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value="Search" style="display:inline-block; float: left;" onfocus="this.value == 'Search' ? this.value = '' : true" onblur="this.value == '' ? this.value = 'Search' : true" /></fieldset> </form> <span id='navEnd'></span> </div> <div id="revision"> <p class="show_diff"> Showing changes from revision #26 to #27: <ins class="diffins">Added</ins> | <del class="diffdel">Removed</del> | <del class="diffmod">Chan</del><ins class="diffmod">ged</ins> </p> <div class='rightHandSide'> <div class='toc clickDown' tabindex='0'> <h3 id='context'>Context</h3> <h4 id='topology'>Topology</h4> <div class='hide'> <p><strong><a class='existingWikiWord' href='/nlab/show/diff/topology'>topology</a></strong> (<a class='existingWikiWord' href='/nlab/show/diff/general+topology'>point-set topology</a>, <a class='existingWikiWord' href='/nlab/show/diff/point-free+topology'>point-free topology</a>)</p> <p>see also <em><a class='existingWikiWord' href='/nlab/show/diff/differential+topology'>differential topology</a></em>, <em><a class='existingWikiWord' href='/nlab/show/diff/algebraic+topology'>algebraic topology</a></em>, <em><a class='existingWikiWord' href='/nlab/show/diff/functional+analysis'>functional analysis</a></em> and <em><a class='existingWikiWord' href='/nlab/show/diff/topological+homotopy+theory'>topological</a> <a class='existingWikiWord' href='/nlab/show/diff/homotopy+theory'>homotopy theory</a></em></p> <p><a class='existingWikiWord' href='/nlab/show/diff/Introduction+to+Topology'>Introduction</a></p> <p><strong>Basic concepts</strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/open+subspace'>open subset</a>, <a class='existingWikiWord' href='/nlab/show/diff/closed+subspace'>closed subset</a>, <a class='existingWikiWord' href='/nlab/show/diff/neighborhood'>neighbourhood</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/topological+space'>topological space</a>, <a class='existingWikiWord' href='/nlab/show/diff/locale'>locale</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/topological+base'>base for the topology</a>, <a class='existingWikiWord' href='/nlab/show/diff/neighborhood+base'>neighbourhood base</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/finer+topology'>finer/coarser topology</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/closed+subspace'>closure</a>, <a class='existingWikiWord' href='/nlab/show/diff/interior'>interior</a>, <a class='existingWikiWord' href='/nlab/show/diff/boundary'>boundary</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/separation+axioms'>separation</a>, <a class='existingWikiWord' href='/nlab/show/diff/sober+topological+space'>sobriety</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/continuous+map'>continuous function</a>, <a class='existingWikiWord' href='/nlab/show/diff/homeomorphism'>homeomorphism</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/uniformly+continuous+map'>uniformly continuous function</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/embedding+of+topological+spaces'>embedding</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/open+map'>open map</a>, <a class='existingWikiWord' href='/nlab/show/diff/closed+map'>closed map</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/sequence'>sequence</a>, <a class='existingWikiWord' href='/nlab/show/diff/net'>net</a>, <a class='existingWikiWord' href='/nlab/show/diff/subnet'>sub-net</a>, <a class='existingWikiWord' href='/nlab/show/diff/filter'>filter</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/convergence'>convergence</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/category'>category</a> <a class='existingWikiWord' href='/nlab/show/diff/Top'>Top</a></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/convenient+category+of+topological+spaces'>convenient category of topological spaces</a></li> </ul> </li> </ul> <p><strong><a href='Top#UniversalConstructions'>Universal constructions</a></strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/weak+topology'>initial topology</a>, <a class='existingWikiWord' href='/nlab/show/diff/weak+topology'>final topology</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/subspace'>subspace</a>, <a class='existingWikiWord' href='/nlab/show/diff/quotient+space'>quotient space</a>,</p> </li> <li> <p>fiber space, <a class='existingWikiWord' href='/nlab/show/diff/space+attachment'>space attachment</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/product+topological+space'>product space</a>, <a class='existingWikiWord' href='/nlab/show/diff/disjoint+union+topological+space'>disjoint union space</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/mapping+cylinder'>mapping cylinder</a>, <a class='existingWikiWord' href='/nlab/show/diff/cocylinder'>mapping cocylinder</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/mapping+cone'>mapping cone</a>, <a class='existingWikiWord' href='/nlab/show/diff/mapping+cocone'>mapping cocone</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/mapping+telescope'>mapping telescope</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/colimits+of+normal+spaces'>colimits of normal spaces</a></p> </li> </ul> <p><strong><a class='existingWikiWord' href='/nlab/show/diff/stuff%2C+structure%2C+property'>Extra stuff, structure, properties</a></strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/nice+topological+space'>nice topological space</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/metric+space'>metric space</a>, <a class='existingWikiWord' href='/nlab/show/diff/metric+topology'>metric topology</a>, <a class='existingWikiWord' href='/nlab/show/diff/metrisable+topological+space'>metrisable space</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Kolmogorov+topological+space'>Kolmogorov space</a>, <a class='existingWikiWord' href='/nlab/show/diff/Hausdorff+space'>Hausdorff space</a>, <a class='existingWikiWord' href='/nlab/show/diff/regular+space'>regular space</a>, <a class='existingWikiWord' href='/nlab/show/diff/normal+space'>normal space</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/sober+topological+space'>sober space</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/compact+space'>compact space</a>, <a class='existingWikiWord' href='/nlab/show/diff/proper+map'>proper map</a></p> <p><a class='existingWikiWord' href='/nlab/show/diff/sequentially+compact+topological+space'>sequentially compact</a>, <a class='existingWikiWord' href='/nlab/show/diff/countably+compact+topological+space'>countably compact</a>, <a class='existingWikiWord' href='/nlab/show/diff/locally+compact+topological+space'>locally compact</a>, <a class='existingWikiWord' href='/nlab/show/diff/sigma-compact+topological+space'>sigma-compact</a>, <a class='existingWikiWord' href='/nlab/show/diff/paracompact+topological+space'>paracompact</a>, <a class='existingWikiWord' href='/nlab/show/diff/countably+paracompact+topological+space'>countably paracompact</a>, <a class='existingWikiWord' href='/nlab/show/diff/strongly+compact+topological+space'>strongly compact</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/compactly+generated+topological+space'>compactly generated space</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/second-countable+space'>second-countable space</a>, <a class='existingWikiWord' href='/nlab/show/diff/first-countable+space'>first-countable space</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/contractible+space'>contractible space</a>, <a class='existingWikiWord' href='/nlab/show/diff/locally+contractible+space'>locally contractible space</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/connected+space'>connected space</a>, <a class='existingWikiWord' href='/nlab/show/diff/locally+connected+topological+space'>locally connected space</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/simply+connected+space'>simply-connected space</a>, <a class='existingWikiWord' href='/nlab/show/diff/semi-locally+simply-connected+topological+space'>locally simply-connected space</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/cell+complex'>cell complex</a>, <a class='existingWikiWord' href='/nlab/show/diff/CW+complex'>CW-complex</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/pointed+topological+space'>pointed space</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/topological+vector+space'>topological vector space</a>, <a class='existingWikiWord' href='/nlab/show/diff/Banach+space'>Banach space</a>, <a class='existingWikiWord' href='/nlab/show/diff/Hilbert+space'>Hilbert space</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/topological+group'>topological group</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/topological+vector+bundle'>topological vector bundle</a>, <a class='existingWikiWord' href='/nlab/show/diff/topological+K-theory'>topological K-theory</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/topological+manifold'>topological manifold</a></p> </li> </ul> <p><strong>Examples</strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/empty+space'>empty space</a>, <a class='existingWikiWord' href='/nlab/show/diff/point+space'>point space</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/discrete+object'>discrete space</a>, <a class='existingWikiWord' href='/nlab/show/diff/codiscrete+space'>codiscrete space</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Sierpinski+space'>Sierpinski space</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/order+topology'>order topology</a>, <a class='existingWikiWord' href='/nlab/show/diff/specialization+topology'>specialization topology</a>, <a class='existingWikiWord' href='/nlab/show/diff/Scott+topology'>Scott topology</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Euclidean+space'>Euclidean space</a></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/real+number'>real line</a>, <a class='existingWikiWord' href='/nlab/show/diff/plane'>plane</a></li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/cylinder+object'>cylinder</a>, <a class='existingWikiWord' href='/nlab/show/diff/cone'>cone</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/sphere'>sphere</a>, <a class='existingWikiWord' href='/nlab/show/diff/ball'>ball</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/circle'>circle</a>, <a class='existingWikiWord' href='/nlab/show/diff/torus'>torus</a>, <a class='existingWikiWord' href='/nlab/show/diff/annulus'>annulus</a>, <a class='existingWikiWord' href='/nlab/show/diff/M%C3%B6bius+strip'>Moebius strip</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/polytope'>polytope</a>, <a class='existingWikiWord' href='/nlab/show/diff/polyhedron'>polyhedron</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/projective+space'>projective space</a> (<a class='existingWikiWord' href='/nlab/show/diff/real+projective+space'>real</a>, <a class='existingWikiWord' href='/nlab/show/diff/complex+projective+space'>complex</a>)</p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/classifying+space'>classifying space</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/configuration+space+of+points'>configuration space</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/path'>path</a>, <a class='existingWikiWord' href='/nlab/show/diff/loop'>loop</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/compact-open+topology'>mapping spaces</a>: <a class='existingWikiWord' href='/nlab/show/diff/compact-open+topology'>compact-open topology</a>, <a class='existingWikiWord' href='/nlab/show/diff/topology+of+uniform+convergence'>topology of uniform convergence</a></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/loop+space'>loop space</a>, <a class='existingWikiWord' href='/nlab/show/diff/path+space'>path space</a></li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Zariski+topology'>Zariski topology</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Cantor+space'>Cantor space</a>, <a class='existingWikiWord' href='/nlab/show/diff/Mandelbrot+set'>Mandelbrot space</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Peano+curve'>Peano curve</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/line+with+two+origins'>line with two origins</a>, <a class='existingWikiWord' href='/nlab/show/diff/long+line'>long line</a>, <a class='existingWikiWord' href='/nlab/show/diff/Sorgenfrey+line'>Sorgenfrey line</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/K-topology'>K-topology</a>, <a class='existingWikiWord' href='/nlab/show/diff/Dowker+space'>Dowker space</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Warsaw+circle'>Warsaw circle</a>, <a class='existingWikiWord' href='/nlab/show/diff/Hawaiian+earring+space'>Hawaiian earring space</a></p> </li> </ul> <p><strong>Basic statements</strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Hausdorff+implies+sober'>Hausdorff spaces are sober</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/schemes+are+sober'>schemes are sober</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/continuous+images+of+compact+spaces+are+compact'>continuous images of compact spaces are compact</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/closed+subspaces+of+compact+Hausdorff+spaces+are+equivalently+compact+subspaces'>closed subspaces of compact Hausdorff spaces are equivalently compact subspaces</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/open+subspaces+of+compact+Hausdorff+spaces+are+locally+compact'>open subspaces of compact Hausdorff spaces are locally compact</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/quotient+projections+out+of+compact+Hausdorff+spaces+are+closed+precisely+if+the+codomain+is+Hausdorff'>quotient projections out of compact Hausdorff spaces are closed precisely if the codomain is Hausdorff</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/compact+spaces+equivalently+have+converging+subnets'>compact spaces equivalently have converging subnet of every net</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Lebesgue+number+lemma'>Lebesgue number lemma</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/sequentially+compact+metric+spaces+are+equivalently+compact+metric+spaces'>sequentially compact metric spaces are equivalently compact metric spaces</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/compact+spaces+equivalently+have+converging+subnets'>compact spaces equivalently have converging subnet of every net</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/sequentially+compact+metric+spaces+are+totally+bounded'>sequentially compact metric spaces are totally bounded</a></p> </li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/continuous+metric+space+valued+function+on+compact+metric+space+is+uniformly+continuous'>continuous metric space valued function on compact metric space is uniformly continuous</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/paracompact+Hausdorff+spaces+are+normal'>paracompact Hausdorff spaces are normal</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/paracompact+Hausdorff+spaces+equivalently+admit+subordinate+partitions+of+unity'>paracompact Hausdorff spaces equivalently admit subordinate partitions of unity</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/closed+injections+are+embeddings'>closed injections are embeddings</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/proper+maps+to+locally+compact+spaces+are+closed'>proper maps to locally compact spaces are closed</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/injective+proper+maps+to+locally+compact+spaces+are+equivalently+the+closed+embeddings'>injective proper maps to locally compact spaces are equivalently the closed embeddings</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/locally+compact+and+sigma-compact+spaces+are+paracompact'>locally compact and sigma-compact spaces are paracompact</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/locally+compact+and+second-countable+spaces+are+sigma-compact'>locally compact and second-countable spaces are sigma-compact</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/second-countable+regular+spaces+are+paracompact'>second-countable regular spaces are paracompact</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/CW-complexes+are+paracompact+Hausdorff+spaces'>CW-complexes are paracompact Hausdorff spaces</a></p> </li> </ul> <p><strong>Theorems</strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Urysohn%27s+lemma'>Urysohn's lemma</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Tietze+extension+theorem'>Tietze extension theorem</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Tychonoff+theorem'>Tychonoff theorem</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/tube+lemma'>tube lemma</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Michael%27s+theorem'>Michael's theorem</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Brouwer%27s+fixed+point+theorem'>Brouwer's fixed point theorem</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/topological+invariance+of+dimension'>topological invariance of dimension</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Jordan+curve+theorem'>Jordan curve theorem</a></p> </li> </ul> <p><strong>Analysis Theorems</strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Heine-Borel+theorem'>Heine-Borel theorem</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/intermediate+value+theorem'>intermediate value theorem</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/extreme+value+theorem'>extreme value theorem</a></p> </li> </ul> <p><strong><a class='existingWikiWord' href='/nlab/show/diff/topological+homotopy+theory'>topological homotopy theory</a></strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/homotopy'>left homotopy</a>, <a class='existingWikiWord' href='/nlab/show/diff/homotopy'>right homotopy</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/homotopy+equivalence'>homotopy equivalence</a>, <a class='existingWikiWord' href='/nlab/show/diff/deformation+retract'>deformation retract</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/fundamental+group'>fundamental group</a>, <a class='existingWikiWord' href='/nlab/show/diff/covering+space'>covering space</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/fundamental+theorem+of+covering+spaces'>fundamental theorem of covering spaces</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/homotopy+group'>homotopy group</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/weak+homotopy+equivalence'>weak homotopy equivalence</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Whitehead+theorem'>Whitehead's theorem</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Freudenthal+suspension+theorem'>Freudenthal suspension theorem</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/nerve+theorem'>nerve theorem</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/homotopy+extension+property'>homotopy extension property</a>, <a class='existingWikiWord' href='/nlab/show/diff/Hurewicz+cofibration'>Hurewicz cofibration</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/topological+cofiber+sequence'>cofiber sequence</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Str%C3%B8m+model+structure'>Strøm model category</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/classical+model+structure+on+topological+spaces'>classical model structure on topological spaces</a></p> </li> </ul> </div> </div> </div> <h1 id='normal_spaces'>Normal spaces</h1> <div class='maruku_toc'><ul><li><a href='#idea'>Idea</a></li><li><a href='#definitions'>Definitions</a></li><li><a href='#Examples'>Examples</a></li><li><a href='#properties'>Properties</a><ul><li><a href='#basic_properties'>Basic properties</a></li><ins class='diffins'><li><a href='#SeparationAxiomInTermsOfLiftingProperties'>In terms of lifting properties</a></li></ins><li><a href='#TietzeEtensionAndLiftingProperty'>Tietze extension and lifting property</a></li><li><a href='#CategoryOfNormalSpaces'>The category of normal spaces</a></li></ul></li><li><a href='#references'>References</a></li></ul></div> <h2 id='idea'>Idea</h2> <p>A normal space is a space (typically a <a class='existingWikiWord' href='/nlab/show/diff/topological+space'>topological space</a>) which satisfies one of the stronger <a class='existingWikiWord' href='/nlab/show/diff/separation+axioms'>separation axioms</a>.</p> <p>the <strong>main <a class='existingWikiWord' href='/nlab/show/diff/separation+axioms'>separation axioms</a></strong></p> <table><thead><tr><th>number</th><th>name</th><th>statement</th><th>reformulation</th></tr></thead><tbody><tr><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_615d55a49ff138fdb807794cf13c867261a6d940_1' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>T</mi> <mn>0</mn></msub></mrow><annotation encoding='application/x-tex'>T_0</annotation></semantics></math></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/Kolmogorov+topological+space'>Kolmogorov</a></td><td style='text-align: left;'>given two distinct points, at least one of them has an <a class='existingWikiWord' href='/nlab/show/diff/neighborhood'>open neighbourhood</a> not containing the other point</td><td style='text-align: left;'>every <a class='existingWikiWord' href='/nlab/show/diff/irreducible+closed+subspace'>irreducible closed subset</a> is the <a class='existingWikiWord' href='/nlab/show/diff/closed+subspace'>closure</a> of at most one point</td></tr> <tr><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_615d55a49ff138fdb807794cf13c867261a6d940_2' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>T</mi> <mn>1</mn></msub></mrow><annotation encoding='application/x-tex'>T_1</annotation></semantics></math></td><td style='text-align: left;' /><td style='text-align: left;'>given two distinct points, both have an <a class='existingWikiWord' href='/nlab/show/diff/neighborhood'>open neighbourhood</a> not containing the other point</td><td style='text-align: left;'>all points are <a class='existingWikiWord' href='/nlab/show/diff/closed+point'>closed</a></td></tr> <tr><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_615d55a49ff138fdb807794cf13c867261a6d940_3' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>T</mi> <mn>2</mn></msub></mrow><annotation encoding='application/x-tex'>T_2</annotation></semantics></math></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/Hausdorff+space'>Hausdorff</a></td><td style='text-align: left;'>given two distinct points, they have <a class='existingWikiWord' href='/nlab/show/diff/disjoint+subsets'>disjoint</a> <a class='existingWikiWord' href='/nlab/show/diff/neighborhood'>open neighbourhoods</a></td><td style='text-align: left;'>the <a class='existingWikiWord' href='/nlab/show/diff/diagonal+morphism'>diagonal</a> is a <a class='existingWikiWord' href='/nlab/show/diff/closed+map'>closed map</a></td></tr> <tr><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_615d55a49ff138fdb807794cf13c867261a6d940_4' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>T</mi> <mrow><mo>></mo><mn>2</mn></mrow></msub></mrow><annotation encoding='application/x-tex'>T_{\gt 2}</annotation></semantics></math></td><td style='text-align: left;' /><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_615d55a49ff138fdb807794cf13c867261a6d940_5' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>T</mi> <mn>1</mn></msub></mrow><annotation encoding='application/x-tex'>T_1</annotation></semantics></math> and…</td><td style='text-align: left;'>all points are <a class='existingWikiWord' href='/nlab/show/diff/closed+point'>closed</a> and…</td></tr> <tr><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_615d55a49ff138fdb807794cf13c867261a6d940_6' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>T</mi> <mn>3</mn></msub></mrow><annotation encoding='application/x-tex'>T_3</annotation></semantics></math></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/regular+space'>regular Hausdorff</a></td><td style='text-align: left;'>…given a point and a <a class='existingWikiWord' href='/nlab/show/diff/closed+subspace'>closed subset</a> not containing it, they have <a class='existingWikiWord' href='/nlab/show/diff/disjoint+subsets'>disjoint</a> <a class='existingWikiWord' href='/nlab/show/diff/neighborhood'>open neighbourhoods</a></td><td style='text-align: left;'>…every <a class='existingWikiWord' href='/nlab/show/diff/neighborhood'>neighbourhood</a> of a point contains the <a class='existingWikiWord' href='/nlab/show/diff/closed+subspace'>closure</a> of an <a class='existingWikiWord' href='/nlab/show/diff/neighborhood'>open neighbourhood</a></td></tr> <tr><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_615d55a49ff138fdb807794cf13c867261a6d940_7' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>T</mi> <mn>4</mn></msub></mrow><annotation encoding='application/x-tex'>T_4</annotation></semantics></math></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/normal+space'>normal Hausdorff</a></td><td style='text-align: left;'>…given two <a class='existingWikiWord' href='/nlab/show/diff/disjoint+subsets'>disjoint</a> <a class='existingWikiWord' href='/nlab/show/diff/closed+subspace'>closed subsets</a>, they have <a class='existingWikiWord' href='/nlab/show/diff/disjoint+subsets'>disjoint</a> <a class='existingWikiWord' href='/nlab/show/diff/neighborhood'>open neighbourhoods</a></td><td style='text-align: left;'>…every <a class='existingWikiWord' href='/nlab/show/diff/neighborhood'>neighbourhood</a> of a <a class='existingWikiWord' href='/nlab/show/diff/closed+subspace'>closed set</a> also contains the <a class='existingWikiWord' href='/nlab/show/diff/closed+subspace'>closure</a> of an <a class='existingWikiWord' href='/nlab/show/diff/neighborhood'>open neighbourhood</a> <br /> … every pair of <a class='existingWikiWord' href='/nlab/show/diff/disjoint+subsets'>disjoint</a> <a class='existingWikiWord' href='/nlab/show/diff/closed+subspace'>closed subsets</a> is separated by an <a class='existingWikiWord' href='/nlab/show/diff/Urysohn%27s+lemma'>Urysohn function</a></td></tr> </tbody></table> <h2 id='definitions'>Definitions</h2> <p>A <a class='existingWikiWord' href='/nlab/show/diff/topological+space'>topological space</a> <math class='maruku-mathml' display='inline' id='mathml_615d55a49ff138fdb807794cf13c867261a6d940_8' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> is <strong>normal</strong> if it satisfies:</p> <ul> <li><math class='maruku-mathml' display='inline' id='mathml_615d55a49ff138fdb807794cf13c867261a6d940_9' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>T</mi> <mn>4</mn></msub></mrow><annotation encoding='application/x-tex'>T_4</annotation></semantics></math>: for every two <a class='existingWikiWord' href='/nlab/show/diff/closed+subspace'>closed</a> <a class='existingWikiWord' href='/nlab/show/diff/disjoint+subsets'>disjoint subsets</a> <math class='maruku-mathml' display='inline' id='mathml_615d55a49ff138fdb807794cf13c867261a6d940_10' xmlns='http://www.w3.org/1998/Math/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> there are (optionally <a class='existingWikiWord' href='/nlab/show/diff/open+subspace'>open</a>) <a class='existingWikiWord' href='/nlab/show/diff/neighborhood'>neighborhoods</a> <math class='maruku-mathml' display='inline' id='mathml_615d55a49ff138fdb807794cf13c867261a6d940_11' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>U</mi><mo>⊃</mo><mi>A</mi></mrow><annotation encoding='application/x-tex'>U \supset A</annotation></semantics></math>, <math class='maruku-mathml' display='inline' id='mathml_615d55a49ff138fdb807794cf13c867261a6d940_12' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>V</mi><mo>⊃</mo><mi>B</mi></mrow><annotation encoding='application/x-tex'>V \supset B</annotation></semantics></math> such that their <a class='existingWikiWord' href='/nlab/show/diff/intersection'>intersection</a> <math class='maruku-mathml' display='inline' id='mathml_615d55a49ff138fdb807794cf13c867261a6d940_13' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>U</mi><mo>∩</mo><mi>V</mi></mrow><annotation encoding='application/x-tex'>U \cap V</annotation></semantics></math> is <a class='existingWikiWord' href='/nlab/show/diff/empty+set'>empty</a>.</li> </ul> <p>By <a class='existingWikiWord' href='/nlab/show/diff/Urysohn%27s+lemma'>Urysohn's lemma</a> this is equivalent to the condition</p> <ul> <li>for every two <a class='existingWikiWord' href='/nlab/show/diff/closed+subspace'>closed</a> <a class='existingWikiWord' href='/nlab/show/diff/disjoint+subsets'>disjoint subsets</a> <math class='maruku-mathml' display='inline' id='mathml_615d55a49ff138fdb807794cf13c867261a6d940_14' xmlns='http://www.w3.org/1998/Math/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> there exists an <a class='existingWikiWord' href='/nlab/show/diff/Urysohn%27s+lemma'>Urysohn function</a> that separates them.</li> </ul> <p>Often one adds the requirement</p> <ul> <li><math class='maruku-mathml' display='inline' id='mathml_615d55a49ff138fdb807794cf13c867261a6d940_15' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>T</mi> <mn>1</mn></msub></mrow><annotation encoding='application/x-tex'>T_1</annotation></semantics></math>: every <a class='existingWikiWord' href='/nlab/show/diff/point'>point</a> in <math class='maruku-mathml' display='inline' id='mathml_615d55a49ff138fdb807794cf13c867261a6d940_16' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> is closed.</li> </ul> <p>(Unlike with <a class='existingWikiWord' href='/nlab/show/diff/regular+space'>regular spaces</a>, <math class='maruku-mathml' display='inline' id='mathml_615d55a49ff138fdb807794cf13c867261a6d940_17' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>T</mi> <mn>0</mn></msub></mrow><annotation encoding='application/x-tex'>T_0</annotation></semantics></math> is not sufficient here.)</p> <p>One may also see terminology where a <strong>normal space</strong> is any space that satisfies <math class='maruku-mathml' display='inline' id='mathml_615d55a49ff138fdb807794cf13c867261a6d940_18' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>T</mi> <mn>4</mn></msub></mrow><annotation encoding='application/x-tex'>T_4</annotation></semantics></math>, while a <strong><math class='maruku-mathml' display='inline' id='mathml_615d55a49ff138fdb807794cf13c867261a6d940_19' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>T</mi> <mn>4</mn></msub></mrow><annotation encoding='application/x-tex'>T_4</annotation></semantics></math>-space</strong> must satisfy both <math class='maruku-mathml' display='inline' id='mathml_615d55a49ff138fdb807794cf13c867261a6d940_20' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>T</mi> <mn>4</mn></msub></mrow><annotation encoding='application/x-tex'>T_4</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_615d55a49ff138fdb807794cf13c867261a6d940_21' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>T</mi> <mn>1</mn></msub></mrow><annotation encoding='application/x-tex'>T_1</annotation></semantics></math>. This has the benefit that a <math class='maruku-mathml' display='inline' id='mathml_615d55a49ff138fdb807794cf13c867261a6d940_22' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>T</mi> <mn>4</mn></msub></mrow><annotation encoding='application/x-tex'>T_4</annotation></semantics></math>-space is always also a <math class='maruku-mathml' display='inline' id='mathml_615d55a49ff138fdb807794cf13c867261a6d940_23' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>T</mi> <mn>3</mn></msub></mrow><annotation encoding='application/x-tex'>T_3</annotation></semantics></math>-space while still having a term available for the weaker notion. On the other hand, the reverse might make more sense, since you would expect any space that satisfies <math class='maruku-mathml' display='inline' id='mathml_615d55a49ff138fdb807794cf13c867261a6d940_24' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>T</mi> <mn>4</mn></msub></mrow><annotation encoding='application/x-tex'>T_4</annotation></semantics></math> to be a <math class='maruku-mathml' display='inline' id='mathml_615d55a49ff138fdb807794cf13c867261a6d940_25' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>T</mi> <mn>4</mn></msub></mrow><annotation encoding='application/x-tex'>T_4</annotation></semantics></math>-space; this convention is also seen.</p> <p>If instead of <math class='maruku-mathml' display='inline' id='mathml_615d55a49ff138fdb807794cf13c867261a6d940_26' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>T</mi> <mn>1</mn></msub></mrow><annotation encoding='application/x-tex'>T_1</annotation></semantics></math>, one requires</p> <ul> <li><math class='maruku-mathml' display='inline' id='mathml_615d55a49ff138fdb807794cf13c867261a6d940_27' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>R</mi> <mn>0</mn></msub></mrow><annotation encoding='application/x-tex'>R_0</annotation></semantics></math>: if <math class='maruku-mathml' display='inline' id='mathml_615d55a49ff138fdb807794cf13c867261a6d940_28' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>x</mi></mrow><annotation encoding='application/x-tex'>x</annotation></semantics></math> is in the <a class='existingWikiWord' href='/nlab/show/diff/closed+subspace'>closure</a> of <math class='maruku-mathml' display='inline' id='mathml_615d55a49ff138fdb807794cf13c867261a6d940_29' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>{</mo><mi>y</mi><mo stretchy='false'>}</mo></mrow><annotation encoding='application/x-tex'>\{y\}</annotation></semantics></math>, then <math class='maruku-mathml' display='inline' id='mathml_615d55a49ff138fdb807794cf13c867261a6d940_30' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>y</mi></mrow><annotation encoding='application/x-tex'>y</annotation></semantics></math> is in the <a class='existingWikiWord' href='/nlab/show/diff/closed+subspace'>closure</a> of <math class='maruku-mathml' display='inline' id='mathml_615d55a49ff138fdb807794cf13c867261a6d940_31' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>{</mo><mi>x</mi><mo stretchy='false'>}</mo></mrow><annotation encoding='application/x-tex'>\{x\}</annotation></semantics></math>,</li> </ul> <p>then the result may be called an <strong><math class='maruku-mathml' display='inline' id='mathml_615d55a49ff138fdb807794cf13c867261a6d940_32' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>R</mi> <mn>3</mn></msub></mrow><annotation encoding='application/x-tex'>R_3</annotation></semantics></math>-space</strong>.</p> <p>Any space that satisfies both <math class='maruku-mathml' display='inline' id='mathml_615d55a49ff138fdb807794cf13c867261a6d940_33' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>T</mi> <mn>4</mn></msub></mrow><annotation encoding='application/x-tex'>T_4</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_615d55a49ff138fdb807794cf13c867261a6d940_34' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>T</mi> <mn>1</mn></msub></mrow><annotation encoding='application/x-tex'>T_1</annotation></semantics></math> must be <a class='existingWikiWord' href='/nlab/show/diff/Hausdorff+space'>Hausdorff</a>, and every Hausdorff space satisfies <math class='maruku-mathml' display='inline' id='mathml_615d55a49ff138fdb807794cf13c867261a6d940_35' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>T</mi> <mn>1</mn></msub></mrow><annotation encoding='application/x-tex'>T_1</annotation></semantics></math>, so one may call such a space a <strong>normal Hausdorff space</strong>; this terminology should be clear to any reader.</p> <p>Any space that satisfies both <math class='maruku-mathml' display='inline' id='mathml_615d55a49ff138fdb807794cf13c867261a6d940_36' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>T</mi> <mn>4</mn></msub></mrow><annotation encoding='application/x-tex'>T_4</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_615d55a49ff138fdb807794cf13c867261a6d940_37' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>R</mi> <mn>0</mn></msub></mrow><annotation encoding='application/x-tex'>R_0</annotation></semantics></math> must be <a class='existingWikiWord' href='/nlab/show/diff/regular+space'>regular</a> (in the weaker sense of that term), and every regular space satisfies <math class='maruku-mathml' display='inline' id='mathml_615d55a49ff138fdb807794cf13c867261a6d940_38' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>R</mi> <mn>0</mn></msub></mrow><annotation encoding='application/x-tex'>R_0</annotation></semantics></math>, so one may call such a space a <strong>normal regular space</strong>; however, those who interpret ‘normal’ to include <math class='maruku-mathml' display='inline' id='mathml_615d55a49ff138fdb807794cf13c867261a6d940_39' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>T</mi> <mn>1</mn></msub></mrow><annotation encoding='application/x-tex'>T_1</annotation></semantics></math> usually also interpret ‘regular’ to include <math class='maruku-mathml' display='inline' id='mathml_615d55a49ff138fdb807794cf13c867261a6d940_40' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>T</mi> <mn>1</mn></msub></mrow><annotation encoding='application/x-tex'>T_1</annotation></semantics></math>, so this term can be ambiguous.</p> <p>Every normal Hausdorff space is an <a class='existingWikiWord' href='/nlab/show/diff/Urysohn+topological+space'>Urysohn space</a>, a fortiori regular and a fortiori Hausdorff.</p> <p>It can be useful to rephrase <math class='maruku-mathml' display='inline' id='mathml_615d55a49ff138fdb807794cf13c867261a6d940_41' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>T</mi> <mn>4</mn></msub></mrow><annotation encoding='application/x-tex'>T_4</annotation></semantics></math> in terms of only open sets instead of also closed ones:</p> <ul> <li><math class='maruku-mathml' display='inline' id='mathml_615d55a49ff138fdb807794cf13c867261a6d940_42' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>T</mi> <mn>4</mn></msub></mrow><annotation encoding='application/x-tex'>T_4</annotation></semantics></math>: if <math class='maruku-mathml' display='inline' id='mathml_615d55a49ff138fdb807794cf13c867261a6d940_43' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>G</mi><mo>,</mo><mi>H</mi><mo>⊂</mo><mi>X</mi></mrow><annotation encoding='application/x-tex'>G,H \subset X</annotation></semantics></math> are <a class='existingWikiWord' href='/nlab/show/diff/open+subspace'>open</a> and <math class='maruku-mathml' display='inline' id='mathml_615d55a49ff138fdb807794cf13c867261a6d940_44' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>G</mi><mo>∪</mo><mi>H</mi><mo>=</mo><mi>X</mi></mrow><annotation encoding='application/x-tex'>G \cup H = X</annotation></semantics></math>, then there exist open sets <math class='maruku-mathml' display='inline' id='mathml_615d55a49ff138fdb807794cf13c867261a6d940_45' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>U</mi><mo>,</mo><mi>V</mi></mrow><annotation encoding='application/x-tex'>U,V</annotation></semantics></math> such that <math class='maruku-mathml' display='inline' id='mathml_615d55a49ff138fdb807794cf13c867261a6d940_46' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>U</mi><mo>∪</mo><mi>G</mi></mrow><annotation encoding='application/x-tex'>U \cup G</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_615d55a49ff138fdb807794cf13c867261a6d940_47' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>V</mi><mo>∪</mo><mi>H</mi></mrow><annotation encoding='application/x-tex'>V \cup H</annotation></semantics></math> are still <math class='maruku-mathml' display='inline' id='mathml_615d55a49ff138fdb807794cf13c867261a6d940_48' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> but <math class='maruku-mathml' display='inline' id='mathml_615d55a49ff138fdb807794cf13c867261a6d940_49' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>U</mi><mo>∩</mo><mi>V</mi></mrow><annotation encoding='application/x-tex'>U \cap V</annotation></semantics></math> is empty.</li> </ul> <p>This definition is suitable for generalisation to <a class='existingWikiWord' href='/nlab/show/diff/locale'>locales</a> and also for use in <a class='existingWikiWord' href='/nlab/show/diff/constructive+mathematics'>constructive mathematics</a> (where it is not equivalent to the usual one).</p> <p>To spell out the localic case, a <strong>normal locale</strong> is a <a class='existingWikiWord' href='/nlab/show/diff/frame'>frame</a> <math class='maruku-mathml' display='inline' id='mathml_615d55a49ff138fdb807794cf13c867261a6d940_50' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>L</mi></mrow><annotation encoding='application/x-tex'>L</annotation></semantics></math> such that</p> <ul> <li><math class='maruku-mathml' display='inline' id='mathml_615d55a49ff138fdb807794cf13c867261a6d940_51' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>T</mi> <mn>4</mn></msub></mrow><annotation encoding='application/x-tex'>T_4</annotation></semantics></math>: if <math class='maruku-mathml' display='inline' id='mathml_615d55a49ff138fdb807794cf13c867261a6d940_52' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>G</mi><mo>,</mo><mi>H</mi><mo>∈</mo><mi>L</mi></mrow><annotation encoding='application/x-tex'>G,H \in L</annotation></semantics></math> are opens and <math class='maruku-mathml' display='inline' id='mathml_615d55a49ff138fdb807794cf13c867261a6d940_53' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>G</mi><mo>∨</mo><mi>H</mi><mo>=</mo><mo>⊤</mo></mrow><annotation encoding='application/x-tex'>G \vee H = \top</annotation></semantics></math>, then there exist opens <math class='maruku-mathml' display='inline' id='mathml_615d55a49ff138fdb807794cf13c867261a6d940_54' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>U</mi><mo>,</mo><mi>V</mi></mrow><annotation encoding='application/x-tex'>U,V</annotation></semantics></math> such that <math class='maruku-mathml' display='inline' id='mathml_615d55a49ff138fdb807794cf13c867261a6d940_55' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>U</mi><mo>∨</mo><mi>G</mi></mrow><annotation encoding='application/x-tex'>U \vee G</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_615d55a49ff138fdb807794cf13c867261a6d940_56' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>V</mi><mo>∨</mo><mi>H</mi></mrow><annotation encoding='application/x-tex'>V \vee H</annotation></semantics></math> are still <math class='maruku-mathml' display='inline' id='mathml_615d55a49ff138fdb807794cf13c867261a6d940_57' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo>⊤</mo></mrow><annotation encoding='application/x-tex'>\top</annotation></semantics></math> but <math class='maruku-mathml' display='inline' id='mathml_615d55a49ff138fdb807794cf13c867261a6d940_58' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>U</mi><mo>∧</mo><mi>V</mi><mo>=</mo><mo>⊥</mo></mrow><annotation encoding='application/x-tex'>U \wedge V = \bot</annotation></semantics></math>.</li> </ul> <h2 id='Examples'>Examples</h2> <div class='num_example'> <h6 id='example'>Example</h6> <p>Let <math class='maruku-mathml' display='inline' id='mathml_615d55a49ff138fdb807794cf13c867261a6d940_59' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mi>X</mi><mo>,</mo><mi>d</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(X,d)</annotation></semantics></math> be a <a class='existingWikiWord' href='/nlab/show/diff/metric+space'>metric space</a> regarded as a <a class='existingWikiWord' href='/nlab/show/diff/topological+space'>topological space</a> via its <a class='existingWikiWord' href='/nlab/show/diff/metric+topology'>metric topology</a>. Then this is a normal Hausdorff space.</p> </div> <div class='proof'> <h6 id='proof'>Proof</h6> <p>We need to show that given two <a class='existingWikiWord' href='/nlab/show/diff/disjoint+subsets'>disjoint</a> <a class='existingWikiWord' href='/nlab/show/diff/closed+subspace'>closed subsets</a> <math class='maruku-mathml' display='inline' id='mathml_615d55a49ff138fdb807794cf13c867261a6d940_60' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>C</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>C</mi> <mn>2</mn></msub><mo>⊂</mo><mi>X</mi></mrow><annotation encoding='application/x-tex'>C_1, C_2 \subset X</annotation></semantics></math>, there exist <a class='existingWikiWord' href='/nlab/show/diff/disjoint+subsets'>disjoint</a> <a class='existingWikiWord' href='/nlab/show/diff/neighborhood'>open neighbourhoods</a> <math class='maruku-mathml' display='inline' id='mathml_615d55a49ff138fdb807794cf13c867261a6d940_61' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>U</mi> <mrow><msub><mi>C</mi> <mn>1</mn></msub></mrow></msub><mo>⊃</mo><msub><mi>C</mi> <mn>1</mn></msub></mrow><annotation encoding='application/x-tex'>U_{C_1} \supset C_1</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_615d55a49ff138fdb807794cf13c867261a6d940_62' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>U</mi> <mrow><msub><mi>C</mi> <mn>2</mn></msub></mrow></msub><mo>⊃</mo><msub><mi>C</mi> <mn>2</mn></msub></mrow><annotation encoding='application/x-tex'>U_{C_2} \supset C_2</annotation></semantics></math>.</p> <p>Consider the function</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_615d55a49ff138fdb807794cf13c867261a6d940_63' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>d</mi><mo stretchy='false'>(</mo><mi>S</mi><mo>,</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mo stretchy='false'>)</mo><mo lspace='verythinmathspace'>:</mo><mi>X</mi><mo>→</mo><mi>ℝ</mi></mrow><annotation encoding='application/x-tex'> d(S,-) \colon X \to \mathbb{R} </annotation></semantics></math></div> <p>which computes <a class='existingWikiWord' href='/nlab/show/diff/metric+space'>distances</a> from a subset <math class='maruku-mathml' display='inline' id='mathml_615d55a49ff138fdb807794cf13c867261a6d940_64' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>S</mi><mo>⊂</mo><mi>X</mi></mrow><annotation encoding='application/x-tex'>S \subset X</annotation></semantics></math>, by forming the <a class='existingWikiWord' href='/nlab/show/diff/meet'>infimum</a> of the distances to all its points:</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_615d55a49ff138fdb807794cf13c867261a6d940_65' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>d</mi><mo stretchy='false'>(</mo><mi>S</mi><mo>,</mo><mi>x</mi><mo stretchy='false'>)</mo><mo>≔</mo><mi>inf</mi><mrow><mo>{</mo><mi>d</mi><mo stretchy='false'>(</mo><mi>s</mi><mo>,</mo><mi>x</mi><mo stretchy='false'>)</mo><mo stretchy='false'>|</mo><mi>s</mi><mo>∈</mo><mi>S</mi><mo>}</mo></mrow><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> d(S,x) \coloneqq inf\left\{ d(s,x) \vert s \in S \right\} \,. </annotation></semantics></math></div> <p>If <math class='maruku-mathml' display='inline' id='mathml_615d55a49ff138fdb807794cf13c867261a6d940_66' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>S</mi></mrow><annotation encoding='application/x-tex'>S</annotation></semantics></math> is closed and <math class='maruku-mathml' display='inline' id='mathml_615d55a49ff138fdb807794cf13c867261a6d940_67' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>x</mi><mo>∉</mo><mi>S</mi></mrow><annotation encoding='application/x-tex'>x \notin S</annotation></semantics></math>, then <math class='maruku-mathml' display='inline' id='mathml_615d55a49ff138fdb807794cf13c867261a6d940_68' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>d</mi><mo stretchy='false'>(</mo><mi>S</mi><mo>,</mo><mi>x</mi><mo stretchy='false'>)</mo><mo>></mo><mn>0</mn></mrow><annotation encoding='application/x-tex'>d(S, x) \gt 0</annotation></semantics></math>. Then the <a class='existingWikiWord' href='/nlab/show/diff/union'>unions</a> of <a class='existingWikiWord' href='/nlab/show/diff/ball'>open balls</a></p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_615d55a49ff138fdb807794cf13c867261a6d940_69' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>U</mi> <mrow><msub><mi>C</mi> <mn>1</mn></msub></mrow></msub><mo>≔</mo><munder><mo lspace='thinmathspace' rspace='thinmathspace'>⋃</mo><mrow><msub><mi>x</mi> <mn>1</mn></msub><mo>∈</mo><msub><mi>C</mi> <mn>1</mn></msub></mrow></munder><msubsup><mi>B</mi> <mrow><msub><mi>x</mi> <mn>1</mn></msub></mrow> <mo>∘</mo></msubsup><mo stretchy='false'>(</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mi>d</mi><mo stretchy='false'>(</mo><msub><mi>C</mi> <mn>2</mn></msub><mo>,</mo><msub><mi>x</mi> <mn>1</mn></msub><mo stretchy='false'>)</mo><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'> U_{C_1} \coloneqq \underset{x_1 \in C_1}{\bigcup} B^\circ_{x_1}( \frac1{2}d(C_2,x_1) ) </annotation></semantics></math></div> <p>and</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_615d55a49ff138fdb807794cf13c867261a6d940_70' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>U</mi> <mrow><msub><mi>C</mi> <mn>2</mn></msub></mrow></msub><mo>≔</mo><munder><mo lspace='thinmathspace' rspace='thinmathspace'>⋃</mo><mrow><msub><mi>x</mi> <mn>2</mn></msub><mo>∈</mo><msub><mi>C</mi> <mn>2</mn></msub></mrow></munder><msubsup><mi>B</mi> <mrow><msub><mi>x</mi> <mn>2</mn></msub></mrow> <mo>∘</mo></msubsup><mo stretchy='false'>(</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mi>d</mi><mo stretchy='false'>(</mo><msub><mi>C</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>x</mi> <mn>2</mn></msub><mo stretchy='false'>)</mo><mo stretchy='false'>)</mo><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> U_{C_2} \coloneqq \underset{x_2 \in C_2}{\bigcup} B^\circ_{x_2}( \frac1{2}d(C_1,x_2) ) \,. </annotation></semantics></math></div> <p>have the required properties. For if there exist <math class='maruku-mathml' display='inline' id='mathml_615d55a49ff138fdb807794cf13c867261a6d940_71' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>x</mi> <mn>1</mn></msub><mo>∈</mo><msub><mi>C</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>x</mi> <mn>2</mn></msub><mo>∈</mo><msub><mi>C</mi> <mn>2</mn></msub></mrow><annotation encoding='application/x-tex'>x_1 \in C_1, x_2 \in C_2</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_615d55a49ff138fdb807794cf13c867261a6d940_72' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>y</mi><mo>∈</mo><msubsup><mi>B</mi> <mrow><msub><mi>x</mi> <mn>1</mn></msub></mrow> <mo>∘</mo></msubsup><mo stretchy='false'>(</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mi>d</mi><mo stretchy='false'>(</mo><msub><mi>C</mi> <mn>2</mn></msub><mo>,</mo><msub><mi>x</mi> <mn>1</mn></msub><mo stretchy='false'>)</mo><mo stretchy='false'>)</mo><mo>∩</mo><msubsup><mi>B</mi> <mrow><msub><mi>x</mi> <mn>2</mn></msub></mrow> <mo>∘</mo></msubsup><mo stretchy='false'>(</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mi>d</mi><mo stretchy='false'>(</mo><msub><mi>C</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>x</mi> <mn>2</mn></msub><mo stretchy='false'>)</mo><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>y \in B^\circ_{x_1}( \frac1{2}d(C_2,x_1) ) \cap B^\circ_{x_2}( \frac1{2}d(C_1,x_2) )</annotation></semantics></math>, then</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_615d55a49ff138fdb807794cf13c867261a6d940_73' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>d</mi><mo stretchy='false'>(</mo><msub><mi>x</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>x</mi> <mn>2</mn></msub><mo stretchy='false'>)</mo><mo>≤</mo><mi>d</mi><mo stretchy='false'>(</mo><msub><mi>x</mi> <mn>1</mn></msub><mo>,</mo><mi>y</mi><mo stretchy='false'>)</mo><mo>+</mo><mi>d</mi><mo stretchy='false'>(</mo><mi>y</mi><mo>,</mo><msub><mi>x</mi> <mn>2</mn></msub><mo stretchy='false'>)</mo><mo><</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo stretchy='false'>(</mo><mi>d</mi><mo stretchy='false'>(</mo><msub><mi>C</mi> <mn>2</mn></msub><mo>,</mo><msub><mi>x</mi> <mn>1</mn></msub><mo stretchy='false'>)</mo><mo>+</mo><mi>d</mi><mo stretchy='false'>(</mo><msub><mi>C</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>x</mi> <mn>2</mn></msub><mo stretchy='false'>)</mo><mo stretchy='false'>)</mo><mo>≤</mo><mi>max</mi><mo stretchy='false'>{</mo><mi>d</mi><mo stretchy='false'>(</mo><msub><mi>C</mi> <mn>2</mn></msub><mo>,</mo><msub><mi>x</mi> <mn>1</mn></msub><mo stretchy='false'>)</mo><mo>,</mo><mi>d</mi><mo stretchy='false'>(</mo><msub><mi>C</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>x</mi> <mn>2</mn></msub><mo stretchy='false'>)</mo><mo stretchy='false'>}</mo></mrow><annotation encoding='application/x-tex'>d(x_1, x_2) \leq d(x_1, y) + d(y, x_2) \lt \frac1{2} (d(C_2, x_1) + d(C_1, x_2)) \leq \max\{d(C_2, x_1), d(C_1, x_2)\}</annotation></semantics></math></div> <p>and if <math class='maruku-mathml' display='inline' id='mathml_615d55a49ff138fdb807794cf13c867261a6d940_74' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>d</mi><mo stretchy='false'>(</mo><msub><mi>C</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>x</mi> <mn>2</mn></msub><mo stretchy='false'>)</mo><mo>≤</mo><mi>d</mi><mo stretchy='false'>(</mo><msub><mi>C</mi> <mn>2</mn></msub><mo>,</mo><msub><mi>x</mi> <mn>1</mn></msub><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>d(C_1, x_2) \leq d(C_2, x_1)</annotation></semantics></math> say, then <math class='maruku-mathml' display='inline' id='mathml_615d55a49ff138fdb807794cf13c867261a6d940_75' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>d</mi><mo stretchy='false'>(</mo><msub><mi>x</mi> <mn>2</mn></msub><mo>,</mo><msub><mi>x</mi> <mn>1</mn></msub><mo stretchy='false'>)</mo><mo>=</mo><mi>d</mi><mo stretchy='false'>(</mo><msub><mi>x</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>x</mi> <mn>2</mn></msub><mo stretchy='false'>)</mo><mo><</mo><mi>d</mi><mo stretchy='false'>(</mo><msub><mi>C</mi> <mn>2</mn></msub><mo>,</mo><msub><mi>x</mi> <mn>1</mn></msub><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>d(x_2, x_1) = d(x_1, x_2) \lt d(C_2, x_1)</annotation></semantics></math>, contradicting the definition of <math class='maruku-mathml' display='inline' id='mathml_615d55a49ff138fdb807794cf13c867261a6d940_76' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>d</mi><mo stretchy='false'>(</mo><msub><mi>C</mi> <mn>2</mn></msub><mo>,</mo><msub><mi>x</mi> <mn>1</mn></msub><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>d(C_2, x_1)</annotation></semantics></math>.</p> </div> <div class='num_prop'> <h6 id='proposition'>Proposition</h6> <p>Every <a class='existingWikiWord' href='/nlab/show/diff/regular+space'>regular</a> <a class='existingWikiWord' href='/nlab/show/diff/second-countable+space'>second countable space</a> is normal.</p> </div> <p>See <a class='existingWikiWord' href='/nlab/show/diff/Urysohn+metrization+theorem'>Urysohn metrization theorem</a> for details.</p> <div class='num_prop'> <h6 id='proposition_2'>Proposition</h6> <p><strong>(Dieudonné‘s theorem)</strong></p> <p>Every <a class='existingWikiWord' href='/nlab/show/diff/paracompact+Hausdorff+space'>paracompact Hausdorff space</a>, in particular every <a class='existingWikiWord' href='/nlab/show/diff/compactum'>compact Hausdorff space</a>, is normal.</p> </div> <p>See <em><a class='existingWikiWord' href='/nlab/show/diff/paracompact+Hausdorff+spaces+are+normal'>paracompact Hausdorff spaces are normal</a></em> for details.</p> <div class='num_example'> <h6 id='nonexample'>Nonexample</h6> <p>The <a class='existingWikiWord' href='/nlab/show/diff/real+number'>real numbers</a> equipped with their <a class='existingWikiWord' href='/nlab/show/diff/K-topology'>K-topology</a> <math class='maruku-mathml' display='inline' id='mathml_615d55a49ff138fdb807794cf13c867261a6d940_77' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>ℝ</mi> <mi>K</mi></msub></mrow><annotation encoding='application/x-tex'>\mathbb{R}_K</annotation></semantics></math> are a <a class='existingWikiWord' href='/nlab/show/diff/Hausdorff+space'>Hausdorff topological space</a> which is not a <a class='existingWikiWord' href='/nlab/show/diff/regular+space'>regular Hausdorff space</a> (hence in particular not a normal Hausdorff space).</p> </div> <div class='proof'> <h6 id='proof_2'>Proof</h6> <p>By construction the K-topology is <a class='existingWikiWord' href='/nlab/show/diff/finer+topology'>finer</a> than the usual <a class='existingWikiWord' href='/nlab/show/diff/Euclidean+space'>euclidean</a> <a class='existingWikiWord' href='/nlab/show/diff/metric+topology'>metric topology</a>. Since the latter is Hausdorff, so is <math class='maruku-mathml' display='inline' id='mathml_615d55a49ff138fdb807794cf13c867261a6d940_78' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>ℝ</mi> <mi>K</mi></msub></mrow><annotation encoding='application/x-tex'>\mathbb{R}_K</annotation></semantics></math>. It remains to see that <math class='maruku-mathml' display='inline' id='mathml_615d55a49ff138fdb807794cf13c867261a6d940_79' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>ℝ</mi> <mi>K</mi></msub></mrow><annotation encoding='application/x-tex'>\mathbb{R}_K</annotation></semantics></math> contains a point and a <a class='existingWikiWord' href='/nlab/show/diff/disjoint+subsets'>disjoint</a> closed subset such that they do not have disjoint <a class='existingWikiWord' href='/nlab/show/diff/neighborhood'>open neighbourhoods</a>.</p> <p>But this is the case essentially by construction: Observe that</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_615d55a49ff138fdb807794cf13c867261a6d940_80' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>ℝ</mi><mo>\</mo><mi>K</mi><mspace width='thickmathspace' /><mo>=</mo><mspace width='thickmathspace' /><mo stretchy='false'>(</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mn>∞</mn><mo>,</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mn>1</mn><mo stretchy='false'>/</mo><mn>2</mn><mo stretchy='false'>)</mo><mo>∪</mo><mrow><mo>(</mo><mo stretchy='false'>(</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo stretchy='false'>)</mo><mo>\</mo><mi>K</mi><mo>)</mo></mrow><mo>∪</mo><mo stretchy='false'>(</mo><mn>1</mn><mo stretchy='false'>/</mo><mn>2</mn><mo>,</mo><mn>∞</mn><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'> \mathbb{R} \backslash K \;=\; (-\infty,-1/2) \cup \left( (-1,1) \backslash K \right) \cup (1/2, \infty) </annotation></semantics></math></div> <p>is an open subset in <math class='maruku-mathml' display='inline' id='mathml_615d55a49ff138fdb807794cf13c867261a6d940_81' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>ℝ</mi> <mi>K</mi></msub></mrow><annotation encoding='application/x-tex'>\mathbb{R}_K</annotation></semantics></math>, whence</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_615d55a49ff138fdb807794cf13c867261a6d940_82' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>K</mi><mo>=</mo><mi>ℝ</mi><mo>\</mo><mo stretchy='false'>(</mo><mi>ℝ</mi><mo>\</mo><mi>K</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'> K = \mathbb{R} \backslash ( \mathbb{R} \backslash K ) </annotation></semantics></math></div> <p>is a <a class='existingWikiWord' href='/nlab/show/diff/closed+subspace'>closed subset</a> of <math class='maruku-mathml' display='inline' id='mathml_615d55a49ff138fdb807794cf13c867261a6d940_83' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>ℝ</mi> <mi>K</mi></msub></mrow><annotation encoding='application/x-tex'>\mathbb{R}_K</annotation></semantics></math>.</p> <p>But every <a class='existingWikiWord' href='/nlab/show/diff/neighborhood'>open neighbourhood</a> of <math class='maruku-mathml' display='inline' id='mathml_615d55a49ff138fdb807794cf13c867261a6d940_84' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>{</mo><mn>0</mn><mo stretchy='false'>}</mo></mrow><annotation encoding='application/x-tex'>\{0\}</annotation></semantics></math> contains at least <math class='maruku-mathml' display='inline' id='mathml_615d55a49ff138fdb807794cf13c867261a6d940_85' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mi>ϵ</mi><mo>,</mo><mi>ϵ</mi><mo stretchy='false'>)</mo><mo>\</mo><mi>K</mi></mrow><annotation encoding='application/x-tex'>(-\epsilon, \epsilon) \backslash K</annotation></semantics></math> for some positive real number <math class='maruku-mathml' display='inline' id='mathml_615d55a49ff138fdb807794cf13c867261a6d940_86' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>ϵ</mi></mrow><annotation encoding='application/x-tex'>\epsilon</annotation></semantics></math>. There exists then <math class='maruku-mathml' display='inline' id='mathml_615d55a49ff138fdb807794cf13c867261a6d940_87' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>n</mi><mo>∈</mo><msub><mi>ℕ</mi> <mrow><mo>≥</mo><mn>0</mn></mrow></msub></mrow><annotation encoding='application/x-tex'>n \in \mathbb{N}_{\geq 0}</annotation></semantics></math> with <math class='maruku-mathml' display='inline' id='mathml_615d55a49ff138fdb807794cf13c867261a6d940_88' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>1</mn><mo stretchy='false'>/</mo><mi>n</mi><mo><</mo><mi>ϵ</mi></mrow><annotation encoding='application/x-tex'>1/n \lt \epsilon</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_615d55a49ff138fdb807794cf13c867261a6d940_89' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>1</mn><mo stretchy='false'>/</mo><mi>n</mi><mo>∈</mo><mi>K</mi></mrow><annotation encoding='application/x-tex'>1/n \in K</annotation></semantics></math>. An open neighbourhood of <math class='maruku-mathml' display='inline' id='mathml_615d55a49ff138fdb807794cf13c867261a6d940_90' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>K</mi></mrow><annotation encoding='application/x-tex'>K</annotation></semantics></math> needs to contain an open interval around <math class='maruku-mathml' display='inline' id='mathml_615d55a49ff138fdb807794cf13c867261a6d940_91' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>1</mn><mo stretchy='false'>/</mo><mi>n</mi></mrow><annotation encoding='application/x-tex'>1/n</annotation></semantics></math>, and hence will have non-trivial intersection with <math class='maruku-mathml' display='inline' id='mathml_615d55a49ff138fdb807794cf13c867261a6d940_92' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mi>ϵ</mi><mo>,</mo><mi>ϵ</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(-\epsilon, \epsilon)</annotation></semantics></math>. Therefore <math class='maruku-mathml' display='inline' id='mathml_615d55a49ff138fdb807794cf13c867261a6d940_93' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>{</mo><mn>0</mn><mo stretchy='false'>}</mo></mrow><annotation encoding='application/x-tex'>\{0\}</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_615d55a49ff138fdb807794cf13c867261a6d940_94' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>K</mi></mrow><annotation encoding='application/x-tex'>K</annotation></semantics></math> may not be separated by disjoint open neighbourhoods, and so <math class='maruku-mathml' display='inline' id='mathml_615d55a49ff138fdb807794cf13c867261a6d940_95' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>ℝ</mi> <mi>K</mi></msub></mrow><annotation encoding='application/x-tex'>\mathbb{R}_K</annotation></semantics></math> is not normal.</p> </div> <div class='num_example' id='Morse'> <h6 id='counterexample'>Counter-Example</h6> <p>If <math class='maruku-mathml' display='inline' id='mathml_615d55a49ff138fdb807794cf13c867261a6d940_96' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>ω</mi> <mn>1</mn></msub></mrow><annotation encoding='application/x-tex'>\omega_1</annotation></semantics></math> is the first un-<a class='existingWikiWord' href='/nlab/show/diff/countable+ordinal'>countable ordinal</a> with the <a class='existingWikiWord' href='/nlab/show/diff/order+topology'>order topology</a>, and <math class='maruku-mathml' display='inline' id='mathml_615d55a49ff138fdb807794cf13c867261a6d940_97' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mover><mrow><msub><mi>ω</mi> <mn>1</mn></msub></mrow><mo>¯</mo></mover></mrow><annotation encoding='application/x-tex'>\widebar{\omega_1}</annotation></semantics></math> its <a class='existingWikiWord' href='/nlab/show/diff/one-point+compactification'>one-point compactification</a>, then <math class='maruku-mathml' display='inline' id='mathml_615d55a49ff138fdb807794cf13c867261a6d940_98' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi><mo>=</mo><msub><mi>ω</mi> <mn>1</mn></msub><mo>×</mo><mover><mrow><msub><mi>ω</mi> <mn>1</mn></msub></mrow><mo>¯</mo></mover></mrow><annotation encoding='application/x-tex'>X = \omega_1 \times \widebar{\omega_1}</annotation></semantics></math> with the <a class='existingWikiWord' href='/nlab/show/diff/product+topological+space'>product topology</a> is not normal.</p> <p>Indeed, let <math class='maruku-mathml' display='inline' id='mathml_615d55a49ff138fdb807794cf13c867261a6d940_99' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>∞</mn><mo>∈</mo><mover><mrow><msub><mi>ω</mi> <mn>1</mn></msub></mrow><mo>¯</mo></mover></mrow><annotation encoding='application/x-tex'>\infty \in \widebar{\omega_1}</annotation></semantics></math> be the unique point in the <a class='existingWikiWord' href='/nlab/show/diff/complement'>complement</a> of <math class='maruku-mathml' display='inline' id='mathml_615d55a49ff138fdb807794cf13c867261a6d940_100' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>ω</mi> <mn>1</mn></msub><mo>↪</mo><mover><mrow><msub><mi>ω</mi> <mn>1</mn></msub></mrow><mo>¯</mo></mover></mrow><annotation encoding='application/x-tex'>\omega_1 \hookrightarrow \widebar{\omega_1}</annotation></semantics></math>; then it may be shown that every open set <math class='maruku-mathml' display='inline' id='mathml_615d55a49ff138fdb807794cf13c867261a6d940_101' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>U</mi></mrow><annotation encoding='application/x-tex'>U</annotation></semantics></math> in <math class='maruku-mathml' display='inline' id='mathml_615d55a49ff138fdb807794cf13c867261a6d940_102' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> that includes the closed set <math class='maruku-mathml' display='inline' id='mathml_615d55a49ff138fdb807794cf13c867261a6d940_103' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>A</mi><mo>=</mo><mo stretchy='false'>{</mo><mo stretchy='false'>(</mo><mi>x</mi><mo>,</mo><mi>x</mi><mo stretchy='false'>)</mo><mo>:</mo><mi>x</mi><mo>≠</mo><mn>∞</mn><mo stretchy='false'>}</mo></mrow><annotation encoding='application/x-tex'>A = \{(x, x): x \neq \infty\}</annotation></semantics></math> in <math class='maruku-mathml' display='inline' id='mathml_615d55a49ff138fdb807794cf13c867261a6d940_104' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> must somewhere intersect the closed set <math class='maruku-mathml' display='inline' id='mathml_615d55a49ff138fdb807794cf13c867261a6d940_105' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>ω</mi> <mn>1</mn></msub><mo>×</mo><mo stretchy='false'>{</mo><mn>∞</mn><mo stretchy='false'>}</mo></mrow><annotation encoding='application/x-tex'>\omega_1 \times \{\infty\}</annotation></semantics></math> which is disjoint from <math class='maruku-mathml' display='inline' id='mathml_615d55a49ff138fdb807794cf13c867261a6d940_106' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>A</mi></mrow><annotation encoding='application/x-tex'>A</annotation></semantics></math>. For if that were false, then we could define an increasing sequence <math class='maruku-mathml' display='inline' id='mathml_615d55a49ff138fdb807794cf13c867261a6d940_107' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>x</mi> <mi>n</mi></msub><mo>∈</mo><msub><mi>ω</mi> <mn>1</mn></msub></mrow><annotation encoding='application/x-tex'>x_n \in \omega_1</annotation></semantics></math> by recursion, letting <math class='maruku-mathml' display='inline' id='mathml_615d55a49ff138fdb807794cf13c867261a6d940_108' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>x</mi> <mn>0</mn></msub><mo>=</mo><mn>0</mn></mrow><annotation encoding='application/x-tex'>x_0 = 0</annotation></semantics></math> and letting <math class='maruku-mathml' display='inline' id='mathml_615d55a49ff138fdb807794cf13c867261a6d940_109' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>x</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>∈</mo><msub><mi>ω</mi> <mn>1</mn></msub></mrow><annotation encoding='application/x-tex'>x_{n+1} \in \omega_1</annotation></semantics></math> be the least element that is greater than <math class='maruku-mathml' display='inline' id='mathml_615d55a49ff138fdb807794cf13c867261a6d940_110' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>x</mi> <mi>n</mi></msub></mrow><annotation encoding='application/x-tex'>x_n</annotation></semantics></math> and such that <math class='maruku-mathml' display='inline' id='mathml_615d55a49ff138fdb807794cf13c867261a6d940_111' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><msub><mi>x</mi> <mi>n</mi></msub><mo>,</mo><msub><mi>x</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mo stretchy='false'>)</mo><mo>∉</mo><mi>U</mi></mrow><annotation encoding='application/x-tex'>(x_n, x_{n+1}) \notin U</annotation></semantics></math>. Then, letting <math class='maruku-mathml' display='inline' id='mathml_615d55a49ff138fdb807794cf13c867261a6d940_112' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>b</mi><mo>∈</mo><msub><mi>ω</mi> <mn>1</mn></msub></mrow><annotation encoding='application/x-tex'>b \in \omega_1</annotation></semantics></math> be the <a class='existingWikiWord' href='/nlab/show/diff/join'>supremum</a> of this increasing sequence, the sequence <math class='maruku-mathml' display='inline' id='mathml_615d55a49ff138fdb807794cf13c867261a6d940_113' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><msub><mi>x</mi> <mi>n</mi></msub><mo>,</mo><msub><mi>x</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(x_n, x_{n+1})</annotation></semantics></math> converges to <math class='maruku-mathml' display='inline' id='mathml_615d55a49ff138fdb807794cf13c867261a6d940_114' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mi>b</mi><mo>,</mo><mi>b</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(b, b)</annotation></semantics></math>, and yet the neighborhood <math class='maruku-mathml' display='inline' id='mathml_615d55a49ff138fdb807794cf13c867261a6d940_115' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>U</mi></mrow><annotation encoding='application/x-tex'>U</annotation></semantics></math> of <math class='maruku-mathml' display='inline' id='mathml_615d55a49ff138fdb807794cf13c867261a6d940_116' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mi>b</mi><mo>,</mo><mi>b</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(b, b)</annotation></semantics></math> contains none of the points of this sequence, which is a contradiction.</p> </div> <p>This example also shows that general subspaces of normal spaces need not be normal, since <math class='maruku-mathml' display='inline' id='mathml_615d55a49ff138fdb807794cf13c867261a6d940_117' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>ω</mi> <mn>1</mn></msub><mo>×</mo><mover><mrow><msub><mi>ω</mi> <mn>1</mn></msub></mrow><mo>¯</mo></mover></mrow><annotation encoding='application/x-tex'>\omega_1 \times \widebar{\omega_1}</annotation></semantics></math> is an open subspace of the compact Hausdorff space <math class='maruku-mathml' display='inline' id='mathml_615d55a49ff138fdb807794cf13c867261a6d940_118' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mover><mrow><msub><mi>ω</mi> <mn>1</mn></msub></mrow><mo>¯</mo></mover><mo>×</mo><mover><mrow><msub><mi>ω</mi> <mn>1</mn></msub></mrow><mo>¯</mo></mover></mrow><annotation encoding='application/x-tex'>\widebar{\omega_1} \times \widebar{\omega_1}</annotation></semantics></math>, which is itself normal.</p> <div class='num_example'> <h6 id='counterexample_2'>Counter-Example</h6> <p>An uncountable product of infinite discrete spaces <math class='maruku-mathml' display='inline' id='mathml_615d55a49ff138fdb807794cf13c867261a6d940_119' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> is not normal. More generally, a product of <math class='maruku-mathml' display='inline' id='mathml_615d55a49ff138fdb807794cf13c867261a6d940_120' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>T</mi> <mn>1</mn></msub></mrow><annotation encoding='application/x-tex'>T_1</annotation></semantics></math> spaces <math class='maruku-mathml' display='inline' id='mathml_615d55a49ff138fdb807794cf13c867261a6d940_121' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>X</mi> <mi>i</mi></msub></mrow><annotation encoding='application/x-tex'>X_i</annotation></semantics></math> uncountably many of which are not <a class='existingWikiWord' href='/nlab/show/diff/limit+point+compact+space'>limit point compact</a> is not normal.</p> <p>Indeed, by a simple application of Remark <a class='maruku-ref' href='#retracts'>1</a> below and the fact that closed subspaces of normal Hausdorff spaces are normal Hausdorff, it suffices to see that the archetypal example <math class='maruku-mathml' display='inline' id='mathml_615d55a49ff138fdb807794cf13c867261a6d940_122' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>ℕ</mi> <mrow><msub><mi>ω</mi> <mn>1</mn></msub></mrow></msup></mrow><annotation encoding='application/x-tex'>\mathbb{N}^{\omega_1}</annotation></semantics></math> is not normal. For a readable and not overly long account of this result, see <a href='https://dantopology.wordpress.com/2009/10/13/the-uncountable-product-of-the-countable-discrete-space-is-not-normal/'>Dan Ma’s blog</a>.</p> </div> <div class='num_example'> <h6 id='example_2'>Example</h6> <p>A <a class='existingWikiWord' href='/nlab/show/diff/Dowker+space'>Dowker space</a> is an example of a normal space which is not <a class='existingWikiWord' href='/nlab/show/diff/countably+paracompact+topological+space'>countably paracompact</a>.</p> </div> <h2 id='properties'>Properties</h2> <h3 id='basic_properties'>Basic properties</h3> <div class='num_prop' id='T4InTermsOfTopologicalClosures'> <h6 id='proposition_3'>Proposition</h6> <p><strong>(normality in terms of topological closures)</strong></p> <p>A <a class='existingWikiWord' href='/nlab/show/diff/topological+space'>topological space</a> <math class='maruku-mathml' display='inline' id='mathml_615d55a49ff138fdb807794cf13c867261a6d940_123' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mi>X</mi><mo>,</mo><mi>τ</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(X,\tau)</annotation></semantics></math> is normal Hausdorff, precisely if all points are closed and for all <a class='existingWikiWord' href='/nlab/show/diff/closed+subspace'>closed subsets</a> <math class='maruku-mathml' display='inline' id='mathml_615d55a49ff138fdb807794cf13c867261a6d940_124' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi><mo>⊂</mo><mi>X</mi></mrow><annotation encoding='application/x-tex'>C \subset X</annotation></semantics></math> with <a class='existingWikiWord' href='/nlab/show/diff/neighborhood'>open neighbourhood</a> <math class='maruku-mathml' display='inline' id='mathml_615d55a49ff138fdb807794cf13c867261a6d940_125' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>U</mi><mo>⊃</mo><mi>C</mi></mrow><annotation encoding='application/x-tex'>U \supset C</annotation></semantics></math> there exists a smaller open neighbourhood <math class='maruku-mathml' display='inline' id='mathml_615d55a49ff138fdb807794cf13c867261a6d940_126' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>V</mi><mo>⊃</mo><mi>C</mi></mrow><annotation encoding='application/x-tex'>V \supset C</annotation></semantics></math> whose <a class='existingWikiWord' href='/nlab/show/diff/closed+subspace'>topological closure</a> <math class='maruku-mathml' display='inline' id='mathml_615d55a49ff138fdb807794cf13c867261a6d940_127' xmlns='http://www.w3.org/1998/Math/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> is still contained in <math class='maruku-mathml' display='inline' id='mathml_615d55a49ff138fdb807794cf13c867261a6d940_128' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>U</mi></mrow><annotation encoding='application/x-tex'>U</annotation></semantics></math>:</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_615d55a49ff138fdb807794cf13c867261a6d940_129' xmlns='http://www.w3.org/1998/Math/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' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> C \subset V \subset Cl(V) \subset U \,. </annotation></semantics></math></div></div> <div class='proof'> <h6 id='proof_3'>Proof</h6> <p>In one direction, assume that <math class='maruku-mathml' display='inline' id='mathml_615d55a49ff138fdb807794cf13c867261a6d940_130' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mi>X</mi><mo>,</mo><mi>τ</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(X,\tau)</annotation></semantics></math> is normal, and consider</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_615d55a49ff138fdb807794cf13c867261a6d940_131' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi><mo>⊂</mo><mi>U</mi><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> C \subset U \,. </annotation></semantics></math></div> <p>It follows that the <a class='existingWikiWord' href='/nlab/show/diff/complement'>complement</a> of the open subset <math class='maruku-mathml' display='inline' id='mathml_615d55a49ff138fdb807794cf13c867261a6d940_132' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>U</mi></mrow><annotation encoding='application/x-tex'>U</annotation></semantics></math> is closed and disjoint from <math class='maruku-mathml' display='inline' id='mathml_615d55a49ff138fdb807794cf13c867261a6d940_133' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi></mrow><annotation encoding='application/x-tex'>C</annotation></semantics></math>:</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_615d55a49ff138fdb807794cf13c867261a6d940_134' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi><mo>∩</mo><mi>X</mi><mo>∖</mo><mi>U</mi><mo>=</mo><mi>∅</mi><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> C \cap X \setminus U = \emptyset \,. </annotation></semantics></math></div> <p>Therefore by assumption of normality of <math class='maruku-mathml' display='inline' id='mathml_615d55a49ff138fdb807794cf13c867261a6d940_135' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mi>X</mi><mo>,</mo><mi>τ</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(X,\tau)</annotation></semantics></math>, there exist open neighbourhoods with</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_615d55a49ff138fdb807794cf13c867261a6d940_136' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>V</mi><mo>⊃</mo><mi>C</mi><mspace width='thinmathspace' /><mo>,</mo><mphantom><mi>AA</mi></mphantom><mi>W</mi><mo>⊃</mo><mi>X</mi><mo>∖</mo><mi>U</mi><mphantom><mi>AA</mi></mphantom><mtext>with</mtext><mphantom><mi>AA</mi></mphantom><mi>V</mi><mo>∩</mo><mi>W</mi><mo>=</mo><mi>∅</mi><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> V \supset C \,, \phantom{AA} W \supset X \setminus U \phantom{AA} \text{with} \phantom{AA} V \cap W = \emptyset \,. </annotation></semantics></math></div> <p>But this means that</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_615d55a49ff138fdb807794cf13c867261a6d940_137' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>V</mi><mo>⊂</mo><mi>X</mi><mo>∖</mo><mi>W</mi></mrow><annotation encoding='application/x-tex'> V \subset X \setminus W </annotation></semantics></math></div> <p>and since the <a class='existingWikiWord' href='/nlab/show/diff/complement'>complement</a> <math class='maruku-mathml' display='inline' id='mathml_615d55a49ff138fdb807794cf13c867261a6d940_138' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi><mo>∖</mo><mi>W</mi></mrow><annotation encoding='application/x-tex'>X \setminus W</annotation></semantics></math> of the open set <math class='maruku-mathml' display='inline' id='mathml_615d55a49ff138fdb807794cf13c867261a6d940_139' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>W</mi></mrow><annotation encoding='application/x-tex'>W</annotation></semantics></math> is closed, it still contains the closure of <math class='maruku-mathml' display='inline' id='mathml_615d55a49ff138fdb807794cf13c867261a6d940_140' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>V</mi></mrow><annotation encoding='application/x-tex'>V</annotation></semantics></math>, so that we have</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_615d55a49ff138fdb807794cf13c867261a6d940_141' xmlns='http://www.w3.org/1998/Math/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>X</mi><mo>∖</mo><mi>W</mi><mo>⊂</mo><mi>U</mi></mrow><annotation encoding='application/x-tex'> C \subset V \subset Cl(V) \subset X \setminus W \subset U </annotation></semantics></math></div> <p>as required.</p> <p>In the other direction, assume that for every open neighbourhood <math class='maruku-mathml' display='inline' id='mathml_615d55a49ff138fdb807794cf13c867261a6d940_142' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_615d55a49ff138fdb807794cf13c867261a6d940_143' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi></mrow><annotation encoding='application/x-tex'>C</annotation></semantics></math> there exists a smaller open neighbourhood <math class='maruku-mathml' display='inline' id='mathml_615d55a49ff138fdb807794cf13c867261a6d940_144' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>V</mi></mrow><annotation encoding='application/x-tex'>V</annotation></semantics></math> with</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_615d55a49ff138fdb807794cf13c867261a6d940_145' xmlns='http://www.w3.org/1998/Math/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' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> C \subset V \subset Cl(V) \subset U \,. </annotation></semantics></math></div> <p>Consider disjoint closed subsets</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_615d55a49ff138fdb807794cf13c867261a6d940_146' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>C</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>C</mi> <mn>2</mn></msub><mo>⊂</mo><mi>X</mi><mspace width='thinmathspace' /><mo>,</mo><mphantom><mi>AAA</mi></mphantom><msub><mi>C</mi> <mn>1</mn></msub><mo>∩</mo><msub><mi>C</mi> <mn>2</mn></msub><mo>=</mo><mi>∅</mi><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> C_1, C_2 \subset X \,, \phantom{AAA} C_1 \cap C_2 = \emptyset \,. </annotation></semantics></math></div> <p>We need to produce disjoint open neighbourhoods for them.</p> <p>From their disjointness it follows that</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_615d55a49ff138fdb807794cf13c867261a6d940_147' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi><mo>∖</mo><msub><mi>C</mi> <mn>2</mn></msub><mo>⊃</mo><msub><mi>C</mi> <mn>1</mn></msub></mrow><annotation encoding='application/x-tex'> X \setminus C_2 \supset C_1 </annotation></semantics></math></div> <p>is an open neighbourhood. Hence by assumption there is an open neighbourhood <math class='maruku-mathml' display='inline' id='mathml_615d55a49ff138fdb807794cf13c867261a6d940_148' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>V</mi></mrow><annotation encoding='application/x-tex'>V</annotation></semantics></math> with</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_615d55a49ff138fdb807794cf13c867261a6d940_149' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>C</mi> <mn>1</mn></msub><mo>⊂</mo><mi>V</mi><mo>⊂</mo><mi>Cl</mi><mo stretchy='false'>(</mo><mi>V</mi><mo stretchy='false'>)</mo><mo>⊂</mo><mi>X</mi><mo>∖</mo><msub><mi>C</mi> <mn>2</mn></msub><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> C_1 \subset V \subset Cl(V) \subset X \setminus C_2 \,. </annotation></semantics></math></div> <p>Thus</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_615d55a49ff138fdb807794cf13c867261a6d940_150' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>V</mi><mo>⊃</mo><msub><mi>C</mi> <mn>1</mn></msub><mspace width='thinmathspace' /><mo>,</mo><mphantom><mi>AAAA</mi></mphantom><mi>X</mi><mo>∖</mo><mi>Cl</mi><mo stretchy='false'>(</mo><mi>V</mi><mo stretchy='false'>)</mo><mo>⊃</mo><msub><mi>C</mi> <mn>2</mn></msub></mrow><annotation encoding='application/x-tex'> V \supset C_1 \,, \phantom{AAAA} X \setminus Cl(V) \supset C_2 </annotation></semantics></math></div> <p>are two disjoint open neighbourhoods, as required.</p> </div> <ins class='diffins'><h3 id='SeparationAxiomInTermsOfLiftingProperties'>In terms of lifting properties</h3></ins><ins class='diffins'> </ins><ins class='diffins'><p>The <a class='existingWikiWord' href='/nlab/show/diff/separation+axioms'>separation conditions</a> <math class='maruku-mathml' display='inline' id='mathml_eaeadcce475f3e0bcf8296591cdfe3f26ea3f56c_151' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>T</mi> <mn>0</mn></msub></mrow><annotation encoding='application/x-tex'>T_0</annotation></semantics></math> to <math class='maruku-mathml' display='inline' id='mathml_eaeadcce475f3e0bcf8296591cdfe3f26ea3f56c_152' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>T</mi> <mn>4</mn></msub></mrow><annotation encoding='application/x-tex'>T_4</annotation></semantics></math> may equivalently be understood as <a class='existingWikiWord' href='/nlab/show/diff/lifting+property'>lifting properties</a> against certain maps of <a class='existingWikiWord' href='/nlab/show/diff/finite+topological+space'>finite topological spaces</a>, among others.</p></ins><ins class='diffins'> </ins><ins class='diffins'><p>This is discussed at <em><a class='existingWikiWord' href='/nlab/show/diff/separation+axioms+in+terms+of+lifting+properties'>separation axioms in terms of lifting properties</a></em>, to which we refer for further details. Here we just briefly indicate the corresponding lifting diagrams.</p></ins><ins class='diffins'> </ins><ins class='diffins'><p>In the following diagrams, the relevant <a class='existingWikiWord' href='/nlab/show/diff/finite+topological+space'>finite topological spaces</a> are indicated explicitly by illustration of their <a class='existingWikiWord' href='/nlab/show/diff/forgetful+functor'>underlying</a> point set and their <a class='existingWikiWord' href='/nlab/show/diff/open+subspace'>open subsets</a>:</p></ins><ins class='diffins'> </ins><ins class='diffins'><ul> <li> <p>points (elements) are denoted by <math class='maruku-mathml' display='inline' id='mathml_eaeadcce475f3e0bcf8296591cdfe3f26ea3f56c_153' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo>•</mo></mrow><annotation encoding='application/x-tex'>\bullet</annotation></semantics></math> with subscripts indicating where the points map to;</p> </li> <li> <p>boxes are put around <a class='existingWikiWord' href='/nlab/show/diff/open+subspace'>open subsets</a>,</p> </li> <li> <p>an arrow <math class='maruku-mathml' display='inline' id='mathml_eaeadcce475f3e0bcf8296591cdfe3f26ea3f56c_154' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mo>•</mo> <mi>u</mi></msub><mo>→</mo><msub><mo>•</mo> <mi>c</mi></msub></mrow><annotation encoding='application/x-tex'>\bullet_u \to \bullet_c</annotation></semantics></math> means that <math class='maruku-mathml' display='inline' id='mathml_eaeadcce475f3e0bcf8296591cdfe3f26ea3f56c_155' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mo>•</mo> <mi>c</mi></msub></mrow><annotation encoding='application/x-tex'>\bullet_c</annotation></semantics></math> is in the <a class='existingWikiWord' href='/nlab/show/diff/closed+subspace'>topological closure</a> of <math class='maruku-mathml' display='inline' id='mathml_eaeadcce475f3e0bcf8296591cdfe3f26ea3f56c_156' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mo>•</mo> <mi>u</mi></msub></mrow><annotation encoding='application/x-tex'>\bullet_u</annotation></semantics></math>.</p> </li> </ul></ins><ins class='diffins'> </ins><ins class='diffins'><p>In the lifting diagrams for <math class='maruku-mathml' display='inline' id='mathml_eaeadcce475f3e0bcf8296591cdfe3f26ea3f56c_157' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>T</mi> <mn>2</mn></msub><mo>−</mo><msub><mi>T</mi> <mn>4</mn></msub></mrow><annotation encoding='application/x-tex'>T_2-T_4</annotation></semantics></math> below, an arrow out of the given <a class='existingWikiWord' href='/nlab/show/diff/topological+space'>topological space</a> <math class='maruku-mathml' display='inline' id='mathml_eaeadcce475f3e0bcf8296591cdfe3f26ea3f56c_158' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> is a <a class='existingWikiWord' href='/nlab/show/diff/map'>map</a> that determines (classifies) a decomposition of <math class='maruku-mathml' display='inline' id='mathml_eaeadcce475f3e0bcf8296591cdfe3f26ea3f56c_159' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> into a <a class='existingWikiWord' href='/nlab/show/diff/union'>union</a> of <a class='existingWikiWord' href='/nlab/show/diff/subset'>subsets</a> with properties indicated by the picture of the finite space.</p></ins><ins class='diffins'> </ins><ins class='diffins'><p>Notice that the diagrams for <math class='maruku-mathml' display='inline' id='mathml_eaeadcce475f3e0bcf8296591cdfe3f26ea3f56c_160' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>T</mi> <mn>2</mn></msub></mrow><annotation encoding='application/x-tex'>T_2</annotation></semantics></math>-<math class='maruku-mathml' display='inline' id='mathml_eaeadcce475f3e0bcf8296591cdfe3f26ea3f56c_161' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>T</mi> <mn>4</mn></msub></mrow><annotation encoding='application/x-tex'>T_4</annotation></semantics></math> below do not in themselves imply <a class='existingWikiWord' href='/nlab/show/diff/accessible+topological+space'>$T_1$</a>.</p></ins><ins class='diffins'> </ins><ins class='diffins'><p>\begin{proposition} <strong>(Lifting property encoding <math class='maruku-mathml' display='inline' id='mathml_eaeadcce475f3e0bcf8296591cdfe3f26ea3f56c_162' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>T</mi> <mn>0</mn></msub></mrow><annotation encoding='application/x-tex'>T_0</annotation></semantics></math>)</strong> \linebreak The following <a class='existingWikiWord' href='/nlab/show/diff/lifting+property'>lifting property</a> in <a class='existingWikiWord' href='/nlab/show/diff/Top'>Top</a> equivalently encodes the <a class='existingWikiWord' href='/nlab/show/diff/separation+axioms'>separation axiom</a> <a class='existingWikiWord' href='/nlab/show/diff/Kolmogorov+topological+space'>$T_0$</a>:</p></ins><ins class='diffins'> </ins><ins class='diffins'><p>\begin{tikzcd} [column sep={between origins, 40pt}, row sep={between origins, 40pt}] {\boxed{\bullet_0\leftrightarrow \bullet_1}} \ar[rr, { \forall }] \ar[dd] && X \ar[dd,{ (T_0) }] \ \ \bullet_{0=1} \ar[rr] \ar[uurr, dashed, { \exists }] && \bullet \end{tikzcd}</p></ins><ins class='diffins'> </ins><ins class='diffins'><p>\end{proposition}</p></ins><ins class='diffins'> </ins><ins class='diffins'><p>\begin{proposition} <strong>(Lifting property encoding <math class='maruku-mathml' display='inline' id='mathml_eaeadcce475f3e0bcf8296591cdfe3f26ea3f56c_163' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>T</mi> <mn>1</mn></msub></mrow><annotation encoding='application/x-tex'>T_1</annotation></semantics></math>)</strong> \linebreak The following <a class='existingWikiWord' href='/nlab/show/diff/lifting+property'>lifting property</a> in <a class='existingWikiWord' href='/nlab/show/diff/Top'>Top</a> equivalently encodes the <a class='existingWikiWord' href='/nlab/show/diff/separation+axioms'>separation axiom</a> <a class='existingWikiWord' href='/nlab/show/diff/accessible+topological+space'>$T_1$</a>:</p></ins><ins class='diffins'> </ins><ins class='diffins'><p>\begin{tikzcd} [column sep={between origins, 40pt}, row sep={between origins, 40pt}] {\boxed{ \overset{\boxed{\bullet_{0}}}{}\searrow\underset{\bullet_1}{} }} % {\boxed{ \boxed{ {\bullet_0} }\rightarrow{\bullet_1} }} \ar[rr, { \forall }] \ar[dd] && X \ar[dd,{ (T_1) }] \ \ \bullet_{0=1} \ar[rr] \ar[uurr, dashed, { \exists }] && \bullet \end{tikzcd} \end{proposition}</p></ins><ins class='diffins'> </ins><ins class='diffins'><p>\begin{proposition} <strong>(Lifting property encoding <math class='maruku-mathml' display='inline' id='mathml_eaeadcce475f3e0bcf8296591cdfe3f26ea3f56c_164' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>T</mi> <mn>2</mn></msub></mrow><annotation encoding='application/x-tex'>T_2</annotation></semantics></math>)</strong> \linebreak The following <a class='existingWikiWord' href='/nlab/show/diff/lifting+property'>lifting property</a> in <a class='existingWikiWord' href='/nlab/show/diff/Top'>Top</a> equivalently encodes the <a class='existingWikiWord' href='/nlab/show/diff/separation+axioms'>separation axiom</a> <a class='existingWikiWord' href='/nlab/show/diff/Hausdorff+space'>$T_2$</a>:</p></ins><ins class='diffins'> </ins><ins class='diffins'><p>\begin{tikzcd} [column sep={between origins, 80pt}, row sep={between origins, 40pt}] \boxed{{\boxed{\bullet_x}, \boxed{\bullet_y}}} \ar[rr, { }] \ar[dd, hook, { \forall }{left}, ,{ (T_2) }{right}] && \boxed{ \overset{ \boxed{ \boxed{\bullet_x} \;\; \, \;\; \boxed{\bullet_y}} }{ \underset{ \bullet_X } { \searrow \;\, \swarrow } } } \ar[dd] \ \ X \ar[rr] \ar[uurr, dashed, { \exists }] && {\boxed{\bullet_{x=X=y}}} \end{tikzcd} \end{proposition}</p></ins><ins class='diffins'> </ins><ins class='diffins'><p>\begin{proposition} <strong>(Lifting property encoding <math class='maruku-mathml' display='inline' id='mathml_eaeadcce475f3e0bcf8296591cdfe3f26ea3f56c_165' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>T</mi> <mn>3</mn></msub></mrow><annotation encoding='application/x-tex'>T_3</annotation></semantics></math>)</strong> \linebreak The following <a class='existingWikiWord' href='/nlab/show/diff/lifting+property'>lifting property</a> in <a class='existingWikiWord' href='/nlab/show/diff/Top'>Top</a> equivalently encodes the <a class='existingWikiWord' href='/nlab/show/diff/separation+axioms'>separation axiom</a> <a class='existingWikiWord' href='/nlab/show/diff/regular+space'>$T_3$</a>:</p></ins><ins class='diffins'> </ins><ins class='diffins'><p>\begin{tikzcd} [column sep={between origins, 60pt}, row sep={between origins, 40pt}] {\boxed{\bullet_x} } \ar[dd,{ \forall }{left}, { (T_3) }{right}] \ar[rr] && { \boxed{ \boxed{ \overset{\boxed{\bullet_x}}{} \searrow \underset{\bullet_X}{} \swarrow \overset{\boxed{\bullet_U}}{} }!!!!!!! {\,\,\,\,\,\,\searrow\underset{\bullet_F}{} } } } \ar[dd] \ \ X \ar[rr] \ar[uurr, dashed, { \exists }] && {\boxed{ \overset{\boxed{\bullet_{x=X=U}}}{}\searrow\underset{\bullet_F}{} }} \end{tikzcd} \end{proposition}</p></ins><ins class='diffins'> </ins><ins class='diffins'><p>\begin{proposition} <strong>(Lifting property encoding <math class='maruku-mathml' display='inline' id='mathml_eaeadcce475f3e0bcf8296591cdfe3f26ea3f56c_166' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>T</mi> <mn>4</mn></msub></mrow><annotation encoding='application/x-tex'>T_4</annotation></semantics></math>)</strong> \linebreak The following <a class='existingWikiWord' href='/nlab/show/diff/lifting+property'>lifting property</a> in <a class='existingWikiWord' href='/nlab/show/diff/Top'>Top</a> equivalently encodes the <a class='existingWikiWord' href='/nlab/show/diff/separation+axioms'>separation axiom</a> <a class='existingWikiWord' href='/nlab/show/diff/normal+space'>$T_4$</a>:</p></ins><ins class='diffins'> </ins><ins class='diffins'><p>\begin{tikzcd} [column sep={between origins, 60pt}, row sep={between origins, 40pt}] \varnothing \ar[dd, { (T_4) }{right}] \ar[rr] && \boxed{ \boxed{\underset{\bullet_x}{}\swarrow \,\,\,\,\,\,\,} !!!!!!!! \,\,\,\,\,\,\,\, \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \,\,\,\, \,\,\,\,\,} !!!!!!! !!!!!!!!!!!!!!!!!!!!!!!!! !!!!!!!! !!!!!!!! !!!! \boxed{ \boxed{ \overset{\boxed{\bullet_u}}{} \searrow \underset{\bullet_X}{} \swarrow \boxed{ \overset{\boxed{\bullet_v}}{} !!!!!!! {\,\,\,\,\,\,\searrow\underset{\bullet_y}{} } }!!!!!!!%%%%%%%% !!!!!} %\,\,\,\,\,\,\,\,\, \,\,\,\,\,\,\,\,\,} \ar[dd] \ \ X \ar[rr] \ar[uurr, dashed, { \exists }] && { \boxed{ \underset{\bullet_x}{}{\swarrow} <br />\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, %\,\, %\,\,\,\,\,\,\, } !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! %!!%!!!!!!! \boxed{\overset{\boxed{\bullet_{U=X=V}}}{} {\searrow} \underset{\bullet_y}{} } } \end{tikzcd} \end{proposition}</p></ins><ins class='diffins'> </ins><h3 id='TietzeEtensionAndLiftingProperty'>Tietze extension and lifting property</h3> <p>The <a class='existingWikiWord' href='/nlab/show/diff/Tietze+extension+theorem'>Tietze extension theorem</a> applies to normal spaces.</p> <p>In fact the Tietze extension theorem can serve as a basis of a <a class='existingWikiWord' href='/nlab/show/diff/category+theory'>category theoretic</a> characterization of normal spaces: a (Hausdorff) space <math class='maruku-mathml' display='inline' id='mathml_615d55a49ff138fdb807794cf13c867261a6d940_151' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> is normal if and only if every function <math class='maruku-mathml' display='inline' id='mathml_615d55a49ff138fdb807794cf13c867261a6d940_152' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>f</mi><mo lspace='verythinmathspace'>:</mo><mi>A</mi><mo>→</mo><mi>ℝ</mi></mrow><annotation encoding='application/x-tex'>f \colon A \to \mathbb{R}</annotation></semantics></math> from a <a class='existingWikiWord' href='/nlab/show/diff/closed+subspace'>closed subspace</a> <math class='maruku-mathml' display='inline' id='mathml_615d55a49ff138fdb807794cf13c867261a6d940_153' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>A</mi><mo>⊂</mo><mi>X</mi></mrow><annotation encoding='application/x-tex'>A \subset X</annotation></semantics></math> admits an <a class='existingWikiWord' href='/nlab/show/diff/extension'>extension</a> <math class='maruku-mathml' display='inline' id='mathml_615d55a49ff138fdb807794cf13c867261a6d940_154' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mover><mi>f</mi><mo stretchy='false'>˜</mo></mover><mo>:</mo><mi>X</mi><mo>→</mo><mi>ℝ</mi></mrow><annotation encoding='application/x-tex'>\tilde{f}: X \to \mathbb{R}</annotation></semantics></math>, or what is the same, every <a class='existingWikiWord' href='/nlab/show/diff/regular+monomorphism'>regular monomorphism</a> into <math class='maruku-mathml' display='inline' id='mathml_615d55a49ff138fdb807794cf13c867261a6d940_155' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> in <math class='maruku-mathml' display='inline' id='mathml_615d55a49ff138fdb807794cf13c867261a6d940_156' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Haus</mi></mrow><annotation encoding='application/x-tex'>Haus</annotation></semantics></math> has the <a class='existingWikiWord' href='/nlab/show/diff/lifting+property'>left lifting property</a> with respect to the map <math class='maruku-mathml' display='inline' id='mathml_615d55a49ff138fdb807794cf13c867261a6d940_157' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>ℝ</mi><mo>→</mo><mn>1</mn></mrow><annotation encoding='application/x-tex'>\mathbb{R} \to 1</annotation></semantics></math>. (See <em><a class='existingWikiWord' href='/nlab/show/diff/separation+axioms+in+terms+of+lifting+properties'>separation axioms in terms of lifting properties</a></em> (<a href='#Gavrilovich14'>Gavrilovich 14</a>) for further categorical characterizations of various topological properties in terms of lifting problems.)</p> <h3 id='CategoryOfNormalSpaces'>The category of normal spaces</h3> <p>Although normal (Hausdorff) spaces are “<a class='existingWikiWord' href='/nlab/show/diff/nice+topological+space'>nice topological spaces</a>” (being for example <a class='existingWikiWord' href='/nlab/show/diff/Tychonoff+space'>Tychonoff spaces</a>, by <a class='existingWikiWord' href='/nlab/show/diff/Urysohn%27s+lemma'>Urysohn's lemma</a>), the <a class='existingWikiWord' href='/nlab/show/diff/category'>category</a> of normal topological spaces with <a class='existingWikiWord' href='/nlab/show/diff/continuous+map'>continuous maps</a> between them seems not to be very well-behaved. (Cf. the rule of thumb expressed in <a class='existingWikiWord' href='/nlab/show/diff/dichotomy+between+nice+objects+and+nice+categories'>dichotomy between nice objects and nice categories</a>.) It admits <a class='existingWikiWord' href='/nlab/show/diff/equalizer'>equalizers</a> of pairs of maps <math class='maruku-mathml' display='inline' id='mathml_615d55a49ff138fdb807794cf13c867261a6d940_158' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>f</mi><mo>,</mo><mi>g</mi><mo>:</mo><mi>X</mi><mo>⇉</mo><mi>Y</mi></mrow><annotation encoding='application/x-tex'>f, g: X \rightrightarrows Y</annotation></semantics></math> (computed as in <math class='maruku-mathml' display='inline' id='mathml_615d55a49ff138fdb807794cf13c867261a6d940_159' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Top</mi></mrow><annotation encoding='application/x-tex'>Top</annotation></semantics></math> or <math class='maruku-mathml' display='inline' id='mathml_615d55a49ff138fdb807794cf13c867261a6d940_160' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Haus</mi></mrow><annotation encoding='application/x-tex'>Haus</annotation></semantics></math>; one uses the easy fact that closed subspaces of normal spaces are normal). However it curiously does not have <a class='existingWikiWord' href='/nlab/show/diff/cartesian+product'>products</a> – or at least it is not closed under products in <math class='maruku-mathml' display='inline' id='mathml_615d55a49ff138fdb807794cf13c867261a6d940_161' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Top</mi></mrow><annotation encoding='application/x-tex'>Top</annotation></semantics></math>, as shown by Counter-Example <a class='maruku-ref' href='#Morse'>3</a>. It follows that this category is not a <a class='existingWikiWord' href='/nlab/show/diff/reflective+subcategory'>reflective subcategory</a> of <math class='maruku-mathml' display='inline' id='mathml_615d55a49ff138fdb807794cf13c867261a6d940_162' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Top</mi></mrow><annotation encoding='application/x-tex'>Top</annotation></semantics></math>, as <math class='maruku-mathml' display='inline' id='mathml_615d55a49ff138fdb807794cf13c867261a6d940_163' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Haus</mi></mrow><annotation encoding='application/x-tex'>Haus</annotation></semantics></math> is.</p> <div class='num_remark' id='retracts'> <h6 id='remark'>Remark</h6> <p>A small saving grace is that the category of normal spaces is <a class='existingWikiWord' href='/nlab/show/diff/Cauchy+complete+category'>Cauchy complete</a>, in fact is closed under retracts in <math class='maruku-mathml' display='inline' id='mathml_615d55a49ff138fdb807794cf13c867261a6d940_164' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Top</mi></mrow><annotation encoding='application/x-tex'>Top</annotation></semantics></math>. This is so whether or not the Hausdorff condition is included. (If <math class='maruku-mathml' display='inline' id='mathml_615d55a49ff138fdb807794cf13c867261a6d940_165' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>r</mi><mo>:</mo><mi>Y</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding='application/x-tex'>r: Y \to X</annotation></semantics></math> retracts <math class='maruku-mathml' display='inline' id='mathml_615d55a49ff138fdb807794cf13c867261a6d940_166' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>i</mi><mo>:</mo><mi>X</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding='application/x-tex'>i: X \to Y</annotation></semantics></math>, then <math class='maruku-mathml' display='inline' id='mathml_615d55a49ff138fdb807794cf13c867261a6d940_167' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>r</mi></mrow><annotation encoding='application/x-tex'>r</annotation></semantics></math> is a quotient map and <math class='maruku-mathml' display='inline' id='mathml_615d55a49ff138fdb807794cf13c867261a6d940_168' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>i</mi></mrow><annotation encoding='application/x-tex'>i</annotation></semantics></math> is a subspace inclusion. If <math class='maruku-mathml' display='inline' id='mathml_615d55a49ff138fdb807794cf13c867261a6d940_169' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>A</mi><mo>,</mo><mi>B</mi></mrow><annotation encoding='application/x-tex'>A, B</annotation></semantics></math> are closed and disjoint in <math class='maruku-mathml' display='inline' id='mathml_615d55a49ff138fdb807794cf13c867261a6d940_170' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math>, then <math class='maruku-mathml' display='inline' id='mathml_615d55a49ff138fdb807794cf13c867261a6d940_171' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>r</mi> <mrow><mo lspace='verythinmathspace' rspace='0em'>−</mo><mn>1</mn></mrow></msup><mo stretchy='false'>(</mo><mi>A</mi><mo stretchy='false'>)</mo><mo>,</mo><msup><mi>r</mi> <mrow><mo lspace='verythinmathspace' rspace='0em'>−</mo><mn>1</mn></mrow></msup><mo stretchy='false'>(</mo><mi>B</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>r^{-1}(A), r^{-1}(B)</annotation></semantics></math> are closed and disjoint in <math class='maruku-mathml' display='inline' id='mathml_615d55a49ff138fdb807794cf13c867261a6d940_172' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Y</mi></mrow><annotation encoding='application/x-tex'>Y</annotation></semantics></math>. Separate them by disjoint open sets <math class='maruku-mathml' display='inline' id='mathml_615d55a49ff138fdb807794cf13c867261a6d940_173' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>U</mi><mo>⊇</mo><msup><mi>r</mi> <mrow><mo lspace='verythinmathspace' rspace='0em'>−</mo><mn>1</mn></mrow></msup><mo stretchy='false'>(</mo><mi>A</mi><mo stretchy='false'>)</mo><mo>,</mo><mi>V</mi><mo>⊇</mo><msup><mi>r</mi> <mrow><mo lspace='verythinmathspace' rspace='0em'>−</mo><mn>1</mn></mrow></msup><mo stretchy='false'>(</mo><mi>B</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>U \supseteq r^{-1}(A), V \supseteq r^{-1}(B)</annotation></semantics></math> in <math class='maruku-mathml' display='inline' id='mathml_615d55a49ff138fdb807794cf13c867261a6d940_174' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Y</mi></mrow><annotation encoding='application/x-tex'>Y</annotation></semantics></math>; then <math class='maruku-mathml' display='inline' id='mathml_615d55a49ff138fdb807794cf13c867261a6d940_175' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>i</mi> <mrow><mo lspace='verythinmathspace' rspace='0em'>−</mo><mn>1</mn></mrow></msup><mo stretchy='false'>(</mo><mi>U</mi><mo stretchy='false'>)</mo><mo>,</mo><msup><mi>i</mi> <mrow><mo lspace='verythinmathspace' rspace='0em'>−</mo><mn>1</mn></mrow></msup><mo stretchy='false'>(</mo><mi>V</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>i^{-1}(U), i^{-1}(V)</annotation></semantics></math> are disjoint open sets separating <math class='maruku-mathml' display='inline' id='mathml_615d55a49ff138fdb807794cf13c867261a6d940_176' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>i</mi> <mrow><mo lspace='verythinmathspace' rspace='0em'>−</mo><mn>1</mn></mrow></msup><msup><mi>r</mi> <mrow><mo lspace='verythinmathspace' rspace='0em'>−</mo><mn>1</mn></mrow></msup><mo stretchy='false'>(</mo><mi>A</mi><mo stretchy='false'>)</mo><mo>=</mo><mi>A</mi><mo>,</mo><msup><mi>i</mi> <mrow><mo lspace='verythinmathspace' rspace='0em'>−</mo><mn>1</mn></mrow></msup><msup><mi>r</mi> <mrow><mo lspace='verythinmathspace' rspace='0em'>−</mo><mn>1</mn></mrow></msup><mo stretchy='false'>(</mo><mi>B</mi><mo stretchy='false'>)</mo><mo>=</mo><mi>B</mi></mrow><annotation encoding='application/x-tex'>i^{-1} r^{-1}(A) = A, i^{-1} r^{-1}(B) = B</annotation></semantics></math> in <math class='maruku-mathml' display='inline' id='mathml_615d55a49ff138fdb807794cf13c867261a6d940_177' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math>.)</p> </div> <p>More at the page <em><a class='existingWikiWord' href='/nlab/show/diff/colimits+of+normal+spaces'>colimits of normal spaces</a></em>.</p> <h2 id='references'>References</h2> <p>The class of normal spaces was introduced by Tietze (1923) and Aleksandrov–Uryson (1924).</p> <ul> <li><span><ins class='diffins'> Ryszard</ins><ins class='diffins'> Engelking,</ins></span><del class='diffmod'><p>Ryszard Engelking, <em>General topology</em>, (Monographie Matematyczne, tom 60) Warszawa 1977; expanded Russian edition Mir 1986.</p></del><ins class='diffmod'><em>General topology</em></ins><span><ins class='diffins'> ,</ins><ins class='diffins'> (Monographie</ins><ins class='diffins'> Matematyczne,</ins><ins class='diffins'> tom</ins><ins class='diffins'> 60)</ins><ins class='diffins'> Warszawa</ins><ins class='diffins'> 1977;</ins><ins class='diffins'> expanded</ins><ins class='diffins'> Russian</ins><ins class='diffins'> edition</ins><ins class='diffins'> Mir</ins><ins class='diffins'> 1986.</ins></span></li><del class='diffdel'> </del><del class='diffdel'><li id='Gavrilovich14'> <p><a class='existingWikiWord' href='/nlab/show/diff/Misha+Gavrilovich'>Misha Gavrilovich</a>, <em>Point set topology as diagram chasing computations</em>, (<a href='https://arxiv.org/abs/1408.6710'>arXiv:1408.6710</a>).</p> </li></del> </ul> <ins class='diffins'><p>Discussion of <a class='existingWikiWord' href='/nlab/show/diff/separation+axioms+in+terms+of+lifting+properties'>separation axioms in terms of lifting properties</a> is due to:</p></ins><ins class='diffins'> </ins><ins class='diffins'><ul> <li id='Gavrilovich14'><a class='existingWikiWord' href='/nlab/show/diff/Misha+Gavrilovich'>Misha Gavrilovich</a>, <em>Point set topology as diagram chasing computations</em>, (<a href='https://arxiv.org/abs/1408.6710'>arXiv:1408.6710</a>).</li> </ul></ins><ins class='diffins'> </ins><p> </p> <p> </p><ins class='diffins'> </ins><ins class='diffins'><p> </p></ins> <p> </p> <p> </p> <p> </p> <p> </p> <p> </p> </div> <div class="revisedby"> <p> Last revised on October 19, 2021 at 10:53:24. 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