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1.1.4-5.2-10-8.2-13.5-10-11.1-5.2-30-15.3-35-27.3-2.5-6 2.8-13.8 9.4-13.6 6.9.2 13.4 7 17.5 12C70.9 34 75 43.8 86.3 47.4z"/> </svg> </span> <span class="webName">nLab</span> Urysohn's lemma </h1> <div class="navigation"> <span class="skipNav"><a href='#navEnd'>Skip the Navigation Links</a> | </span> <span style="display:inline-block; width: 0.3em;"></span> <a href="/nlab/show/HomePage" accesskey="H" title="Home page">Home Page</a> | <a href="/nlab/all_pages" accesskey="A" title="List of all pages">All Pages</a> | <a href="/nlab/latest_revisions" accesskey="U" title="Latest edits and page creations">Latest Revisions</a> | <a href="https://nforum.ncatlab.org/discussion/7713/#Item_7" 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" 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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='#statement'>Statement</a></li> <li><a href='#Proof'>Proof</a></li> <li><a href='#Implications'>Implications</a></li> <li><a href='#related_statements'>Related statements</a></li> <li><a href='#references'>References</a></li> </ul> </div> <h2 id="idea">Idea</h2> <p><em>Urysohn’s lemma</em> (prop. <a class="maruku-ref" href="#UrysohnLemma"></a> below) states that on a <a class="existingWikiWord" href="/nlab/show/normal+topological+space">normal topological space</a> disjoint <a class="existingWikiWord" href="/nlab/show/closed+subsets">closed subsets</a> may be separated by <a class="existingWikiWord" href="/nlab/show/continuous+functions">continuous functions</a> in the sense that a continuous function exists which takes value 0 on one of the two subsets and value 1 on the other (called an “Urysohn function”, def. <a class="maruku-ref" href="#UrysohnFunction"></a>) below. In fact the existence of such functions is equivalent to a space being normal (remark <a class="maruku-ref" href="#Equivalence"></a> below).</p> <p>Urysohn’s lemma is a key ingredient for instance in the proof of the <a class="existingWikiWord" href="/nlab/show/Tietze+extension+theorem">Tietze extension theorem</a> and in the proof of the existence of <a class="existingWikiWord" href="/nlab/show/partitions+of+unity">partitions of unity</a> on <a class="existingWikiWord" href="/nlab/show/paracompact+topological+spaces">paracompact topological spaces</a>. See the list of implications <a href="#Implications">below</a>.</p> <h2 id="statement">Statement</h2> <div class="num_defn" id="UrysohnFunction"> <h6 id="definition">Definition</h6> <p><strong>(Urysohn function)</strong></p> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a>, and let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>,</mo><mi>B</mi><mo>⊂</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">A,B \subset X</annotation></semantics></math> be disjoint <a class="existingWikiWord" href="/nlab/show/closed+subsets">closed subsets</a>. Then an <em>Urysohn function</em> for this situation is</p> <ul> <li>a <a class="existingWikiWord" href="/nlab/show/continuous+function">continuous function</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo lspace="verythinmathspace">:</mo><mi>X</mi><mo>→</mo><mo stretchy="false">[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">f \colon X \to [0,1]</annotation></semantics></math></li> </ul> <p>to the <a class="existingWikiWord" href="/nlab/show/closed+interval">closed interval</a> equipped with its <a class="existingWikiWord" href="/nlab/show/Euclidean+space">Euclidean</a> <a class="existingWikiWord" href="/nlab/show/metric+topology">metric topology</a>, such that</p> <ul> <li> <p>it takes the value 0 on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> and the value 1 on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</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>f</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">{</mo><mn>0</mn><mo stretchy="false">}</mo><mphantom><mi>AAA</mi></mphantom><mtext>and</mtext><mphantom><mi>AAA</mi></mphantom><mi>f</mi><mo stretchy="false">(</mo><mi>B</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">{</mo><mn>1</mn><mo stretchy="false">}</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> f(A) = \{0\} \phantom{AAA} \text{and} \phantom{AAA} f(B) = \{1\} \,. </annotation></semantics></math></div></li> </ul> </div> <div class="num_prop" id="UrysohnLemma"> <h6 id="proposition">Proposition</h6> <p><strong>(Urysohn’s lemma)</strong></p> <p>Assuming <a class="existingWikiWord" href="/nlab/show/excluded+middle">excluded middle</a> then:</p> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/normal+topological+space">normal</a> (or <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>T</mi> <mn>4</mn></msub></mrow><annotation encoding="application/x-tex">T_4</annotation></semantics></math>) <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a>, and let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>,</mo><mi>B</mi><mo>⊂</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">A,B \subset X</annotation></semantics></math> be two disjoint <a class="existingWikiWord" href="/nlab/show/closed+subsets">closed subsets</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>. Then there exists an Urysohn function (def. <a class="maruku-ref" href="#UrysohnFunction"></a>).</p> </div> <div class="num_remark"> <h6 id="remark">Remark</h6> <p>Beware that the function in prop. <a class="maruku-ref" href="#UrysohnLemma"></a> may take the values 0 or 1 even outside of the two subsets. The condition that the function takes value 0 or 1, respectively, <em>precisely</em> on the two subsets corresponds to <em><a class="existingWikiWord" href="/nlab/show/perfectly+normal+spaces">perfectly normal spaces</a></em>.</p> </div> <div class="num_remark" id="Equivalence"> <h6 id="remark_2">Remark</h6> <p>It is immediate that, conversely, the existence of an Urysohn function (def. <a class="maruku-ref" href="#UrysohnFunction"></a>) implies that the topological space is <a class="existingWikiWord" href="/nlab/show/normal+topological+space">normal</a>. For let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>,</mo><mi>B</mi><mo>⊂</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">A, B \subset X</annotation></semantics></math> be disjoint closed subsets, and consider a continuous function <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><mo stretchy="false">[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">f \colon X \to [0,1]</annotation></semantics></math> with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">{</mo><mn>0</mn><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">f(A) = \{0\}</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>B</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">{</mo><mn>1</mn><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">f(B) = \{1\}</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><msub><mi>U</mi> <mi>A</mi></msub><mo>≔</mo><msup><mi>f</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mo stretchy="false">[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy="false">/</mo><mn>3</mn><mo stretchy="false">)</mo><mphantom><mi>AAA</mi></mphantom><msub><mi>U</mi> <mi>B</mi></msub><mo>≔</mo><msup><mi>f</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mo stretchy="false">(</mo><mn>2</mn><mo stretchy="false">/</mo><mn>3</mn><mo>,</mo><mn>1</mn><mo stretchy="false">]</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> U_A \coloneqq f^{-1}([0,1/3) \phantom{AAA} U_B \coloneqq f^{-1}((2/3,1]) </annotation></semantics></math></div> <p>are disjoint open neighbourhoods of these subsets.</p> <p>Hence Urysohn’s lemma shows that a topological space being normal is <em>equivalent</em> to it admitting Urysohn functions.</p> </div> <h2 id="Proof">Proof</h2> <div class="proof"> <h6 id="proof_2">Proof</h6> <p>of Urysohn’s lemma, prop. <a class="maruku-ref" href="#UrysohnLemma"></a></p> <p>Set</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>C</mi> <mn>0</mn></msub><mo>≔</mo><mi>A</mi><mphantom><mi>AAA</mi></mphantom><msub><mi>U</mi> <mn>1</mn></msub><mo>≔</mo><mi>X</mi><mo>\</mo><mi>B</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> C_0 \coloneqq A \phantom{AAA} U_1 \coloneqq X \backslash B \,. </annotation></semantics></math></div> <p>Since by assumption</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>∩</mo><mi>B</mi><mo>=</mo><mi>∅</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> A \cap B = \emptyset \,. </annotation></semantics></math></div> <p>we have</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>C</mi> <mn>0</mn></msub><mo>⊂</mo><msub><mi>U</mi> <mn>1</mn></msub><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> C_0 \subset U_1 \,. </annotation></semantics></math></div> <p>Notice that (by <a href="separation+axioms#T4InTermsOfTopologicalClosures">this lemma</a>) if a space is normal then every open neighbourhood <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo>⊃</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">U \supset C</annotation></semantics></math> of a closed subset <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> contains a smaller neighbourhood <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math> together with its closure <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Cl</mi><mo stretchy="false">(</mo><mi>V</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Cl(V)</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>C</mi><mo>⊂</mo><mi>V</mi><mo>⊂</mo><mi>Cl</mi><mo stretchy="false">(</mo><mi>V</mi><mo stretchy="false">)</mo><mo>⊂</mo><mi>U</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> C \subset V \subset Cl(V) \subset U \,. </annotation></semantics></math></div> <p>Apply this fact successively to the above situation to obtain the following infinite sequence of nested open subsets <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>U</mi> <mi>r</mi></msub></mrow><annotation encoding="application/x-tex">U_r</annotation></semantics></math> and closed subsets <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>C</mi> <mi>r</mi></msub></mrow><annotation encoding="application/x-tex">C_r</annotation></semantics></math></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>C</mi> <mn>0</mn></msub></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><mo>⊂</mo></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><msub><mi>U</mi> <mn>1</mn></msub></mtd></mtr> <mtr><mtd><msub><mi>C</mi> <mn>0</mn></msub></mtd> <mtd></mtd> <mtd></mtd> <mtd><mo>⊂</mo></mtd> <mtd></mtd> <mtd></mtd> <mtd><msub><mi>U</mi> <mrow><mn>1</mn><mo stretchy="false">/</mo><mn>2</mn></mrow></msub></mtd> <mtd><mo>⊂</mo></mtd> <mtd><msub><mi>C</mi> <mrow><mn>1</mn><mo stretchy="false">/</mo><mn>2</mn></mrow></msub></mtd> <mtd></mtd> <mtd></mtd> <mtd><mo>⊂</mo></mtd> <mtd></mtd> <mtd></mtd> <mtd><msub><mi>U</mi> <mn>1</mn></msub></mtd></mtr> <mtr><mtd><msub><mi>C</mi> <mn>0</mn></msub></mtd> <mtd><mo>⊂</mo></mtd> <mtd><msub><mi>U</mi> <mrow><mn>1</mn><mo stretchy="false">/</mo><mn>4</mn></mrow></msub></mtd> <mtd><mo>⊂</mo></mtd> <mtd><msub><mi>C</mi> <mrow><mn>1</mn><mo stretchy="false">/</mo><mn>4</mn></mrow></msub></mtd> <mtd><mo>⊂</mo></mtd> <mtd><msub><mi>U</mi> <mrow><mn>1</mn><mo stretchy="false">/</mo><mn>2</mn></mrow></msub></mtd> <mtd><mo>⊂</mo></mtd> <mtd><msub><mi>C</mi> <mrow><mn>1</mn><mo stretchy="false">/</mo><mn>2</mn></mrow></msub></mtd> <mtd><mo>⊂</mo></mtd> <mtd><msub><mi>U</mi> <mrow><mn>3</mn><mo stretchy="false">/</mo><mn>4</mn></mrow></msub></mtd> <mtd><mo>⊂</mo></mtd> <mtd><msub><mi>C</mi> <mrow><mn>3</mn><mo stretchy="false">/</mo><mn>4</mn></mrow></msub></mtd> <mtd><mo>⊂</mo></mtd> <mtd><msub><mi>U</mi> <mn>1</mn></msub></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ C_0 && && && &\subset& && && && U_1 \\ C_0 && &\subset& && U_{1/2} &\subset& C_{1/2} && &\subset& && U_1 \\ C_0 &\subset& U_{1/4} &\subset& C_{1/4} &\subset& U_{1/2} &\subset& C_{1/2} &\subset& U_{3/4} &\subset& C_{3/4} &\subset& U_1 } </annotation></semantics></math></div> <p>and so on, labeled by the <a class="existingWikiWord" href="/nlab/show/dyadic+rational+numbers">dyadic rational numbers</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ℚ</mi> <mi>dy</mi></msub><mo>⊂</mo><mi>ℚ</mi></mrow><annotation encoding="application/x-tex">\mathbb{Q}_{dy} \subset \mathbb{Q}</annotation></semantics></math> within <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></p> <div style="float:right;margin:0 10px 10px 0;"> <img src="https://ncatlab.org/nlab/files/UrysohnConstruction.png" width="400" /> </div><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> <mi>r</mi></msub><mo>⊂</mo><mi>X</mi><msub><mo stretchy="false">}</mo> <mrow><mi>r</mi><mo>∈</mo><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy="false">]</mo><mo>∩</mo><msub><mi>ℚ</mi> <mi>dy</mi></msub></mrow></msub></mrow><annotation encoding="application/x-tex"> \{ U_{r} \subset X \}_{r \in (0,1] \cap \mathbb{Q}_{dy}} </annotation></semantics></math></div> <p>with the property</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><munder><mo>∀</mo><mrow><msub><mi>r</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>r</mi> <mn>2</mn></msub><mo>∈</mo><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy="false">]</mo><mo>∩</mo><msub><mi>ℚ</mi> <mi>dy</mi></msub></mrow></munder><mrow><mo>(</mo><mrow><mo>(</mo><msub><mi>r</mi> <mn>1</mn></msub><mo><</mo><msub><mi>r</mi> <mn>2</mn></msub><mo>)</mo></mrow><mo>⇒</mo><mrow><mo>(</mo><msub><mi>U</mi> <mrow><msub><mi>r</mi> <mn>1</mn></msub></mrow></msub><mo>⊂</mo><mi>Cl</mi><mo stretchy="false">(</mo><msub><mi>U</mi> <mrow><msub><mi>r</mi> <mn>1</mn></msub></mrow></msub><mo stretchy="false">)</mo><mo>⊂</mo><msub><mi>U</mi> <mrow><msub><mi>r</mi> <mn>2</mn></msub></mrow></msub><mo>)</mo></mrow><mo>)</mo></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \underset{r_1,r_2 \in (0,1] \cap \mathbb{Q}_{dy}}{\forall} \left( \left( r_1 \lt r_2 \right) \Rightarrow \left( U_{r_1} \subset Cl(U_{r_1}) \subset U_{r_2} \right) \right) \,. </annotation></semantics></math></div> <p>Define then the function</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>f</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>X</mi><mo>⟶</mo><mo stretchy="false">[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex"> f \;\colon\; X \longrightarrow [0,1] </annotation></semantics></math></div> <p>to assign to a point <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>∈</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">x \in X</annotation></semantics></math> the <a class="existingWikiWord" href="/nlab/show/infimum">infimum</a> of the labels of those open subsets in this sequence that contain <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>≔</mo><munder><mi>lim</mi><mrow><msub><mi>U</mi> <mi>r</mi></msub><mo>⊃</mo><mo stretchy="false">{</mo><mi>x</mi><mo stretchy="false">}</mo></mrow></munder><mi>r</mi></mrow><annotation encoding="application/x-tex"> f(x) \coloneqq \underset{U_r \supset \{x\}}{\lim} r </annotation></semantics></math></div> <p>Here the <a class="existingWikiWord" href="/nlab/show/limit+of+a+net">limit</a> is over the <a class="existingWikiWord" href="/nlab/show/directed+set">directed set</a> of those <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>U</mi> <mi>r</mi></msub></mrow><annotation encoding="application/x-tex">U_r</annotation></semantics></math> that contain <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>, ordered by reverse inclusion.</p> <p>This function clearly has the property that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">{</mo><mn>0</mn><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">f(A) = \{0\}</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>B</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">{</mo><mn>1</mn><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">f(B) = \{1\}</annotation></semantics></math>. It only remains to see that it is continuous.</p> <p>To this end, first observe that</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><mo>⋆</mo><mo stretchy="false">)</mo></mtd> <mtd></mtd> <mtd><mrow><mo>(</mo><mi>x</mi><mo>∈</mo><mi>Cl</mi><mo stretchy="false">(</mo><msub><mi>U</mi> <mi>r</mi></msub><mo stretchy="false">)</mo><mo>)</mo></mrow></mtd> <mtd><mo>⇒</mo></mtd> <mtd><mrow><mo>(</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>≤</mo><mi>r</mi><mo>)</mo></mrow></mtd></mtr> <mtr><mtd><mo stretchy="false">(</mo><mo>⋆</mo><mo>⋆</mo><mo stretchy="false">)</mo></mtd> <mtd></mtd> <mtd><mrow><mo>(</mo><mi>x</mi><mo>∈</mo><msub><mi>U</mi> <mi>r</mi></msub><mo>)</mo></mrow></mtd> <mtd><mo>⇐</mo></mtd> <mtd><mrow><mo>(</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo><</mo><mi>r</mi><mo>)</mo></mrow></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ (\star) && \left( x \in Cl(U_r) \right) &\Rightarrow& \left( f(x) \leq r \right) \\ (\star\star) && \left( x \in U_r \right) &\Leftarrow& \left( f(x) \lt r \right) } \,. </annotation></semantics></math></div> <p>Here it is immediate from the definition that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>∈</mo><msub><mi>U</mi> <mi>r</mi></msub><mo stretchy="false">)</mo><mo>⇒</mo><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>≤</mo><mi>r</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(x \in U_r) \Rightarrow (f(x) \leq r)</annotation></semantics></math> and that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo><</mo><mi>r</mi><mo stretchy="false">)</mo><mo>⇒</mo><mo stretchy="false">(</mo><mi>x</mi><mo>∈</mo><msub><mi>U</mi> <mi>r</mi></msub><mo>⊂</mo><mi>Cl</mi><mo stretchy="false">(</mo><msub><mi>U</mi> <mi>r</mi></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(f(x) \lt r) \Rightarrow (x \in U_r \subset Cl(U_r))</annotation></semantics></math>. For the remaining implication, it is sufficient to observe that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>∈</mo><mo>∂</mo><msub><mi>U</mi> <mi>r</mi></msub><mo stretchy="false">)</mo><mo>⇒</mo><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mi>r</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> (x \in \partial U_r) \Rightarrow (f(x) = r) \,, </annotation></semantics></math></div> <p>where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>∂</mo><msub><mi>U</mi> <mi>r</mi></msub><mo>≔</mo><mi>Cl</mi><mo stretchy="false">(</mo><msub><mi>U</mi> <mi>r</mi></msub><mo stretchy="false">)</mo><mo>\</mo><msub><mi>U</mi> <mi>r</mi></msub></mrow><annotation encoding="application/x-tex">\partial U_r \coloneqq Cl(U_r) \backslash U_r</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/boundary">boundary</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>U</mi> <mi>r</mi></msub></mrow><annotation encoding="application/x-tex">U_r</annotation></semantics></math>.</p> <p>This holds because the <a class="existingWikiWord" href="/nlab/show/dyadic+numbers">dyadic numbers</a> are <a class="existingWikiWord" href="/nlab/show/dense+subset">dense</a> in <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">\mathbb{R}</annotation></semantics></math>. (And this would fail if we stopped the above decomposition into <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>U</mi> <mrow><mi>a</mi><mo stretchy="false">/</mo><msup><mn>2</mn> <mi>n</mi></msup></mrow></msub></mrow><annotation encoding="application/x-tex">U_{a/2^n}</annotation></semantics></math>-s at some finite <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>.) Namely, in one direction, if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>∈</mo><mo>∂</mo><msub><mi>U</mi> <mi>r</mi></msub></mrow><annotation encoding="application/x-tex">x \in \partial U_r</annotation></semantics></math> then for every small positive real number <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϵ</mi></mrow><annotation encoding="application/x-tex">\epsilon</annotation></semantics></math> there exists a dyadic rational number <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>r</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">r'</annotation></semantics></math> with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>r</mi><mo><</mo><mi>r</mi><mo>′</mo><mo><</mo><mi>r</mi><mo>+</mo><mi>ϵ</mi></mrow><annotation encoding="application/x-tex">r \lt r' \lt r + \epsilon</annotation></semantics></math>, and by construction <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>U</mi> <mrow><mi>r</mi><mo>′</mo></mrow></msub><mo>⊃</mo><mi>Cl</mi><mo stretchy="false">(</mo><msub><mi>U</mi> <mi>r</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">U_{r'} \supset Cl(U_r)</annotation></semantics></math> hence <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> <mrow><mi>r</mi><mo>′</mo></mrow></msub></mrow><annotation encoding="application/x-tex">x \in U_{r'}</annotation></semantics></math>. This implies that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><munder><mi>lim</mi><mrow><msub><mi>U</mi> <mi>r</mi></msub><mo>⊃</mo><mo stretchy="false">{</mo><mi>x</mi><mo stretchy="false">}</mo></mrow></munder><mo>=</mo><mi>r</mi></mrow><annotation encoding="application/x-tex">\underset{U_r \supset \{x\}}{\lim} = r</annotation></semantics></math>.</p> <p id="PreimagesOfTheSubbaseOpens"> Now we claim that for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>α</mi><mo>∈</mo><mo stretchy="false">[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">\alpha \in [0,1]</annotation></semantics></math> then</p> <ol> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>f</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mspace width="thinmathspace"></mspace><mo stretchy="false">(</mo><mi>α</mi><mo>,</mo><mn>1</mn><mo stretchy="false">]</mo><mspace width="thinmathspace"></mspace><mo stretchy="false">)</mo><mo>=</mo><munder><mo>∪</mo><mrow><mi>r</mi><mo>></mo><mi>α</mi></mrow></munder><mrow><mo>(</mo><mi>X</mi><mo>\</mo><mi>Cl</mi><mo stretchy="false">(</mo><msub><mi>U</mi> <mi>r</mi></msub><mo stretchy="false">)</mo><mo>)</mo></mrow></mrow><annotation encoding="application/x-tex">f^{-1}(\,(\alpha, 1]\,) = \underset{r \gt \alpha}{\cup} \left( X \backslash Cl(U_r) \right)</annotation></semantics></math></p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>f</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mspace width="thinmathspace"></mspace><mo stretchy="false">[</mo><mn>0</mn><mo>,</mo><mi>α</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo stretchy="false">)</mo><mo>=</mo><munder><mo>∪</mo><mrow><mi>r</mi><mo><</mo><mi>α</mi></mrow></munder><msub><mi>U</mi> <mi>r</mi></msub></mrow><annotation encoding="application/x-tex">f^{-1}(\,[0,\alpha)\,) = \underset{r \lt \alpha}{\cup} U_r</annotation></semantics></math></p> </li> </ol> <p>Thereby <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>f</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mspace width="thinmathspace"></mspace><mo stretchy="false">(</mo><mi>α</mi><mo>,</mo><mn>1</mn><mo stretchy="false">]</mo><mspace width="thinmathspace"></mspace><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f^{-1}(\,(\alpha, 1]\,)</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>f</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mspace width="thinmathspace"></mspace><mo stretchy="false">[</mo><mn>0</mn><mo>,</mo><mi>α</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f^{-1}(\,[0,\alpha)\,)</annotation></semantics></math> are exhibited as unions of open subsets, and hence they are open.</p> <p>Regarding the first point:</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></mtd> <mtd><mi>x</mi><mo>∈</mo><msup><mi>f</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mspace width="thinmathspace"></mspace><mo stretchy="false">(</mo><mi>α</mi><mo>,</mo><mn>1</mn><mo stretchy="false">]</mo><mspace width="thinmathspace"></mspace><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><mo>⇔</mo><mspace width="thinmathspace"></mspace></mtd> <mtd><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>></mo><mi>α</mi></mtd></mtr> <mtr><mtd><mo>⇔</mo><mspace width="thinmathspace"></mspace></mtd> <mtd><munder><mo>∃</mo><mrow><mi>r</mi><mo>></mo><mi>α</mi></mrow></munder><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>></mo><mi>r</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><mover><mo>⇒</mo><mrow><mo stretchy="false">(</mo><mo>⋆</mo><mo stretchy="false">)</mo></mrow></mover><mspace width="thinmathspace"></mspace></mtd> <mtd><munder><mo>∃</mo><mrow><mi>r</mi><mo>></mo><mi>α</mi></mrow></munder><mrow><mo>(</mo><mi>x</mi><mo>∉</mo><mi>Cl</mi><mo stretchy="false">(</mo><msub><mi>U</mi> <mi>r</mi></msub><mo stretchy="false">)</mo><mo>)</mo></mrow></mtd></mtr> <mtr><mtd><mo>⇔</mo><mspace width="thinmathspace"></mspace></mtd> <mtd><mi>x</mi><mo>∈</mo><munder><mo>∪</mo><mrow><mi>r</mi><mo>></mo><mi>α</mi></mrow></munder><mrow><mo>(</mo><mi>X</mi><mo>\</mo><mi>Cl</mi><mo stretchy="false">(</mo><msub><mi>U</mi> <mi>r</mi></msub><mo stretchy="false">)</mo><mo>)</mo></mrow></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \begin{aligned} & x \in f^{-1}( \,(\alpha,1]\, ) \\ \Leftrightarrow\, & f(x) \gt \alpha \\ \Leftrightarrow\, & \underset{r \gt \alpha}{\exists} (f(x) \gt r) \\ \overset{(\star)}{\Rightarrow}\, & \underset{r \gt \alpha}{\exists} \left( x \notin Cl(U_r) \right) \\ \Leftrightarrow\, & x \in \underset{r \gt \alpha}{\cup} \left(X \backslash Cl(U_r)\right) \end{aligned} </annotation></semantics></math></div> <p>and</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></mtd> <mtd><mi>x</mi><mo>∈</mo><munder><mo>∪</mo><mrow><mi>r</mi><mo>></mo><mi>α</mi></mrow></munder><mrow><mo>(</mo><mi>X</mi><mo>\</mo><mi>Cl</mi><mo stretchy="false">(</mo><msub><mi>U</mi> <mi>r</mi></msub><mo stretchy="false">)</mo><mo>)</mo></mrow></mtd></mtr> <mtr><mtd><mo>⇔</mo><mspace width="thinmathspace"></mspace></mtd> <mtd><munder><mo>∃</mo><mrow><mi>r</mi><mo>></mo><mi>α</mi></mrow></munder><mrow><mo>(</mo><mi>x</mi><mo>∉</mo><mi>Cl</mi><mo stretchy="false">(</mo><msub><mi>U</mi> <mi>r</mi></msub><mo stretchy="false">)</mo><mo>)</mo></mrow></mtd></mtr> <mtr><mtd><mo>⇒</mo><mspace width="thinmathspace"></mspace></mtd> <mtd><munder><mo>∃</mo><mrow><mi>r</mi><mo>></mo><mi>α</mi></mrow></munder><mrow><mo>(</mo><mi>x</mi><mo>∉</mo><msub><mi>U</mi> <mi>r</mi></msub><mo>)</mo></mrow></mtd></mtr> <mtr><mtd><mover><mo>⇒</mo><mrow><mo stretchy="false">(</mo><mo>⋆</mo><mo>⋆</mo><mo stretchy="false">)</mo></mrow></mover><mspace width="thinmathspace"></mspace></mtd> <mtd><munder><mo>∃</mo><mrow><mi>r</mi><mo>></mo><mi>α</mi></mrow></munder><mrow><mo>(</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>≥</mo><mi>r</mi><mo>)</mo></mrow></mtd></mtr> <mtr><mtd><mo>⇔</mo><mspace width="thinmathspace"></mspace></mtd> <mtd><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>></mo><mi>α</mi></mtd></mtr> <mtr><mtd><mo>⇔</mo><mspace width="thinmathspace"></mspace></mtd> <mtd><mi>x</mi><mo>∈</mo><msup><mi>f</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mspace width="thinmathspace"></mspace><mo stretchy="false">(</mo><mi>α</mi><mo>,</mo><mn>1</mn><mo stretchy="false">]</mo><mspace width="thinmathspace"></mspace><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \begin{aligned} & x \in \underset{r \gt \alpha}{\cup} \left(X \backslash Cl(U_r)\right) \\ \Leftrightarrow\, & \underset{r \gt \alpha}{\exists} \left( x \notin Cl(U_r) \right) \\ \Rightarrow\, & \underset{r \gt \alpha}{\exists} \left( x \notin U_r \right) \\ \overset{(\star \star)}{\Rightarrow}\, & \underset{r \gt \alpha}{\exists} \left( f(x) \geq r \right) \\ \Leftrightarrow\, & f(x) \gt \alpha \\ \Leftrightarrow\, & x \in f^{-1}(\, (\alpha,1] \,) \end{aligned} \,. </annotation></semantics></math></div> <p>Regarding the second point:</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></mtd> <mtd><mi>x</mi><mo>∈</mo><msup><mi>f</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mspace width="thinmathspace"></mspace><mo stretchy="false">[</mo><mn>0</mn><mo>,</mo><mi>α</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><mo>⇔</mo><mspace width="thinmathspace"></mspace></mtd> <mtd><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo><</mo><mi>α</mi></mtd></mtr> <mtr><mtd><mo>⇔</mo><mspace width="thinmathspace"></mspace></mtd> <mtd><munder><mo>∃</mo><mrow><mi>r</mi><mo><</mo><mi>α</mi></mrow></munder><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo><</mo><mi>r</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><mover><mo>⇒</mo><mrow><mo stretchy="false">(</mo><mo>⋆</mo><mo>⋆</mo><mo stretchy="false">)</mo></mrow></mover><mspace width="thinmathspace"></mspace></mtd> <mtd><munder><mo>∃</mo><mrow><mi>r</mi><mo><</mo><mi>α</mi></mrow></munder><mo stretchy="false">(</mo><mi>x</mi><mo>∈</mo><msub><mi>U</mi> <mi>r</mi></msub><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><mo>⇔</mo><mspace width="thinmathspace"></mspace></mtd> <mtd><mi>x</mi><mo>∈</mo><munder><mo>∪</mo><mrow><mi>r</mi><mo><</mo><mi>α</mi></mrow></munder><msub><mi>U</mi> <mi>r</mi></msub></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \begin{aligned} & x \in f^{-1}(\, [0,\alpha) \,) \\ \Leftrightarrow\, & f(x) \lt \alpha \\ \Leftrightarrow\, & \underset{r \lt \alpha}{\exists}( f(x) \lt r ) \\ \overset{(\star \star)}{\Rightarrow}\, & \underset{r \lt \alpha}{\exists }( x \in U_r ) \\ \Leftrightarrow\, & x \in \underset{r \lt \alpha}{\cup} U_r \end{aligned} </annotation></semantics></math></div> <p>and</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></mtd> <mtd><mi>x</mi><mo>∈</mo><munder><mo>∪</mo><mrow><mi>r</mi><mo><</mo><mi>α</mi></mrow></munder><msub><mi>U</mi> <mi>r</mi></msub></mtd></mtr> <mtr><mtd><mo>⇔</mo><mspace width="thinmathspace"></mspace></mtd> <mtd><munder><mo>∃</mo><mrow><mi>r</mi><mo><</mo><mi>α</mi></mrow></munder><mo stretchy="false">(</mo><mi>x</mi><mo>∈</mo><msub><mi>U</mi> <mi>r</mi></msub><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><mover><mo>⇒</mo><mrow></mrow></mover><mspace width="thinmathspace"></mspace></mtd> <mtd><munder><mo>∃</mo><mrow><mi>r</mi><mo><</mo><mi>α</mi></mrow></munder><mo stretchy="false">(</mo><mi>x</mi><mo>∈</mo><mi>Cl</mi><mo stretchy="false">(</mo><msub><mi>U</mi> <mi>r</mi></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><mover><mo>⇒</mo><mrow><mo stretchy="false">(</mo><mo>⋆</mo><mo stretchy="false">)</mo></mrow></mover><mspace width="thinmathspace"></mspace></mtd> <mtd><munder><mo>∃</mo><mrow><mi>r</mi><mo><</mo><mi>α</mi></mrow></munder><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>≤</mo><mi>r</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><mo>⇔</mo><mspace width="thinmathspace"></mspace></mtd> <mtd><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo><</mo><mi>α</mi></mtd></mtr> <mtr><mtd><mo>⇔</mo><mspace width="thinmathspace"></mspace></mtd> <mtd><mi>x</mi><mo>∈</mo><msup><mi>f</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mspace width="thinmathspace"></mspace><mo stretchy="false">[</mo><mn>0</mn><mo>,</mo><mi>α</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \begin{aligned} & x \in \underset{r \lt \alpha}{\cup} U_r \\ \Leftrightarrow\, & \underset{r \lt \alpha}{\exists }( x \in U_r ) \\ \overset{}{\Rightarrow}\, & \underset{r \lt \alpha}{\exists }( x \in Cl(U_r) ) \\ \overset{(\star)}{\Rightarrow}\, & \underset{r \lt \alpha}{\exists }( f(x) \leq r ) \\ \Leftrightarrow\, & f(x) \lt \alpha \\ \Leftrightarrow\, & x \in f^{-1}(\, [0,\alpha) \,) \end{aligned} \,. </annotation></semantics></math></div> <p>(In these derivations we repeatedly use that <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><mo>∩</mo><msub><mi>ℚ</mi> <mi>dy</mi></msub></mrow><annotation encoding="application/x-tex">(0,1] \cap \mathbb{Q}_{dy}</annotation></semantics></math> is dense in <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>, and we use the <a class="existingWikiWord" href="/nlab/show/contrapositions">contrapositions</a> of <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">(\star)</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mo>⋆</mo><mo>⋆</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\star \star)</annotation></semantics></math>, by <a class="existingWikiWord" href="/nlab/show/excluded+middle">excluded middle</a>.)</p> <p>Now since the subsets <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><mo stretchy="false">[</mo><mn>0</mn><mo>,</mo><mi>α</mi><mo stretchy="false">)</mo><mo>,</mo><mo stretchy="false">(</mo><mi>α</mi><mo>,</mo><mn>1</mn><mo stretchy="false">]</mo><msub><mo stretchy="false">}</mo> <mrow><mi>α</mi><mo>∈</mo><mo stretchy="false">[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy="false">]</mo></mrow></msub></mrow><annotation encoding="application/x-tex">\{ [0,\alpha), (\alpha,1]\}_{\alpha \in [0,1]}</annotation></semantics></math> form a <a class="existingWikiWord" href="/nlab/show/topological+subbase">sub-base</a> for the Euclidean metric topology on <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>, it follows that all pre-images 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> are open, hence 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 continuous.</p> </div> <h2 id="Implications">Implications</h2> <p>Urysohn’s lemma is key in the proof of many other theorems, for instance</p> <ul> <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/paracompact+Hausdorff+spaces+equivalently+admit+subordinate+partitions+of+unity">paracompact Hausdorff spaces equivalently admit subordinate partitions of unity</a></p> </li> </ul> <h2 id="related_statements">Related statements</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/paracompact+Hausdorff+spaces+are+normal">paracompact Hausdorff spaces are normal</a></li> </ul> <h2 id="references">References</h2> <p>Due to <a class="existingWikiWord" href="/nlab/show/Pavel+Urysohn">Pavel Urysohn</a>.</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Terence+Tao">Terence Tao</a>, <em><a href="https://terrytao.wordpress.com/2009/03/02/245b-notes-12-continuous-functions-on-locally-compact-hausdorff-spaces/#more-1844">245B, Notes 12: Continuous functions on locally compact Hausdorff spaces</a></em></p> </li> <li> <p><a href="http://planetmath.org/proofofurysohnslemma">Proof at planetmath</a></p> </li> <li> <p>Wikipedia, <em><a href="https://en.wikipedia.org/wiki/Urysohn%27s_lemma">Urysohn’s lemma</a></em></p> </li> </ul> <p>Lectures notes include</p> <ul> <li>Tarun Chitra, section 2.1 of <em>The Stone-Cech Compactification</em> 2009 <a href="httP://www.math.cornell.edu/~riley/Teaching/Topology2009/essays/chitra.pdf">pdf</a></li> </ul> 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