{\mathrlap{s_{S X}}} \\ (S X, \tau_{S X}) &\underset{S f}{\longrightarrow}& (S S X, \tau_{S S X}) } \,. </annotation></semantics></math></div> <p>By lemma <a class="maruku-ref" href="#SoberificationIsIndeedSober"></a> and lemma <a class="maruku-ref" href="#UnitIntoSXDetectsT0AndSoberity"></a>, the right vertical morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>s</mi> <mrow><mi>S</mi><mi>X</mi></mrow></msub></mrow><annotation encoding="application/x-tex">s_{S X}</annotation></semantics></math> is an isomorphism (a <a class="existingWikiWord" href="/nlab/show/homeomorphism">homeomorphism</a>), hence has an <a class="existingWikiWord" href="/nlab/show/inverse+morphism">inverse morphism</a>. This defines the diagonal morphism, which is the desired factorization.</p> <p>To see that this factorization is unique, consider two factorizations <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>f</mi><mo stretchy="false">˜</mo></mover><mo>,</mo><mover><mi>f</mi><mo>¯</mo></mover><mo lspace="verythinmathspace">:</mo><mo lspace="verythinmathspace">:</mo><mo stretchy="false">(</mo><mi>S</mi><mi>X</mi><mo>,</mo><msub><mi>τ</mi> <mrow><mi>S</mi><mi>X</mi></mrow></msub><mo stretchy="false">)</mo><mo>→</mo><mo stretchy="false">(</mo><mi>Y</mi><mo>,</mo><msubsup><mi>τ</mi> <mi>Y</mi> <mi>sob</mi></msubsup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\tilde f, \overline{f} \colon \colon (S X, \tau_{S X}) \to (Y, \tau_Y^{sob})</annotation></semantics></math> and apply the soberification construction once more to the triangles</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><msub><mi>τ</mi> <mi>X</mi></msub><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>⟶</mo><mi>f</mi></mover></mtd> <mtd><mo stretchy="false">(</mo><mi>Y</mi><mo>,</mo><msubsup><mi>τ</mi> <mi>Y</mi> <mi>sob</mi></msubsup><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><msub><mi>s</mi> <mi>X</mi></msub></mrow></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd><msub><mo>↗</mo> <mrow><mover><mi>f</mi><mo stretchy="false">˜</mo></mover><mo>,</mo><mover><mi>f</mi><mo>¯</mo></mover></mrow></msub></mtd></mtr> <mtr><mtd><mo stretchy="false">(</mo><mi>S</mi><mi>X</mi><mo>,</mo><msub><mi>τ</mi> <mrow><mi>S</mi><mi>X</mi></mrow></msub><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mphantom><mi>AAA</mi></mphantom><mo>↦</mo><mphantom><mi>AAA</mi></mphantom><mrow><mtable><mtr><mtd><mo stretchy="false">(</mo><mi>S</mi><mi>X</mi><mo>,</mo><msub><mi>τ</mi> <mrow><mi>S</mi><mi>X</mi></mrow></msub><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>⟶</mo><mrow><mi>S</mi><mi>f</mi></mrow></mover></mtd> <mtd><mo stretchy="false">(</mo><mi>Y</mi><mo>,</mo><msubsup><mi>τ</mi> <mi>Y</mi> <mi>sob</mi></msubsup><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mo>≃</mo></msup><mo stretchy="false">↓</mo></mtd> <mtd><msub><mo>↗</mo> <mrow><mover><mi>f</mi><mo stretchy="false">˜</mo></mover><mo>,</mo><mover><mi>f</mi><mo>¯</mo></mover></mrow></msub></mtd></mtr> <mtr><mtd><mo stretchy="false">(</mo><mi>S</mi><mi>X</mi><mo>,</mo><msub><mi>τ</mi> <mrow><mi>S</mi><mi>X</mi></mrow></msub><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ (X, \tau_X) &\overset{f}{\longrightarrow}& (Y, \tau_Y^{sob}) \\ {}^{\mathllap{s_X}}\downarrow & \nearrow_{ \tilde f, \overline{f}} \\ (S X , \tau_{S X}) } \phantom{AAA} \mapsto \phantom{AAA} \array{ (S X, \tau_{S X}) &\overset{S f}{\longrightarrow}& (Y, \tau_Y^{sob}) \\ {}^{\simeq}\downarrow & \nearrow_{ \tilde f, \overline{f}} \\ (S X , \tau_{S X}) } \,. </annotation></semantics></math></div> <p>Here on the right we used again lemma <a class="maruku-ref" href="#UnitIntoSXDetectsT0AndSoberity"></a> to find that the vertical morphism is an isomorphism, and that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>f</mi><mo stretchy="false">˜</mo></mover></mrow><annotation encoding="application/x-tex">\tilde f</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>f</mi><mo>¯</mo></mover></mrow><annotation encoding="application/x-tex">\overline{f}</annotation></semantics></math> do not change under soberification, as they already map between sober spaces. But now that the left vertical morphism is an isomorphism, the commutativity of this triangle for both <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>f</mi><mo stretchy="false">˜</mo></mover></mrow><annotation encoding="application/x-tex">\tilde f</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>f</mi><mo>¯</mo></mover></mrow><annotation encoding="application/x-tex">\overline{f}</annotation></semantics></math> implies that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>f</mi><mo stretchy="false">˜</mo></mover><mo>=</mo><mover><mi>f</mi><mo>¯</mo></mover></mrow><annotation encoding="application/x-tex">\tilde f = \overline{f}</annotation></semantics></math>.</p> </div> <h3 id="enough_points">Enough points</h3> <p>A topological space has enough points in the sense of def. <a class="maruku-ref" href="#EnoughPointsOfATopologicalSpace"></a> if and only if its <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> quotient is sober. The category of topological spaces with enough points is a <a class="existingWikiWord" href="/nlab/show/reflective+subcategory">reflective subcategory</a> of the category <a class="existingWikiWord" href="/nlab/show/Top">Top</a> of all topological spaces, and a topological space is <a class="existingWikiWord" href="/nlab/show/T0">T0</a> iff this reflection is sober.</p> <h2 id="examples_and_nonexamples">Examples and Non-examples</h2> <ul> <li> <p>With <a class="existingWikiWord" href="/nlab/show/classical+logic">classical logic</a>, every <a class="existingWikiWord" href="/nlab/show/Hausdorff+space">Hausdorff space</a> is sober, but this can fail <a class="existingWikiWord" href="/nlab/show/constructive+mathematics">constructively</a>. See at <em><a class="existingWikiWord" href="/nlab/show/Hausdorff+implies+sober">Hausdorff implies sober</a></em>.</p> <p id="NonComparisonToT1"> Since the Hausdorff condition implies the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>T</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">T_1</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/separation+axiom">separation axiom</a>, this means that (<a class="existingWikiWord" href="/nlab/show/classical+logic">classically</a>) there is a large <a class="existingWikiWord" href="/nlab/show/intersection">intersection</a> of the <a class="existingWikiWord" href="/nlab/show/classes">classes</a> of <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>-topological spaces and sober topological spaces. But neither class is contained in the other, as the following counter-examples show:</p> <ul> <li> <p>The <a class="existingWikiWord" href="/nlab/show/Sierpinski+space">Sierpinski space</a> is sober, but not <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>.</p> </li> <li> <p>The <a class="existingWikiWord" href="/nlab/show/cofinite+topology">cofinite topology</a> on a non-<a class="existingWikiWord" href="/nlab/show/finite+set">finite set</a> is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>T</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">T_1</annotation></semantics></math> but not sober.</p> </li> </ul> </li> <li> <p>The topological space underlying any <a class="existingWikiWord" href="/nlab/show/scheme">scheme</a> is sober. See at <em><a class="existingWikiWord" href="/nlab/show/schemes+are+sober">schemes are sober</a></em>.</p> </li> </ul> <p>Further examples of spaces that are <em>not</em> sober includes the following:</p> <ul> <li>Any nontrivial <a class="existingWikiWord" href="/nlab/show/indiscrete+space">indiscrete space</a> is not sober, since it is not <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>.</li> </ul> <p>More interestingly:</p> <ul> <li>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/commutative+ring">commutative ring</a>, then the space <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>R</mi> <mn>2</mn></msup></mrow><annotation encoding="application/x-tex">R^2</annotation></semantics></math> with the <a class="existingWikiWord" href="/nlab/show/Zariski+topology">Zariski topology</a> is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>T</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">T_1</annotation></semantics></math> but not sober, since every subvariety is an irreducible closed set which is not the <a class="existingWikiWord" href="/nlab/show/topological+closure">closure</a> of a point. Its <em>soberification</em> is, unsurprisingly, the <a class="existingWikiWord" href="/nlab/show/scheme">scheme</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Spec</mi><mo stretchy="false">(</mo><mi>R</mi><mo stretchy="false">[</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">]</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Spec(R[x,y])</annotation></semantics></math>, which contains “generic points” whose closures are the subvarieties.</li> </ul> <p>The following non-example shows that sobriety is not a hereditary separation property, i.e., <a class="existingWikiWord" href="/nlab/show/topological+subspaces">topological subspaces</a> of sober spaces need not be sober:</p> <ul> <li>The <a class="existingWikiWord" href="/nlab/show/Alexandroff+topology">Alexandroff topology</a> on a <a class="existingWikiWord" href="/nlab/show/poset">poset</a> is also not, in general, sober. For instance, if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi></mrow><annotation encoding="application/x-tex">P</annotation></semantics></math> is the infinite binary tree (the set of finite <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>-words (<a class="existingWikiWord" href="/nlab/show/lists">lists</a>) with the “extends” preorder), then the soberification of its Alexandroff topology is <span class="newWikiWord">Wilson space<a href="/nlab/new/Wilson+space">?</a></span>, the space of finite or infinite <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>-words (<a class="existingWikiWord" href="/nlab/show/streams">streams</a>). Generally, the soberification has the set of <a class="existingWikiWord" href="/nlab/show/filters">filters</a> on the poset as its points.</li> </ul> <h2 id="related_concepts">Related concepts</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/nice+topological+space">nice topological space</a></li> <li><a class="existingWikiWord" href="/nlab/show/sober+measurable+space">sober measurable space</a></li> <li><a class="existingWikiWord" href="/nlab/show/Cauchy-complete+category">Cauchy-complete category</a></li> </ul> <h2 id="references">References</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Francis+Borceux">Francis Borceux</a>, volume 3, section 1.9 of <em><a class="existingWikiWord" href="/nlab/show/Handbook+of+Categorical+Algebra">Handbook of Categorical Algebra</a></em>, Cambridge University Press (1994)</p> </li> <li id="Johnstone82"> <p><a class="existingWikiWord" href="/nlab/show/Peter+Johnstone">Peter Johnstone</a>, section II.1, from II.1 on in <em><a class="existingWikiWord" href="/nlab/show/Stone+Spaces">Stone Spaces</a></em>, Cambridge Studies in Advanced Mathematics <strong>3</strong>, Cambridge University Press 1982. xxi+370 pp. <a href="http://www.ams.org/mathscinet-getitem?mr=698074">MR85f:54002</a>, reprinted 1986.</p> </li> <li id="MacLaneMoerdijk"> <p><a class="existingWikiWord" href="/nlab/show/Saunders+MacLane">Saunders MacLane</a>, <a class="existingWikiWord" href="/nlab/show/Ieke+Moerdijk">Ieke Moerdijk</a>, around definition IX.3.2 in <em><a class="existingWikiWord" href="/nlab/show/Sheaves+in+Geometry+and+Logic">Sheaves in Geometry and Logic</a></em></p> </li> <li> <p>Sober spaces in the <a href="http://topology.jdabbs.com/properties/73"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>π</mi></mrow><annotation encoding="application/x-tex">\pi</annotation></semantics></math>-base</a>.</p> </li> </ul> <p>A <a class="existingWikiWord" href="/nlab/show/model+category">model category</a> of <a class="existingWikiWord" href="/nlab/show/locales">locales</a> which makes the <a class="existingWikiWord" href="/nlab/show/reflective+subcategory">reflection</a> of <a class="existingWikiWord" href="/nlab/show/sober+topological+spaces">sober topological spaces</a> a <a class="existingWikiWord" href="/nlab/show/Quillen+adjunction">Quillen adjunction</a> to the sober-restriction of the <a class="existingWikiWord" href="/nlab/show/classical+model+structure+on+topological+spaces">classical model structure on topological spaces</a>:</p> <ul> <li id="SterlingKapulkin23"><a class="existingWikiWord" href="/nlab/show/Sterling+Ebel">Sterling Ebel</a>, <a class="existingWikiWord" href="/nlab/show/Chris+Kapulkin">Chris Kapulkin</a>, Thm. 4.45 in: <em>Synthetic approach to the Quillen model structure on topological spaces</em> [<a href="https://arxiv.org/abs/2310.14235">arXiv:2310.14235</a>]</li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on October 13, 2024 at 21:25:46. 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