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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 #62 to #63: <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='contents'>Contents</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='#SeparationAxiomInTermsOfLiftingProperties'>In terms of lifting properties</a></li><li><a href='#HausdorffReflections'>Hausdorff reflection</a></li><li><a href='#monadicity'>Monadicity</a></li><li><a href='#sobriety'>Sobriety</a></li><li><a href='#relation_to_compact_spaces'>Relation to compact spaces</a></li><li><a href='#dense_subspaces_and_epimorphisms'>Dense subspaces and Epimorphisms</a></li></ul></li><li><a href='#related_notions'>Related notions</a></li><li><a href='#BeyondTopologicalSpaces'>Beyond topological spaces</a><ul><li><a href='#HausdorffLocale'>Hausdorff locales</a></li><li><a href='#separated_toposes_and_schemes'>Separated toposes and schemes</a></li></ul></li><li><a href='#in_constructive_mathematics'>In constructive mathematics</a></li><li><a href='#history'>History</a></li><li><a href='#related_concepts'>Related concepts</a></li><li><a href='#references'>References</a></li></ul></div> <h2 id='idea'>Idea</h2> <p>A <a class='existingWikiWord' href='/nlab/show/diff/topological+space'>topological space</a> (or more generally, a <a class='existingWikiWord' href='/nlab/show/diff/convergence+space'>convergence space</a>) is <em>Hausdorff</em> if <a class='existingWikiWord' href='/nlab/show/diff/convergence'>convergence</a> is unique. The concept can also be defined for <a class='existingWikiWord' href='/nlab/show/diff/locale'>locales</a> (see Definition <a class='maruku-ref' href='#proper'>3</a> below) and <a class='existingWikiWord' href='/nlab/show/diff/vertical+categorification'>categorified</a> (see <a href='#BeyondTopologicalSpaces'>Beyond topological spaces</a> below). A Hausdorff space is often called <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_1' 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>, since this condition came second in the original list of four <a class='existingWikiWord' href='/nlab/show/diff/separation+axioms'>separation axioms</a> (there are more now) satisfied by <a class='existingWikiWord' href='/nlab/show/diff/metric+space'>metric spaces</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_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_2' 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_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_3' 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_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_4' 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_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_5' 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_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_6' 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_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_7' 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_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_8' 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> <p>Hausdorff spaces are a kind of <a class='existingWikiWord' href='/nlab/show/diff/nice+topological+space'>nice topological space</a>; they do not form a particularly <a class='existingWikiWord' href='/nlab/show/diff/nice+category+of+spaces'>nice category of spaces</a> themselves, but many such nice categories consist of only Hausdorff spaces. In fact, <a class='existingWikiWord' href='/nlab/show/diff/Felix+Hausdorff'>Felix Hausdorff</a>'s original definition of ‘topological space’ actually required the space to be Hausdorff, hence the name. Certainly <a class='existingWikiWord' href='/nlab/show/diff/homotopy+theory'>homotopy theory</a> (up to <a class='existingWikiWord' href='/nlab/show/diff/weak+homotopy+equivalence'>weak homotopy equivalence</a>) needs only Hausdorff spaces. It is also common in analysis to assume that all spaces encountered are Hausdorff; if necessary, this can be arranged since every space has a Hausdorff quotient (in fact, the Hausdorff spaces form a <a class='existingWikiWord' href='/nlab/show/diff/reflective+subcategory'>reflective subcategory</a> of <a class='existingWikiWord' href='/nlab/show/diff/Top'>Top</a>), although usually an easier method is available than this sledgehammer.</p> <h2 id='Definitions'>Definitions</h2> <p>There are many equivalent ways of characterizing a space <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_9' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>S</mi></mrow><annotation encoding='application/x-tex'>S</annotation></semantics></math> as <strong>Hausdorff</strong>. The traditional definition is this:</p> <div class='num_defn' id='classical'> <h6 id='definition'>Definition</h6> <p>Given points <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_10' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>x</mi></mrow><annotation encoding='application/x-tex'>x</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_11' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>y</mi></mrow><annotation encoding='application/x-tex'>y</annotation></semantics></math> of <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_12' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>S</mi></mrow><annotation encoding='application/x-tex'>S</annotation></semantics></math>, if <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_13' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>x</mi><mo>≠</mo><mi>y</mi></mrow><annotation encoding='application/x-tex'>x \neq y</annotation></semantics></math>, then there exist <a class='existingWikiWord' href='/nlab/show/diff/neighborhood'>open neighbourhoods</a> <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_14' 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_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_15' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>x</mi></mrow><annotation encoding='application/x-tex'>x</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_16' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>V</mi></mrow><annotation encoding='application/x-tex'>V</annotation></semantics></math> of <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_17' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>y</mi></mrow><annotation encoding='application/x-tex'>y</annotation></semantics></math> in <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_18' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>S</mi></mrow><annotation encoding='application/x-tex'>S</annotation></semantics></math> that are <a class='existingWikiWord' href='/nlab/show/diff/disjoint+subsets'>disjoint</a>: such that their <a class='existingWikiWord' href='/nlab/show/diff/intersection'>intersection</a> <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_19' 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 the <a class='existingWikiWord' href='/nlab/show/diff/empty+set'>empty set</a> (or explicitly, such that <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_20' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>x</mi><mo>′</mo><mo>≠</mo><mi>y</mi><mo>′</mo></mrow><annotation encoding='application/x-tex'>x' \ne y'</annotation></semantics></math> whenever <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_21' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>x</mi><mo>′</mo><mo>∈</mo><mi>U</mi></mrow><annotation encoding='application/x-tex'>x' \in U</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_22' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>y</mi><mo>′</mo><mo>∈</mo><mi>V</mi></mrow><annotation encoding='application/x-tex'>y' \in V</annotation></semantics></math>).</p> </div> <p>That is, any two distinct points can be <em>separated</em> by open neighbourhoods, and it is simply a mundane way of saying that <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_23' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo>≠</mo></mrow><annotation encoding='application/x-tex'>\ne</annotation></semantics></math> is <a class='existingWikiWord' href='/nlab/show/diff/open+subspace'>open</a> in the <a class='existingWikiWord' href='/nlab/show/diff/product+topological+space'>product topology</a> on <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_24' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>S</mi><mo>×</mo><mi>S</mi></mrow><annotation encoding='application/x-tex'>S \times S</annotation></semantics></math>.</p> <p>Here is a classically equivalent definition that is more suitable for <a class='existingWikiWord' href='/nlab/show/diff/constructive+mathematics'>constructive mathematics</a>:</p> <div class='num_defn' id='constructive'> <h6 id='definition_2'>Definition</h6> <p>Given points <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_25' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>x</mi></mrow><annotation encoding='application/x-tex'>x</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_26' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>y</mi></mrow><annotation encoding='application/x-tex'>y</annotation></semantics></math> of <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_27' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>S</mi></mrow><annotation encoding='application/x-tex'>S</annotation></semantics></math>, if every neighbourhood <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_28' 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_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_29' 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_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_30' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>S</mi></mrow><annotation encoding='application/x-tex'>S</annotation></semantics></math> meets every neighbourhood <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_31' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>V</mi></mrow><annotation encoding='application/x-tex'>V</annotation></semantics></math> of <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_32' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>y</mi></mrow><annotation encoding='application/x-tex'>y</annotation></semantics></math> in <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_33' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>S</mi></mrow><annotation encoding='application/x-tex'>S</annotation></semantics></math> (which means that <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_34' 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/inhabited+set'>inhabited</a>), then <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_35' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>x</mi><mo>=</mo><mi>y</mi></mrow><annotation encoding='application/x-tex'>x = y</annotation></semantics></math>.</p> </div> <p>This is the mundane way of saying that <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_36' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo>=</mo></mrow><annotation encoding='application/x-tex'>=</annotation></semantics></math> is <a class='existingWikiWord' href='/nlab/show/diff/closed+subspace'>closed</a> in <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_37' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>S</mi><mo>×</mo><mi>S</mi></mrow><annotation encoding='application/x-tex'>S \times S</annotation></semantics></math>.</p> <p>Another way of saying this, which makes sense also for <a class='existingWikiWord' href='/nlab/show/diff/locale'>locales</a>, is the following:</p> <div class='num_defn' id='proper'> <h6 id='definition_3'>Definition</h6> <p>The <a class='existingWikiWord' href='/nlab/show/diff/diagonal+morphism'>diagonal</a> embedding <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_38' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>S</mi><mo>→</mo><mi>S</mi><mo>×</mo><mi>S</mi></mrow><annotation encoding='application/x-tex'>S \to S \times S</annotation></semantics></math> is a <a class='existingWikiWord' href='/nlab/show/diff/proper+map'>proper map</a> (or equivalently a <a class='existingWikiWord' href='/nlab/show/diff/closed+map'>closed map</a>, since any closed subspace inclusion is proper).</p> </div> <p>This way of stating the definition generalizes to <a class='existingWikiWord' href='/nlab/show/diff/sheaf+and+topos+theory'>topos theory</a> and thus to many other contexts; but it is not always a faithful generalization of the classical notion for topological spaces. See <em><a href='#BeyondTopologicalSpaces'>Beyond topological spaces</a></em> below for more.</p> <p>Here is an equivalent definition (constructively equivalent to Definition <a class='maruku-ref' href='#constructive'>2</a>) that makes sense for arbitrary <a class='existingWikiWord' href='/nlab/show/diff/convergence+space'>convergence spaces</a> or <a class='existingWikiWord' href='/nlab/show/diff/preconvergence+space'><span><del class='diffmod'> preconvergence</del><ins class='diffmod'> generalised</ins><ins class='diffins'> filter</ins> spaces</span></a>:</p> <div class='num_defn' id='convergence'> <h6 id='definition_4'>Definition</h6> <p>Given a <a class='existingWikiWord' href='/nlab/show/diff/net'>net</a> (or equivalently, a proper <a class='existingWikiWord' href='/nlab/show/diff/filter'>filter</a>) <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_39' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>F</mi></mrow><annotation encoding='application/x-tex'>F</annotation></semantics></math> in <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_40' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>S</mi></mrow><annotation encoding='application/x-tex'>S</annotation></semantics></math>, if it converges to both <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_41' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>x</mi></mrow><annotation encoding='application/x-tex'>x</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_42' 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_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_43' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>x</mi><mo>=</mo><mi>y</mi></mrow><annotation encoding='application/x-tex'>x = y</annotation></semantics></math>.</p> </div> <p>Or equivalently,</p> <p>\begin{definition} Given a <a class='existingWikiWord' href='/nlab/show/diff/net'>net</a> (or equivalently, a proper <a class='existingWikiWord' href='/nlab/show/diff/filter'>filter</a>) <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_44' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>F</mi></mrow><annotation encoding='application/x-tex'>F</annotation></semantics></math> in <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_45' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>S</mi></mrow><annotation encoding='application/x-tex'>S</annotation></semantics></math>, the set of all limits of <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_46' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>F</mi></mrow><annotation encoding='application/x-tex'>F</annotation></semantics></math> is a <a class='existingWikiWord' href='/nlab/show/diff/subsingleton'>subsingleton</a>. \end{definition}</p> <p>That is, convergence in a Hausdorff space is unique.</p> <h2 id='examples'>Examples</h2> <div class='num_prop'> <h6 id='proposition'>Proposition</h6> <p><a class='existingWikiWord' href='/nlab/show/diff/CW-complexes+are+paracompact+Hausdorff+spaces'>a CW-complex is a Hausdorff space</a>.</p> </div> <h2 id='Properties'>Properties</h2> <h3 id='SeparationAxiomInTermsOfLiftingProperties'>In terms of lifting properties</h3> <p>The <a class='existingWikiWord' href='/nlab/show/diff/separation+axioms'>separation conditions</a> <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_47' 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_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_48' 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> <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> <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> <ul> <li> <p>points (elements) are denoted by <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_49' 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_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_50' 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_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_51' 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_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_52' 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> <p>In the lifting diagrams for <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_53' 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_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_54' 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_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_55' 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> <p>Notice that the diagrams for <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_56' 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_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_57' 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> <p>\begin{proposition} <strong>(Lifting property encoding <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_58' 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> <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> <p>\end{proposition}</p> <p>\begin{proposition} <strong>(Lifting property encoding <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_59' 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> <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> <p>\begin{proposition} <strong>(Lifting property encoding <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_60' 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> <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> <p>\begin{proposition} <strong>(Lifting property encoding <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_61' 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> <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> <p>\begin{proposition} <strong>(Lifting property encoding <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_62' 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> <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> <h3 id='HausdorffReflections'>Hausdorff reflection</h3> <div class='num_prop' id='HausdorffReflection'> <h6 id='proposition_2'>Proposition</h6> <p><strong>(Hausdorff reflection)</strong></p> <p>For every <a class='existingWikiWord' href='/nlab/show/diff/topological+space'>topological space</a> <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_63' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> there exists a <a class='existingWikiWord' href='/nlab/show/diff/Hausdorff+space'>Hausdorff topological space</a> <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_64' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>H</mi><mi>X</mi></mrow><annotation encoding='application/x-tex'>H X</annotation></semantics></math> and a <a class='existingWikiWord' href='/nlab/show/diff/continuous+map'>continuous function</a></p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_65' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>h</mi> <mi>X</mi></msub><mspace width='thickmathspace' /><mo lspace='verythinmathspace'>:</mo><mspace width='thickmathspace' /><mi>X</mi><mo>⟶</mo><mi>H</mi><mi>X</mi></mrow><annotation encoding='application/x-tex'> h_X \;\colon\; X \longrightarrow H X </annotation></semantics></math></div> <p>which is the “closest approximation from the left” to <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_66' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> by a Hausdorff topological space, in that for <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_67' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Y</mi></mrow><annotation encoding='application/x-tex'>Y</annotation></semantics></math> any <a class='existingWikiWord' href='/nlab/show/diff/Hausdorff+space'>Hausdorff topological space</a>, then <a class='existingWikiWord' href='/nlab/show/diff/continuous+map'>continuous functions</a> of the form</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_68' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>f</mi><mspace width='thickmathspace' /><mo lspace='verythinmathspace'>:</mo><mspace width='thickmathspace' /><mi>X</mi><mo>⟶</mo><mi>Y</mi></mrow><annotation encoding='application/x-tex'> f \;\colon\; X \longrightarrow Y </annotation></semantics></math></div> <p>are in <a class='existingWikiWord' href='/nlab/show/diff/bijection'>bijection</a> with <a class='existingWikiWord' href='/nlab/show/diff/continuous+map'>continuous function</a> of the form</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_69' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mover><mi>f</mi><mo stretchy='false'>˜</mo></mover><mspace width='thickmathspace' /><mo lspace='verythinmathspace'>:</mo><mspace width='thickmathspace' /><mi>H</mi><mi>X</mi><mo>⟶</mo><mi>Y</mi></mrow><annotation encoding='application/x-tex'> \tilde f \;\colon\; H X \longrightarrow Y </annotation></semantics></math></div> <p>and such that the bijection is constituted by</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_70' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>f</mi><mo>=</mo><mover><mi>f</mi><mo stretchy='false'>˜</mo></mover><mo>∘</mo><msub><mi>h</mi> <mi>X</mi></msub><mspace width='thickmathspace' /><mo lspace='verythinmathspace'>:</mo><mspace width='thickmathspace' /><mi>X</mi><mover><mo>⟶</mo><mrow><msub><mi>h</mi> <mi>X</mi></msub></mrow></mover><mi>H</mi><mi>X</mi><mover><mo>⟶</mo><mover><mi>f</mi><mo stretchy='false'>˜</mo></mover></mover><mi>Y</mi><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> f = \tilde f \circ h_X \;\colon\; X \overset{h_X}{\longrightarrow} H X \overset{\tilde f}{\longrightarrow} Y \,. </annotation></semantics></math></div> <p>Here <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_71' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>H</mi><mi>X</mi></mrow><annotation encoding='application/x-tex'>H X</annotation></semantics></math> (or more precisely <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_72' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>h</mi> <mi>X</mi></msub></mrow><annotation encoding='application/x-tex'>h_X</annotation></semantics></math>) is also called the <em>Hausdorff reflection</em> (or <em>Hausdorffication</em> or similar) of <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_73' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math>.</p> <p>Moreover, the operation <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_74' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>H</mi><mo stretchy='false'>(</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>H(-)</annotation></semantics></math> extends to <a class='existingWikiWord' href='/nlab/show/diff/continuous+map'>continuous functions</a> <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_75' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>f</mi><mo lspace='verythinmathspace'>:</mo><mi>X</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding='application/x-tex'>f \colon X \to Y</annotation></semantics></math></p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_76' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mi>X</mi><mover><mo>→</mo><mi>f</mi></mover><mi>Y</mi><mo stretchy='false'>)</mo><mspace width='thickmathspace' /><mo>↦</mo><mspace width='thickmathspace' /><mo stretchy='false'>(</mo><mi>H</mi><mi>X</mi><mover><mo>→</mo><mrow><mi>H</mi><mi>f</mi></mrow></mover><mi>H</mi><mi>Y</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'> (X \overset{f}{\to} Y) \;\mapsto\; (H X \overset{H f}{\to} H Y) </annotation></semantics></math></div> <p>by setting</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_77' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>H</mi><mi>f</mi><mspace width='thickmathspace' /><mo lspace='verythinmathspace'>:</mo><mspace width='thickmathspace' /><mo stretchy='false'>[</mo><mi>x</mi><mo stretchy='false'>]</mo><mo>↦</mo><mo stretchy='false'>[</mo><mi>f</mi><mo stretchy='false'>(</mo><mi>x</mi><mo stretchy='false'>)</mo><mo stretchy='false'>]</mo><mspace width='thinmathspace' /><mo>,</mo></mrow><annotation encoding='application/x-tex'> H f \;\colon\; [x] \mapsto [f(x)] \,, </annotation></semantics></math></div> <p>where <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_78' 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> denotes the <a class='existingWikiWord' href='/nlab/show/diff/equivalence+class'>equivalence class</a> under <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_79' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mo>∼</mo> <mi>X</mi></msub></mrow><annotation encoding='application/x-tex'>\sim_X</annotation></semantics></math> of any <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_80' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>x</mi><mo>∈</mo><mi>X</mi></mrow><annotation encoding='application/x-tex'>x \in X</annotation></semantics></math>.</p> <p>Finally, the comparison map is compatible with this in that the follows <a class='existingWikiWord' href='/nlab/show/diff/commutative+square'>squares commute</a>:</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_81' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mrow><mtable><mtr><mtd><mi>X</mi></mtd> <mtd><mover><mo>⟶</mo><mi>f</mi></mover></mtd> <mtd><mi>Y</mi></mtd></mtr> <mtr><mtd><msup><mrow /> <mpadded lspace='-100%width' width='0'><mrow><msub><mi>h</mi> <mi>X</mi></msub></mrow></mpadded></msup><mo stretchy='false'>↓</mo></mtd> <mtd /> <mtd><msup><mo stretchy='false'>↓</mo> <mpadded width='0'><mrow><msub><mi>h</mi> <mi>Y</mi></msub></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><mi>H</mi><mi>X</mi></mtd> <mtd><munder><mo>⟶</mo><mrow><mi>H</mi><mi>f</mi></mrow></munder></mtd> <mtd><mi>H</mi><mi>Y</mi></mtd></mtr></mtable></mrow><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> \array{ X &\overset{f}{\longrightarrow}& Y \\ {}^{\mathllap{h_X}}\downarrow && \downarrow^{\mathrlap{h_Y}} \\ H X &\underset{H f}{\longrightarrow}& H Y } \,. </annotation></semantics></math></div></div> <div class='proof'> <h6 id='proof'>Proof</h6> <p>There are various ways to construct <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_82' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>h</mi> <mi>X</mi></msub></mrow><annotation encoding='application/x-tex'>h_X</annotation></semantics></math>, see below prop. <a class='maruku-ref' href='#HausdorffReflectionViaHomsIntoAllHausdorffSpaces'>3</a> and prop. <a class='maruku-ref' href='#HausdorffReflectionViaTransitiveClosureOfDiagonal'>4</a></p> </div> <div class='num_remark'> <h6 id='remark'>Remark</h6> <p>In the language of <a class='existingWikiWord' href='/nlab/show/diff/category+theory'>category theory</a> the Hausdorff reflection of prop. <a class='maruku-ref' href='#HausdorffReflection'>2</a> says that</p> <ol> <li> <p><math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_83' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>H</mi></mrow><annotation encoding='application/x-tex'>H</annotation></semantics></math> is a <em><a class='existingWikiWord' href='/nlab/show/diff/functor'>functor</a></em> <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_84' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>H</mi><mspace width='thickmathspace' /><mo lspace='verythinmathspace'>:</mo><mspace width='thickmathspace' /><mi>Top</mi><mo>⟶</mo><msub><mi>Top</mi> <mi>Haus</mi></msub></mrow><annotation encoding='application/x-tex'>H \;\colon\; Top \longrightarrow Top_{Haus}</annotation></semantics></math> from the <a class='existingWikiWord' href='/nlab/show/diff/category'>category</a> <a class='existingWikiWord' href='/nlab/show/diff/Top'>Top</a> of <a class='existingWikiWord' href='/nlab/show/diff/topological+space'>topological spaces</a> to the <a class='existingWikiWord' href='/nlab/show/diff/full+subcategory'>full subcategory</a> <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_85' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>Top</mi> <mi>Haus</mi></msub><mover><mo>↪</mo><mi>ι</mi></mover><mi>Top</mi></mrow><annotation encoding='application/x-tex'>Top_{Haus} \overset{\iota}{\hookrightarrow} Top</annotation></semantics></math> of Hausdorff topological spaces;</p> </li> <li> <p><math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_86' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>h</mi> <mi>X</mi></msub><mo lspace='verythinmathspace'>:</mo><mi>X</mi><mo>→</mo><mi>H</mi><mi>X</mi></mrow><annotation encoding='application/x-tex'>h_X \colon X \to H X</annotation></semantics></math> is a <a class='existingWikiWord' href='/nlab/show/diff/natural+transformation'>natural transformation</a> from the <a class='existingWikiWord' href='/nlab/show/diff/identity+functor'>identity functor</a> on <a class='existingWikiWord' href='/nlab/show/diff/Top'>Top</a> to the functor <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_87' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>ι</mi><mo>∘</mo><mi>H</mi></mrow><annotation encoding='application/x-tex'>\iota \circ H</annotation></semantics></math></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Hausdorff+space'>Hausdorff topological spaces</a> form a <a class='existingWikiWord' href='/nlab/show/diff/reflective+subcategory'>reflective subcategory</a> of all <a class='existingWikiWord' href='/nlab/show/diff/topological+space'>topological spaces</a> in that <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_88' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>H</mi></mrow><annotation encoding='application/x-tex'>H</annotation></semantics></math> is <a class='existingWikiWord' href='/nlab/show/diff/left+adjoint'>left adjoint</a> to the inclusion functor <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_89' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>ι</mi></mrow><annotation encoding='application/x-tex'>\iota</annotation></semantics></math></p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_90' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>Top</mi> <mi>Haus</mi></msub><munderover><mo>⊥</mo><munder><mo>↪</mo><mi>ι</mi></munder><mover><mo>⟵</mo><mi>H</mi></mover></munderover><mi>Top</mi><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> Top_{Haus} \underoverset{\underset{\iota}{\hookrightarrow}}{\overset{H}{\longleftarrow}}{\bot} Top \,. </annotation></semantics></math></div></li> </ol> </div> <p>There are various ways to see the existence and to construct the Hausdorff reflection. The following is maybe the quickest way to see the existence, even though it leaves the actual construction rather implicit.</p> <div class='num_prop' id='HausdorffReflectionViaHomsIntoAllHausdorffSpaces'> <h6 id='proposition_3'>Proposition</h6> <p>Let <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_91' 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> be a <a class='existingWikiWord' href='/nlab/show/diff/topological+space'>topological space</a> and consider the <a class='existingWikiWord' href='/nlab/show/diff/equivalence+relation'>equivalence relation</a> <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_92' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo>∼</mo></mrow><annotation encoding='application/x-tex'>\sim</annotation></semantics></math> on the underlying set <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_93' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> for which <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_94' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>x</mi><mo>∼</mo><mi>y</mi></mrow><annotation encoding='application/x-tex'>x \sim y</annotation></semantics></math> precisely if for every <a class='existingWikiWord' href='/nlab/show/diff/surjection'>surjective</a> <a class='existingWikiWord' href='/nlab/show/diff/continuous+map'>continuous function</a> <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_95' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>f</mi><mo lspace='verythinmathspace'>:</mo><mi>X</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding='application/x-tex'>f \colon X \to Y</annotation></semantics></math> into any <a class='existingWikiWord' href='/nlab/show/diff/Hausdorff+space'>Hausdorff topological space</a> <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_96' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Y</mi></mrow><annotation encoding='application/x-tex'>Y</annotation></semantics></math> we have <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_97' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>f</mi><mo stretchy='false'>(</mo><mi>x</mi><mo stretchy='false'>)</mo><mo>=</mo><mi>f</mi><mo stretchy='false'>(</mo><mi>y</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>f(x) = f(y)</annotation></semantics></math>.</p> <p>Then the set of <a class='existingWikiWord' href='/nlab/show/diff/equivalence+class'>equivalence classes</a></p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_98' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>H</mi><mi>X</mi><mo>≔</mo><mi>X</mi><mo stretchy='false'>/</mo><mo>∼</mo></mrow><annotation encoding='application/x-tex'> H X \coloneqq X /{\sim} </annotation></semantics></math></div> <p>equipped with the <a class='existingWikiWord' href='/nlab/show/diff/quotient+space'>quotient topology</a> is a <a class='existingWikiWord' href='/nlab/show/diff/Hausdorff+space'>Hausdorff topological space</a> and the quotient map <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_99' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>h</mi> <mi>X</mi></msub><mspace width='thickmathspace' /><mo lspace='verythinmathspace'>:</mo><mspace width='thickmathspace' /><mi>X</mi><mo>→</mo><mi>X</mi><mo stretchy='false'>/</mo><mo>∼</mo></mrow><annotation encoding='application/x-tex'>h_X \;\colon\; X \to X/{\sim}</annotation></semantics></math> exhibits the Hausdorff reflection of <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_100' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math>, according to prop. <a class='maruku-ref' href='#HausdorffReflection'>2</a>.</p> </div> <div class='proof'> <h6 id='proof_2'>Proof</h6> <p>First observe that every continuous function <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_101' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>f</mi><mo lspace='verythinmathspace'>:</mo><mi>X</mi><mo>⟶</mo><mi>Y</mi></mrow><annotation encoding='application/x-tex'>f \colon X \longrightarrow Y</annotation></semantics></math> into a Hausdorff space <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_102' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Y</mi></mrow><annotation encoding='application/x-tex'>Y</annotation></semantics></math> factors uniquely via <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_103' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>h</mi> <mi>X</mi></msub></mrow><annotation encoding='application/x-tex'>h_X</annotation></semantics></math> through a continuous function <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_104' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mover><mi>f</mi><mo stretchy='false'>˜</mo></mover></mrow><annotation encoding='application/x-tex'>\tilde f</annotation></semantics></math></p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_105' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>f</mi><mo>=</mo><mover><mi>f</mi><mo stretchy='false'>˜</mo></mover><mo>∘</mo><msub><mi>h</mi> <mi>X</mi></msub></mrow><annotation encoding='application/x-tex'> f = \tilde f \circ h_X </annotation></semantics></math></div> <p>where</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_106' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mover><mi>f</mi><mo stretchy='false'>˜</mo></mover><mo lspace='verythinmathspace'>:</mo><mo stretchy='false'>[</mo><mi>x</mi><mo stretchy='false'>]</mo><mo>↦</mo><mi>f</mi><mo stretchy='false'>(</mo><mi>x</mi><mo stretchy='false'>)</mo><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> \tilde f \colon [x] \mapsto f(x) \,. </annotation></semantics></math></div> <p>That this is well defined and continuous follows directly from the definitions.</p> <p>What remains to be seen is that <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_107' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>H</mi><mi>X</mi></mrow><annotation encoding='application/x-tex'>H X</annotation></semantics></math> is indeed a Hausdorff space. Hence assume that <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_108' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>[</mo><mi>x</mi><mo stretchy='false'>]</mo><mo>≠</mo><mo stretchy='false'>[</mo><mi>y</mi><mo stretchy='false'>]</mo><mo>∈</mo><mi>H</mi><mi>X</mi></mrow><annotation encoding='application/x-tex'>[x] \neq [y] \in H X</annotation></semantics></math>. By construction of <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_109' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>H</mi><mi>X</mi></mrow><annotation encoding='application/x-tex'>H X</annotation></semantics></math> this means that there exists a Hausdorff space <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_110' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Y</mi></mrow><annotation encoding='application/x-tex'>Y</annotation></semantics></math> and a surjective continuous function <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_111' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>f</mi><mo lspace='verythinmathspace'>:</mo><mi>X</mi><mo>⟶</mo><mi>Y</mi></mrow><annotation encoding='application/x-tex'>f \colon X \longrightarrow Y</annotation></semantics></math> such that <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_112' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>f</mi><mo stretchy='false'>(</mo><mi>x</mi><mo stretchy='false'>)</mo><mo>≠</mo><mi>f</mi><mo stretchy='false'>(</mo><mi>y</mi><mo stretchy='false'>)</mo><mo>∈</mo><mi>Y</mi></mrow><annotation encoding='application/x-tex'>f(x) \neq f(y) \in Y</annotation></semantics></math>. Accordingly, since <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_113' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Y</mi></mrow><annotation encoding='application/x-tex'>Y</annotation></semantics></math> is Hausdorff, there exist disjoint open neighbourhoods <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_114' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>U</mi> <mi>x</mi></msub><mo>,</mo><msub><mi>U</mi> <mi>y</mi></msub><mo>∈</mo><msub><mi>τ</mi> <mi>Y</mi></msub></mrow><annotation encoding='application/x-tex'>U_x, U_y \in \tau_Y</annotation></semantics></math>. Moreover, by the previous statement there exists a continuous function <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_115' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mover><mi>f</mi><mo stretchy='false'>˜</mo></mover><mo lspace='verythinmathspace'>:</mo><mi>H</mi><mi>X</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding='application/x-tex'>\tilde f \colon H X \to Y</annotation></semantics></math> with <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_116' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mover><mi>f</mi><mo stretchy='false'>˜</mo></mover><mo stretchy='false'>(</mo><mo stretchy='false'>[</mo><mi>x</mi><mo stretchy='false'>]</mo><mo stretchy='false'>)</mo><mo>=</mo><mi>f</mi><mo stretchy='false'>(</mo><mi>x</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>\tilde f([x]) = f(x)</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_117' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mover><mi>f</mi><mo stretchy='false'>˜</mo></mover><mo stretchy='false'>(</mo><mo stretchy='false'>[</mo><mi>y</mi><mo stretchy='false'>]</mo><mo stretchy='false'>)</mo><mo>=</mo><mi>f</mi><mo stretchy='false'>(</mo><mi>y</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>\tilde f([y]) = f(y)</annotation></semantics></math>. Since, by the nature of continuous functions, the pre-images <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_118' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mover><mi>f</mi><mo stretchy='false'>˜</mo></mover> <mrow><mo lspace='verythinmathspace' rspace='0em'>−</mo><mn>1</mn></mrow></msup><mo stretchy='false'>(</mo><msub><mi>U</mi> <mi>x</mi></msub><mo stretchy='false'>)</mo><mo>,</mo><msup><mover><mi>f</mi><mo stretchy='false'>˜</mo></mover> <mrow><mo lspace='verythinmathspace' rspace='0em'>−</mo><mn>1</mn></mrow></msup><mo stretchy='false'>(</mo><msub><mi>U</mi> <mi>y</mi></msub><mo stretchy='false'>)</mo><mo>⊂</mo><mi>H</mi><mi>X</mi></mrow><annotation encoding='application/x-tex'>\tilde f^{-1}( U_x ), \tilde f^{-1}(U_y) \subset H X</annotation></semantics></math> are still disjoint and open, we have found disjoint neighbourhoods of <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_119' 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> and <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_120' 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>. Hence <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_121' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>H</mi><mi>X</mi></mrow><annotation encoding='application/x-tex'>H X</annotation></semantics></math> is Hausdorff.</p> </div> <p>Some readers may find the following a more direct way of constructing the Hausdorff reflection:</p> <div class='num_prop' id='HausdorffReflectionViaTransitiveClosureOfDiagonal'> <h6 id='proposition_4'>Proposition</h6> <p>For <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_122' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mi>Y</mi><mo>,</mo><msub><mi>τ</mi> <mi>Y</mi></msub><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(Y,\tau_Y)</annotation></semantics></math> a <a class='existingWikiWord' href='/nlab/show/diff/topological+space'>topological space</a>, write <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_123' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>r</mi> <mi>Y</mi></msub><mo>⊂</mo><mi>Y</mi><mo>×</mo><mi>Y</mi></mrow><annotation encoding='application/x-tex'>r_Y \subset Y \times Y</annotation></semantics></math> for the <a class='existingWikiWord' href='/nlab/show/diff/transitive+relation'>transitive closure</a> of the <a class='existingWikiWord' href='/nlab/show/diff/relation'>relation</a> given by the <a class='existingWikiWord' href='/nlab/show/diff/closed+subspace'>topological closure</a> <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_124' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Cl</mi><mo stretchy='false'>(</mo><msub><mi>Δ</mi> <mi>Y</mi></msub><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>Cl(\Delta_Y)</annotation></semantics></math> of the <a class='existingWikiWord' href='/nlab/show/diff/image'>image</a> of the <a class='existingWikiWord' href='/nlab/show/diff/diagonal+morphism'>diagonal</a> <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_125' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>Δ</mi> <mi>Y</mi></msub><mo lspace='verythinmathspace'>:</mo><mi>Y</mi><mo>↪</mo><mi>Y</mi><mo>×</mo><mi>Y</mi></mrow><annotation encoding='application/x-tex'>\Delta_Y \colon Y \hookrightarrow Y \times Y</annotation></semantics></math>.</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_126' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>r</mi> <mi>Y</mi></msub><mo>≔</mo><mi>Trans</mi><mo stretchy='false'>(</mo><mi>Cl</mi><mo stretchy='false'>(</mo><msub><mi>Delta</mi> <mi>Y</mi></msub><mo stretchy='false'>)</mo><mo stretchy='false'>)</mo><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> r_Y \coloneqq Trans(Cl(Delta_Y)) \,. </annotation></semantics></math></div> <p>Now for <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_127' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mi>X</mi><mo>,</mo><msub><mi>τ</mi> <mi>X</mi></msub><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(X,\tau_X)</annotation></semantics></math> a <a class='existingWikiWord' href='/nlab/show/diff/topological+space'>topological space</a>, define by <a class='existingWikiWord' href='/nlab/show/diff/induction'>induction</a> for each <a class='existingWikiWord' href='/nlab/show/diff/ordinal+number'>ordinal number</a> <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_128' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>α</mi></mrow><annotation encoding='application/x-tex'>\alpha</annotation></semantics></math> an <a class='existingWikiWord' href='/nlab/show/diff/equivalence+relation'>equivalence relation</a> <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_129' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>r</mi> <mi>α</mi></msup></mrow><annotation encoding='application/x-tex'>r^\alpha</annotation></semantics></math> on <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_130' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> as follows, where we write <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_131' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>q</mi> <mi>α</mi></msup><mo lspace='verythinmathspace'>:</mo><mi>X</mi><mo>→</mo><msup><mi>H</mi> <mi>α</mi></msup><mo stretchy='false'>(</mo><mi>X</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>q^\alpha \colon X \to H^\alpha(X)</annotation></semantics></math> for the corresponding <a class='existingWikiWord' href='/nlab/show/diff/quotient+space'>quotient topological space</a> projection:</p> <p>We start the induction with the trivial equivalence relation:</p> <ul> <li><math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_132' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msubsup><mi>r</mi> <mi>X</mi> <mn>0</mn></msubsup><mo>≔</mo><msub><mi>Δ</mi> <mi>X</mi></msub></mrow><annotation encoding='application/x-tex'>r^0_X \coloneqq \Delta_X</annotation></semantics></math>;</li> </ul> <p>For a <a class='existingWikiWord' href='/nlab/show/diff/successor'>successor ordinal</a> we set</p> <ul> <li><math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_133' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msubsup><mi>r</mi> <mi>X</mi> <mrow><mi>α</mi><mo>+</mo><mn>1</mn></mrow></msubsup><mo>≔</mo><mrow><mo>{</mo><mo stretchy='false'>(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo stretchy='false'>)</mo><mo>∈</mo><mi>X</mi><mo>×</mo><mi>X</mi><mspace width='thinmathspace' /><mo stretchy='false'>|</mo><mspace width='thinmathspace' /><mo stretchy='false'>(</mo><msup><mi>q</mi> <mi>α</mi></msup><mo stretchy='false'>(</mo><mi>a</mi><mo stretchy='false'>)</mo><mo>,</mo><msup><mi>q</mi> <mi>α</mi></msup><mo stretchy='false'>(</mo><mi>b</mi><mo stretchy='false'>)</mo><mo stretchy='false'>)</mo><mo>∈</mo><msub><mi>r</mi> <mrow><msup><mi>H</mi> <mi>α</mi></msup><mo stretchy='false'>(</mo><mi>X</mi><mo stretchy='false'>)</mo></mrow></msub><mo>}</mo></mrow></mrow><annotation encoding='application/x-tex'>r_X^{\alpha+1} \coloneqq \left\{ (a,b) \in X \times X \,\vert\, (q^\alpha(a), q^\alpha(b)) \in r_{H^\alpha(X)} \right\}</annotation></semantics></math></li> </ul> <p>and for a <a class='existingWikiWord' href='/nlab/show/diff/ordinal+number'>limit ordinal</a> <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_134' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>α</mi></mrow><annotation encoding='application/x-tex'>\alpha</annotation></semantics></math> we set</p> <ul> <li><math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_135' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msubsup><mi>r</mi> <mi>X</mi> <mi>α</mi></msubsup><mo>≔</mo><munder><mo>∪</mo><mrow><mi>β</mi><mo><</mo><mi>α</mi></mrow></munder><msubsup><mi>r</mi> <mi>X</mi> <mi>β</mi></msubsup></mrow><annotation encoding='application/x-tex'>r_X^\alpha \coloneqq \underset{\beta \lt \alpha}{\cup} r_X^\beta</annotation></semantics></math>.</li> </ul> <p>Then:</p> <ol> <li> <p>there exists an ordinal <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_136' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>α</mi></mrow><annotation encoding='application/x-tex'>\alpha</annotation></semantics></math> such that <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_137' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msubsup><mi>r</mi> <mi>X</mi> <mi>α</mi></msubsup><mo>=</mo><msubsup><mi>r</mi> <mi>X</mi> <mrow><mi>α</mi><mo>+</mo><mn>1</mn></mrow></msubsup></mrow><annotation encoding='application/x-tex'>r_X^\alpha = r_X^{\alpha+1}</annotation></semantics></math></p> </li> <li> <p>for this <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_138' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>α</mi></mrow><annotation encoding='application/x-tex'>\alpha</annotation></semantics></math> then <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_139' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>H</mi> <mi>α</mi></msup><mo stretchy='false'>(</mo><mi>X</mi><mo stretchy='false'>)</mo><mo>=</mo><mi>H</mi><mo stretchy='false'>(</mo><mi>X</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>H^\alpha(X) = H(X)</annotation></semantics></math> is the Hausdorff reflection from prop. <a class='maruku-ref' href='#HausdorffReflectionViaHomsIntoAllHausdorffSpaces'>3</a>.</p> </li> </ol> </div> <p>(<a href='#vanMunster14'>vanMunster 14, section 4</a>)</p> <h3 id='monadicity'>Monadicity</h3> <p>The topology on a <a class='existingWikiWord' href='/nlab/show/diff/compact+space'>compact</a> Hausdorff space is given precisely by the (existent because compact, unique because Hausdorff) limit of each <a class='existingWikiWord' href='/nlab/show/diff/ultrafilter'>ultrafilter</a> on the space. Accordingly, compact Hausdorff topological spaces are (perhaps surprisingly) described by a (large) <a class='existingWikiWord' href='/nlab/show/diff/algebraic+theory'>algebraic theory</a>. In fact, the category of compact Hausdorff spaces is <a class='existingWikiWord' href='/nlab/show/diff/monadic+functor'>monadic</a> (over <a class='existingWikiWord' href='/nlab/show/diff/Set'>Set</a>); the <a class='existingWikiWord' href='/nlab/show/diff/monad'>monad</a> in question maps each set to the set ultrafilters on it. (The results of this paragraph require the <a class='existingWikiWord' href='/nlab/show/diff/ultrafilter+theorem'>ultrafilter theorem</a>, a weak form of the <a class='existingWikiWord' href='/nlab/show/diff/axiom+of+choice'>axiom of choice</a>; see <a class='existingWikiWord' href='/nlab/show/diff/ultrafilter'>ultrafilter monad</a>.)</p> <p>A compact Hausdorff <a class='existingWikiWord' href='/nlab/show/diff/locale'>locale</a> (or space) is necessarily <a class='existingWikiWord' href='/nlab/show/diff/regular+space'>regular</a>; a regular locale (or <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_140' 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> space) is necessarily Hausdorff. Accordingly, <a class='existingWikiWord' href='/nlab/show/diff/locale'>locale</a> theory usually speaks of ‘compact regular’ locales instead of ‘compact Hausdorff’ locales, since the definition of regularity is easier and more natural. Then a version of the previous paragraph works for compact regular locales <em>without</em> the ultrafilter theorem, and indeed <a class='existingWikiWord' href='/nlab/show/diff/constructive+mathematics'>constructively</a> over any <a class='existingWikiWord' href='/nlab/show/diff/topos'>topos</a>.</p> <h3 id='sobriety'>Sobriety</h3> <p>Using <a class='existingWikiWord' href='/nlab/show/diff/classical+logic'>classical logic</a> (but not in <a class='existingWikiWord' href='/nlab/show/diff/constructive+mathematics'>constructive logic</a>) every Hausdorff space is a <a class='existingWikiWord' href='/nlab/show/diff/sober+topological+space'>sober topological space</a>: <em><a class='existingWikiWord' href='/nlab/show/diff/Hausdorff+implies+sober'>Hausdorff implies sober</a></em>.</p> <h3 id='relation_to_compact_spaces'>Relation to compact spaces</h3> <div class='num_prop'> <h6 id='proposition_5'>Proposition</h6> <p><a class='existingWikiWord' href='/nlab/show/diff/compact+subspaces+of+Hausdorff+spaces+are+closed'>compact subspaces of Hausdorff spaces are closed</a>.</p> </div> <div class='num_prop'> <h6 id='proposition_6'>Proposition</h6> <p><a class='existingWikiWord' href='/nlab/show/diff/maps+from+compact+spaces+to+Hausdorff+spaces+are+closed+and+proper'>maps from compact spaces to Hausdorff spaces are closed and proper</a></p> </div> <h3 id='dense_subspaces_and_epimorphisms'>Dense subspaces and Epimorphisms</h3> <p>\begin{proposition} \label{DenseSubspaceInclusionsAreEpimorphismsAmongHausdorffSpaces}</p> <p>In the <a class='existingWikiWord' href='/nlab/show/diff/category'>category</a> of <a class='existingWikiWord' href='/nlab/show/diff/Hausdorff+space'>Hausdorff topological spaces</a> (with <a class='existingWikiWord' href='/nlab/show/diff/continuous+map'>continuous functions</a> between them), the inclusion of a <a class='existingWikiWord' href='/nlab/show/diff/dense+subspace'>dense subspace</a></p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_141' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>A</mi><mover><mo>↪</mo><mrow><mspace width='thickmathspace' /><mspace width='thickmathspace' /><mi>i</mi><mspace width='thickmathspace' /><mspace width='thickmathspace' /></mrow></mover><mi>X</mi></mrow><annotation encoding='application/x-tex'> A \overset{\;\;i\;\;}{\hookrightarrow} X </annotation></semantics></math></div> <p>is an <a class='existingWikiWord' href='/nlab/show/diff/epimorphism'>epimorphism</a>.</p> <p>\end{proposition}</p> <p>Here is a proof in the language of <a class='existingWikiWord' href='/nlab/show/diff/category+theory'>category theory</a>:</p> <p>\begin{proof}</p> <p>We have to show that for <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_142' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mi>f</mi><mo>,</mo><mi>g</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(f,g)</annotation></semantics></math> any <a class='existingWikiWord' href='/nlab/show/diff/pair'>pair</a> of <a class='existingWikiWord' href='/nlab/show/diff/parallel+morphisms'>parallel morphisms</a> out of <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_143' 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 class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_144' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>A</mi><mover><mo>↪</mo><mrow><mspace width='thickmathspace' /><mspace width='thickmathspace' /><mi>i</mi><mspace width='thickmathspace' /><mspace width='thickmathspace' /></mrow></mover><mi>X</mi><munderover><mo>⇉</mo><mrow><mspace width='thickmathspace' /><mspace width='thickmathspace' /><mi>g</mi><mspace width='thickmathspace' /><mspace width='thickmathspace' /></mrow><mrow><mspace width='thickmathspace' /><mspace width='thickmathspace' /><mi>f</mi><mspace width='thickmathspace' /><mspace width='thickmathspace' /></mrow></munderover><mi>Y</mi></mrow><annotation encoding='application/x-tex'> A \overset{\;\;i\;\;}{\hookrightarrow} X \underoverset {\;\;g\;\;} {\;\;f\;\;} {\rightrightarrows} Y </annotation></semantics></math></div> <p>into a <a class='existingWikiWord' href='/nlab/show/diff/Hausdorff+space'>Hausdorff space</a> <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_145' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Y</mi></mrow><annotation encoding='application/x-tex'>Y</annotation></semantics></math>, the <a class='existingWikiWord' href='/nlab/show/diff/equality'>equality</a> <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_146' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>f</mi><mo>∘</mo><mi>i</mi><mo>=</mo><mi>g</mi><mo>∘</mo><mi>i</mi></mrow><annotation encoding='application/x-tex'>f \circ i = g \circ i</annotation></semantics></math> implies <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_147' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>f</mi><mo>=</mo><mi>g</mi></mrow><annotation encoding='application/x-tex'>f = g</annotation></semantics></math>. Equivalently, that <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_148' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>f</mi><mo>∘</mo><mi>i</mi><mo>=</mo><mi>g</mi><mo>∘</mo><mi>i</mi></mrow><annotation encoding='application/x-tex'>f \circ i = g \circ i</annotation></semantics></math> implies that <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_149' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mn>1</mn> <mi>X</mi></msub><mo lspace='verythinmathspace'>:</mo><mi>X</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding='application/x-tex'>1_X \colon X \to X</annotation></semantics></math> is the <a class='existingWikiWord' href='/nlab/show/diff/equalizer'>equalizer</a> of the pair <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_150' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mi>f</mi><mo>,</mo><mi>g</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(f, g)</annotation></semantics></math>. But the equalizer <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_151' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>E</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding='application/x-tex'>E \to X</annotation></semantics></math> is formed by taking a <a class='existingWikiWord' href='/nlab/show/diff/pullback'>pullback</a> of the <a class='existingWikiWord' href='/nlab/show/diff/diagonal+morphism'>diagonal map</a> <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_152' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Δ</mi><mo lspace='verythinmathspace'>:</mo><mi>Y</mi><mo>→</mo><mi>Y</mi><mo>×</mo><mi>Y</mi></mrow><annotation encoding='application/x-tex'>\Delta \colon Y \to Y \times Y</annotation></semantics></math> along <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_153' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mi>f</mi><mo>,</mo><mi>g</mi><mo stretchy='false'>)</mo><mo lspace='verythinmathspace'>:</mo><mi>X</mi><mo>→</mo><mi>Y</mi><mo>×</mo><mi>Y</mi></mrow><annotation encoding='application/x-tex'>(f, g) \colon X \to Y \times Y</annotation></semantics></math>:</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_154' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mrow><mtable><mtr><mtd><mi>E</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>Y</mi></mtd></mtr> <mtr><mtd><mo stretchy='false'>↓</mo></mtd> <mtd /> <mtd><mo stretchy='false'>↓</mo><mpadded width='0'><mi>Δ</mi></mpadded></mtd></mtr> <mtr><mtd><mi>X</mi></mtd> <mtd><munder><mo>→</mo><mrow><mo stretchy='false'>(</mo><mi>f</mi><mo>,</mo><mi>g</mi><mo stretchy='false'>)</mo></mrow></munder></mtd> <mtd><mi>Y</mi><mo>×</mo><mi>Y</mi><mo>.</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding='application/x-tex'>\array{ E & \to & Y \\ \downarrow & & \downarrow \mathrlap{\Delta} \\ X & \underset{(f, g)}{\to} & Y \times Y. } </annotation></semantics></math></div> <p>Since <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_155' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Y</mi></mrow><annotation encoding='application/x-tex'>Y</annotation></semantics></math> is Hausdorff, the subset <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_156' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Δ</mi><mo>:</mo><mi>Y</mi><mo>→</mo><mi>Y</mi><mo>×</mo><mi>Y</mi></mrow><annotation encoding='application/x-tex'>\Delta: Y \to Y \times Y</annotation></semantics></math> is closed, and the pullback <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_157' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>E</mi><mo>↪</mo><mi>X</mi></mrow><annotation encoding='application/x-tex'>E \hookrightarrow X</annotation></semantics></math> of this closed subset along the continuous map <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_158' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>h</mi><mo>=</mo><mo stretchy='false'>(</mo><mi>f</mi><mo>,</mo><mi>g</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>h = (f, g)</annotation></semantics></math>, which is <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_159' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>E</mi><mo>=</mo><msup><mi>h</mi> <mrow><mo lspace='verythinmathspace' rspace='0em'>−</mo><mn>1</mn></mrow></msup><mo stretchy='false'>(</mo><mi>Δ</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>E = h^{-1}(\Delta)</annotation></semantics></math>, is also closed. Since <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_160' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>E</mi></mrow><annotation encoding='application/x-tex'>E</annotation></semantics></math> is a closed subset of <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_161' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> and contains a dense subspace <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_162' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>i</mi><mo>:</mo><mi>A</mi><mo>↪</mo><mi>X</mi></mrow><annotation encoding='application/x-tex'>i: A \hookrightarrow X</annotation></semantics></math>, it must be all of <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_163' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> (as a subset of itself). \end{proof}</p> <p>Note, incidentally, that <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_164' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> itself doesn’t have to be Hausdorff for the argument to go through.</p> <p>Alternatively, here is a proof in the language of <a class='existingWikiWord' href='/nlab/show/diff/general+topology'>basic topology</a>:</p> <p>\begin{proof}</p> <p>We have to show that for <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_165' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mi>f</mi><mo>,</mo><mi>g</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(f,g)</annotation></semantics></math> any <a class='existingWikiWord' href='/nlab/show/diff/pair'>pair</a> of <a class='existingWikiWord' href='/nlab/show/diff/parallel+morphisms'>parallel morphisms</a> out of <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_166' 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 class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_167' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>A</mi><mover><mo>↪</mo><mrow><mspace width='thickmathspace' /><mspace width='thickmathspace' /><mi>i</mi><mspace width='thickmathspace' /><mspace width='thickmathspace' /></mrow></mover><mi>X</mi><munderover><mo>⇉</mo><mrow><mspace width='thickmathspace' /><mspace width='thickmathspace' /><mi>g</mi><mspace width='thickmathspace' /><mspace width='thickmathspace' /></mrow><mrow><mspace width='thickmathspace' /><mspace width='thickmathspace' /><mi>f</mi><mspace width='thickmathspace' /><mspace width='thickmathspace' /></mrow></munderover><mi>Y</mi></mrow><annotation encoding='application/x-tex'> A \overset{\;\;i\;\;}{\hookrightarrow} X \underoverset {\;\;g\;\;} {\;\;f\;\;} {\rightrightarrows} Y </annotation></semantics></math></div> <p>into a <a class='existingWikiWord' href='/nlab/show/diff/Hausdorff+space'>Hausdorff space</a> <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_168' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Y</mi></mrow><annotation encoding='application/x-tex'>Y</annotation></semantics></math>, the <a class='existingWikiWord' href='/nlab/show/diff/equality'>equality</a> <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_169' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>f</mi><mo>∘</mo><mi>i</mi><mo>=</mo><mi>g</mi><mo>∘</mo><mi>i</mi></mrow><annotation encoding='application/x-tex'>f \circ i = g \circ i</annotation></semantics></math> implies <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_170' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>f</mi><mo>=</mo><mi>g</mi></mrow><annotation encoding='application/x-tex'>f = g</annotation></semantics></math>. With <a class='existingWikiWord' href='/nlab/show/diff/classical+logic'>classical logic</a> we may equivalently show the <a class='existingWikiWord' href='/nlab/show/diff/contrapositive'>contrapositive</a>: That <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_171' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>f</mi><mo>≠</mo><mi>g</mi></mrow><annotation encoding='application/x-tex'>f \neq g</annotation></semantics></math> implies <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_172' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>f</mi><mo>∘</mo><mi>i</mi><mo>≠</mo><mi>g</mi><mo>∘</mo><mi>i</mi></mrow><annotation encoding='application/x-tex'>f \circ i \neq g \circ i</annotation></semantics></math>.</p> <p>So assume that <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_173' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>f</mi><mo>≠</mo><mi>g</mi></mrow><annotation encoding='application/x-tex'>f \neq g</annotation></semantics></math>. This means that there exists <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_174' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>x</mi><mo>∈</mo><mi>X</mi></mrow><annotation encoding='application/x-tex'>x \in X</annotation></semantics></math> with <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_175' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>f</mi><mo stretchy='false'>(</mo><mi>x</mi><mo stretchy='false'>)</mo><mo>≠</mo><mi>g</mi><mo stretchy='false'>(</mo><mi>x</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>f(x) \neq g(x)</annotation></semantics></math>. Since <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_176' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Y</mi></mrow><annotation encoding='application/x-tex'>Y</annotation></semantics></math> is Hausdorff, there exist then <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_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_177' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>O</mi> <mrow><mi>f</mi><mo stretchy='false'>(</mo><mi>x</mi><mo stretchy='false'>)</mo></mrow></msub><mo>,</mo><mspace width='thickmathspace' /><msub><mi>O</mi> <mrow><mi>g</mi><mo stretchy='false'>(</mo><mi>x</mi><mo stretchy='false'>)</mo></mrow></msub><mo>⊂</mo><mi>Y</mi></mrow><annotation encoding='application/x-tex'>O_{f(x)},\;O_{g(x)} \subset Y</annotation></semantics></math>, i.e. <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_178' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>f</mi><mo stretchy='false'>(</mo><mi>x</mi><mo stretchy='false'>)</mo><mo>∈</mo><msub><mi>O</mi> <mrow><mi>f</mi><mo stretchy='false'>(</mo><mi>x</mi><mo stretchy='false'>)</mo></mrow></msub></mrow><annotation encoding='application/x-tex'>f(x) \in O_{f(x)}</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_179' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>g</mi><mo stretchy='false'>(</mo><mi>x</mi><mo stretchy='false'>)</mo><mo>∈</mo><msub><mi>O</mi> <mrow><mi>g</mi><mo stretchy='false'>(</mo><mi>x</mi><mo stretchy='false'>)</mo></mrow></msub></mrow><annotation encoding='application/x-tex'>g(x) \in O_{g(x)}</annotation></semantics></math> with <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_180' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>O</mi> <mrow><mi>f</mi><mo stretchy='false'>(</mo><mi>x</mi><mo stretchy='false'>)</mo></mrow></msub><mo>∩</mo><msub><mi>O</mi> <mrow><mi>g</mi><mo stretchy='false'>(</mo><mi>x</mi><mo stretchy='false'>)</mo></mrow></msub><mo>=</mo><mo>∅</mo></mrow><annotation encoding='application/x-tex'>O_{f(x)} \cap O_{g(x)} = \varnothing</annotation></semantics></math>.</p> <p>But their <a class='existingWikiWord' href='/nlab/show/diff/preimage'>preimages</a> must intersect at least in <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_181' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>x</mi><mo>∈</mo><msup><mi>f</mi> <mrow><mo lspace='verythinmathspace' rspace='0em'>−</mo><mn>1</mn></mrow></msup><mo maxsize='1.2em' minsize='1.2em'>(</mo><msub><mi>O</mi> <mrow><mi>f</mi><mo stretchy='false'>(</mo><mi>x</mi><mo stretchy='false'>)</mo></mrow></msub><mo maxsize='1.2em' minsize='1.2em'>)</mo><mo>∩</mo><msup><mi>g</mi> <mrow><mo lspace='verythinmathspace' rspace='0em'>−</mo><mn>1</mn></mrow></msup><mo maxsize='1.2em' minsize='1.2em'>(</mo><msub><mi>O</mi> <mrow><mi>g</mi><mo stretchy='false'>(</mo><mi>x</mi><mo stretchy='false'>)</mo></mrow></msub><mo maxsize='1.2em' minsize='1.2em'>)</mo></mrow><annotation encoding='application/x-tex'>x \in f^{-1}\big( O_{f(x)} \big) \cap g^{-1}\big( O_{g(x)} \big)</annotation></semantics></math>. Since this intersection is an <a class='existingWikiWord' href='/nlab/show/diff/open+subspace'>open subset</a> (as preimages of open subsets are open by definition of <a class='existingWikiWord' href='/nlab/show/diff/continuous+map'>continuous functions</a>, and since finite <a class='existingWikiWord' href='/nlab/show/diff/intersection'>intersections</a> of open subsets are open by the definition of <a class='existingWikiWord' href='/nlab/show/diff/topological+space'>topological spaces</a>) there exists a point <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_182' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>a</mi><mo>∈</mo><mi>A</mi></mrow><annotation encoding='application/x-tex'>a \in A</annotation></semantics></math> with <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_183' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>i</mi><mo stretchy='false'>(</mo><mi>a</mi><mo stretchy='false'>)</mo><mo>∈</mo><msup><mi>f</mi> <mrow><mo lspace='verythinmathspace' rspace='0em'>−</mo><mn>1</mn></mrow></msup><mo maxsize='1.2em' minsize='1.2em'>(</mo><msub><mi>O</mi> <mrow><mi>f</mi><mo stretchy='false'>(</mo><mi>x</mi><mo stretchy='false'>)</mo></mrow></msub><mo maxsize='1.2em' minsize='1.2em'>)</mo><mo>∩</mo><msup><mi>g</mi> <mrow><mo lspace='verythinmathspace' rspace='0em'>−</mo><mn>1</mn></mrow></msup><mo maxsize='1.2em' minsize='1.2em'>(</mo><msub><mi>O</mi> <mrow><mi>g</mi><mo stretchy='false'>(</mo><mi>x</mi><mo stretchy='false'>)</mo></mrow></msub><mo maxsize='1.2em' minsize='1.2em'>)</mo></mrow><annotation encoding='application/x-tex'>i(a) \in f^{-1}\big( O_{f(x)} \big) \cap g^{-1}\big( O_{g(x)} \big)</annotation></semantics></math> (by definition of <a class='existingWikiWord' href='/nlab/show/diff/dense+subspace'>dense subset</a>). But since then <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_184' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>f</mi><mo stretchy='false'>(</mo><mi>i</mi><mo stretchy='false'>(</mo><mi>a</mi><mo stretchy='false'>)</mo><mo stretchy='false'>)</mo><mo>∈</mo><msub><mi>O</mi> <mrow><mi>f</mi><mo stretchy='false'>(</mo><mi>x</mi><mo stretchy='false'>)</mo></mrow></msub></mrow><annotation encoding='application/x-tex'>f(i(a)) \in O_{f(x)}</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_185' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>g</mi><mo stretchy='false'>(</mo><mi>i</mi><mo stretchy='false'>(</mo><mi>a</mi><mo stretchy='false'>)</mo><mo stretchy='false'>)</mo><mo>∈</mo><msub><mi>O</mi> <mrow><mi>g</mi><mo stretchy='false'>(</mo><mi>x</mi><mo stretchy='false'>)</mo></mrow></msub></mrow><annotation encoding='application/x-tex'>g(i(a)) \in O_{g(x)}</annotation></semantics></math> while <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_186' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>O</mi> <mrow><mi>f</mi><mo stretchy='false'>(</mo><mi>x</mi><mo stretchy='false'>)</mo></mrow></msub></mrow><annotation encoding='application/x-tex'>O_{f(x)}</annotation></semantics></math> is <a class='existingWikiWord' href='/nlab/show/diff/disjoint+subsets'>disjoint</a> from <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_187' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>O</mi> <mrow><mi>g</mi><mo stretchy='false'>(</mo><mi>x</mi><mo stretchy='false'>)</mo></mrow></msub></mrow><annotation encoding='application/x-tex'>O_{g(x)}</annotation></semantics></math>, it follows that <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_188' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>f</mi><mo stretchy='false'>(</mo><mi>i</mi><mo stretchy='false'>(</mo><mi>a</mi><mo stretchy='false'>)</mo><mo stretchy='false'>)</mo><mo>≠</mo><mi>g</mi><mo stretchy='false'>(</mo><mi>i</mi><mo stretchy='false'>(</mo><mi>a</mi><mo stretchy='false'>)</mo><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>f(i(a)) \neq g(i(a))</annotation></semantics></math>. This means that <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_189' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>f</mi><mo>∘</mo><mi>i</mi><mo>≠</mo><mi>g</mi><mo>∘</mo><mi>i</mi></mrow><annotation encoding='application/x-tex'>f \circ i \neq g \circ i</annotation></semantics></math>. \end{proof}</p> <p>\linebreak</p> <p>In fact the converse holds: any <a class='existingWikiWord' href='/nlab/show/diff/epimorphism'>epimorphism</a> in the category of Hausdorff spaces has dense image (e.g. <a href='#LL18'>LL 18</a>).</p> <h2 id='related_notions'>Related notions</h2> <p>Arguably, the desire to make spaces Hausdorff (<math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_190' 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>) in analysis is really a desire to make them <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_191' 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>; nearly every space that arises in analysis is at least <a class='existingWikiWord' href='/nlab/show/diff/regular+space'>regular</a>, and a regular <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_192' 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> space must be Hausdorff. Forcing a space to be <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_193' 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 like forcing a <a class='existingWikiWord' href='/nlab/show/diff/category'>category</a> to be <a class='existingWikiWord' href='/nlab/show/diff/skeleton'>skeletal</a>; indeed, forcing a <a class='existingWikiWord' href='/nlab/show/diff/preorder'>preorder</a> to be a <a class='existingWikiWord' href='/nlab/show/diff/partial+order'>partial order</a> is a special case of both (see <a class='existingWikiWord' href='/nlab/show/diff/specialization+topology'>specialisation topology</a> for how). It may be nice to assume, when working with a particular space, that it is <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_194' 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> but not to assume, when working with a particular underlying set, that every topology on it is <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_195' 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>.</p> <p>Whatever one thinks of that, there is a non-<math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_196' 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> version of Hausdorff space, an <strong><math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_197' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>R</mi> <mn>1</mn></msub></mrow><annotation encoding='application/x-tex'>R_1</annotation></semantics></math> space</strong>. (The symbol here comes from being a weak version of a <strong>r</strong>egular space; in general a <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_198' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>T</mi> <mi>i</mi></msub></mrow><annotation encoding='application/x-tex'>T_i</annotation></semantics></math> space is precisely both <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_199' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>R</mi> <mrow><mi>i</mi><mo>−</mo><mn>1</mn></mrow></msub></mrow><annotation encoding='application/x-tex'>R_{i-1}</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_200' 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>). This is also called a <strong>preregular space</strong> (in <em><a class='existingWikiWord' href='/nlab/show/diff/Handbook+of+Analysis+and+its+Foundations'>HAF</a></em>) and a <strong>reciprocal space</strong> (in convergence theory).</p> <div class='un_defn'> <h6 id='definition_of_'>Definition (of <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_201' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>R</mi> <mn>1</mn></msub></mrow><annotation encoding='application/x-tex'>R_1</annotation></semantics></math>)</h6> <p>Given points <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_202' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>a</mi></mrow><annotation encoding='application/x-tex'>a</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_203' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>b</mi></mrow><annotation encoding='application/x-tex'>b</annotation></semantics></math>, if every neighbourhood of <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_204' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>a</mi></mrow><annotation encoding='application/x-tex'>a</annotation></semantics></math> meets every neighbourhood of <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_205' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>b</mi></mrow><annotation encoding='application/x-tex'>b</annotation></semantics></math>, then every neighbourhood of <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_206' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>a</mi></mrow><annotation encoding='application/x-tex'>a</annotation></semantics></math> is a neighbourhood of <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_207' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>b</mi></mrow><annotation encoding='application/x-tex'>b</annotation></semantics></math>. Equivalently, if any net (or proper filter) converges to both <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_208' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>a</mi></mrow><annotation encoding='application/x-tex'>a</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_209' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>b</mi></mrow><annotation encoding='application/x-tex'>b</annotation></semantics></math>, then every net (or filter) that converges to <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_210' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>a</mi></mrow><annotation encoding='application/x-tex'>a</annotation></semantics></math> also converges to <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_211' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>b</mi></mrow><annotation encoding='application/x-tex'>b</annotation></semantics></math>.</p> </div> <p>There is also a notion of <strong>sequentially Hausdorff space</strong>:</p> <div class='un_defn'> <h6 id='definition_of_sequentially_hausdorff'>Definition (of sequentially Hausdorff)</h6> <p>Whenever a <a class='existingWikiWord' href='/nlab/show/diff/sequence'>sequence</a> converges to both <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_212' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>x</mi></mrow><annotation encoding='application/x-tex'>x</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_213' 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_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_214' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>x</mi><mo>=</mo><mi>y</mi></mrow><annotation encoding='application/x-tex'>x = y</annotation></semantics></math>.</p> </div> <p>Some forms of <a class='existingWikiWord' href='/nlab/show/diff/predicative+mathematics'>predicative mathematics</a> find this concept more useful. Hausdorffness implies sequential Hausdorffness, but the converse is false even for <a class='existingWikiWord' href='/nlab/show/diff/sequential+topological+space'>sequential space</a>s (although it is true for <a class='existingWikiWord' href='/nlab/show/diff/first-countable+space'>first-countable space</a>s).</p> <p>The reader can now easily define a <em>sequentially <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_215' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>R</mi> <mn>1</mn></msub></mrow><annotation encoding='application/x-tex'>R_1</annotation></semantics></math> space</em>.</p> <h2 id='BeyondTopologicalSpaces'>Beyond topological spaces</h2> <h3 id='HausdorffLocale'>Hausdorff locales</h3> <p>The most obvious definition for a <a class='existingWikiWord' href='/nlab/show/diff/locale'>locale</a> <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_216' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> to be <strong>Hausdorff</strong> is that its <a class='existingWikiWord' href='/nlab/show/diff/diagonal+morphism'>diagonal</a> <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_217' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi><mo>→</mo><mi>X</mi><mo>×</mo><mi>X</mi></mrow><annotation encoding='application/x-tex'>X\to X\times X</annotation></semantics></math> is a <a class='existingWikiWord' href='/nlab/show/diff/closed+map'>closed</a> (and hence <a class='existingWikiWord' href='/nlab/show/diff/proper+map'>proper</a>) inclusion. However, if <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_218' 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/sober+topological+space'>sober space</a> regarded as a locale, this might not coincide with the condition for <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_219' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> to be Hausdorff as a space, since the <a class='existingWikiWord' href='/nlab/show/diff/cartesian+product'>Cartesian product</a> <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_220' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi><mo>×</mo><mi>X</mi></mrow><annotation encoding='application/x-tex'>X\times X</annotation></semantics></math> in the <a class='existingWikiWord' href='/nlab/show/diff/category'>category</a> <a class='existingWikiWord' href='/nlab/show/diff/Loc'>Loc</a> of locales might not coincide with the product in the category <a class='existingWikiWord' href='/nlab/show/diff/Top'>Top</a> of topological spaces (the <a class='existingWikiWord' href='/nlab/show/diff/product+topological+space'>Tychonoff product</a>). But it does coincide if <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_221' 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/locally+compact+locale'>locally compact locale</a>, so in that case the two notions of Hausdorff are the same.</p> <h3 id='separated_toposes_and_schemes'>Separated toposes and schemes</h3> <p>This notion of a <em>Hausdorff locale</em> is a special case of that of <em><a class='existingWikiWord' href='/nlab/show/diff/separated+geometric+morphism'>Hausdorff topos</a></em> in <a class='existingWikiWord' href='/nlab/show/diff/sheaf+and+topos+theory'>topos theory</a>. This also is formally similar to notions such as a <em><a class='existingWikiWord' href='/nlab/show/diff/separated+morphism+of+schemes'>separated scheme</a></em> etc. The corresponding relative notion (over an arbitrary <a class='existingWikiWord' href='/nlab/show/diff/base+topos'>base topos</a>) is that of <em><a class='existingWikiWord' href='/nlab/show/diff/separated+geometric+morphism'>separated geometric morphism</a></em>. For schemes see <em><a class='existingWikiWord' href='/nlab/show/diff/separated+morphism+of+schemes'>separated morphism of schemes</a></em>.</p> <h2 id='in_constructive_mathematics'>In constructive mathematics</h2> <p>In <a class='existingWikiWord' href='/nlab/show/diff/constructive+mathematics'>constructive mathematics</a>, the Hausdorff notion multifurcates further, due to the variety of possible meanings of <a class='existingWikiWord' href='/nlab/show/diff/closed+subspace'>closed subspace</a>. If we ask the diagonal to be <em>weakly</em> <a class='existingWikiWord' href='/nlab/show/diff/closed+subspace'>closed</a>, then in the spatial case, this means that it contains all its limit points, giving Definition <a class='maruku-ref' href='#constructive'>2</a> above. But if we ask the diagonal to be <em>strongly</em> closed, i.e. the complement of an open set, then in the spatial case this means that there is a <a class='existingWikiWord' href='/nlab/show/diff/inequality'>tight inequality</a> <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_222' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo>≠</mo></mrow><annotation encoding='application/x-tex'>\ne</annotation></semantics></math> (the <a class='existingWikiWord' href='/nlab/show/diff/exterior'>exterior</a> of <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_223' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo>=</mo></mrow><annotation encoding='application/x-tex'>=</annotation></semantics></math>) relative to which Definition <a class='maruku-ref' href='#classical'>1</a> holds. (We use <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_224' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo>≠</mo></mrow><annotation encoding='application/x-tex'>\ne</annotation></semantics></math> twice in that definition: in the hypothesis that <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_225' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>x</mi><mo>≠</mo><mi>y</mi></mrow><annotation encoding='application/x-tex'>x \ne y</annotation></semantics></math> and in the conclusion that <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_226' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>x</mi><mo>′</mo><mo>≠</mo><mi>y</mi><mo>′</mo></mrow><annotation encoding='application/x-tex'>x' \ne y'</annotation></semantics></math>.)</p> <p>It is natural to call these conditions <em>weakly Hausdorff</em> and <em>strongly Hausdorff</em>, but one should be aware of terminological clashes: in classical mathematics there is a different notion of a <a class='existingWikiWord' href='/nlab/show/diff/weakly+Hausdorff+topological+space'>weak Hausdorff space</a>, whereas (strong) Hausdorffness for locales has by some authors been called “strongly Hausdorff” only to contrast it with Hausdorffness for spaces.</p> <p>As a simple example, consider a <a class='existingWikiWord' href='/nlab/show/diff/discrete+object'>discrete space</a> <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_227' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> regarded as a locale. Since it is locally compact, the locale product <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_228' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi><mo>×</mo><mi>X</mi></mrow><annotation encoding='application/x-tex'>X\times X</annotation></semantics></math> coincides with the space product (a theorem that is valid constructively); but nevertheless we have:</p> <ol> <li>The diagonal <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_229' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi><mo>→</mo><mi>X</mi><mo>×</mo><mi>X</mi></mrow><annotation encoding='application/x-tex'>X\to X\times X</annotation></semantics></math> always has an open complement.</li> <li>Definition <a class='maruku-ref' href='#constructive'>2</a> above always holds, since <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_230' 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> and <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_231' 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> are neighborhoods of <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_232' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>x</mi></mrow><annotation encoding='application/x-tex'>x</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_233' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>y</mi></mrow><annotation encoding='application/x-tex'>y</annotation></semantics></math>, and if they intersect then <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_234' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>x</mi><mo>=</mo><mi>y</mi></mrow><annotation encoding='application/x-tex'>x=y</annotation></semantics></math>. That is, <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_235' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> is spatially weakly Hausdorff.</li> <li>The diagonal <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_236' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi><mo>→</mo><mi>X</mi><mo>×</mo><mi>X</mi></mrow><annotation encoding='application/x-tex'>X\to X\times X</annotation></semantics></math> is the complement of an open subset (i.e. <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_237' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> is spatially strongly Hausdorff) if and only if equality in <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_238' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> is stable under <a class='existingWikiWord' href='/nlab/show/diff/double+negation'>double negation</a>, in other words if <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_239' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> admits a tight <a class='existingWikiWord' href='/nlab/show/diff/inequality'>inequality relation</a>.</li> <li>The locale diagonal <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_240' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Δ</mi><mo>:</mo><mi>X</mi><mo>→</mo><mi>X</mi><mo>×</mo><mi>X</mi></mrow><annotation encoding='application/x-tex'>\Delta:X\to X\times X</annotation></semantics></math> is a closed sublocale (i.e. <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_241' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> is localically strongly Hausdorff) if and only if <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_242' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> has <a class='existingWikiWord' href='/nlab/show/diff/decidable+equality'>decidable equality</a>. For closedness of <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_243' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Δ</mi></mrow><annotation encoding='application/x-tex'>\Delta</annotation></semantics></math> means that <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_244' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>Δ</mi> <mo>*</mo></msub><mo stretchy='false'>(</mo><mi>U</mi><mo>∪</mo><msup><mi>Δ</mi> <mo>*</mo></msup><mo stretchy='false'>(</mo><mi>V</mi><mo stretchy='false'>)</mo><mo stretchy='false'>)</mo><mo>⊆</mo><msub><mi>Δ</mi> <mo>*</mo></msub><mo stretchy='false'>(</mo><mi>U</mi><mo stretchy='false'>)</mo><mo>∪</mo><mi>V</mi></mrow><annotation encoding='application/x-tex'>\Delta_\ast(U\cup \Delta^\ast(V)) \subseteq \Delta_\ast(U) \cup V</annotation></semantics></math> for any <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_245' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>U</mi><mo>∈</mo><mi>O</mi><mo stretchy='false'>(</mo><mi>X</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>U\in O(X)</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_246' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>V</mi><mo>∈</mo><mi>O</mi><mo stretchy='false'>(</mo><mi>X</mi><mo>×</mo><mi>X</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>V\in O(X\times X)</annotation></semantics></math>, where <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_247' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>x</mi><mo>∈</mo><msup><mi>Δ</mi> <mo>*</mo></msup><mo stretchy='false'>(</mo><mi>V</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>x\in \Delta^\ast(V)</annotation></semantics></math> means <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_248' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mi>x</mi><mo>,</mo><mi>x</mi><mo stretchy='false'>)</mo><mo>∈</mo><mi>V</mi></mrow><annotation encoding='application/x-tex'>(x,x)\in V</annotation></semantics></math>, while <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_249' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy='false'>)</mo><mo>∈</mo><msub><mi>Δ</mi> <mo>*</mo></msub><mo stretchy='false'>(</mo><mi>U</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(x,y)\in \Delta_\ast(U)</annotation></semantics></math> means <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_250' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mi>x</mi><mo>=</mo><mi>y</mi><mo stretchy='false'>)</mo><mo>→</mo><mo stretchy='false'>(</mo><mi>x</mi><mo>∈</mo><mi>U</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(x=y)\to (x\in U)</annotation></semantics></math>. Now let <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_251' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>U</mi><mo>=</mo><mi>∅</mi></mrow><annotation encoding='application/x-tex'>U = \emptyset</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_252' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>V</mi><mo>=</mo><mo stretchy='false'>{</mo><mo stretchy='false'>(</mo><mi>x</mi><mo>,</mo><mi>x</mi><mo stretchy='false'>)</mo><mo lspace='mediummathspace' rspace='mediummathspace'>∣</mo><mi>x</mi><mo>∈</mo><mi>X</mi><mo stretchy='false'>}</mo></mrow><annotation encoding='application/x-tex'>V = \{ (x,x) \mid x\in X \}</annotation></semantics></math>. Then <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_253' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy='false'>)</mo><mo>∈</mo><msub><mi>Δ</mi> <mo>*</mo></msub><mo stretchy='false'>(</mo><mi>U</mi><mo>∪</mo><msup><mi>Δ</mi> <mo>*</mo></msup><mo stretchy='false'>(</mo><mi>V</mi><mo stretchy='false'>)</mo><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(x,y) \in \Delta_\ast(U\cup \Delta^\ast(V))</annotation></semantics></math> means <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_254' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mi>x</mi><mo>=</mo><mi>y</mi><mo stretchy='false'>)</mo><mo>→</mo><mo stretchy='false'>(</mo><mo>⊥</mo><mo>∨</mo><mo stretchy='false'>(</mo><mi>x</mi><mo>=</mo><mi>x</mi><mo stretchy='false'>)</mo><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(x=y) \to (\bot \vee (x=x))</annotation></semantics></math>, which is a tautology; while <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_255' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy='false'>)</mo><mo>∈</mo><msub><mi>Δ</mi> <mo>*</mo></msub><mo stretchy='false'>(</mo><mi>U</mi><mo stretchy='false'>)</mo><mo>∪</mo><mi>V</mi></mrow><annotation encoding='application/x-tex'>(x,y) \in \Delta_\ast(U) \cup V</annotation></semantics></math> means <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_256' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mo stretchy='false'>(</mo><mi>x</mi><mo>=</mo><mi>y</mi><mo stretchy='false'>)</mo><mo>→</mo><mo>⊥</mo><mo stretchy='false'>)</mo><mo>∨</mo><mo stretchy='false'>(</mo><mi>x</mi><mo>=</mo><mi>y</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>((x=y)\to \bot) \vee (x=y)</annotation></semantics></math>, i.e. <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_257' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo>¬</mo><mo stretchy='false'>(</mo><mi>x</mi><mo>=</mo><mi>y</mi><mo stretchy='false'>)</mo><mo>∨</mo><mo stretchy='false'>(</mo><mi>x</mi><mo>=</mo><mi>y</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>\neg(x=y) \vee (x=y)</annotation></semantics></math>.</li> <li>I don’t know what it means for <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_258' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> to be localically weakly Hausdorff. (Weak closure in locales is very inexplicit.)</li> </ol> <p>In particular, the statement “all discrete locales are localically strongly Hausdorff” is equivalent to <a class='existingWikiWord' href='/nlab/show/diff/excluded+middle'>excluded middle</a>.</p> <p>However, non-discrete spaces can constructively be localically strongly Hausdorff without having decidable equality. For instance, any <a class='existingWikiWord' href='/nlab/show/diff/regular+space'>regular space</a> is also regular as a locale, and hence localically strongly Hausdorff. We can also say:</p> <div class='num_theorem' id='Apartness'> <h6 id='theorem'>Theorem</h6> <p>In any topological space <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_259' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math>, let <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_260' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>x</mi><mo>#</mo><mi>y</mi></mrow><annotation encoding='application/x-tex'>x\#y</annotation></semantics></math> mean that there exist opens <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_261' 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> with <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_262' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>x</mi><mo>∈</mo><mi>U</mi></mrow><annotation encoding='application/x-tex'>x\in U</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_263' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>y</mi><mo>∈</mo><mi>V</mi></mrow><annotation encoding='application/x-tex'>y\in V</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_264' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>U</mi><mo>∩</mo><mi>V</mi><mo>=</mo><mi>∅</mi></mrow><annotation encoding='application/x-tex'>U\cap V = \emptyset</annotation></semantics></math>; then <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_265' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo>#</mo></mrow><annotation encoding='application/x-tex'>\#</annotation></semantics></math> is always an <a class='existingWikiWord' href='/nlab/show/diff/inequality'>inequality relation</a>. If the spatial product <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_266' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi><mo>×</mo><mi>X</mi></mrow><annotation encoding='application/x-tex'>X\times X</annotation></semantics></math> coincides with the locale product (such as if <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_267' 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 class='existingWikiWord' href='/nlab/show/diff/locally+compact+topological+space'>locally compact</a>), then <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_268' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> is localically strongly Hausdorff if and only if <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_269' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo>#</mo></mrow><annotation encoding='application/x-tex'>\#</annotation></semantics></math> is an <a class='existingWikiWord' href='/nlab/show/diff/apartness+relation'>apartness relation</a> and every open set in <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_270' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> is <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_271' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo>#</mo></mrow><annotation encoding='application/x-tex'>\#</annotation></semantics></math>-open (i.e. for any <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_272' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>x</mi><mo>∈</mo><mi>U</mi></mrow><annotation encoding='application/x-tex'>x\in U</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_273' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>y</mi><mo>∈</mo><mi>X</mi></mrow><annotation encoding='application/x-tex'>y\in X</annotation></semantics></math> we have <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_274' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>y</mi><mo>∈</mo><mi>U</mi><mo>∨</mo><mi>x</mi><mo>#</mo><mi>y</mi></mrow><annotation encoding='application/x-tex'>y\in U \vee x\#y</annotation></semantics></math>).</p> </div> <div class='proof'> <h6 id='proof_3'>Proof</h6> <p>Note that <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_275' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo>#</mo></mrow><annotation encoding='application/x-tex'>\#</annotation></semantics></math> is, as a subset <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_276' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>W</mi> <mo>#</mo></msub><mo>⊆</mo><mi>X</mi><mo>×</mo><mi>X</mi></mrow><annotation encoding='application/x-tex'>W_\# \subseteq X\times X</annotation></semantics></math>, the exterior of the diagonal in the product topology, mentioned above. If <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_277' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> is localically strongly Hausdorff, then <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_278' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>W</mi> <mo>#</mo></msub></mrow><annotation encoding='application/x-tex'>W_\#</annotation></semantics></math> <em>must</em> be the open set of which the diagonal is the complementary closed sublocale, since it is the largest open set disjoint from the diagonal.</p> <p>To say that the diagonal <em>is</em> its complementary closed sublocale implies in particular that for any open set <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_279' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>U</mi><mo>⊆</mo><mi>X</mi></mrow><annotation encoding='application/x-tex'>U\subseteq X</annotation></semantics></math>, the open set <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_280' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mi>U</mi><mo>×</mo><mi>U</mi><mo stretchy='false'>)</mo><mo>∪</mo><msub><mi>W</mi> <mo>#</mo></msub></mrow><annotation encoding='application/x-tex'>(U\times U) \cup W_\#</annotation></semantics></math> is the largest open subset of <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_281' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi><mo>×</mo><mi>X</mi></mrow><annotation encoding='application/x-tex'>X\times X</annotation></semantics></math> whose intersection with the diagonal is contained in <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_282' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>U</mi><mo>∩</mo><mi>U</mi><mo>=</mo><mi>U</mi></mrow><annotation encoding='application/x-tex'>U\cap U = U</annotation></semantics></math>. Specifically, therefore, <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_283' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mi>U</mi><mo>×</mo><mi>U</mi><mo stretchy='false'>)</mo><mo>∪</mo><msub><mi>W</mi> <mo>#</mo></msub></mrow><annotation encoding='application/x-tex'>(U\times U) \cup W_\#</annotation></semantics></math> contains <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_284' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>U</mi><mo>×</mo><mi>X</mi></mrow><annotation encoding='application/x-tex'>U\times X</annotation></semantics></math> (since <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_285' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>U</mi><mo>×</mo><mi>X</mi></mrow><annotation encoding='application/x-tex'>U\times X</annotation></semantics></math> is an open subset of <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_286' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi><mo>×</mo><mi>X</mi></mrow><annotation encoding='application/x-tex'>X\times X</annotation></semantics></math> whose intersection with the diagonal is <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_287' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>U</mi></mrow><annotation encoding='application/x-tex'>U</annotation></semantics></math>). That is, if <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_288' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>x</mi><mo>∈</mo><mi>U</mi></mrow><annotation encoding='application/x-tex'>x\in U</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_289' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>y</mi><mo>∈</mo><mi>X</mi></mrow><annotation encoding='application/x-tex'>y\in X</annotation></semantics></math>, then either <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_290' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy='false'>)</mo><mo>∈</mo><mi>U</mi><mo>×</mo><mi>U</mi></mrow><annotation encoding='application/x-tex'>(x,y)\in U\times U</annotation></semantics></math> (i.e. <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_291' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>y</mi><mo>∈</mo><mi>U</mi></mrow><annotation encoding='application/x-tex'>y\in U</annotation></semantics></math>) or <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_292' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy='false'>)</mo><mo>∈</mo><msub><mi>W</mi> <mo>#</mo></msub></mrow><annotation encoding='application/x-tex'>(x,y)\in W_\#</annotation></semantics></math> (i.e. <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_293' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>x</mi><mo>#</mo><mi>y</mi></mrow><annotation encoding='application/x-tex'>x\#y</annotation></semantics></math>). This shows that <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_294' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>U</mi></mrow><annotation encoding='application/x-tex'>U</annotation></semantics></math> is <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_295' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo>#</mo></mrow><annotation encoding='application/x-tex'>\#</annotation></semantics></math>-open.</p> <p>To show that <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_296' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo>#</mo></mrow><annotation encoding='application/x-tex'>\#</annotation></semantics></math> is an apartness, note that for any <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_297' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>x</mi></mrow><annotation encoding='application/x-tex'>x</annotation></semantics></math> the set <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_298' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>{</mo><mi>z</mi><mo lspace='mediummathspace' rspace='mediummathspace'>∣</mo><mi>x</mi><mo>#</mo><mi>z</mi><mo stretchy='false'>}</mo></mrow><annotation encoding='application/x-tex'>\{ z \mid x\# z \}</annotation></semantics></math> is open, since it is the preimage of <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_299' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>W</mi> <mo>#</mo></msub></mrow><annotation encoding='application/x-tex'>W_\#</annotation></semantics></math> under a section of the second projection <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_300' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi><mo>×</mo><mi>X</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding='application/x-tex'>X\times X \to X</annotation></semantics></math>. Thus, it is <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_301' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo>#</mo></mrow><annotation encoding='application/x-tex'>\#</annotation></semantics></math>-open, which is to say that if <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_302' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>x</mi><mo>#</mo><mi>z</mi></mrow><annotation encoding='application/x-tex'>x\# z</annotation></semantics></math> then for any <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_303' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>y</mi></mrow><annotation encoding='application/x-tex'>y</annotation></semantics></math> either <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_304' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>x</mi><mo>#</mo><mi>y</mi></mrow><annotation encoding='application/x-tex'>x\#y</annotation></semantics></math> or <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_305' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>y</mi><mo>#</mo><mi>z</mi></mrow><annotation encoding='application/x-tex'>y\#z</annotation></semantics></math>, which is the missing <a class='existingWikiWord' href='/nlab/show/diff/comparison'>comparison</a> axiom for <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_306' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo>#</mo></mrow><annotation encoding='application/x-tex'>\#</annotation></semantics></math> to be an apartness.</p> <p>Conversely, suppose <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_307' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo>#</mo></mrow><annotation encoding='application/x-tex'>\#</annotation></semantics></math> is an apartness and every open set is <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_308' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo>#</mo></mrow><annotation encoding='application/x-tex'>\#</annotation></semantics></math>-open (i.e. the apartness topology refines the given topology on <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_309' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math>). Let <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_310' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>A</mi><mo>⊆</mo><mi>X</mi><mo>×</mo><mi>X</mi></mrow><annotation encoding='application/x-tex'>A\subseteq X\times X</annotation></semantics></math> be an open set; we must show that <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_311' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>A</mi><mo>∪</mo><msub><mi>W</mi> <mo>#</mo></msub></mrow><annotation encoding='application/x-tex'>A\cup W_\#</annotation></semantics></math> is the largest open subset of <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_312' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi><mo>×</mo><mi>X</mi></mrow><annotation encoding='application/x-tex'>X\times X</annotation></semantics></math> whose intersection with the diagonal is contained in <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_313' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>A</mi><mo>∩</mo><msub><mi>Δ</mi> <mi>X</mi></msub></mrow><annotation encoding='application/x-tex'>A\cap \Delta_X</annotation></semantics></math>. In other words, suppose <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_314' 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\times V</annotation></semantics></math> is a basic open in <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_315' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi><mo>×</mo><mi>X</mi></mrow><annotation encoding='application/x-tex'>X\times X</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_316' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mi>U</mi><mo>×</mo><mi>V</mi><mo stretchy='false'>)</mo><mo>∩</mo><msub><mi>Δ</mi> <mi>X</mi></msub></mrow><annotation encoding='application/x-tex'>(U\times V)\cap \Delta_X</annotation></semantics></math> (which is <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_317' 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 contained in <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_318' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>A</mi><mo>∩</mo><msub><mi>Δ</mi> <mi>X</mi></msub></mrow><annotation encoding='application/x-tex'>A\cap \Delta_X</annotation></semantics></math>; we must show <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_319' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>U</mi><mo>×</mo><mi>V</mi><mo>⊆</mo><mi>A</mi><mo>∪</mo><msub><mi>W</mi> <mo>#</mo></msub></mrow><annotation encoding='application/x-tex'>U\times V\subseteq A\cup W_\#</annotation></semantics></math>. In terms of elements, we assume that if <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_320' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>x</mi><mo>∈</mo><mi>U</mi></mrow><annotation encoding='application/x-tex'>x\in U</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_321' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>x</mi><mo>∈</mo><mi>V</mi></mrow><annotation encoding='application/x-tex'>x\in V</annotation></semantics></math> then <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_322' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mi>x</mi><mo>,</mo><mi>x</mi><mo stretchy='false'>)</mo><mo>∈</mo><mi>A</mi></mrow><annotation encoding='application/x-tex'>(x,x)\in A</annotation></semantics></math>, and we must show that if <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_323' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>x</mi><mo>∈</mo><mi>U</mi></mrow><annotation encoding='application/x-tex'>x\in U</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_324' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>y</mi><mo>∈</mo><mi>V</mi></mrow><annotation encoding='application/x-tex'>y\in V</annotation></semantics></math> then <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_325' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy='false'>)</mo><mo>∈</mo><mi>A</mi><mo>∨</mo><mi>x</mi><mo>#</mo><mi>y</mi></mrow><annotation encoding='application/x-tex'>(x,y)\in A \vee x\#y</annotation></semantics></math>.</p> <p>Assuming <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_326' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>x</mi><mo>∈</mo><mi>U</mi></mrow><annotation encoding='application/x-tex'>x\in U</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_327' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>y</mi><mo>∈</mo><mi>V</mi></mrow><annotation encoding='application/x-tex'>y\in V</annotation></semantics></math>, since <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_328' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>U</mi></mrow><annotation encoding='application/x-tex'>U</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_329' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>V</mi></mrow><annotation encoding='application/x-tex'>V</annotation></semantics></math> are <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_330' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo>#</mo></mrow><annotation encoding='application/x-tex'>\#</annotation></semantics></math>-open we have either <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_331' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>y</mi><mo>∈</mo><mi>U</mi></mrow><annotation encoding='application/x-tex'>y\in U</annotation></semantics></math> or <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_332' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>x</mi><mo>#</mo><mi>y</mi></mrow><annotation encoding='application/x-tex'>x\# y</annotation></semantics></math>, and either <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_333' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>x</mi><mo>∈</mo><mi>V</mi></mrow><annotation encoding='application/x-tex'>x\in V</annotation></semantics></math> or <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_334' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>x</mi><mo>#</mo><mi>y</mi></mrow><annotation encoding='application/x-tex'>x\#y</annotation></semantics></math>. Since we are done if <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_335' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>x</mi><mo>#</mo><mi>y</mi></mrow><annotation encoding='application/x-tex'>x\#y</annotation></semantics></math>, it suffices to assume <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_336' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>y</mi><mo>∈</mo><mi>U</mi></mrow><annotation encoding='application/x-tex'>y\in U</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_337' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>x</mi><mo>∈</mo><mi>V</mi></mrow><annotation encoding='application/x-tex'>x\in V</annotation></semantics></math>. Therefore, by assumption, <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_338' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mi>x</mi><mo>,</mo><mi>x</mi><mo stretchy='false'>)</mo><mo>∈</mo><mi>A</mi></mrow><annotation encoding='application/x-tex'>(x,x)\in A</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_339' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mi>y</mi><mo>,</mo><mi>y</mi><mo stretchy='false'>)</mo><mo>∈</mo><mi>A</mi></mrow><annotation encoding='application/x-tex'>(y,y)\in A</annotation></semantics></math>. Since <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_340' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>A</mi></mrow><annotation encoding='application/x-tex'>A</annotation></semantics></math> is open in the product topology, we have opens <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_341' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>U</mi><mo>′</mo><mo>,</mo><mi>V</mi><mo>′</mo></mrow><annotation encoding='application/x-tex'>U',V'</annotation></semantics></math> with <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_342' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>x</mi><mo>∈</mo><mi>U</mi><mo>′</mo></mrow><annotation encoding='application/x-tex'>x\in U'</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_343' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>y</mi><mo>∈</mo><mi>V</mi><mo>′</mo></mrow><annotation encoding='application/x-tex'>y\in V'</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_344' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>U</mi><mo>′</mo><mo>×</mo><mi>U</mi><mo>′</mo><mo>⊆</mo><mi>A</mi></mrow><annotation encoding='application/x-tex'>U'\times U'\subseteq A</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_345' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>V</mi><mo>′</mo><mo>×</mo><mi>V</mi><mo>′</mo><mo>⊆</mo><mi>A</mi></mrow><annotation encoding='application/x-tex'>V'\times V' \subseteq A</annotation></semantics></math>. But now <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_346' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo>#</mo></mrow><annotation encoding='application/x-tex'>\#</annotation></semantics></math>-openness of <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_347' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>U</mi><mo>′</mo></mrow><annotation encoding='application/x-tex'>U'</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_348' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>V</mi><mo>′</mo></mrow><annotation encoding='application/x-tex'>V'</annotation></semantics></math> tells us again that either <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_349' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>x</mi><mo>#</mo><mi>y</mi></mrow><annotation encoding='application/x-tex'>x\#y</annotation></semantics></math> (in which case we are done) or <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_350' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>y</mi><mo>∈</mo><mi>U</mi><mo>′</mo></mrow><annotation encoding='application/x-tex'>y\in U'</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_351' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>x</mi><mo>∈</mo><mi>V</mi><mo>′</mo></mrow><annotation encoding='application/x-tex'>x\in V'</annotation></semantics></math>. In the latter case, <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_352' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy='false'>)</mo><mo>∈</mo><mi>U</mi><mo>′</mo><mo>×</mo><mi>U</mi><mo>′</mo></mrow><annotation encoding='application/x-tex'>(x,y)\in U'\times U'</annotation></semantics></math> (and also <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_353' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>V</mi><mo>′</mo><mo>×</mo><mi>V</mi><mo>′</mo></mrow><annotation encoding='application/x-tex'>V'\times V'</annotation></semantics></math>), and hence is also in <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_354' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>A</mi></mrow><annotation encoding='application/x-tex'>A</annotation></semantics></math>.</p> </div> <p>Note that the apartness <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_355' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo>#</mo></mrow><annotation encoding='application/x-tex'>\#</annotation></semantics></math> need not be <a class='existingWikiWord' href='/nlab/show/diff/tight+relation'>tight</a>, and in particular <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_356' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> need not be spatially Hausdorff. In particular, if <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_357' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> might not even be <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_358' 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>: since localic Hausdorffness is (of course) only a property of the open-set lattice, it only “sees” the <a class='existingWikiWord' href='/nlab/show/diff/sober+topological+space'>sobrification</a> and in particular the <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_359' 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> quotient (<a class='existingWikiWord' href='/nlab/show/diff/Kolmogorov+topological+space'>Kolmogorov quotient</a>). However, this is all that can go wrong: if <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_360' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo>¬</mo><mo stretchy='false'>(</mo><mi>x</mi><mo>#</mo><mi>y</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>\neg(x\# y)</annotation></semantics></math>, then by <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_361' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo>#</mo></mrow><annotation encoding='application/x-tex'>\#</annotation></semantics></math>-openness every open set containing <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_362' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>x</mi></mrow><annotation encoding='application/x-tex'>x</annotation></semantics></math> must also contain <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_363' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>y</mi></mrow><annotation encoding='application/x-tex'>y</annotation></semantics></math> and vice versa, so if <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_364' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> is <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_365' 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> then <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_366' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>x</mi><mo>=</mo><mi>y</mi></mrow><annotation encoding='application/x-tex'>x=y</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_367' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo>#</mo></mrow><annotation encoding='application/x-tex'>\#</annotation></semantics></math> is tight.</p> <p>If the locale product <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_368' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi><mo>×</mo><mi>X</mi></mrow><annotation encoding='application/x-tex'>X\times X</annotation></semantics></math> does not coincide with the spatial product, then the “only if” direction of the above proof still works, if we define <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_369' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>W</mi> <mo>#</mo></msub></mrow><annotation encoding='application/x-tex'>W_\#</annotation></semantics></math> to be the open part of the locale product <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_370' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi><mo>×</mo><mi>X</mi></mrow><annotation encoding='application/x-tex'>X\times X</annotation></semantics></math> given by <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_371' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>W</mi> <mo>#</mo></msub><mo>=</mo><mo lspace='thinmathspace' rspace='thinmathspace'>⋁</mo><mo stretchy='false'>{</mo><mi>U</mi><mo>⊗</mo><mi>V</mi><mo lspace='mediummathspace' rspace='mediummathspace'>∣</mo><mi>U</mi><mo>∩</mo><mi>V</mi><mo>=</mo><mi>∅</mi><mo stretchy='false'>}</mo></mrow><annotation encoding='application/x-tex'>W_\# = \bigvee \{ U\otimes V \mid U\cap V = \emptyset \}</annotation></semantics></math>. A different proof is to recall that by <a href='/nlab/show/apartness+relation#ClosedLocalicEquivalenceRelation'>this theorem</a>, an apartness relation is the same as a (strongly) closed equivalence relation on a discrete locale, and the quotient of such an equivalence relation is the <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_372' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo>#</mo></mrow><annotation encoding='application/x-tex'>\#</annotation></semantics></math>-topology. Thus, if <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_373' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> is localically strongly Hausdorff, its diagonal is a closed equivalence relation, which yields by pullback a closed equivalence relation on the discrete locale <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_374' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>X</mi> <mi>d</mi></msub></mrow><annotation encoding='application/x-tex'>X_d</annotation></semantics></math> on the same set of points. This is the <a class='existingWikiWord' href='/nlab/show/diff/kernel+pair'>kernel pair</a> of the canonical surjection <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_375' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>X</mi> <mi>d</mi></msub><mo>→</mo><mi>X</mi></mrow><annotation encoding='application/x-tex'>X_d \to X</annotation></semantics></math>, and hence its quotient (the <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_376' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo>#</mo></mrow><annotation encoding='application/x-tex'>\#</annotation></semantics></math>-topology) maps to <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_377' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math>, i.e. refines the topology of <math class='maruku-mathml' display='inline' id='mathml_8125fbb1e04a1936d9c1077f4d85e96a5f020d7f_378' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math>.</p> <h2 id='history'>History</h2> <p>First introduced by <a class='existingWikiWord' href='/nlab/show/diff/Hausdorff+space'>Hausdorff</a> in <a class='existingWikiWord' href='/nlab/show/diff/Grundz%C3%BCge+der+Mengenlehre'>Grundzüge der Mengenlehre</a>, Hausdorff spaces were the original concept of a <a class='existingWikiWord' href='/nlab/show/diff/topological+space'>topological space</a>. Later the Hausdorffness condition was dropped, apparently first by <a class='existingWikiWord' href='/nlab/show/diff/Kazimierz+Kuratowski'>Kuratowski</a> in 1920s.</p> <h2 id='related_concepts'>Related concepts</h2> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/non-Hausdorff+topological+space'>non-Hausdorff topological space</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/locally+Hausdorff+topological+space'>locally Hausdorff topological space</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Kolmogorov+topological+space'>Kolmogorov topological space</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/nice+topological+space'>nice topological space</a></p> </li> </ul> <h2 id='references'>References</h2> <p>The notion is due to</p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/Felix+Hausdorff'>Felix Hausdorff</a>, <em><a class='existingWikiWord' href='/nlab/show/diff/Grundz%C3%BCge+der+Mengenlehre'>Grundzüge der Mengenlehre</a></em>, Leipzig: Veit (1914) [ISBN:978-0-8284-0061-9] Reprinted by Chelsea Publishing Company (1944, 1949, 1965) [[ark:/13960/t2891gn8g](https://archive.org/details/grundzgedermen00hausuoft/page/n5/mode/2up)]</li> </ul> <p>See also</p> <ul> <li>Wikipedia, <em><a href='https://en.wikipedia.org/wiki/Hausdorff_space'>Hausdorff space</a></em></li> </ul> <p>and the references at <em><a class='existingWikiWord' href='/nlab/show/diff/topology'>topology</a></em>.</p> <p>A detailed discussion of Hausdorff reflection is in</p> <ul> <li id='vanMunster14'>Bart van Munster, <em>The Hausdorff quotient</em>, 2014 (<a href='https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.728.5810'>pdf</a>)</li> </ul> <p>On epimorphisms of Hausdorff spaces:</p> <ul> <li id='LL18'>Jérôme Lapuyade-Lahorgue, <em>The epimorphisms of the category Haus are exactly the image-dense morphisms</em> (<a href='https://arxiv.org/abs/1810.00778'>arXiv:1810.00778</a>)</li> </ul> <p>Comments on the relation to <a class='existingWikiWord' href='/nlab/show/diff/sheaf+and+topos+theory'>topos theory</a>:</p> <ul> <li>S. Niefield, <em>A note on the locally Hausdorff property</em>, Cahiers TGDC (1983) (<a href='http://www.numdam.org/item?id=CTGDC_1983__24_1_87_0'>numdam</a>)</li> </ul> <p> </p> <p> </p> <p> </p> <p> </p> <p> </p> <p> </p> <p> </p> <p> </p> <p> </p> <p> </p> <p> </p> </div> <div class="revisedby"> <p> Last revised on November 18, 2024 at 13:44:58. 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