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width: 0.3em;"></span> <a href="/nlab/show/diff/HomePage" accesskey="H" title="Home page">Home Page</a> | <a href="/nlab/all_pages" accesskey="A" title="List of all pages">All Pages</a> | <a href="/nlab/latest_revisions" accesskey="U" title="Latest edits and page creations">Latest Revisions</a> | <a href="https://nforum.ncatlab.org/discussion/10335/#Item_13" title="Discuss this page in its dedicated thread on the nForum" style="color: black">Discuss this page</a> | <form accept-charset="utf-8" action="/nlab/search" id="navigationSearchForm" method="get"> <fieldset class="search"><input type="text" id="searchField" name="query" value="Search" style="display:inline-block; float: left;" onfocus="this.value == 'Search' ? this.value = '' : true" onblur="this.value == '' ? this.value = 'Search' : true" /></fieldset> </form> <span id='navEnd'></span> </div> <div id="revision"> <p class="show_diff"> Showing changes from revision #57 to #58: <ins class="diffins">Added</ins> | <del class="diffdel">Removed</del> | <del class="diffmod">Chan</del><ins class="diffmod">ged</ins> </p> <blockquote> <p>This is about the <a class='existingWikiWord' href='/nlab/show/diff/separation+axioms'>separation axioms</a> in <a class='existingWikiWord' href='/nlab/show/diff/topology'>topology</a>. For the axiom in <a class='existingWikiWord' href='/nlab/show/diff/set+theory'>set theory</a> also called “separation”, see <a class='existingWikiWord' href='/nlab/show/diff/axiom+of+separation'>axiom of separation</a>.</p> </blockquote> <hr /> <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='the_separation_axioms'>The separation axioms</h1> <div class='maruku_toc'><ul><li><a href='#Idea'>Idea</a><ul><li><a href='#SeparationAxiomInTermsOfLiftingProperties'>In terms of lifting properties</a></li></ul></li><li><a href='#TheClassicalTheory'>The classical theory</a><ul><li><a href='#separation_conditions'>Separation conditions</a></li><li><a href='#separation_axioms'>Separation axioms</a></li><li><a href='#EquivalentIncarnationsOfTheAxioms'>Reformulation in terms of topological closures</a></li><li><a href='#relations_between_the_axioms'>Relations between the axioms</a></li><li><a href='#Reflection'>Reflection</a></li><li><a href='#other_axioms'>Other axioms</a></li></ul></li><li><a href='#beyond_the_classical_theory'>Beyond the classical theory</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>The plain definition of <em><a class='existingWikiWord' href='/nlab/show/diff/topological+space'>topological space</a></em> happens to allow examples where distinct points or distinct subsets of the underlying set of a topological space appear as as more-or-less unseparable as seen by the topology on that set. In many applications one wants to exclude at least some of such degenerate examples from the discussion and instead focus on sufficiently <a class='existingWikiWord' href='/nlab/show/diff/nice+topological+space'>nice topological spaces</a>. The relevant conditions to be imposed on top of the plain <a class='existingWikiWord' href='/nlab/show/diff/axiom'>axioms</a> of a <a class='existingWikiWord' href='/nlab/show/diff/topological+space'>topological space</a> are hence known as <em>separation axioms</em>.</p> <p>These axioms are all of the form of saying that two <a class='existingWikiWord' href='/nlab/show/diff/subset'>subsets</a> (of certain forms) in the topological space are ‘separated’ from each other in one sense if they are ‘separated’ in a (generally) weaker sense. Most of these conditions can be expressed as <a class='existingWikiWord' href='/nlab/show/diff/separation+axioms+in+terms+of+lifting+properties'>lifting properties with respect to maps of finite topological spaces or the real line</a>. For example the weakest axiom (called <math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_10' 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>) demands that if two points are distinct as elements of the <a class='existingWikiWord' href='/nlab/show/diff/forgetful+functor'>underlying</a> set, then there exists at least one <a class='existingWikiWord' href='/nlab/show/diff/open+subspace'>open subset</a> that contains one but not the other, i.e. the function determined by two distinct points is not <a class='existingWikiWord' href='/nlab/show/diff/continuous+map'>continuous</a> as a map from the <a class='existingWikiWord' href='/nlab/show/diff/discrete+and+indiscrete+topology'>indiscrete space</a>. In other words, <math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_11' 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> says that each continuous function from the indiscrete space is necessarily trivial. As a <a class='existingWikiWord' href='/nlab/show/diff/lifting+property'>lifting property</a> this is expressed as: the map collapsing the indiscrete space with two points into a single point, has the left lifting property with respect to the map collapsing the underlying space into a single point.</p> <p>In this fashion one may impose a hierarchy of stronger axioms. For example demanding that given two distinct points, then each of them is contained in some open subset not containing the other (<math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_12' 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>), i.e. any map from the space with one open point and one closed is necessarily trivial, or that such a pair of open subsets around two distinct points may in addition be chosen to be disjoint (<math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_13' 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>); such a pair of open subsets can be equivalently described as a continuous function to the space with two open points and one closed, and <math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_14' 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> as saying that the inclusion of two open points into the space with two points and one closed, factors via any injective function from the space with two points, to the underlying space. <br />This last condition, <math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_15' 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>, also called the <em><a class='existingWikiWord' href='/nlab/show/diff/Hausdorff+space'>Hausdorff condition</a></em> is the most common among all separation axioms. Often (but by far not always) this is considered by default.</p> <p>As a lifting property, <math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_16' 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> is expressed as: the map collapsing the space with one point open and one point closed into a single point, has the left lifting property with respect to the map collapsing the underlying space into a single point. <math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_17' 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> is expressed as: any injective map from the discerete space with two points into the space, has the left lifting proprety with respect to the map collapsing the space with one closed point and two open points into a single point.</p> <p>Rewriting the <a class='existingWikiWord' href='/nlab/show/diff/separation+axioms+in+terms+of+lifting+properties'>separation axioms in terms of lifting properties</a> with respect to maps of finite topological spaces, i.e. monotone maps of preorders, provides a combinatorial point of view on this hierachy, and simplifies certain universal constructions such as <a href='#Reflection'>reflection</a> or <a href='#KolmogorovQuotient'>Kolmogorov quotient</a>.</p> <p>The main separation axioms are these:</p> <p id='TableOfMainSeparationAxioms'>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_e1760b45eef4a61baf80f64f3a3292d3cfff7533_18' 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_e1760b45eef4a61baf80f64f3a3292d3cfff7533_19' 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_e1760b45eef4a61baf80f64f3a3292d3cfff7533_20' 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_e1760b45eef4a61baf80f64f3a3292d3cfff7533_21' 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_e1760b45eef4a61baf80f64f3a3292d3cfff7533_22' 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_e1760b45eef4a61baf80f64f3a3292d3cfff7533_23' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>T</mi> <mn>3</mn></msub></mrow><annotation encoding='application/x-tex'>T_3</annotation></semantics></math></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_e1760b45eef4a61baf80f64f3a3292d3cfff7533_24' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>T</mi> <mn>4</mn></msub></mrow><annotation encoding='application/x-tex'>T_4</annotation></semantics></math></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> <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_e1760b45eef4a61baf80f64f3a3292d3cfff7533_25' 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_e1760b45eef4a61baf80f64f3a3292d3cfff7533_26' 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_e1760b45eef4a61baf80f64f3a3292d3cfff7533_27' 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_e1760b45eef4a61baf80f64f3a3292d3cfff7533_28' 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_e1760b45eef4a61baf80f64f3a3292d3cfff7533_29' 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_e1760b45eef4a61baf80f64f3a3292d3cfff7533_30' 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_e1760b45eef4a61baf80f64f3a3292d3cfff7533_31' 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_e1760b45eef4a61baf80f64f3a3292d3cfff7533_32' 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_e1760b45eef4a61baf80f64f3a3292d3cfff7533_33' 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_e1760b45eef4a61baf80f64f3a3292d3cfff7533_34' 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_e1760b45eef4a61baf80f64f3a3292d3cfff7533_35' 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_e1760b45eef4a61baf80f64f3a3292d3cfff7533_36' 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_e1760b45eef4a61baf80f64f3a3292d3cfff7533_37' 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_e1760b45eef4a61baf80f64f3a3292d3cfff7533_38' 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_e1760b45eef4a61baf80f64f3a3292d3cfff7533_39' 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_e1760b45eef4a61baf80f64f3a3292d3cfff7533_40' 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> <p>Originally in <a href='#Tietze23'>Tietze 23</a> the four separation axioms <math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_41' 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>3</mn></msub><mo>,</mo><msub><mi>T</mi> <mn>4</mn></msub><mo>,</mo><msub><mi>T</mi> <mn>5</mn></msub></mrow><annotation encoding='application/x-tex'>T_2, T_3, T_4, T_5</annotation></semantics></math> were considered (see at <em><a href='#History'>History</a></em> below for more); nowadays one considers various more. Besides the extrapolation of the original sequence from <math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_42' 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> through <math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_43' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>T</mi> <mn>6</mn></msub></mrow><annotation encoding='application/x-tex'>T_6</annotation></semantics></math> (with <math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_44' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>T</mi> <mrow><mn>2</mn><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></msub></mrow><annotation encoding='application/x-tex'>T_{2\frac{1}{2}}</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_45' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>T</mi> <mrow><mn>3</mn><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></msub></mrow><annotation encoding='application/x-tex'>T_{3\frac{1}{2}}</annotation></semantics></math> interpolated), there is a similar sequence of axioms called <math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_46' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>R</mi> <mn>0</mn></msub><mo>,</mo><msub><mi>R</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>R</mi> <mn>2</mn></msub><mo>,</mo><msub><mi>R</mi> <mn>3</mn></msub></mrow><annotation encoding='application/x-tex'>R_0, R_1, R_2, R_3</annotation></semantics></math> (with their extrapolations and interpolations) of the same form, except that they do not start with mentioning two set-theoretically distinct points, but two points satisfying the conclusion of <math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_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>. This and more is spelled out <a href='#TheClassicalTheory'>below</a>.</p> <p>There are also axioms that do not follow the pattern of “if certain two subsets are separated in some weak sense, then they are also separated in some stronger sense”, but that still axiomatize some kind of separatedness. For example the condition on a <a class='existingWikiWord' href='/nlab/show/diff/topological+space'>topological space</a> being <a class='existingWikiWord' href='/nlab/show/diff/sober+topological+space'>sober</a> is of a different nature, but is implied by <math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_48' 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> and implies <math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_49' 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>. Notice that via their <a class='existingWikiWord' href='/nlab/show/diff/full+subcategory'>full embedding</a> into <a class='existingWikiWord' href='/nlab/show/diff/locale'>locales</a>, <a class='existingWikiWord' href='/nlab/show/diff/sober+topological+space'>sober topological spaces</a> may be understood without reference to their underlying set of points at all.</p> <p>All separation axioms are satisfied by <a class='existingWikiWord' href='/nlab/show/diff/metric+space'>metric spaces</a>, from whom the concept of topological space was originally abstracted. Hence imposing some of them may also be understood as gauging just how far one allows topological spaces to generalize away from metric spaces.</p> <p>Several separation axioms may also be interpreted in broader contexts that plain topological spaces, for instance for <a class='existingWikiWord' href='/nlab/show/diff/convergence+space'>convergence space</a> or for <a class='existingWikiWord' href='/nlab/show/diff/locale'>locales</a>; or the may be considered under weaker assumptions, such as those of <a class='existingWikiWord' href='/nlab/show/diff/constructive+mathematics'>constructive mathematics</a> and <a class='existingWikiWord' href='/nlab/show/diff/predicative+mathematics'>predicative mathematics</a>.</p> <h2 id='TheClassicalTheory'>The classical theory</h2> <p>First, we will consider how, for topological spaces in <a class='existingWikiWord' href='/nlab/show/diff/classical+mathematics'>classical mathematics</a>, the separation axioms are about sets' being ‘separated’ as stated above. Throughout, fix a <a class='existingWikiWord' href='/nlab/show/diff/topological+space'>topological space</a> <math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_50' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>S</mi></mrow><annotation encoding='application/x-tex'>S</annotation></semantics></math>.</p> <h3 id='separation_conditions'>Separation conditions</h3> <p>Fix two sets (<a class='existingWikiWord' href='/nlab/show/diff/subset'>subsets</a>) <math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_51' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>F</mi></mrow><annotation encoding='application/x-tex'>F</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_52' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>G</mi></mrow><annotation encoding='application/x-tex'>G</annotation></semantics></math> of <math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_53' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>S</mi></mrow><annotation encoding='application/x-tex'>S</annotation></semantics></math>.</p> <ul> <li> <p>The sets <math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_54' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>F</mi></mrow><annotation encoding='application/x-tex'>F</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_55' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>G</mi></mrow><annotation encoding='application/x-tex'>G</annotation></semantics></math> are <strong><a class='existingWikiWord' href='/nlab/show/diff/disjoint+subsets'>disjoint</a></strong> if their <a class='existingWikiWord' href='/nlab/show/diff/intersection'>intersection</a> is <a class='existingWikiWord' href='/nlab/show/diff/empty+set'>empty</a>:</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_56' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>F</mi><mo>∩</mo><mi>G</mi><mo>=</mo><mi>∅</mi><mo>.</mo></mrow><annotation encoding='application/x-tex'> F \cap G = \empty .</annotation></semantics></math></div></li> <li> <p>They are <strong>topologically disjoint</strong> if there exists a <a class='existingWikiWord' href='/nlab/show/diff/neighborhood'>neighbourhood</a> of one set that is disjoint from the other set:</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_57' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mo>∃</mo><mspace width='thickmathspace' /><mi>U</mi><mover><mo>⊇</mo><mo>∘</mo></mover><mi>F</mi><mo>,</mo><mspace width='thickmathspace' /><mi>U</mi><mo>∩</mo><mi>G</mi><mo>=</mo><mi>∅</mi><mo stretchy='false'>)</mo><mspace width='thickmathspace' /><mo>∨</mo><mspace width='thickmathspace' /><mo stretchy='false'>(</mo><mo>∃</mo><mspace width='thickmathspace' /><mi>V</mi><mover><mo>⊇</mo><mo>∘</mo></mover><mi>G</mi><mo>,</mo><mspace width='thickmathspace' /><mi>F</mi><mo>∩</mo><mi>V</mi><mo>=</mo><mi>∅</mi><mo stretchy='false'>)</mo><mo>.</mo></mrow><annotation encoding='application/x-tex'> (\exists\; U \stackrel{\circ}\supseteq F,\; U \cap G = \empty) \;\vee\; (\exists\; V \stackrel{\circ}\supseteq G,\; F \cap V = \empty) .</annotation></semantics></math></div> <p>Notice that topologically disjoint sets must be disjoint.</p> </li> <li> <p>They are <strong>separated</strong> if each set has a neighbourhood that is disjoint from the other set:</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_58' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mo>∃</mo><mspace width='thickmathspace' /><mi>U</mi><mover><mo>⊇</mo><mo>∘</mo></mover><mi>F</mi><mo>,</mo><mspace width='thickmathspace' /><mi>U</mi><mo>∩</mo><mi>G</mi><mo>=</mo><mi>∅</mi><mo stretchy='false'>)</mo><mspace width='thickmathspace' /><mo>∧</mo><mspace width='thickmathspace' /><mo stretchy='false'>(</mo><mo>∃</mo><mspace width='thickmathspace' /><mi>V</mi><mover><mo>⊇</mo><mo>∘</mo></mover><mi>G</mi><mo>,</mo><mspace width='thickmathspace' /><mi>F</mi><mo>∩</mo><mi>V</mi><mo>=</mo><mi>∅</mi><mo stretchy='false'>)</mo><mspace width='thickmathspace' /><mspace width='thickmathspace' /><mo>≡</mo><mspace width='thickmathspace' /><mspace width='thickmathspace' /><mo>∃</mo><mspace width='thickmathspace' /><mi>U</mi><mover><mo>⊇</mo><mo>∘</mo></mover><mi>F</mi><mo>,</mo><mspace width='thickmathspace' /><mo>∃</mo><mspace width='thickmathspace' /><mi>V</mi><mover><mo>⊇</mo><mo>∘</mo></mover><mi>G</mi><mo>,</mo><mspace width='thickmathspace' /><mi>U</mi><mo>∩</mo><mi>G</mi><mo>=</mo><mi>∅</mi><mspace width='thickmathspace' /><mo>∧</mo><mspace width='thickmathspace' /><mi>F</mi><mo>∩</mo><mi>V</mi><mo>=</mo><mi>∅</mi><mo>.</mo></mrow><annotation encoding='application/x-tex'> (\exists\; U \stackrel{\circ}\supseteq F,\; U \cap G = \empty) \;\wedge\; (\exists\; V \stackrel{\circ}\supseteq G,\; F \cap V = \empty) \;\;\equiv\;\; \exists\; U \stackrel{\circ}\supseteq F,\; \exists\; V \stackrel{\circ}\supseteq G,\; U \cap G = \empty \;\wedge\; F \cap V = \empty . </annotation></semantics></math></div> <p>Notice that separated sets must be topologically disjoint.</p> </li> <li id='SeparatedByNeighbourhoods'> <p>They are <strong>separated by neighbourhoods</strong> if they have disjoint neighbourhoods:</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_59' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo>∃</mo><mspace width='thickmathspace' /><mi>U</mi><mover><mo>⊇</mo><mo>∘</mo></mover><mi>F</mi><mo>,</mo><mspace width='thickmathspace' /><mo>∃</mo><mspace width='thickmathspace' /><mi>V</mi><mover><mo>⊇</mo><mo>∘</mo></mover><mi>G</mi><mo>,</mo><mspace width='thickmathspace' /><mi>U</mi><mo>∩</mo><mi>V</mi><mo>=</mo><mi>∅</mi><mo>.</mo></mrow><annotation encoding='application/x-tex'> \exists\; U \stackrel{\circ}\supseteq F,\; \exists\; V \stackrel{\circ}\supseteq G,\; U \cap V = \empty .</annotation></semantics></math></div> <p>Notice that sets separated by neighbourhoods must be separated.</p> </li> <li> <p>They are <strong>separated by closed neighbourhoods</strong> if they have disjoint closed neighbourhoods:</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_60' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo>∃</mo><mspace width='thickmathspace' /><mi>U</mi><mover><mo>⊇</mo><mo>∘</mo></mover><mi>F</mi><mo>,</mo><mspace width='thickmathspace' /><mo>∃</mo><mspace width='thickmathspace' /><mi>V</mi><mover><mo>⊇</mo><mo>∘</mo></mover><mi>G</mi><mo>,</mo><mspace width='thickmathspace' /><mi>Cl</mi><mo stretchy='false'>(</mo><mi>U</mi><mo stretchy='false'>)</mo><mo>∩</mo><mi>Cl</mi><mo stretchy='false'>(</mo><mi>V</mi><mo stretchy='false'>)</mo><mo>=</mo><mi>∅</mi><mo>.</mo></mrow><annotation encoding='application/x-tex'> \exists\; U \stackrel{\circ}\supseteq F,\; \exists\; V \stackrel{\circ}\supseteq G,\; Cl(U) \cap Cl(V) = \empty .</annotation></semantics></math></div> <p>Notice that sets separated by closed neighbourhoods must be separated by neighbourhoods.</p> </li> <li> <p>They are <strong>separated by a function</strong> if there exists a continuous <a class='existingWikiWord' href='/nlab/show/diff/real+number'>real</a>-valued <a class='existingWikiWord' href='/nlab/show/diff/function'>function</a> on the space that maps <math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_61' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>F</mi></mrow><annotation encoding='application/x-tex'>F</annotation></semantics></math> to <math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_62' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>0</mn></mrow><annotation encoding='application/x-tex'>0</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_63' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>G</mi></mrow><annotation encoding='application/x-tex'>G</annotation></semantics></math> to <math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_64' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>1</mn></mrow><annotation encoding='application/x-tex'>1</annotation></semantics></math>:</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_65' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo>∃</mo><mspace width='thickmathspace' /><mi>f</mi><mo>:</mo><mi>S</mi><mo>→</mo><mstyle mathvariant='bold'><mi>R</mi></mstyle><mo>,</mo><mspace width='thickmathspace' /><mi>F</mi><mo>⊆</mo><msup><mi>f</mi> <mo>*</mo></msup><mo stretchy='false'>(</mo><mo stretchy='false'>{</mo><mn>0</mn><mo stretchy='false'>}</mo><mo stretchy='false'>)</mo><mspace width='thickmathspace' /><mo>∧</mo><mspace width='thickmathspace' /><mi>G</mi><mo>⊆</mo><msup><mi>f</mi> <mo>*</mo></msup><mo stretchy='false'>(</mo><mo stretchy='false'>{</mo><mn>1</mn><mo stretchy='false'>}</mo><mo stretchy='false'>)</mo><mo>.</mo></mrow><annotation encoding='application/x-tex'> \exists\; f: S \to \mathbf{R},\; F \subseteq f^*(\{0\}) \;\wedge\; G \subseteq f^*(\{1\}) .</annotation></semantics></math></div> <p>Notice that sets separated by a function must be separated by closed neighbourhoods (the preimages of <math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_66' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>[</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mi>ϵ</mi><mo>,</mo><mi>ϵ</mi><mo stretchy='false'>]</mo></mrow><annotation encoding='application/x-tex'>[-\epsilon, \epsilon]</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_67' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>[</mo><mn>1</mn><mo>−</mo><mi>ϵ</mi><mo>,</mo><mn>1</mn><mo>+</mo><mi>ϵ</mi><mo stretchy='false'>]</mo></mrow><annotation encoding='application/x-tex'>[1-\epsilon, 1+\epsilon]</annotation></semantics></math>).</p> </li> <li> <p>Finally, they are <strong>precisely separated by a function</strong> if there exists a continuous real-valued function on the space that maps precisely <math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_68' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>F</mi></mrow><annotation encoding='application/x-tex'>F</annotation></semantics></math> to <math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_69' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>0</mn></mrow><annotation encoding='application/x-tex'>0</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_70' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>G</mi></mrow><annotation encoding='application/x-tex'>G</annotation></semantics></math> to <math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_71' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>1</mn></mrow><annotation encoding='application/x-tex'>1</annotation></semantics></math>:</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_72' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo>∃</mo><mspace width='thickmathspace' /><mi>f</mi><mo>:</mo><mi>S</mi><mo>→</mo><mstyle mathvariant='bold'><mi>R</mi></mstyle><mo>,</mo><mspace width='thickmathspace' /><mi>F</mi><mo>=</mo><msup><mi>f</mi> <mo>*</mo></msup><mo stretchy='false'>(</mo><mo stretchy='false'>{</mo><mn>0</mn><mo stretchy='false'>}</mo><mo stretchy='false'>)</mo><mspace width='thickmathspace' /><mo>∧</mo><mspace width='thickmathspace' /><mi>G</mi><mo>=</mo><msup><mi>f</mi> <mo>*</mo></msup><mo stretchy='false'>(</mo><mo stretchy='false'>{</mo><mn>1</mn><mo stretchy='false'>}</mo><mo stretchy='false'>)</mo><mo>.</mo></mrow><annotation encoding='application/x-tex'> \exists\; f: S \to \mathbf{R},\; F = f^*(\{0\}) \;\wedge\; G = f^*(\{1\}) .</annotation></semantics></math></div> <p>Notice that sets precisely separated by a function must be separated by a function.</p> </li> </ul> <p>Often <math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_73' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>F</mi></mrow><annotation encoding='application/x-tex'>F</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_74' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>G</mi></mrow><annotation encoding='application/x-tex'>G</annotation></semantics></math> will be points (identified with their <a class='existingWikiWord' href='/nlab/show/diff/singleton'>singleton</a> subsets); in that case, one usually says <em>distinct</em> in place of <em>disjoint</em>.</p> <p>Often <math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_75' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>F</mi></mrow><annotation encoding='application/x-tex'>F</annotation></semantics></math> or <math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_76' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>G</mi></mrow><annotation encoding='application/x-tex'>G</annotation></semantics></math> will be closed sets; notice that disjoint closed sets are automatically separated, while a closed set and a point, if disjoint, are automatically topologically disjoint.</p> <h3 id='separation_axioms'>Separation axioms</h3> <p>The classical separation axioms are all statements of the form</p> <ul> <li>When <math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_77' 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 (point/closed) set and <math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_78' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>G</mi></mrow><annotation encoding='application/x-tex'>G</annotation></semantics></math> is a (point/closed) set, if <math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_79' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>F</mi></mrow><annotation encoding='application/x-tex'>F</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_80' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>G</mi></mrow><annotation encoding='application/x-tex'>G</annotation></semantics></math> are (separated in some weak sense), then they are (separated in some strong sense).</li> </ul> <p>The axioms with names (at least with known to the authors so far of this article) are summarised in the tables below. When a row or column is missing from a table, either no name is known or the implication follows from the converses mentioned after the separation conditions above in the context of that table; there are two potential tables that are completely blank for the latter reason. When an entry in a table is repeated, that corresponds to a theorem that one separation axiom implies another.</p> <p>When both sets are points:</p> <table><tr><th>Stronger condition ↓\Weaker condition →</th> <th>Distinct</th> <th>Topologically distinct</th></tr> <tr><th>Topologically distinct</th> <td><math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_1' xmlns='http://www.w3.org/1998/Math/MathML'> <semantics> <mrow> <msub><mi>T</mi> <mn>0</mn></msub> </mrow> <annotation encoding='application/x-tex'>T_0</annotation> </semantics> </math></td></tr> <tr><th>Separated</th> <td><math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_2' xmlns='http://www.w3.org/1998/Math/MathML'> <semantics> <mrow> <msub><mi>T</mi> <mn>1</mn></msub> </mrow> <annotation encoding='application/x-tex'>T_1</annotation> </semantics> </math></td> <td><math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_3' xmlns='http://www.w3.org/1998/Math/MathML'> <semantics> <mrow> <msub><mi>R</mi> <mn>0</mn></msub> </mrow> <annotation encoding='application/x-tex'>R_0</annotation> </semantics> </math></td></tr> <tr><th>Separated by neighbourhoods</th> <td><math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_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><math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_5' 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></td></tr> <tr><th>Separated by closed neighbourhoods</th> <td><math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_6' xmlns='http://www.w3.org/1998/Math/MathML'> <semantics> <mrow> <msub><mi>T</mi> <mrow><mn>2</mn><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></msub> </mrow> <annotation encoding='application/x-tex'>T_{2\frac{1}{2}}</annotation> </semantics> </math></td> <td><math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_7' xmlns='http://www.w3.org/1998/Math/MathML'> <semantics> <mrow> <msub><mi>R</mi> <mrow><mn>1</mn><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></msub> </mrow> <annotation encoding='application/x-tex'>R_{1\frac{1}{2}}</annotation> </semantics> </math></td></tr> <tr><th>Separated by a function</th> <td>Completely <math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_8' 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>Completely <math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_9' 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></td></tr></table> <p>When one set is a point and the other is closed:</p> <table><tr><th>Stronger condition ↓\Weaker condition →</th> <th>Disjoint</th></tr> <tr><th>Separated by neighbourhoods</th> <td>Regular</td></tr> <tr><th>Separated by closed neighbourhoods</th> <td>Regular</td></tr> <tr><th>Separated by a function</th> <td>Completely regular</td></tr></table> <p>When both sets are closed:</p> <table><tr><th>Stronger condition ↓\Weaker condition →</th> <th>Disjoint</th></tr> <tr><th>Separated by neighbourhoods</th> <td>Normal</td></tr> <tr><th>Separated by closed neighbourhoods</th> <td>Normal</td></tr> <tr><th>Separated by a function</th> <td>Normal</td></tr> <tr><th>Precisely separated by a function</th> <td>Perfectly normal</td></tr></table> <p>When the sets are arbitrary:</p> <table><tr><th>Stronger condition ↓\Weaker condition →</th> <th>Separated</th></tr> <tr><th>Separated by neighbourhoods</th> <td>Completely normal ($T_5$)</td></tr></table> <h3 id='EquivalentIncarnationsOfTheAxioms'>Reformulation in terms of topological closures</h3> <p>Many of the separation axioms have a useful equivalent formulation in terms of certain <a class='existingWikiWord' href='/nlab/show/diff/closed+subspace'>topological closures</a>.</p> <div class='num_prop' id='T0InTermsOfClosureOfPoints'> <h6 id='proposition'>Proposition</h6> <p><strong>(<math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_81' 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> in terms of topological closures)</strong></p> <p>A <a class='existingWikiWord' href='/nlab/show/diff/topological+space'>topological space</a> <math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_82' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mi>X</mi><mo>,</mo><mi>τ</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(X,\tau)</annotation></semantics></math> is <math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_83' 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> precisely if the function <math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_84' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Cl</mi><mo stretchy='false'>(</mo><mo stretchy='false'>{</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mo stretchy='false'>}</mo><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>Cl(\{-\})</annotation></semantics></math> from the underlying set of <math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_85' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> to the set of <a class='existingWikiWord' href='/nlab/show/diff/irreducible+closed+subspace'>irreducible closed subsets</a> of <math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_86' 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/injection'>injective</a>:</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_87' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Cl</mi><mo stretchy='false'>(</mo><mo stretchy='false'>{</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mo stretchy='false'>}</mo><mo stretchy='false'>)</mo><mspace width='thickmathspace' /><mo lspace='verythinmathspace'>:</mo><mspace width='thickmathspace' /><mi>X</mi><mo>↪</mo><mi>IrrClSub</mi><mo stretchy='false'>(</mo><mi>X</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'> Cl(\{-\}) \;\colon\; X \hookrightarrow IrrClSub(X) </annotation></semantics></math></div></div> <p>(This statement also motivates the definition of <a class='existingWikiWord' href='/nlab/show/diff/sober+topological+space'>sober topological spaces</a>, for which <math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_88' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Cl</mi><mo stretchy='false'>(</mo><mo stretchy='false'>{</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mo stretchy='false'>}</mo><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>Cl(\{-\})</annotation></semantics></math> is required to be a <a class='existingWikiWord' href='/nlab/show/diff/bijection'>bijection</a>).</p> <div class='proof'> <h6 id='proof'>Proof</h6> <p>Assume first that <math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_89' 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_e1760b45eef4a61baf80f64f3a3292d3cfff7533_90' 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 we need to show that if <math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_91' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>x</mi><mo>,</mo><mi>y</mi><mo>∈</mo><mi>X</mi></mrow><annotation encoding='application/x-tex'>x,y \in X</annotation></semantics></math> are such that <math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_92' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Cl</mi><mo stretchy='false'>(</mo><mo stretchy='false'>{</mo><mi>x</mi><mo stretchy='false'>}</mo><mo stretchy='false'>)</mo><mo>=</mo><mi>Cl</mi><mo stretchy='false'>(</mo><mo stretchy='false'>{</mo><mi>y</mi><mo stretchy='false'>}</mo><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>Cl(\{x\}) = Cl(\{y\})</annotation></semantics></math> then <math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_93' 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>. Hence assume that <math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_94' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Cl</mi><mo stretchy='false'>(</mo><mo stretchy='false'>{</mo><mi>x</mi><mo stretchy='false'>}</mo><mo stretchy='false'>)</mo><mo>=</mo><mi>Cl</mi><mo stretchy='false'>(</mo><mo stretchy='false'>{</mo><mi>y</mi><mo stretchy='false'>}</mo><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>Cl(\{x\}) = Cl(\{y\})</annotation></semantics></math>. Since the closure of a point is the <a class='existingWikiWord' href='/nlab/show/diff/complement'>complement</a> of the union of the open subsets not containing the point (lemma \ref{UnionOfOpensGivesClosure}), this means that the union of open subsets that do not contain <math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_95' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>x</mi></mrow><annotation encoding='application/x-tex'>x</annotation></semantics></math> is the same as the union of open subsets that do not contain <math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_96' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>y</mi></mrow><annotation encoding='application/x-tex'>y</annotation></semantics></math>:</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_97' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><munder><mo>∪</mo><mfrac linethickness='0'><mrow><mrow><mi>U</mi><mo>⊂</mo><mi>X</mi><mspace width='thinmathspace' /><mtext>open</mtext></mrow></mrow><mrow><mrow><mi>U</mi><mo>⊂</mo><mi>X</mi><mo>\</mo><mo stretchy='false'>{</mo><mi>x</mi><mo stretchy='false'>}</mo></mrow></mrow></mfrac></munder><mrow><mo>(</mo><mi>U</mi><mo>)</mo></mrow><mspace width='thickmathspace' /><mo>=</mo><mspace width='thickmathspace' /><munder><mo>∪</mo><mfrac linethickness='0'><mrow><mrow><mi>U</mi><mo>⊂</mo><mi>X</mi><mspace width='thinmathspace' /><mtext>open</mtext></mrow></mrow><mrow><mrow><mi>U</mi><mo>⊂</mo><mi>X</mi><mo>\</mo><mo stretchy='false'>{</mo><mi>y</mi><mo stretchy='false'>}</mo></mrow></mrow></mfrac></munder><mrow><mo>(</mo><mi>U</mi><mo>)</mo></mrow></mrow><annotation encoding='application/x-tex'> \underset{ {U \subset X \, \text{open}} \atop { U \subset X\backslash \{x\} } }{\cup} \left( U \right) \;=\; \underset{ {U \subset X \, \text{open}} \atop { U \subset X\backslash \{y\} } }{\cup} \left( U \right) </annotation></semantics></math></div> <p>But if the two points were distinct, <math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_98' 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 by <math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_99' 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> one of the above unions would contain <math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_100' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>x</mi></mrow><annotation encoding='application/x-tex'>x</annotation></semantics></math> or <math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_101' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>y</mi></mrow><annotation encoding='application/x-tex'>y</annotation></semantics></math>, while the other would not, in contradiction to the above equality. Hence we have a <a class='existingWikiWord' href='/nlab/show/diff/proof+by+contradiction'>proof by contradiction</a>.</p> <p>Conversely, assume that <math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_102' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mrow><mo>(</mo><mi>Cl</mi><mo stretchy='false'>{</mo><mi>x</mi><mo stretchy='false'>}</mo><mo>=</mo><mi>Cl</mi><mo stretchy='false'>{</mo><mi>y</mi><mo stretchy='false'>}</mo><mo>)</mo></mrow><mo>⇒</mo><mrow><mo>(</mo><mi>x</mi><mo>=</mo><mi>y</mi><mo>)</mo></mrow></mrow><annotation encoding='application/x-tex'>\left( Cl\{x\} = Cl\{y\}\right) \Rightarrow \left( x = y\right)</annotation></semantics></math>, and assume that <math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_103' 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>. Hence by <a class='existingWikiWord' href='/nlab/show/diff/contrapositive'>contraposition</a> <math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_104' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi mathvariant='normal'>Cl</mi><mo stretchy='false'>(</mo><mo stretchy='false'>{</mo><mi>x</mi><mo stretchy='false'>}</mo><mo stretchy='false'>)</mo><mo>≠</mo><mi mathvariant='normal'>Cl</mi><mo stretchy='false'>(</mo><mo stretchy='false'>{</mo><mi>y</mi><mo stretchy='false'>}</mo><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>\mathrm{Cl}(\{x\}) \neq \mathrm{Cl}(\{y\})</annotation></semantics></math>. We need to show that there exists an open set which contains one of the two points, but not the other.</p> <p>Assume there were no such open subset. By lemma <a href='Introduction+to+Topology+--+1#UnionOfOpensGivesClosure'>this lemma</a> this would mean that <math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_105' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>x</mi><mo>∈</mo><mi mathvariant='normal'>Cl</mi><mo stretchy='false'>(</mo><mo stretchy='false'>{</mo><mi>y</mi><mo stretchy='false'>}</mo><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>x \in \mathrm{Cl}(\{y\})</annotation></semantics></math> and that <math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_106' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>y</mi><mo>∈</mo><mi mathvariant='normal'>Cl</mi><mo stretchy='false'>(</mo><mo stretchy='false'>{</mo><mi>x</mi><mo stretchy='false'>}</mo><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>y \in \mathrm{Cl}(\{x\})</annotation></semantics></math>. But this would imply that <math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_107' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Cl</mi><mo stretchy='false'>(</mo><mo stretchy='false'>{</mo><mi>x</mi><mo stretchy='false'>}</mo><mo stretchy='false'>)</mo><mo>⊂</mo><mi mathvariant='normal'>Cl</mi><mo stretchy='false'>(</mo><mo stretchy='false'>{</mo><mi>y</mi><mo stretchy='false'>}</mo><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>Cl(\{x\}) \subset \mathrm{Cl}(\{y\})</annotation></semantics></math> and that <math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_108' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi mathvariant='normal'>Cl</mi><mo stretchy='false'>(</mo><mo stretchy='false'>{</mo><mi>y</mi><mo stretchy='false'>}</mo><mo stretchy='false'>)</mo><mo>⊂</mo><mi mathvariant='normal'>Cl</mi><mo stretchy='false'>(</mo><mo stretchy='false'>{</mo><mi>x</mi><mo stretchy='false'>}</mo><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>\mathrm{Cl}(\{y\}) \subset \mathrm{Cl}(\{x\})</annotation></semantics></math>, hence that <math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_109' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi mathvariant='normal'>Cl</mi><mo stretchy='false'>(</mo><mo stretchy='false'>{</mo><mi>x</mi><mo stretchy='false'>}</mo><mo stretchy='false'>)</mo><mo>=</mo><mi mathvariant='normal'>Cl</mi><mo stretchy='false'>(</mo><mo stretchy='false'>{</mo><mi>y</mi><mo stretchy='false'>}</mo><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>\mathrm{Cl}(\{x\}) = \mathrm{Cl}(\{y\})</annotation></semantics></math>. This is a <a class='existingWikiWord' href='/nlab/show/diff/proof+by+contradiction'>proof by contradiction</a>.</p> </div> <div class='num_prop' id='T1InTermsOfClosureOfPoints'> <h6 id='proposition_2'>Proposition</h6> <p><strong>(<math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_110' 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> in terms of topological closures)</strong></p> <p>A <a class='existingWikiWord' href='/nlab/show/diff/topological+space'>topological space</a> <math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_111' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mi>X</mi><mo>,</mo><mi>τ</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(X,\tau)</annotation></semantics></math> is <math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_112' 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> precisely if all its points are <a class='existingWikiWord' href='/nlab/show/diff/closed+point'>closed points</a>.</p> </div> <div class='proof'> <h6 id='proof_2'>Proof</h6> <p>Assume first that <math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_113' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mi>X</mi><mo>,</mo><mi>τ</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(X,\tau)</annotation></semantics></math> is <math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_114' 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>. We need to show that for every point <math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_115' 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> we have <math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_116' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Cl</mi><mo stretchy='false'>(</mo><mo stretchy='false'>{</mo><mi>x</mi><mo stretchy='false'>}</mo><mo stretchy='false'>)</mo><mo>=</mo><mo stretchy='false'>{</mo><mi>x</mi><mo stretchy='false'>}</mo></mrow><annotation encoding='application/x-tex'>Cl(\{x\}) = \{x\}</annotation></semantics></math>. Since the closure of a point is the <a class='existingWikiWord' href='/nlab/show/diff/complement'>complement</a> of the union of all open subsets not containing this point, this is the case precisely if the union of all open subsets not containing <math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_117' 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_e1760b45eef4a61baf80f64f3a3292d3cfff7533_118' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi><mo>\</mo><mo stretchy='false'>{</mo><mi>x</mi><mo stretchy='false'>}</mo></mrow><annotation encoding='application/x-tex'>X \backslash \{x\}</annotation></semantics></math>, hence if every point <math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_119' 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 \neq x</annotation></semantics></math> is member of at least one open subset not containing <math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_120' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>x</mi></mrow><annotation encoding='application/x-tex'>x</annotation></semantics></math>. This is true by <math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_121' 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>.</p> <p>Conversely, assume that for all <math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_122' 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> then <math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_123' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Cl</mi><mo stretchy='false'>(</mo><mo stretchy='false'>{</mo><mi>x</mi><mo stretchy='false'>}</mo><mo stretchy='false'>)</mo><mo>=</mo><mo stretchy='false'>{</mo><mi>x</mi><mo stretchy='false'>}</mo></mrow><annotation encoding='application/x-tex'>Cl(\{x\}) = \{x\}</annotation></semantics></math>. Then for <math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_124' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>x</mi><mo>≠</mo><mi>y</mi><mo>∈</mo><mi>X</mi></mrow><annotation encoding='application/x-tex'>x \neq y \in X</annotation></semantics></math> two distinct points we need to produce an open subset of <math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_125' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>y</mi></mrow><annotation encoding='application/x-tex'>y</annotation></semantics></math> that does not contain <math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_126' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>x</mi></mrow><annotation encoding='application/x-tex'>x</annotation></semantics></math>. But as before, since <math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_127' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Cl</mi><mo stretchy='false'>(</mo><mo stretchy='false'>{</mo><mi>x</mi><mo stretchy='false'>}</mo><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>Cl(\{x\})</annotation></semantics></math> is the complement of the union of all open subsets that do not contain <math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_128' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>x</mi></mrow><annotation encoding='application/x-tex'>x</annotation></semantics></math>, and the assumption <math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_129' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Cl</mi><mo stretchy='false'>{</mo><mi>x</mi><mo stretchy='false'>}</mo><mo>=</mo><mo stretchy='false'>{</mo><mi>x</mi><mo stretchy='false'>}</mo></mrow><annotation encoding='application/x-tex'>Cl\{x\} = \{x\}</annotation></semantics></math> means that <math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_130' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>y</mi></mrow><annotation encoding='application/x-tex'>y</annotation></semantics></math> is member of one of these open subsets that do not contain <math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_131' 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> <div class='num_prop' id='T2InTermsOfClosedDiagonal'> <h6 id='proposition_3'>Proposition</h6> <p><strong>(<math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_132' 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 terms of topological closures)</strong></p> <p>A <a class='existingWikiWord' href='/nlab/show/diff/topological+space'>topological space</a> <math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_133' 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> is <math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_134' 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>=<a class='existingWikiWord' href='/nlab/show/diff/Hausdorff+space'>Hausdorff</a> precisely if the <a class='existingWikiWord' href='/nlab/show/diff/image'>image</a> of the <a class='existingWikiWord' href='/nlab/show/diff/diagonal+morphism'>diagonal</a></p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_135' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mrow><mtable><mtr><mtd><mi>X</mi></mtd> <mtd><mover><mo>⟶</mo><mrow><msub><mi>Δ</mi> <mi>X</mi></msub></mrow></mover></mtd> <mtd><mi>X</mi><mo>×</mo><mi>X</mi></mtd></mtr> <mtr><mtd><mi>x</mi></mtd> <mtd><mover><mo>↦</mo><mphantom><mi>AAA</mi></mphantom></mover></mtd> <mtd><mo stretchy='false'>(</mo><mi>x</mi><mo>,</mo><mi>x</mi><mo stretchy='false'>)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding='application/x-tex'> \array{ X &\overset{\Delta_X}{\longrightarrow}& X \times X \\ x &\overset{\phantom{AAA}}{\mapsto}& (x,x) } </annotation></semantics></math></div> <p>is a <a class='existingWikiWord' href='/nlab/show/diff/closed+subspace'>closed subset</a> in the <a class='existingWikiWord' href='/nlab/show/diff/product+topological+space'>product topological space</a> <math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_136' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mi>X</mi><mo>×</mo><mi>X</mi><mo>,</mo><msub><mi>τ</mi> <mrow><mi>X</mi><mo>×</mo><mi>X</mi></mrow></msub><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(X \times X, \tau_{X \times X})</annotation></semantics></math>.</p> </div> <div class='proof'> <h6 id='proof_3'>Proof</h6> <p>The Hausdorff condition, that for <math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_137' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>x</mi><mo>≠</mo><mi>y</mi><mo>∈</mo><mi>X</mi></mrow><annotation encoding='application/x-tex'>x \neq y \in X</annotation></semantics></math> then there exist disjoint open neighbourhood <math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_138' 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><mi>X</mi></mrow><annotation encoding='application/x-tex'>U_x, U_y \subset X</annotation></semantics></math>, is equivalently rephrased in terms of the product topology as: Every point <math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_139' 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>X</mi></mrow><annotation encoding='application/x-tex'>(x,y) \in X</annotation></semantics></math> which is not on the diagonal has an open neighbourhood <math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_140' 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></mrow><annotation encoding='application/x-tex'>U_x \times U_y</annotation></semantics></math> which still does not intersect the diagonal.</p> <p>Hence if <math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_141' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> is Hausdorff, then the diagonal <math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_142' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>Δ</mi> <mi>X</mi></msub><mo stretchy='false'>(</mo><mi>X</mi><mo stretchy='false'>)</mo><mo>⊂</mo><mi>X</mi><mo>×</mo><mi>X</mi></mrow><annotation encoding='application/x-tex'>\Delta_X(X) \subset X \times X</annotation></semantics></math> is the complement of a union of such open sets, and hence is closed.</p> <p>Conversely, if the diagonal is closed, then (by <a href='Introduction+to+Topology+--+1#UnionOfOpensGivesClosure'>this lemma</a>) every point <math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_143' 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></mrow><annotation encoding='application/x-tex'>(x,y)</annotation></semantics></math> not on the diagonal, hence with <math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_144' 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>, has an open neighbourhood <math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_145' 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></mrow><annotation encoding='application/x-tex'>U_x \times U_y</annotation></semantics></math> still not intersecting the diagonal, hence so that <math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_146' 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><mi>∅</mi></mrow><annotation encoding='application/x-tex'>U_x \cap U_y = \emptyset</annotation></semantics></math>. Thus <math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_147' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mi>X</mi><mo>,</mo><mi>τ</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(X,\tau)</annotation></semantics></math> is Hausdorff.</p> </div> <div class='num_remark'> <h6 id='remark'>Remark</h6> <p>The characterization of the Hausdorff separation condition via the closure of the diagonal in prop. <a class='maruku-ref' href='#T2InTermsOfClosedDiagonal'>3</a> is the basis for the definition of <em><a class='existingWikiWord' href='/nlab/show/diff/separated+morphism+of+schemes'>separated scheme</a></em>.</p> </div> <div class='num_prop' id='T3InTermsOfTopologicalClosures'> <h6 id='proposition_4'>Proposition</h6> <p><strong>(<math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_148' 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> in terms of topological closures)</strong></p> <p>A <a class='existingWikiWord' href='/nlab/show/diff/topological+space'>topological space</a> <math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_149' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mi>X</mi><mo>,</mo><mi>τ</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(X,\tau)</annotation></semantics></math> is <a class='existingWikiWord' href='/nlab/show/diff/regular+space'>regular</a>, precisely if for all <a class='existingWikiWord' href='/nlab/show/diff/closed+subspace'>closed subsets</a> <math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_150' 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 <a class='existingWikiWord' href='/nlab/show/diff/neighborhood'>open neighbourhood</a> <math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_151' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>U</mi><mo>⊃</mo><mo stretchy='false'>{</mo><mi>x</mi><mo stretchy='false'>}</mo></mrow><annotation encoding='application/x-tex'>U \supset \{x\}</annotation></semantics></math> there exists a smaller open neighbourhood <math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_152' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>V</mi><mo>⊃</mo><mo stretchy='false'>{</mo><mi>x</mi><mo stretchy='false'>}</mo></mrow><annotation encoding='application/x-tex'>V \supset \{x\}</annotation></semantics></math> whose <a class='existingWikiWord' href='/nlab/show/diff/closed+subspace'>topological closure</a> <math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_153' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Cl</mi><mo stretchy='false'>(</mo><mi>V</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>Cl(V)</annotation></semantics></math> is still contained in <math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_154' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>U</mi></mrow><annotation encoding='application/x-tex'>U</annotation></semantics></math>:</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_155' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>{</mo><mi>x</mi><mo stretchy='false'>}</mo><mo>⊂</mo><mi>V</mi><mo>⊂</mo><mi>Cl</mi><mo stretchy='false'>(</mo><mi>V</mi><mo stretchy='false'>)</mo><mo>⊂</mo><mi>U</mi><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> \{x\} \subset V \subset Cl(V) \subset U \,. </annotation></semantics></math></div></div> <p>The <strong>proof</strong> of prop. \re{T3InTermsOfTopologicalClosures} is the direct specialization of the following proof for prop. <a class='maruku-ref' href='#T4InTermsOfTopologicalClosures'>5</a> to the case that <math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_156' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi><mo>=</mo><mo stretchy='false'>{</mo><mi>x</mi><mo stretchy='false'>}</mo></mrow><annotation encoding='application/x-tex'>C = \{x\}</annotation></semantics></math> (using that by <math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_157' 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>, which is part of the definition of <math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_158' 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>, the singleton subset is indeed closed by prop. <a class='maruku-ref' href='#T1InTermsOfClosureOfPoints'>2</a>).</p> <div class='num_prop' id='T4InTermsOfTopologicalClosures'> <h6 id='proposition_5'>Proposition</h6> <p><strong>(<math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_159' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>T</mi> <mn>4</mn></msub></mrow><annotation encoding='application/x-tex'>T_4</annotation></semantics></math> in terms of topological closures)</strong></p> <p>A <a class='existingWikiWord' href='/nlab/show/diff/topological+space'>topological space</a> <math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_160' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mi>X</mi><mo>,</mo><mi>τ</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(X,\tau)</annotation></semantics></math> is <a class='existingWikiWord' href='/nlab/show/diff/normal+space'>normal</a>, precisely if for all <a class='existingWikiWord' href='/nlab/show/diff/closed+subspace'>closed subsets</a> <math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_161' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi><mo>⊂</mo><mi>X</mi></mrow><annotation encoding='application/x-tex'>C \subset X</annotation></semantics></math> with <a class='existingWikiWord' href='/nlab/show/diff/neighborhood'>open neighbourhood</a> <math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_162' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>U</mi><mo>⊃</mo><mi>C</mi></mrow><annotation encoding='application/x-tex'>U \supset C</annotation></semantics></math> there exists a smaller open neighbourhood <math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_163' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>V</mi><mo>⊃</mo><mi>C</mi></mrow><annotation encoding='application/x-tex'>V \supset C</annotation></semantics></math> whose <a class='existingWikiWord' href='/nlab/show/diff/closed+subspace'>topological closure</a> <math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_164' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Cl</mi><mo stretchy='false'>(</mo><mi>V</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>Cl(V)</annotation></semantics></math> is still contained in <math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_165' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>U</mi></mrow><annotation encoding='application/x-tex'>U</annotation></semantics></math>:</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_166' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi><mo>⊂</mo><mi>V</mi><mo>⊂</mo><mi>Cl</mi><mo stretchy='false'>(</mo><mi>V</mi><mo stretchy='false'>)</mo><mo>⊂</mo><mi>U</mi><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> C \subset V \subset Cl(V) \subset U \,. </annotation></semantics></math></div></div> <div class='proof'> <h6 id='proof_4'>Proof</h6> <p>In one direction, assume that <math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_167' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mi>X</mi><mo>,</mo><mi>τ</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(X,\tau)</annotation></semantics></math> is normal, and consider <math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_168' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi><mo>⊂</mo><mi>U</mi></mrow><annotation encoding='application/x-tex'>C \subset U</annotation></semantics></math>. It follows that the <a class='existingWikiWord' href='/nlab/show/diff/complement'>complement</a> of the open subset <math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_169' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>U</mi></mrow><annotation encoding='application/x-tex'>U</annotation></semantics></math> is closed and disjoint from <math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_170' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi></mrow><annotation encoding='application/x-tex'>C</annotation></semantics></math>:</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_171' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi><mo>∩</mo><mi>X</mi><mo>\</mo><mi>U</mi><mo>=</mo><mi>∅</mi><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> C \cap X \backslash U = \emptyset \,. </annotation></semantics></math></div> <p>Therefore by assumption of normality of <math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_172' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mi>X</mi><mo>,</mo><mi>τ</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(X,\tau)</annotation></semantics></math>, there exists open neighbourhoods <math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_173' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>V</mi><mo>⊃</mo><mi>C</mi></mrow><annotation encoding='application/x-tex'>V \supset C</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_174' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>W</mi><mo>⊃</mo><mi>X</mi><mo>\</mo><mi>U</mi></mrow><annotation encoding='application/x-tex'>W \supset X \backslash U</annotation></semantics></math> with</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_175' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>V</mi><mo>∩</mo><mi>W</mi><mo>=</mo><mi>∅</mi><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> V \cap W = \emptyset \,. </annotation></semantics></math></div> <p>But this means that</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_176' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>V</mi><mo>⊂</mo><mi>X</mi><mo>\</mo><mi>W</mi></mrow><annotation encoding='application/x-tex'> V \subset X \backslash W </annotation></semantics></math></div> <p>and since the <a class='existingWikiWord' href='/nlab/show/diff/complement'>complement</a> <math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_177' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi><mo>\</mo><mi>W</mi></mrow><annotation encoding='application/x-tex'>X \backslash W</annotation></semantics></math> of the open set <math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_178' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>W</mi></mrow><annotation encoding='application/x-tex'>W</annotation></semantics></math> is closed, it still contains the closure of <math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_179' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>V</mi></mrow><annotation encoding='application/x-tex'>V</annotation></semantics></math>, so that we have</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_180' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi><mo>⊂</mo><mi>V</mi><mo>⊂</mo><mi>Cl</mi><mo stretchy='false'>(</mo><mi>V</mi><mo stretchy='false'>)</mo><mo>⊂</mo><mi>X</mi><mo>\</mo><mi>W</mi><mo>⊂</mo><mi>U</mi><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> C \subset V \subset Cl(V) \subset X \backslash W \subset U \,. </annotation></semantics></math></div> <p>In the other direction, assume that for every open neighbourhood <math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_181' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>U</mi><mo>⊃</mo><mi>C</mi></mrow><annotation encoding='application/x-tex'>U \supset C</annotation></semantics></math> of a closed subset <math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_182' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi></mrow><annotation encoding='application/x-tex'>C</annotation></semantics></math> there exists a smaller open neighbourhood <math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_183' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>V</mi></mrow><annotation encoding='application/x-tex'>V</annotation></semantics></math> with <math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_184' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi><mo>⊂</mo><mi>V</mi><mo>⊂</mo><mi>Cl</mi><mo stretchy='false'>(</mo><mi>V</mi><mo stretchy='false'>)</mo><mo>⊂</mo><mi>U</mi></mrow><annotation encoding='application/x-tex'>C \subset V \subset Cl(V) \subset U</annotation></semantics></math>. Consider disjoint closed subsets <math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_185' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>C</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>C</mi> <mn>2</mn></msub><mo>⊂</mo><mi>X</mi></mrow><annotation encoding='application/x-tex'>C_1, C_2 \subset X</annotation></semantics></math>. We need to produce disjoint open neighbourhoods for them.</p> <p>From their disjointness it follows that <math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_186' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi><mo>\</mo><msub><mi>C</mi> <mn>2</mn></msub><mo>⊃</mo><msub><mi>C</mi> <mn>1</mn></msub></mrow><annotation encoding='application/x-tex'>X \backslash C_2 \supset C_1</annotation></semantics></math> is an open neighbourhood. Hence by assumption there is an open neighbourhood <math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_187' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>V</mi></mrow><annotation encoding='application/x-tex'>V</annotation></semantics></math> with</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_188' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>C</mi> <mn>1</mn></msub><mo>⊂</mo><mi>V</mi><mo>⊂</mo><mi>Cl</mi><mo stretchy='false'>(</mo><mi>V</mi><mo stretchy='false'>)</mo><mo>⊂</mo><mi>X</mi><mo>\</mo><msub><mi>C</mi> <mn>2</mn></msub><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> C_1 \subset V \subset Cl(V) \subset X \backslash C_2 \,. </annotation></semantics></math></div> <p>Thus <math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_189' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>V</mi><mo>⊃</mo><msub><mi>C</mi> <mn>1</mn></msub></mrow><annotation encoding='application/x-tex'>V \supset C_1</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_190' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi><mo>\</mo><mi>Cl</mi><mo stretchy='false'>(</mo><mi>X</mi><mo stretchy='false'>)</mo><mo>⊃</mo><msub><mi>C</mi> <mn>2</mn></msub></mrow><annotation encoding='application/x-tex'>X \backslash Cl(X) \supset C_2</annotation></semantics></math> are two disjoint open neighbourhoods, as required.</p> </div> <h3 id='relations_between_the_axioms'>Relations between the axioms</h3> <p>First of all, notice (prop. <a class='maruku-ref' href='#T1InTermsOfClosureOfPoints'>2</a>) that the <math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_191' 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> condition, saying that distinct points are separated, is equivalent to the condition that every point is closed. Thus, <math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_192' 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> serves as a linchpin between conditions on points and conditions on closed sets.</p> <p>Many implications between separation axioms can be seen in the following Hasse diagram:</p> <p><img alt='' src='http://ncatlab.org/nlab/files/separation.png' width='472' /></p> <p>Here, there are two entries at each node; the one on the right includes the <math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_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> axiom, while the one on the left does not. This diagram shows the separation axioms as a meet sub-<a class='existingWikiWord' href='/nlab/show/diff/semilattice'>semilattice</a> of the lattice of all conditions on topological spaces; for example, you can see, by following the diagram upwards, that any space that is both <strong>n</strong>ormal and <strong>r</strong>egular must be <math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_194' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>R</mi> <mn>3</mn></msub></mrow><annotation encoding='application/x-tex'>R_3</annotation></semantics></math>. And since <math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_195' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>R</mi> <mn>3</mn></msub></mrow><annotation encoding='application/x-tex'>R_3</annotation></semantics></math> never appears in the tables above, you can take this as a <em>definition</em> of <math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_196' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>R</mi> <mn>3</mn></msub></mrow><annotation encoding='application/x-tex'>R_3</annotation></semantics></math>.</p> <p>In general, the names in this diagram are:</p> <ul> <li>‘P’ for ‘perfectly’,</li> <li>‘C’ for ‘complete’,</li> <li>‘N’ for ‘<a class='existingWikiWord' href='/nlab/show/diff/normal+space'>normal</a>’,</li> <li>‘R’ (without a subscript) for ‘<a class='existingWikiWord' href='/nlab/show/diff/regular+space'>regular</a>’</li> <li>‘T’ or ‘R’ with a subscript are written and pronounced that way.</li> </ul> <p>Warning: <math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_197' 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> for <math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_198' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>i</mi><mo>≥</mo><mn>3</mn></mrow><annotation encoding='application/x-tex'>i \geq 3</annotation></semantics></math> has been used in different ways in the past, and perhaps by some schools still. Also, all of the <math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_199' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>R</mi> <mi>i</mi></msub></mrow><annotation encoding='application/x-tex'>R_i</annotation></semantics></math> terms are rare. It is safest to say, for example, ‘normal Hausdorff’ for <math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_200' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>T</mi> <mn>4</mn></msub></mrow><annotation encoding='application/x-tex'>T_4</annotation></semantics></math> and clearer to say, for example, ‘normal regular’ for <math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_201' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>R</mi> <mn>3</mn></msub></mrow><annotation encoding='application/x-tex'>R_3</annotation></semantics></math>. If you want to avoid the subscript terms entirely, then you can, by doing the above and the following:</p> <ul> <li><math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_202' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>T</mi> <mrow><mn>2</mn><mstyle displaystyle='false'><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle></mrow></msub></mrow><annotation encoding='application/x-tex'>T_{2\tfrac{1}{2}}</annotation></semantics></math> = <a class='existingWikiWord' href='/nlab/show/diff/Urysohn+topological+space'>Urysohn</a></li> <li><math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_203' 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> = <a class='existingWikiWord' href='/nlab/show/diff/Hausdorff+space'>Hausdorff</a>,</li> <li><math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_204' 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> = accessible,</li> <li><math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_205' 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> = <a class='existingWikiWord' href='/nlab/show/diff/Kolmogorov+topological+space'>Kolmogorov</a>,</li> <li><math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_206' 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> = reciprocal,</li> <li><math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_207' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>R</mi> <mn>0</mn></msub></mrow><annotation encoding='application/x-tex'>R_0</annotation></semantics></math> = <a class='existingWikiWord' href='/nlab/show/diff/symmetric+topological+space'>symmetric</a>.</li> </ul> <p>On the other hand, if you want to use <em>more</em> symbols, then you can:</p> <ul> <li><math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_208' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>R</mi> <mn>2</mn></msub></mrow><annotation encoding='application/x-tex'>R_2</annotation></semantics></math> = regular,</li> <li><math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_209' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>R</mi> <mrow><mn>2</mn><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></msub></mrow><annotation encoding='application/x-tex'>R_{2\frac{1}{2}}</annotation></semantics></math> = completely regular.</li> </ul> <p>It would be easy to invent an <math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_210' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>N</mi> <mi>i</mi></msub></mrow><annotation encoding='application/x-tex'>N_i</annotation></semantics></math> series for the various kinds of normal spaces, but nobody seems to have done so yet.</p> <p>Other terms are also in use, principally ‘Tychonoff’ for <a class='existingWikiWord' href='/nlab/show/diff/Tychonoff+space'>completely regular Hausdorff</a> (<math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_211' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>T</mi> <mrow><mn>3</mn><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></msub></mrow><annotation encoding='application/x-tex'>T_{3\frac{1}{2}}</annotation></semantics></math>).</p> <h3 id='Reflection'>Reflection</h3> <p>(cf. <a href='#HerrlichStrecker71'>Herrlich & Strecker 1971</a>)</p> <div class='num_prop' id='HausdorffReflection'> <h6 id='proposition_6'>Proposition</h6> <p><strong>(<math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_212' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>T</mi> <mi>n</mi></msub></mrow><annotation encoding='application/x-tex'>T_n</annotation></semantics></math>-reflection)</strong></p> <p>Let <math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_213' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>n</mi><mo>∈</mo><mo stretchy='false'>{</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo stretchy='false'>}</mo></mrow><annotation encoding='application/x-tex'>n \in \{0,1,2\}</annotation></semantics></math>. Then for every <a class='existingWikiWord' href='/nlab/show/diff/topological+space'>topological space</a> <math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_214' 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 <math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_215' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>T</mi> <mi>n</mi></msub></mrow><annotation encoding='application/x-tex'>T_n</annotation></semantics></math>-topological space <math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_216' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>T</mi> <mi>n</mi></msub><mi>X</mi></mrow><annotation encoding='application/x-tex'>T_n X</annotation></semantics></math> and a <a class='existingWikiWord' href='/nlab/show/diff/continuous+map'>continuous function</a> of the forma</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_217' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>t</mi> <mi>n</mi></msub><mo stretchy='false'>(</mo><mi>X</mi><mo stretchy='false'>)</mo><mspace width='thickmathspace' /><mo lspace='verythinmathspace'>:</mo><mspace width='thickmathspace' /><mi>X</mi><mo>⟶</mo><msub><mi>T</mi> <mi>n</mi></msub><mi>X</mi></mrow><annotation encoding='application/x-tex'> t_n(X) \;\colon\; X \longrightarrow T_n X </annotation></semantics></math></div> <p>which is the “closest approximation from the left” to <math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_218' 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 <math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_219' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>T</mi> <mi>n</mi></msub></mrow><annotation encoding='application/x-tex'>T_n</annotation></semantics></math>-topological space, in that for <math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_220' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Y</mi></mrow><annotation encoding='application/x-tex'>Y</annotation></semantics></math> any <math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_221' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>T</mi> <mi>n</mi></msub></mrow><annotation encoding='application/x-tex'>T_n</annotation></semantics></math>-space, 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_e1760b45eef4a61baf80f64f3a3292d3cfff7533_222' 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_e1760b45eef4a61baf80f64f3a3292d3cfff7533_223' 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' /><msub><mi>T</mi> <mi>n</mi></msub><mi>X</mi><mo>⟶</mo><mi>Y</mi></mrow><annotation encoding='application/x-tex'> \tilde f \;\colon\; T_n 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_e1760b45eef4a61baf80f64f3a3292d3cfff7533_224' 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>t</mi> <mi>n</mi></msub><mo stretchy='false'>(</mo><mi>X</mi><mo stretchy='false'>)</mo><mspace width='thickmathspace' /><mo lspace='verythinmathspace'>:</mo><mspace width='thickmathspace' /><mi>X</mi><mover><mo>⟶</mo><mrow><msub><mi>t</mi> <mi>n</mi></msub><mo stretchy='false'>(</mo><mi>X</mi><mo stretchy='false'>)</mo></mrow></mover><msub><mi>T</mi> <mi>n</mi></msub><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 t_n(X) \;\colon\; X \overset{t_n(X)}{\longrightarrow} T_n X \overset{\tilde f}{\longrightarrow} Y \,. </annotation></semantics></math></div> <p>Here <math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_225' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi><mover><mo>⟶</mo><mrow><msub><mi>t</mi> <mi>n</mi></msub><mo stretchy='false'>(</mo><mi>X</mi><mo stretchy='false'>)</mo></mrow></mover><msub><mi>T</mi> <mi>n</mi></msub><mo stretchy='false'>(</mo><mi>X</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>X \overset{t_n(X)}{\longrightarrow} T_n(X)</annotation></semantics></math> is called the <em><math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_226' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>T</mi> <mi>n</mi></msub></mrow><annotation encoding='application/x-tex'>T_n</annotation></semantics></math>-reflection</em> of <math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_227' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math>.</p> <ul> <li> <p>For <math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_228' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>n</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding='application/x-tex'>n = 0</annotation></semantics></math> this is known as the <em><a class='existingWikiWord' href='/nlab/show/diff/Kolmogorov+topological+space'>Kolmogorov quotient</a></em> construction (see prop. <a class='maruku-ref' href='#KolmogorovQuotient'>8</a> below).</p> </li> <li> <p>For <math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_229' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>n</mi><mo>=</mo><mn>2</mn></mrow><annotation encoding='application/x-tex'>n = 2</annotation></semantics></math> this is known as <em><a class='existingWikiWord' href='/nlab/show/diff/Hausdorff+space'>Hausdorff reflection</a></em> or <em>Hausdorffication</em> or similar.</p> </li> </ul> <p>Moreover, the operation <math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_230' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>T</mi> <mi>n</mi></msub><mo stretchy='false'>(</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>T_n(-)</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_e1760b45eef4a61baf80f64f3a3292d3cfff7533_231' 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_e1760b45eef4a61baf80f64f3a3292d3cfff7533_232' 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><msub><mi>T</mi> <mi>n</mi></msub><mi>X</mi><mover><mo>→</mo><mrow><msub><mi>T</mi> <mi>n</mi></msub><mi>f</mi></mrow></mover><msub><mi>T</mi> <mi>n</mi></msub><mi>Y</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'> (X \overset{f}{\to} Y) \;\mapsto\; (T_n X \overset{T_n f}{\to} T_n Y) </annotation></semantics></math></div> <p>such as to preserve <a class='existingWikiWord' href='/nlab/show/diff/composition'>composition</a> of functions as well as <a class='existingWikiWord' href='/nlab/show/diff/identity+function'>identity functions</a>:</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_233' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><msub><mi>T</mi> <mi>n</mi></msub><mi>g</mi><mo stretchy='false'>)</mo><mo>∘</mo><mo stretchy='false'>(</mo><msub><mi>T</mi> <mi>n</mi></msub><mi>f</mi><mo stretchy='false'>)</mo><mo>=</mo><msub><mi>T</mi> <mi>n</mi></msub><mo stretchy='false'>(</mo><mi>g</mi><mo>∘</mo><mi>f</mi><mo stretchy='false'>)</mo><mphantom><mi>AA</mi></mphantom><mspace width='thinmathspace' /><mo>,</mo><mphantom><mi>AA</mi></mphantom><msub><mi>T</mi> <mi>n</mi></msub><mo stretchy='false'>(</mo><msub><mi>id</mi> <mi>X</mi></msub><mo stretchy='false'>)</mo><mo>=</mo><msub><mi>id</mi> <mrow><msub><mi>T</mi> <mi>n</mi></msub><mi>X</mi></mrow></msub><mspace width='thinmathspace' /></mrow><annotation encoding='application/x-tex'> (T_n g) \circ (T_n f) = T_n(g \circ f) \phantom{AA} \,, \phantom{AA} T_n (id_X) = id_{T_n X} \, </annotation></semantics></math></div> <p>Finally, the comparison map is compatible with this in that for all continuous functions <math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_234' 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> then</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_235' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>t</mi> <mi>n</mi></msub><mo stretchy='false'>(</mo><mi>Y</mi><mo stretchy='false'>)</mo><mo>∘</mo><mi>f</mi><mo>=</mo><msub><mi>T</mi> <mi>n</mi></msub><mo stretchy='false'>(</mo><mi>f</mi><mo stretchy='false'>)</mo><mo>∘</mo><msub><mi>t</mi> <mi>n</mi></msub><mo stretchy='false'>(</mo><mi>X</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'> t_n(Y) \circ f = T_n(f)\circ t_n(X) </annotation></semantics></math></div> <p>hence then follows <a class='existingWikiWord' href='/nlab/show/diff/commutative+square'>squares commutes</a>:</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_236' 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>t</mi> <mi>n</mi></msub><mo stretchy='false'>(</mo><mi>X</mi><mo stretchy='false'>)</mo></mrow></mpadded></msup><mo stretchy='false'>↓</mo></mtd> <mtd /> <mtd><msup><mo stretchy='false'>↓</mo> <mpadded width='0'><mrow><msub><mi>t</mi> <mi>n</mi></msub><mo stretchy='false'>(</mo><mi>Y</mi><mo stretchy='false'>)</mo></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><msub><mi>T</mi> <mi>n</mi></msub><mi>X</mi></mtd> <mtd><munder><mo>⟶</mo><mrow><msub><mi>T</mi> <mi>n</mi></msub><mi>f</mi></mrow></munder></mtd> <mtd><msub><mi>T</mi> <mi>n</mi></msub><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{t_n(X)}}\downarrow && \downarrow^{\mathrlap{t_n(Y)}} \\ T_n X &\underset{T_n f}{\longrightarrow}& T_n Y } \,. </annotation></semantics></math></div></div> <p>We give a <strong>proof</strong> of the existence of this reflection below as the proof of prop. <a class='maruku-ref' href='#HausdorffReflectionViaHomsIntoAllHausdorffSpaces'>7</a>.</p> <div class='num_remark'> <h6 id='remark_2'>Remark</h6> <p><strong>(<a class='existingWikiWord' href='/nlab/show/diff/reflective+subcategory'>reflective subcategories</a>)</strong></p> <p>In the language of <a class='existingWikiWord' href='/nlab/show/diff/category+theory'>category theory</a> the <math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_237' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>T</mi> <mi>n</mi></msub></mrow><annotation encoding='application/x-tex'>T_n</annotation></semantics></math>-reflection of prop. <a class='maruku-ref' href='#HausdorffReflection'>6</a> says that</p> <ol> <li> <p><math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_238' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>T</mi> <mi>n</mi></msub><mo stretchy='false'>(</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>T_n(-)</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_e1760b45eef4a61baf80f64f3a3292d3cfff7533_239' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>T</mi> <mi>n</mi></msub><mspace width='thickmathspace' /><mo lspace='verythinmathspace'>:</mo><mspace width='thickmathspace' /><mi>Top</mi><mo>⟶</mo><msub><mi>Top</mi> <mrow><msub><mi>T</mi> <mi>n</mi></msub></mrow></msub></mrow><annotation encoding='application/x-tex'>T_n \;\colon\; Top \longrightarrow Top_{T_n}</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_e1760b45eef4a61baf80f64f3a3292d3cfff7533_240' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>Top</mi> <mrow><msub><mi>T</mi> <mi>n</mi></msub></mrow></msub><mover><mo>↪</mo><mi>ι</mi></mover><mi>Top</mi></mrow><annotation encoding='application/x-tex'>Top_{T_n} \overset{\iota}{\hookrightarrow} Top</annotation></semantics></math> of Hausdorff topological spaces;</p> </li> <li> <p><math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_241' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>t</mi> <mi>n</mi></msub><mo stretchy='false'>(</mo><mi>X</mi><mo stretchy='false'>)</mo><mo lspace='verythinmathspace'>:</mo><mi>X</mi><mo>→</mo><msub><mi>T</mi> <mi>n</mi></msub><mi>X</mi></mrow><annotation encoding='application/x-tex'>t_n(X) \colon X \to T_n X</annotation></semantics></math> is a <em><a class='existingWikiWord' href='/nlab/show/diff/natural+transformation'>natural transformation</a></em> 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_e1760b45eef4a61baf80f64f3a3292d3cfff7533_242' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>ι</mi><mo>∘</mo><msub><mi>T</mi> <mi>n</mi></msub></mrow><annotation encoding='application/x-tex'>\iota \circ T_n</annotation></semantics></math></p> </li> <li> <p><math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_243' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>T</mi> <mi>n</mi></msub></mrow><annotation encoding='application/x-tex'>T_n</annotation></semantics></math>-topological spaces 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_e1760b45eef4a61baf80f64f3a3292d3cfff7533_244' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>T</mi> <mi>n</mi></msub></mrow><annotation encoding='application/x-tex'>T_n</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_e1760b45eef4a61baf80f64f3a3292d3cfff7533_245' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>ι</mi></mrow><annotation encoding='application/x-tex'>\iota</annotation></semantics></math>; this situation is denoted as follows:</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_246' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>Top</mi> <mrow><msub><mi>T</mi> <mi>n</mi></msub></mrow></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_{T_n} \underoverset{\underset{\iota}{\hookrightarrow}}{\overset{H}{\longleftarrow}}{\bot} Top \,. </annotation></semantics></math></div></li> </ol> <p>Generally, an <em><a class='existingWikiWord' href='/nlab/show/diff/adjoint+functor'>adjunction</a></em> between two <a class='existingWikiWord' href='/nlab/show/diff/functor'>functors</a></p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_247' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>L</mi><mspace width='thickmathspace' /><mo lspace='verythinmathspace'>:</mo><mspace width='thickmathspace' /><mi>𝒞</mi><mo>↔</mo><mi>𝒟</mi><mspace width='thickmathspace' /><mo lspace='verythinmathspace'>:</mo><mspace width='thickmathspace' /><mi>R</mi></mrow><annotation encoding='application/x-tex'> L \;\colon\; \mathcal{C} \leftrightarrow \mathcal{D} \;\colon\; R </annotation></semantics></math></div> <p>is for all pairs of objects <math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_248' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>c</mi><mo>∈</mo><mi>𝒞</mi></mrow><annotation encoding='application/x-tex'>c \in \mathcal{C}</annotation></semantics></math>, <math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_249' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>d</mi><mo>∈</mo><mi>𝒟</mi></mrow><annotation encoding='application/x-tex'>d \in \mathcal{D}</annotation></semantics></math> a <a class='existingWikiWord' href='/nlab/show/diff/bijection'>bijection</a> between sets of <a class='existingWikiWord' href='/nlab/show/diff/morphism'>morphisms</a> of the form</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_250' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mrow><mo>{</mo><mi>L</mi><mo stretchy='false'>(</mo><mi>c</mi><mo stretchy='false'>)</mo><mo>⟶</mo><mi>d</mi><mo>}</mo></mrow><mo>≃</mo><mrow><mo>{</mo><mi>c</mi><mo>⟶</mo><mi>R</mi><mo stretchy='false'>(</mo><mi>d</mi><mo stretchy='false'>)</mo><mo>}</mo></mrow><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> \left\{ L(c) \longrightarrow d \right\} \simeq \left\{ c \longrightarrow R(d) \right\} \,. </annotation></semantics></math></div> <p>i.e.</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_251' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>Hom</mi> <mi>𝒟</mi></msub><mo stretchy='false'>(</mo><mi>L</mi><mo stretchy='false'>(</mo><mi>c</mi><mo stretchy='false'>)</mo><mo>,</mo><mi>d</mi><mo stretchy='false'>)</mo><munderover><mo>⟶</mo><mo>≃</mo><mrow><msub><mi>ϕ</mi> <mrow><mi>c</mi><mo>,</mo><mi>d</mi></mrow></msub></mrow></munderover><msub><mi>Hom</mi> <mi>𝒞</mi></msub><mo stretchy='false'>(</mo><mi>c</mi><mo>,</mo><mi>R</mi><mo stretchy='false'>(</mo><mi>d</mi><mo stretchy='false'>)</mo><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'> Hom_{\mathcal{D}}(L(c), d) \underoverset{\simeq}{\phi_{c,d}}{\longrightarrow} Hom_{\mathcal{C}}(c, R(d)) </annotation></semantics></math></div> <p>and such that these bijections are “<a class='existingWikiWord' href='/nlab/show/diff/natural+bijection'>natural</a>” in that they for all pairs of morphisms <math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_252' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>f</mi><mo lspace='verythinmathspace'>:</mo><mi>c</mi><mo>′</mo><mo>→</mo><mi>c</mi></mrow><annotation encoding='application/x-tex'>f \colon c' \to c</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_253' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>g</mi><mo lspace='verythinmathspace'>:</mo><mi>d</mi><mo>→</mo><mi>d</mi><mo>′</mo></mrow><annotation encoding='application/x-tex'>g \colon d \to d'</annotation></semantics></math> then the folowing <a class='existingWikiWord' href='/nlab/show/diff/commutative+diagram'>diagram commutes</a>:</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_254' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mrow><mtable><mtr><mtd><msub><mi>Hom</mi> <mi>𝒟</mi></msub><mo stretchy='false'>(</mo><mi>L</mi><mo stretchy='false'>(</mo><mi>c</mi><mo stretchy='false'>)</mo><mo>,</mo><mi>d</mi><mo stretchy='false'>)</mo></mtd> <mtd><munderover><mo>⟶</mo><mo>≃</mo><mrow><msub><mi>ϕ</mi> <mrow><mi>c</mi><mo>,</mo><mi>d</mi></mrow></msub></mrow></munderover></mtd> <mtd><msub><mi>Hom</mi> <mi>𝒞</mi></msub><mo stretchy='false'>(</mo><mi>c</mi><mo>,</mo><mi>R</mi><mo stretchy='false'>(</mo><mi>d</mi><mo stretchy='false'>)</mo><mo stretchy='false'>)</mo></mtd></mtr> <mtr><mtd><mpadded lspace='-100%width' width='0'><mrow><mi>g</mi><mo>∘</mo><mo stretchy='false'>(</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mo stretchy='false'>)</mo><mo>∘</mo><mi>L</mi><mo stretchy='false'>(</mo><mi>f</mi><mo stretchy='false'>)</mo></mrow></mpadded><mo stretchy='false'>↓</mo></mtd> <mtd /> <mtd><mo stretchy='false'>↓</mo><mpadded width='0'><mrow><mi>R</mi><mo stretchy='false'>(</mo><mi>g</mi><mo stretchy='false'>)</mo><mo>∘</mo><mo stretchy='false'>(</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mo stretchy='false'>)</mo><mo>∘</mo><mi>f</mi></mrow></mpadded></mtd></mtr> <mtr><mtd><msub><mi>Hom</mi> <mi>𝒞</mi></msub><mo stretchy='false'>(</mo><mi>L</mi><mo stretchy='false'>(</mo><mi>c</mi><mo>′</mo><mo stretchy='false'>)</mo><mo>,</mo><mi>d</mi><mo>′</mo><mo stretchy='false'>)</mo></mtd> <mtd><munderover><mo>⟶</mo><mo>≃</mo><mrow><msub><mi>ϕ</mi> <mrow><mi>c</mi><mo>′</mo><mo>,</mo><mi>d</mi><mo>′</mo></mrow></msub></mrow></munderover></mtd> <mtd><msub><mi>Hom</mi> <mi>𝒟</mi></msub><mo stretchy='false'>(</mo><mi>c</mi><mo>′</mo><mo>,</mo><mi>R</mi><mo stretchy='false'>(</mo><mi>d</mi><mo>′</mo><mo stretchy='false'>)</mo><mo stretchy='false'>)</mo></mtd></mtr></mtable></mrow><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> \array{ Hom_{\mathcal{D}}(L(c), d) &\underoverset{\simeq}{\phi_{c,d}}{\longrightarrow}& Hom_{\mathcal{C}}(c, R(d)) \\ {\mathllap{g \circ (-) \circ L(f)}}\downarrow && \downarrow{\mathrlap{ R(g) \circ (-) \circ f }} \\ Hom_{\mathcal{C}}(L(c'), d') &\underoverset{\simeq}{\phi_{c',d'}}{\longrightarrow}& Hom_{\mathcal{D}}(c', R(d')) } \,. </annotation></semantics></math></div></div> <p>There are various ways to see the existence and to construct the <math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_255' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>T</mi> <mi>n</mi></msub></mrow><annotation encoding='application/x-tex'>T_n</annotation></semantics></math>-reflections. The following is the quickest way to see the existence, even if to some tastes the construction seems more implicit or abstract than the previous one.</p> <div class='num_prop' id='HausdorffReflectionViaHomsIntoAllHausdorffSpaces'> <h6 id='proposition_7'>Proposition</h6> <p><strong>(<math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_256' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>T</mi> <mi>n</mi></msub></mrow><annotation encoding='application/x-tex'>T_n</annotation></semantics></math>-reflection via surjections into <math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_257' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>T</mi> <mi>n</mi></msub></mrow><annotation encoding='application/x-tex'>T_n</annotation></semantics></math>-spaces)</strong></p> <p>Let <math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_258' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>n</mi><mo>∈</mo><mo stretchy='false'>{</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo stretchy='false'>}</mo></mrow><annotation encoding='application/x-tex'>n \in \{0,1,2\}</annotation></semantics></math>. Let <math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_259' 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_e1760b45eef4a61baf80f64f3a3292d3cfff7533_260' 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_e1760b45eef4a61baf80f64f3a3292d3cfff7533_261' 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_e1760b45eef4a61baf80f64f3a3292d3cfff7533_262' 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_e1760b45eef4a61baf80f64f3a3292d3cfff7533_263' 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 <math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_264' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>T</mi> <mi>n</mi></msub></mrow><annotation encoding='application/x-tex'>T_n</annotation></semantics></math>-topological space <math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_265' 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_e1760b45eef4a61baf80f64f3a3292d3cfff7533_266' 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_e1760b45eef4a61baf80f64f3a3292d3cfff7533_267' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>T</mi> <mi>n</mi></msub><mi>X</mi><mo>≔</mo><mi>X</mi><mo stretchy='false'>/</mo><mo>∼</mo></mrow><annotation encoding='application/x-tex'> T_n 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 <math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_268' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>T</mi> <mi>n</mi></msub></mrow><annotation encoding='application/x-tex'>T_n</annotation></semantics></math>-topological space, and the quotient map <math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_269' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>t</mi> <mi>n</mi></msub><mo stretchy='false'>(</mo><mi>X</mi><mo stretchy='false'>)</mo><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'>t_n(X) \;\colon\; X \to X/{\sim}</annotation></semantics></math> exhibits the <math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_270' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>T</mi> <mi>n</mi></msub></mrow><annotation encoding='application/x-tex'>T_n</annotation></semantics></math>-reflection of <math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_271' 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'>6</a>.</p> </div> <div class='proof'> <h6 id='proof_5'>Proof</h6> <p>First we observe that every continuous function <math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_272' 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 <math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_273' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>T</mi> <mi>n</mi></msub></mrow><annotation encoding='application/x-tex'>T_n</annotation></semantics></math>-topological space <math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_274' 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_e1760b45eef4a61baf80f64f3a3292d3cfff7533_275' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>t</mi> <mi>n</mi></msub><mo stretchy='false'>(</mo><mi>X</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>t_n(X)</annotation></semantics></math> through a continuous function <math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_276' 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_e1760b45eef4a61baf80f64f3a3292d3cfff7533_277' 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_e1760b45eef4a61baf80f64f3a3292d3cfff7533_278' 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>To see this, first factor <math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_279' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>f</mi></mrow><annotation encoding='application/x-tex'>f</annotation></semantics></math> through its <a class='existingWikiWord' href='/nlab/show/diff/image'>image</a> <math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_280' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>f</mi><mo stretchy='false'>(</mo><mi>X</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>f(X)</annotation></semantics></math></p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_281' 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>f</mi><mo stretchy='false'>(</mo><mi>X</mi><mo stretchy='false'>)</mo><mo>↪</mo><mi>Y</mi></mrow><annotation encoding='application/x-tex'> f \;\colon\; X \longrightarrow f(X) \hookrightarrow Y </annotation></semantics></math></div> <p>equipped with its <a class='existingWikiWord' href='/nlab/show/diff/subspace+topology'>subspace topology</a> as a subspace of <math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_282' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Y</mi></mrow><annotation encoding='application/x-tex'>Y</annotation></semantics></math>. It follows that <math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_283' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>f</mi><mo stretchy='false'>(</mo><mi>X</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>f(X)</annotation></semantics></math> is a <math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_284' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>T</mi> <mi>n</mi></msub></mrow><annotation encoding='application/x-tex'>T_n</annotation></semantics></math>-topological space if <math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_285' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Y</mi></mrow><annotation encoding='application/x-tex'>Y</annotation></semantics></math> is.</p> <p>It follows by definition of <math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_286' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>t</mi> <mi>n</mi></msub><mo stretchy='false'>(</mo><mi>X</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>t_n(X)</annotation></semantics></math> that the factorization exists at the level of sets as stated, since if <math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_287' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>x</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>x</mi> <mn>2</mn></msub><mo>∈</mo><mi>X</mi></mrow><annotation encoding='application/x-tex'>x_1, x_2 \in X</annotation></semantics></math> have the same <a class='existingWikiWord' href='/nlab/show/diff/equivalence+class'>equivalence class</a> <math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_288' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>[</mo><msub><mi>x</mi> <mn>1</mn></msub><mo stretchy='false'>]</mo><mo>=</mo><mo stretchy='false'>[</mo><msub><mi>x</mi> <mn>2</mn></msub><mo stretchy='false'>]</mo></mrow><annotation encoding='application/x-tex'>[x_1] = [x_2]</annotation></semantics></math> in <math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_289' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>T</mi> <mi>n</mi></msub><mi>X</mi></mrow><annotation encoding='application/x-tex'>T_n X</annotation></semantics></math>, then by definition they have the same image under all continuous surjective functions to a <math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_290' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>T</mi> <mi>n</mi></msub></mrow><annotation encoding='application/x-tex'>T_n</annotation></semantics></math>-space, hence in particular under <math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_291' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi><mo>→</mo><mi>f</mi><mo stretchy='false'>(</mo><mi>X</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>X \to f(X)</annotation></semantics></math>. This means that <math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_292' 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> as above is well defined. Moreover, it is clear that this is the unique factorization.</p> <p>To see that <math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_293' 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> is continuous, consider <math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_294' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>U</mi><mo>∈</mo><mi>Y</mi></mrow><annotation encoding='application/x-tex'>U \in Y</annotation></semantics></math> an open subset. We need to show that <math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_295' 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><mi>U</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>\tilde f^{-1}(U)</annotation></semantics></math> is open in <math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_296' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi><mo stretchy='false'>/</mo><mo>∼</mo></mrow><annotation encoding='application/x-tex'>X/\sim</annotation></semantics></math>. But by definition of the <a class='existingWikiWord' href='/nlab/show/diff/quotient+space'>quotient topology</a>, this is open precisely if its pre-image under the quotient projection <math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_297' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>t</mi> <mi>n</mi></msub><mo stretchy='false'>(</mo><mi>X</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>t_n(X)</annotation></semantics></math> is open, hence precisely if</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_298' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><msub><mi>t</mi> <mi>n</mi></msub><mo stretchy='false'>(</mo><mi>X</mi><mo stretchy='false'>)</mo><msup><mo stretchy='false'>)</mo> <mrow><mo lspace='verythinmathspace' rspace='0em'>−</mo><mn>1</mn></mrow></msup><mo stretchy='false'>(</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><mi>U</mi><mo stretchy='false'>)</mo><mo stretchy='false'>)</mo><mo>=</mo><mo stretchy='false'>(</mo><mover><mi>f</mi><mo stretchy='false'>˜</mo></mover><mo>∘</mo><msub><mi>t</mi> <mi>n</mi></msub><mo stretchy='false'>(</mo><mi>X</mi><mo stretchy='false'>)</mo><msup><mo stretchy='false'>)</mo> <mrow><mo lspace='verythinmathspace' rspace='0em'>−</mo><mn>1</mn></mrow></msup><mo stretchy='false'>(</mo><mi>U</mi><mo stretchy='false'>)</mo><mo>=</mo><mi>f</mi><mo stretchy='false'>(</mo><mi>U</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'> (t_n(X))^{-1}(\tilde f^{-1}(U)) = ( \tilde f \circ t_n(X) )^{-1}(U) = f(U) </annotation></semantics></math></div> <p>is open in <math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_299' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math>. But this is the case by the assumption that <math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_300' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>f</mi></mrow><annotation encoding='application/x-tex'>f</annotation></semantics></math> is continuous.</p> <p>What remains to be seen is that <math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_301' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>T</mi> <mi>n</mi></msub><mi>X</mi></mrow><annotation encoding='application/x-tex'>T_n X</annotation></semantics></math> as constructed is indeed a <math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_302' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>T</mi> <mi>n</mi></msub></mrow><annotation encoding='application/x-tex'>T_n</annotation></semantics></math>-topological space. Hence assume that <math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_303' 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><msub><mi>T</mi> <mi>n</mi></msub><mi>X</mi></mrow><annotation encoding='application/x-tex'>[x] \neq [y] \in T_n X</annotation></semantics></math> are two distinct points. We need to open neighbourhoods around one or both of these point not containing the other point and possibly disjoint to each other.</p> <p>Now by definition of <math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_304' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>T</mi> <mi>n</mi></msub><mi>X</mi></mrow><annotation encoding='application/x-tex'>T_n X</annotation></semantics></math> this means that there exists a <math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_305' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>T</mi> <mi>n</mi></msub></mrow><annotation encoding='application/x-tex'>T_n</annotation></semantics></math>-topological space <math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_306' 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_e1760b45eef4a61baf80f64f3a3292d3cfff7533_307' 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_e1760b45eef4a61baf80f64f3a3292d3cfff7533_308' 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_e1760b45eef4a61baf80f64f3a3292d3cfff7533_309' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Y</mi></mrow><annotation encoding='application/x-tex'>Y</annotation></semantics></math> is <math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_310' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>T</mi> <mi>n</mi></msub></mrow><annotation encoding='application/x-tex'>T_n</annotation></semantics></math>, there exist the respective kinds of neighbourhoods around these image points in <math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_311' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Y</mi></mrow><annotation encoding='application/x-tex'>Y</annotation></semantics></math>. Moreover, by the previous statement there exists a continuous function <math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_312' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mover><mi>f</mi><mo stretchy='false'>˜</mo></mover><mo lspace='verythinmathspace'>:</mo><msub><mi>T</mi> <mi>n</mi></msub><mi>X</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding='application/x-tex'>\tilde f \colon T_n X \to Y</annotation></semantics></math> with <math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_313' 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_e1760b45eef4a61baf80f64f3a3292d3cfff7533_314' 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>. By the nature of continuous functions, the pre-images of these open neighbourhoods in <math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_315' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Y</mi></mrow><annotation encoding='application/x-tex'>Y</annotation></semantics></math> are still open in <math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_316' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> and still satisfy the required disjunction properties. Therefore <math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_317' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>T</mi> <mi>n</mi></msub><mi>X</mi></mrow><annotation encoding='application/x-tex'>T_n X</annotation></semantics></math> is a <math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_318' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>T</mi> <mi>n</mi></msub></mrow><annotation encoding='application/x-tex'>T_n</annotation></semantics></math>-space.</p> </div> <p>Here are alternative constructions of the reflections:</p> <div class='num_prop' id='KolmogorovQuotient'> <h6 id='proposition_8'>Proposition</h6> <p><strong>(<a class='existingWikiWord' href='/nlab/show/diff/Kolmogorov+topological+space'>Kolmogorov quotient</a>)</strong></p> <p>Let <math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_319' 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>. Consider the <a class='existingWikiWord' href='/nlab/show/diff/relation'>relation</a> on the underlying set by which <math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_320' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>x</mi> <mn>1</mn></msub><mo>∼</mo><msub><mi>x</mi> <mn>1</mn></msub></mrow><annotation encoding='application/x-tex'>x_1 \sim x_1</annotation></semantics></math> precisely if neighther <math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_321' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>x</mi> <mi>i</mi></msub></mrow><annotation encoding='application/x-tex'>x_i</annotation></semantics></math> has an <a class='existingWikiWord' href='/nlab/show/diff/neighborhood'>open neighbourhood</a> not containing the other. This is an <a class='existingWikiWord' href='/nlab/show/diff/equivalence+relation'>equivalence relation</a>. The <a class='existingWikiWord' href='/nlab/show/diff/quotient+space'>quotient topological space</a> <math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_322' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi><mo>→</mo><mi>X</mi><mo stretchy='false'>/</mo><mo>∼</mo></mrow><annotation encoding='application/x-tex'>X \to X/\sim</annotation></semantics></math> by this equivalence relation exhibits the <math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_323' 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>-reflection of <math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_324' 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'>6</a>.</p> </div> <div class='num_prop' id='HausdorffReflectionViaTransitiveClosureOfDiagonal'> <h6 id='proposition_9'>Proposition</h6> <p><strong>(<a class='existingWikiWord' href='/nlab/show/diff/Hausdorff+space'>Hausdorff reflection</a>)</strong></p> <p>For <math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_325' 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_e1760b45eef4a61baf80f64f3a3292d3cfff7533_326' 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 tthe <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_e1760b45eef4a61baf80f64f3a3292d3cfff7533_327' 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_e1760b45eef4a61baf80f64f3a3292d3cfff7533_328' 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_e1760b45eef4a61baf80f64f3a3292d3cfff7533_329' 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_e1760b45eef4a61baf80f64f3a3292d3cfff7533_330' 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_e1760b45eef4a61baf80f64f3a3292d3cfff7533_331' 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_e1760b45eef4a61baf80f64f3a3292d3cfff7533_332' 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_e1760b45eef4a61baf80f64f3a3292d3cfff7533_333' 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_e1760b45eef4a61baf80f64f3a3292d3cfff7533_334' 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_e1760b45eef4a61baf80f64f3a3292d3cfff7533_335' 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_e1760b45eef4a61baf80f64f3a3292d3cfff7533_336' 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_e1760b45eef4a61baf80f64f3a3292d3cfff7533_337' 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_e1760b45eef4a61baf80f64f3a3292d3cfff7533_338' 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_e1760b45eef4a61baf80f64f3a3292d3cfff7533_339' 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_e1760b45eef4a61baf80f64f3a3292d3cfff7533_340' 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_e1760b45eef4a61baf80f64f3a3292d3cfff7533_341' 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_e1760b45eef4a61baf80f64f3a3292d3cfff7533_342' 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'>7</a>.</p> </li> </ol> </div> <p>(<a href='Hausdorff+space#vanMunster14'>vanMunster 14, section 4</a>)</p> <h3 id='other_axioms'>Other axioms</h3> <p>There are other axioms sometimes included among the separation axioms that don't fit the preceding pattern; but like the others, they all hold of a metric space:</p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/sober+topological+space'>sober</a> and having enough points,</li> <li><a class='existingWikiWord' href='/nlab/show/diff/regular+space'>semiregular</a>,</li> <li><a class='existingWikiWord' href='/nlab/show/diff/fully+normal+topological+space'>fully normal</a> and fully <math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_343' 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>, which are related to <a class='existingWikiWord' href='/nlab/show/diff/paracompact+topological+space'>paracompactness</a> (see at <em><a class='existingWikiWord' href='/nlab/show/diff/fully+normal+spaces+are+equivalently+paracompact'>fully normal spaces are equivalently paracompact</a></em>).</li> </ul> <h2 id='beyond_the_classical_theory'>Beyond the classical theory</h2> <p>The axioms <math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_344' 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 below can be phrased entirely in terms of the <a class='existingWikiWord' href='/nlab/show/diff/specialization+order'>specialisation order</a>, as follows:</p> <ul> <li>In general, the specialisation order is a <a class='existingWikiWord' href='/nlab/show/diff/preorder'>preorder</a>.</li> <li>The space is <math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_345' 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> if and only if the specialisation order is a <a class='existingWikiWord' href='/nlab/show/diff/partial+order'>partial order</a>.</li> <li>The space is <math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_346' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>R</mi> <mn>0</mn></msub></mrow><annotation encoding='application/x-tex'>R_0</annotation></semantics></math> if and only if the specialisation order is an <a class='existingWikiWord' href='/nlab/show/diff/equivalence+relation'>equivalence relation</a>.</li> <li>The space is <math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_347' 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> if and only if the specialisation order is the <a class='existingWikiWord' href='/nlab/show/diff/equality'>equality relation</a>.</li> </ul> <p>Note that <em>any</em> preorder is the specialisation order for its own <a class='existingWikiWord' href='/nlab/show/diff/specialization+topology'>specialisation topology</a>.</p> <p>The separation conditions that appear in <math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_348' 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> and below, or rather their <a class='existingWikiWord' href='/nlab/show/diff/negation'>negation</a>s, can be easily phrased in terms of the <a class='existingWikiWord' href='/nlab/show/diff/convergence+space'>convergence structure</a>, as follows:</p> <ul> <li>Two points are not distinct if and only if they are equal (of course).</li> <li>They are <strong>topologically indistinguishable</strong> (that is, not topologically distinct) if and only if every <a class='existingWikiWord' href='/nlab/show/diff/net'>net</a> (or <a class='existingWikiWord' href='/nlab/show/diff/filter'>filter</a>) that <a class='existingWikiWord' href='/nlab/show/diff/convergence'>converges</a> to one must also converge to the other; it's enough to check the <a class='existingWikiWord' href='/nlab/show/diff/ultrafilter'>ultrafilters</a> generated by the two points.</li> <li>They are not separated if and only there exists a net (or proper filter) that converges to both.</li> </ul> <p>So by taking contrapositives, it's easy to generalise <math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_349' 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> and below to <a class='existingWikiWord' href='/nlab/show/diff/convergence+space'>convergence spaces</a>. (All of the axioms <em>can</em> be generalised to convergence spaces, since the convergence structure determines the topology, but there are several ways to do so, and it's not clear in general which is best.)</p> <p>For <a class='existingWikiWord' href='/nlab/show/diff/locale'>locales</a>, the axioms at the other end are clearest. Here we want to put everything in terms of open sets, so we simply work with the <a class='existingWikiWord' href='/nlab/show/diff/complement'>complements</a> of the closed sets that appear in those axioms. Rather than talk about a closed set <math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_350' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>F</mi></mrow><annotation encoding='application/x-tex'>F</annotation></semantics></math> and a neighbourhood <math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_351' 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_e1760b45eef4a61baf80f64f3a3292d3cfff7533_352' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>F</mi></mrow><annotation encoding='application/x-tex'>F</annotation></semantics></math>, we talk about an open set <math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_353' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>G</mi></mrow><annotation encoding='application/x-tex'>G</annotation></semantics></math> and an open set <math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_354' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>U</mi></mrow><annotation encoding='application/x-tex'>U</annotation></semantics></math> such that <math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_355' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>G</mi><mo>∪</mo><mi>U</mi></mrow><annotation encoding='application/x-tex'>G \cup U</annotation></semantics></math> is the entire space. Now the axioms at the low end are tricky, although there is a standard answer as far down as <math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_356' 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>. (Note that every locale is <math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_357' 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>, indeed sober.)</p> <p>In <a class='existingWikiWord' href='/nlab/show/diff/constructive+mathematics'>constructive mathematics</a>, while the classical definitions all make sense, they are never quite what is wanted. For the low axioms, one may use, as with convergence spaces, conditions that are classically the negations of the separation conditions; for the high axioms, one may use the open sets that are classically the complements of the closed sets in the axioms. In the middle axioms, these work together; for example, the condition that a point <math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_358' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>x</mi></mrow><annotation encoding='application/x-tex'>x</annotation></semantics></math> is disjoint from a closed set <math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_359' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>F</mi></mrow><annotation encoding='application/x-tex'>F</annotation></semantics></math> becomes the condition that <math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_360' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>x</mi></mrow><annotation encoding='application/x-tex'>x</annotation></semantics></math> belongs to an open set <math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_361' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>G</mi></mrow><annotation encoding='application/x-tex'>G</annotation></semantics></math>.</p> <p>Specific examples should be found on the pages for specific separation axioms.</p> <h2 id='History'>History</h2> <p>In <a href='#Tietze23'>Tietze 23, part B, starting on page 300</a>. 4 axioms are discussed, called (in words, not numbers) the first, second, third, and fourth separation axioms (<em>erstes, zweites, drittes, und viertes Trennbarkeitsaxiom</em>). The first of these is <math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_362' 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>, the second is <math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_363' 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>, the third is <math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_364' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>T</mi> <mn>4</mn></msub></mrow><annotation encoding='application/x-tex'>T_4</annotation></semantics></math>, and the fourth is <math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_365' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>T</mi> <mn>5</mn></msub></mrow><annotation encoding='application/x-tex'>T_5</annotation></semantics></math>. So while this paper may be the first to consider a hierarchy of separation axioms, it is not the source of our <math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_366' 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> notation, and it does not number them in the same way.</p> <p>All of these after the first are stated in such a way as to not imply the first, and those after the second are similarly stated in such a way as not to imply the second. However, Tietze does seem to want them to be a hierarchy. For one thing, his general definition of topological space — stated in multiple equivalent ways in part A — includes the first separation axiom, so in context it seems that the others are meant to be postulated only of spaces that are already Hausdorff.</p> <p>Also, after stating the axioms, he immediately provides examples of spaces that satisfy one property but not a higher one, taking care to list both the first and the second separation axiom among those that the third is independent of, but leaving out the second, listing only the first and the third, when listing the axioms that the fourth is independent of. He does eventually give an example of a space that satisfies the second and third but not the first (so a normal regular space that is not Hausdorff), but it is later and more of an afterthought (and the only non-Hausdorff space in this paper). He never asks whether there exists of a regular space that is not normal.</p> <p>Immediately after this example of a non-Hausdorff space, <a href='#Tietze23'>Tietze 23</a> lists the <math class='maruku-mathml' display='inline' id='mathml_e1760b45eef4a61baf80f64f3a3292d3cfff7533_367' 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> axiom after all! But it is not on the same level as the others to him. Instead, it is merely an alternative form of the first separation axiom that may be used in the presence of the second. It still appears that non-Hausdorff spaces are not considered to be real topological spaces worthy of one's attention.</p> <h2 id='related_concepts'>Related concepts</h2> <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/separation+axioms+in+terms+of+lifting+properties'>separation axioms in terms of lifting properties</a></p> </li> </ul> <h2 id='references'>References</h2> <p>An original article is</p> <ul> <li id='Tietze23'>Heinrich Tietze, <em>Beitrage zur allgemeinen Topologie. I. Axiome für verschiedene Fassungen des Umgebungsbegriffs</em>, Mathematische Annalen, vol 88, pages 290-311 (1923) (<a href='https://www.digizeitschriften.de/en/dms/img/?PID=GDZPPN00226921X&physid=phys298#navi'>online scan</a>)</li> </ul> <p>Lecture notes:</p> <ul> <li id='Naik'><a class='existingWikiWord' href='/nlab/show/diff/Vipul+Naik'>Vipul Naik</a>, <em>Topology: The journey into separation axioms</em> [<a class='existingWikiWord' href='/nlab/files/Naik-SeparationAxioms.pdf' title='pdf'>pdf</a>]</li> </ul> <p>See also:</p> <ul> <li> <p>Wikipedia, <em><a href='http://secure.wikimedia.org/wikipedia/en/wiki/Separation_axiom'>Separation axiom</a></em></p> <p>(This is not really an independent reference, since one of the main authors of the present entry is also one of the main authors of the Wikipedia entry.)</p> </li> </ul> <p>Emphasis on the <a class='existingWikiWord' href='/nlab/show/diff/reflective+subcategory'>reflective</a>-<a class='existingWikiWord' href='/nlab/show/diff/category+theory'>category theory</a>:</p> <ul> <li id='HerrlichStrecker71'><a class='existingWikiWord' href='/nlab/show/diff/Horst+Herrlich'>Horst Herrlich</a>, <a class='existingWikiWord' href='/nlab/show/diff/George+Strecker'>George Strecker</a>, <em>Categorical topology – Its origins as exemplified by the unfolding of the theory of topological reflections and coreflections before 1971</em> (<a href='https://link.springer.com/content/pdf/10.1007%2F978-94-017-0468-7_15.pdf'>pdf</a>), pages 255-341 in: C. E. Aull, R Lowen (eds.), <em>Handbook of the History of General Topology. Vol. 1</em>, Kluwer 1997 (<a href='https://link.springer.com/book/10.1007/978-94-017-0468-7'>doi:10.1007/978-94-017-0468-7</a>)</li> </ul> <p> </p> <p> </p> <p> </p> <p> </p> <p> </p><del class='diffdel'> </del><del class='diffdel'><p> </p></del><del class='diffdel'> </del><del class='diffdel'><p> </p></del><del class='diffdel'> </del><del class='diffdel'><p> </p></del> <p> </p> </div> <div class="revisedby"> <p> Last revised on July 16, 2023 at 20:32:36. See the <a href="/nlab/history/separation+axioms" style="color: #005c19">history</a> of this page for a list of all contributions to it. </p> </div> <div class="navigation navfoot"> <a href="/nlab/edit/separation+axioms" accesskey="E" class="navlink" id="edit" rel="nofollow">Edit</a><a href="https://nforum.ncatlab.org/discussion/10335/#Item_13">Discuss</a><span class="backintime"><a href="/nlab/revision/diff/separation+axioms/57" accesskey="B" class="navlinkbackintime" id="to_previous_revision" rel="nofollow">Previous revision</a></span><a href="/nlab/show/separation+axioms" accesskey="C" class="navlink" id="see_changes" rel="nofollow">Hide changes</a><a href="/nlab/history/separation+axioms" accesskey="S" class="navlink" id="history" rel="nofollow">History (57 revisions)</a> <a href="/nlab/show/separation+axioms/cite" style="color: black">Cite</a> <a href="/nlab/print/separation+axioms" accesskey="p" id="view_print" rel="nofollow">Print</a> <a href="/nlab/source/separation+axioms" id="view_source" rel="nofollow">Source</a> </div> </div>  </div>  </body> </html>