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float: left;" onfocus="this.value == 'Search' ? this.value = '' : true" onblur="this.value == '' ? this.value = 'Search' : true" /></fieldset> </form> <span id='navEnd'></span> </div> <div id="revision"> <html xmlns="http://www.w3.org/1999/xhtml" xmlns:svg="http://www.w3.org/2000/svg" xml:lang="en" lang="en"> <head><meta http-equiv="Content-type" content="application/xhtml+xml;charset=utf-8" /><title>Contents</title></head> <body> <div class="rightHandSide"> <div class="toc clickDown" tabindex="0"> <h3 id="context">Context</h3> <h4 id="topology">Topology</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/topology">topology</a></strong> (<a class="existingWikiWord" href="/nlab/show/point-set+topology">point-set topology</a>, <a class="existingWikiWord" href="/nlab/show/point-free+topology">point-free topology</a>)</p> <p>see also <em><a class="existingWikiWord" href="/nlab/show/differential+topology">differential topology</a></em>, <em><a class="existingWikiWord" href="/nlab/show/algebraic+topology">algebraic topology</a></em>, <em><a class="existingWikiWord" href="/nlab/show/functional+analysis">functional analysis</a></em> and <em><a class="existingWikiWord" href="/nlab/show/topological+homotopy+theory">topological</a> <a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopy theory</a></em></p> <p><a class="existingWikiWord" href="/nlab/show/Introduction+to+Topology">Introduction</a></p> <p><strong>Basic concepts</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/open+subset">open subset</a>, <a class="existingWikiWord" href="/nlab/show/closed+subset">closed subset</a>, <a class="existingWikiWord" href="/nlab/show/neighbourhood">neighbourhood</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a>, <a class="existingWikiWord" href="/nlab/show/locale">locale</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/base+for+the+topology">base for the topology</a>, <a class="existingWikiWord" href="/nlab/show/neighbourhood+base">neighbourhood base</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/finer+topology">finer/coarser topology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+closure">closure</a>, <a class="existingWikiWord" href="/nlab/show/topological+interior">interior</a>, <a class="existingWikiWord" href="/nlab/show/topological+boundary">boundary</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/separation+axiom">separation</a>, <a class="existingWikiWord" href="/nlab/show/sober+topological+space">sobriety</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/continuous+function">continuous function</a>, <a class="existingWikiWord" href="/nlab/show/homeomorphism">homeomorphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/uniformly+continuous+function">uniformly continuous function</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+embedding">embedding</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/open+map">open map</a>, <a class="existingWikiWord" href="/nlab/show/closed+map">closed map</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/sequence">sequence</a>, <a class="existingWikiWord" href="/nlab/show/net">net</a>, <a class="existingWikiWord" href="/nlab/show/sub-net">sub-net</a>, <a class="existingWikiWord" href="/nlab/show/filter">filter</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/convergence">convergence</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/category">category</a><a class="existingWikiWord" href="/nlab/show/Top">Top</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/convenient+category+of+topological+spaces">convenient category of topological spaces</a></li> </ul> </li> </ul> <p><strong><a href="Top#UniversalConstructions">Universal constructions</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/initial+topology">initial topology</a>, <a class="existingWikiWord" href="/nlab/show/final+topology">final topology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/subspace">subspace</a>, <a class="existingWikiWord" href="/nlab/show/quotient+space">quotient space</a>,</p> </li> <li> <p>fiber space, <a class="existingWikiWord" href="/nlab/show/space+attachment">space attachment</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/product+space">product space</a>, <a class="existingWikiWord" href="/nlab/show/disjoint+union+space">disjoint union space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/mapping+cylinder">mapping cylinder</a>, <a class="existingWikiWord" href="/nlab/show/mapping+cocylinder">mapping cocylinder</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/mapping+cone">mapping cone</a>, <a class="existingWikiWord" href="/nlab/show/mapping+cocone">mapping cocone</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/mapping+telescope">mapping telescope</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/colimits+of+normal+spaces">colimits of normal spaces</a></p> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/stuff%2C+structure%2C+property">Extra stuff, structure, properties</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/nice+topological+space">nice topological space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/metric+space">metric space</a>, <a class="existingWikiWord" href="/nlab/show/metric+topology">metric topology</a>, <a class="existingWikiWord" href="/nlab/show/metrisable+space">metrisable space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Kolmogorov+space">Kolmogorov space</a>, <a class="existingWikiWord" href="/nlab/show/Hausdorff+space">Hausdorff space</a>, <a class="existingWikiWord" href="/nlab/show/regular+space">regular space</a>, <a class="existingWikiWord" href="/nlab/show/normal+space">normal space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/sober+space">sober space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/compact+space">compact space</a>, <a class="existingWikiWord" href="/nlab/show/proper+map">proper map</a></p> <p><a class="existingWikiWord" href="/nlab/show/sequentially+compact+topological+space">sequentially compact</a>, <a class="existingWikiWord" href="/nlab/show/countably+compact+topological+space">countably compact</a>, <a class="existingWikiWord" href="/nlab/show/locally+compact+topological+space">locally compact</a>, <a class="existingWikiWord" href="/nlab/show/sigma-compact+topological+space">sigma-compact</a>, <a class="existingWikiWord" href="/nlab/show/paracompact+space">paracompact</a>, <a class="existingWikiWord" href="/nlab/show/countably+paracompact+topological+space">countably paracompact</a>, <a class="existingWikiWord" href="/nlab/show/strongly+compact+topological+space">strongly compact</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/compactly+generated+space">compactly generated space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/second-countable+space">second-countable space</a>, <a class="existingWikiWord" href="/nlab/show/first-countable+space">first-countable space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/contractible+space">contractible space</a>, <a class="existingWikiWord" href="/nlab/show/locally+contractible+space">locally contractible space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/connected+space">connected space</a>, <a class="existingWikiWord" href="/nlab/show/locally+connected+space">locally connected space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/simply-connected+space">simply-connected space</a>, <a class="existingWikiWord" href="/nlab/show/locally+simply-connected+space">locally simply-connected space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cell+complex">cell complex</a>, <a class="existingWikiWord" href="/nlab/show/CW-complex">CW-complex</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/pointed+topological+space">pointed space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+vector+space">topological vector space</a>, <a class="existingWikiWord" href="/nlab/show/Banach+space">Banach space</a>, <a class="existingWikiWord" href="/nlab/show/Hilbert+space">Hilbert space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+group">topological group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+vector+bundle">topological vector bundle</a>, <a class="existingWikiWord" href="/nlab/show/topological+K-theory">topological K-theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+manifold">topological manifold</a></p> </li> </ul> <p><strong>Examples</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/empty+space">empty space</a>, <a class="existingWikiWord" href="/nlab/show/point+space">point space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/discrete+space">discrete space</a>, <a class="existingWikiWord" href="/nlab/show/codiscrete+space">codiscrete space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Sierpinski+space">Sierpinski space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/order+topology">order topology</a>, <a class="existingWikiWord" href="/nlab/show/specialization+topology">specialization topology</a>, <a class="existingWikiWord" href="/nlab/show/Scott+topology">Scott topology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Euclidean+space">Euclidean space</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/real+line">real line</a>, <a class="existingWikiWord" href="/nlab/show/plane">plane</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cylinder">cylinder</a>, <a class="existingWikiWord" href="/nlab/show/cone">cone</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/sphere">sphere</a>, <a class="existingWikiWord" href="/nlab/show/ball">ball</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/circle">circle</a>, <a class="existingWikiWord" href="/nlab/show/torus">torus</a>, <a class="existingWikiWord" href="/nlab/show/annulus">annulus</a>, <a class="existingWikiWord" href="/nlab/show/Moebius+strip">Moebius strip</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/polytope">polytope</a>, <a class="existingWikiWord" href="/nlab/show/polyhedron">polyhedron</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/projective+space">projective space</a> (<a class="existingWikiWord" href="/nlab/show/real+projective+space">real</a>, <a class="existingWikiWord" href="/nlab/show/complex+projective+space">complex</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/classifying+space">classifying space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/configuration+space+%28mathematics%29">configuration space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/path">path</a>, <a class="existingWikiWord" href="/nlab/show/loop">loop</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/mapping+spaces">mapping spaces</a>: <a class="existingWikiWord" href="/nlab/show/compact-open+topology">compact-open topology</a>, <a class="existingWikiWord" href="/nlab/show/topology+of+uniform+convergence">topology of uniform convergence</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/loop+space">loop space</a>, <a class="existingWikiWord" href="/nlab/show/path+space">path space</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Zariski+topology">Zariski topology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Cantor+space">Cantor space</a>, <a class="existingWikiWord" href="/nlab/show/Mandelbrot+space">Mandelbrot space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Peano+curve">Peano curve</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/line+with+two+origins">line with two origins</a>, <a class="existingWikiWord" href="/nlab/show/long+line">long line</a>, <a class="existingWikiWord" href="/nlab/show/Sorgenfrey+line">Sorgenfrey line</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/K-topology">K-topology</a>, <a class="existingWikiWord" href="/nlab/show/Dowker+space">Dowker space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Warsaw+circle">Warsaw circle</a>, <a class="existingWikiWord" href="/nlab/show/Hawaiian+earring+space">Hawaiian earring space</a></p> </li> </ul> <p><strong>Basic statements</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Hausdorff+spaces+are+sober">Hausdorff spaces are sober</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/schemes+are+sober">schemes are sober</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/continuous+images+of+compact+spaces+are+compact">continuous images of compact spaces are compact</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/closed+subspaces+of+compact+Hausdorff+spaces+are+equivalently+compact+subspaces">closed subspaces of compact Hausdorff spaces are equivalently compact subspaces</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/open+subspaces+of+compact+Hausdorff+spaces+are+locally+compact">open subspaces of compact Hausdorff spaces are locally compact</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quotient+projections+out+of+compact+Hausdorff+spaces+are+closed+precisely+if+the+codomain+is+Hausdorff">quotient projections out of compact Hausdorff spaces are closed precisely if the codomain is Hausdorff</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/compact+spaces+equivalently+have+converging+subnet+of+every+net">compact spaces equivalently have converging subnet of every net</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Lebesgue+number+lemma">Lebesgue number lemma</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/sequentially+compact+metric+spaces+are+equivalently+compact+metric+spaces">sequentially compact metric spaces are equivalently compact metric spaces</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/compact+spaces+equivalently+have+converging+subnet+of+every+net">compact spaces equivalently have converging subnet of every net</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/sequentially+compact+metric+spaces+are+totally+bounded">sequentially compact metric spaces are totally bounded</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/continuous+metric+space+valued+function+on+compact+metric+space+is+uniformly+continuous">continuous metric space valued function on compact metric space is uniformly continuous</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/paracompact+Hausdorff+spaces+are+normal">paracompact Hausdorff spaces are normal</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/paracompact+Hausdorff+spaces+equivalently+admit+subordinate+partitions+of+unity">paracompact Hausdorff spaces equivalently admit subordinate partitions of unity</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/closed+injections+are+embeddings">closed injections are embeddings</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/proper+maps+to+locally+compact+spaces+are+closed">proper maps to locally compact spaces are closed</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/injective+proper+maps+to+locally+compact+spaces+are+equivalently+the+closed+embeddings">injective proper maps to locally compact spaces are equivalently the closed embeddings</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/locally+compact+and+sigma-compact+spaces+are+paracompact">locally compact and sigma-compact spaces are paracompact</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/locally+compact+and+second-countable+spaces+are+sigma-compact">locally compact and second-countable spaces are sigma-compact</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/second-countable+regular+spaces+are+paracompact">second-countable regular spaces are paracompact</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/CW-complexes+are+paracompact+Hausdorff+spaces">CW-complexes are paracompact Hausdorff spaces</a></p> </li> </ul> <p><strong>Theorems</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Urysohn%27s+lemma">Urysohn's lemma</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Tietze+extension+theorem">Tietze extension theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Tychonoff+theorem">Tychonoff theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/tube+lemma">tube lemma</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Michael%27s+theorem">Michael's theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Brouwer%27s+fixed+point+theorem">Brouwer's fixed point theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+invariance+of+dimension">topological invariance of dimension</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Jordan+curve+theorem">Jordan curve theorem</a></p> </li> </ul> <p><strong>Analysis Theorems</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Heine-Borel+theorem">Heine-Borel theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/intermediate+value+theorem">intermediate value theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/extreme+value+theorem">extreme value theorem</a></p> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/topological+homotopy+theory">topological homotopy theory</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/left+homotopy">left homotopy</a>, <a class="existingWikiWord" href="/nlab/show/right+homotopy">right homotopy</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+equivalence">homotopy equivalence</a>, <a class="existingWikiWord" href="/nlab/show/deformation+retract">deformation retract</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+group">fundamental group</a>, <a class="existingWikiWord" href="/nlab/show/covering+space">covering space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+theorem+of+covering+spaces">fundamental theorem of covering spaces</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+group">homotopy group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/weak+homotopy+equivalence">weak homotopy equivalence</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Whitehead%27s+theorem">Whitehead's theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Freudenthal+suspension+theorem">Freudenthal suspension theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/nerve+theorem">nerve theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+extension+property">homotopy extension property</a>, <a class="existingWikiWord" href="/nlab/show/Hurewicz+cofibration">Hurewicz cofibration</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+cofiber+sequence">cofiber sequence</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Str%C3%B8m+model+category">Strøm model category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/classical+model+structure+on+topological+spaces">classical model structure on topological spaces</a></p> </li> </ul> </div></div> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#definition'>Definition</a></li> <li><a href='#properties'>Properties</a></li> <ul> <li><a href='#UniversalConstructions'>Universal constructions</a></li> <li><a href='#relation_with_'>Relation with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Set</mi></mrow><annotation encoding="application/x-tex">Set</annotation></semantics></math></a></li> <li><a href='#MonoEpiMorphisms'>Mono-/Epimorphisms</a></li> <li><a href='#intersections_and_quotients'>Intersections and quotients</a></li> <li><a href='#closed_monoidal_structure'>Closed monoidal structure</a></li> </ul> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> </ul> </div> <h2 id="definition">Definition</h2> <p><strong>Top</strong> denotes the <a class="existingWikiWord" href="/nlab/show/category">category</a> whose <a class="existingWikiWord" href="/nlab/show/objects">objects</a> are <a class="existingWikiWord" href="/nlab/show/topological+spaces">topological spaces</a> and whose <a class="existingWikiWord" href="/nlab/show/morphisms">morphisms</a> are <a class="existingWikiWord" href="/nlab/show/continuous+functions">continuous functions</a> between them. Its <a class="existingWikiWord" href="/nlab/show/isomorphisms">isomorphisms</a> are the <a class="existingWikiWord" href="/nlab/show/homeomorphisms">homeomorphisms</a>.</p> <p>For exposition see <em><a class="existingWikiWord" href="/nlab/show/Introduction+to+Topology+--+1">Introduction to point-set topology</a></em>.</p> <p>Often one considers (sometimes by default) <a class="existingWikiWord" href="/nlab/show/subcategories">subcategories</a> of <a class="existingWikiWord" href="/nlab/show/nice+topological+spaces">nice topological spaces</a> such as <a class="existingWikiWord" href="/nlab/show/compactly+generated+topological+spaces">compactly generated topological spaces</a>, notably because these are <a class="existingWikiWord" href="/nlab/show/cartesian+closed+category">cartesian closed</a>. There other other <a class="existingWikiWord" href="/nlab/show/convenient+categories+of+topological+spaces">convenient categories of topological spaces</a>. With any one such choice understood, it is often useful to regard it as “the” category of topological spaces.</p> <p>The <a class="existingWikiWord" href="/nlab/show/homotopy+category">homotopy category</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Top</mi></mrow><annotation encoding="application/x-tex">Top</annotation></semantics></math> given by its <a class="existingWikiWord" href="/nlab/show/localization">localization</a> at the <a class="existingWikiWord" href="/nlab/show/weak+homotopy+equivalences">weak homotopy equivalences</a> is the <a class="existingWikiWord" href="/nlab/show/classical+homotopy+category">classical homotopy category</a> <a class="existingWikiWord" href="/nlab/show/Ho%28Top%29">Ho(Top)</a>. This is the central object of study in <a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopy theory</a>, see also at <em><a class="existingWikiWord" href="/nlab/show/classical+model+structure+on+topological+spaces">classical model structure on topological spaces</a></em>. The <a class="existingWikiWord" href="/nlab/show/simplicial+localization">simplicial localization</a> of <a class="existingWikiWord" href="/nlab/show/Top">Top</a> at the <a class="existingWikiWord" href="/nlab/show/weak+homotopy+equivalences">weak homotopy equivalences</a> is the archetypical <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category">(∞,1)-category</a>, <a class="existingWikiWord" href="/nlab/show/equivalence+of+%28infinity%2C1%29-categories">equivalent</a> to <a class="existingWikiWord" href="/nlab/show/%E2%88%9EGrpd">∞Grpd</a> (see at <em><a class="existingWikiWord" href="/nlab/show/homotopy+hypothesis">homotopy hypothesis</a></em>).</p> <h2 id="properties">Properties</h2> <h3 id="UniversalConstructions">Universal constructions</h3> <p>We discuss <a class="existingWikiWord" href="/nlab/show/universal+constructions">universal constructions</a> in <a class="existingWikiWord" href="/nlab/show/Top">Top</a>, such as <a class="existingWikiWord" href="/nlab/show/limits">limits</a>/<a class="existingWikiWord" href="/nlab/show/colimits">colimits</a>, etc. The following definition suggests that universal constructions be seen in the context of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Top</mi></mrow><annotation encoding="application/x-tex">Top</annotation></semantics></math> as a <a class="existingWikiWord" href="/nlab/show/topological+concrete+category">topological concrete category</a> (see Proposition <a class="maruku-ref" href="#topcat"></a> below).</p> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></p> <div> <p><strong>examples of <a href="Top#UniversalConstructions">universal constructions of topological spaces</a>:</strong></p> <table><thead><tr><th><math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline" id="mathml_7ec4608812513a5e534c550385bc5d1ae5fa5eaf_1"><semantics><mrow><mphantom><mi>AAAA</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{AAAA}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/limits">limits</a></th><th><math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline" id="mathml_7ec4608812513a5e534c550385bc5d1ae5fa5eaf_2"><semantics><mrow><mphantom><mi>AAAA</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{AAAA}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/colimits">colimits</a></th></tr></thead><tbody><tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline" id="mathml_7ec4608812513a5e534c550385bc5d1ae5fa5eaf_3"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/point+space">point space</a><math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline" id="mathml_7ec4608812513a5e534c550385bc5d1ae5fa5eaf_4"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline" id="mathml_7ec4608812513a5e534c550385bc5d1ae5fa5eaf_5"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/empty+space">empty space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline" id="mathml_7ec4608812513a5e534c550385bc5d1ae5fa5eaf_6"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline" id="mathml_7ec4608812513a5e534c550385bc5d1ae5fa5eaf_7"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/product+topological+space">product topological space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline" id="mathml_7ec4608812513a5e534c550385bc5d1ae5fa5eaf_8"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline" id="mathml_7ec4608812513a5e534c550385bc5d1ae5fa5eaf_9"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/disjoint+union+topological+space">disjoint union topological space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline" id="mathml_7ec4608812513a5e534c550385bc5d1ae5fa5eaf_10"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline" id="mathml_7ec4608812513a5e534c550385bc5d1ae5fa5eaf_11"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/topological+subspace">topological subspace</a> <math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline" id="mathml_7ec4608812513a5e534c550385bc5d1ae5fa5eaf_12"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline" id="mathml_7ec4608812513a5e534c550385bc5d1ae5fa5eaf_13"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/quotient+topological+space">quotient topological space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline" id="mathml_7ec4608812513a5e534c550385bc5d1ae5fa5eaf_14"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline" id="mathml_7ec4608812513a5e534c550385bc5d1ae5fa5eaf_15"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math> fiber space <math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline" id="mathml_7ec4608812513a5e534c550385bc5d1ae5fa5eaf_16"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline" id="mathml_7ec4608812513a5e534c550385bc5d1ae5fa5eaf_17"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/space+attachment">space attachment</a> <math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline" id="mathml_7ec4608812513a5e534c550385bc5d1ae5fa5eaf_18"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline" id="mathml_7ec4608812513a5e534c550385bc5d1ae5fa5eaf_19"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/mapping+cocylinder">mapping cocylinder</a>, <a class="existingWikiWord" href="/nlab/show/mapping+cocone">mapping cocone</a> <math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline" id="mathml_7ec4608812513a5e534c550385bc5d1ae5fa5eaf_20"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline" id="mathml_7ec4608812513a5e534c550385bc5d1ae5fa5eaf_21"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/mapping+cylinder">mapping cylinder</a>, <a class="existingWikiWord" href="/nlab/show/mapping+cone">mapping cone</a>, <a class="existingWikiWord" href="/nlab/show/mapping+telescope">mapping telescope</a> <math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline" id="mathml_7ec4608812513a5e534c550385bc5d1ae5fa5eaf_22"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline" id="mathml_7ec4608812513a5e534c550385bc5d1ae5fa5eaf_23"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/cell+complex">cell complex</a>, <a class="existingWikiWord" href="/nlab/show/CW-complex">CW-complex</a> <math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline" id="mathml_7ec4608812513a5e534c550385bc5d1ae5fa5eaf_24"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td></tr> </tbody></table> </div> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></p> <div class="num_defn" id="InitialAndFinalTopologies"> <h6 id="definition_2">Definition</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/weak+topology">weak topology</a> and <a class="existingWikiWord" href="/nlab/show/strong+topology">strong topology</a>)</strong></p> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><msub><mi>X</mi> <mi>i</mi></msub><mo>=</mo><mo stretchy="false">(</mo><msub><mi>S</mi> <mi>i</mi></msub><mo>,</mo><msub><mi>τ</mi> <mi>i</mi></msub><mo stretchy="false">)</mo><mo>∈</mo><mi>Top</mi><msub><mo stretchy="false">}</mo> <mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\{X_i = (S_i,\tau_i) \in Top\}_{i \in I}</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/class">class</a> of <a class="existingWikiWord" href="/nlab/show/topological+spaces">topological spaces</a>, and let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi><mo>∈</mo><mi>Set</mi></mrow><annotation encoding="application/x-tex">S \in Set</annotation></semantics></math> be a bare <a class="existingWikiWord" href="/nlab/show/set">set</a>. Then</p> <ul> <li> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><mi>S</mi><mover><mo>→</mo><mrow><msub><mi>f</mi> <mi>i</mi></msub></mrow></mover><msub><mi>S</mi> <mi>i</mi></msub><msub><mo stretchy="false">}</mo> <mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\{S \stackrel{f_i}{\to} S_i \}_{i \in I}</annotation></semantics></math> a set of <a class="existingWikiWord" href="/nlab/show/functions">functions</a> out of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math>, the <em><a class="existingWikiWord" href="/nlab/show/initial+topology">initial topology</a></em> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>τ</mi> <mi>initial</mi></msub><mo stretchy="false">(</mo><mo stretchy="false">{</mo><msub><mi>f</mi> <mi>i</mi></msub><msub><mo stretchy="false">}</mo> <mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\tau_{initial}(\{f_i\}_{i \in I})</annotation></semantics></math> is the topology on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math> with the <a class="existingWikiWord" href="/nlab/show/minimum">minimum</a> collection of <a class="existingWikiWord" href="/nlab/show/open+subsets">open subsets</a> such that all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>f</mi> <mi>i</mi></msub><mo lspace="verythinmathspace">:</mo><mo stretchy="false">(</mo><mi>S</mi><mo>,</mo><msub><mi>τ</mi> <mi>initial</mi></msub><mo stretchy="false">(</mo><mo stretchy="false">{</mo><msub><mi>f</mi> <mi>i</mi></msub><msub><mo stretchy="false">}</mo> <mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>→</mo><msub><mi>X</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">f_i \colon (S,\tau_{initial}(\{f_i\}_{i \in I}))\to X_i</annotation></semantics></math> are <a class="existingWikiWord" href="/nlab/show/continuous+function">continuous</a>.</p> </li> <li> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><msub><mi>S</mi> <mi>i</mi></msub><mover><mo>→</mo><mrow><msub><mi>f</mi> <mi>i</mi></msub></mrow></mover><mi>S</mi><msub><mo stretchy="false">}</mo> <mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\{S_i \stackrel{f_i}{\to} S\}_{i \in I}</annotation></semantics></math> a set of <a class="existingWikiWord" href="/nlab/show/functions">functions</a> into <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math>, the <em><a class="existingWikiWord" href="/nlab/show/final+topology">final topology</a></em> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>τ</mi> <mi>final</mi></msub><mo stretchy="false">(</mo><mo stretchy="false">{</mo><msub><mi>f</mi> <mi>i</mi></msub><msub><mo stretchy="false">}</mo> <mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\tau_{final}(\{f_i\}_{i \in I})</annotation></semantics></math> is the topology on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math> with the <a class="existingWikiWord" href="/nlab/show/maximum">maximum</a> collection of <a class="existingWikiWord" href="/nlab/show/open+subsets">open subsets</a> such that all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>f</mi> <mi>i</mi></msub><mo lspace="verythinmathspace">:</mo><msub><mi>X</mi> <mi>i</mi></msub><mo>→</mo><mo stretchy="false">(</mo><mi>S</mi><mo>,</mo><msub><mi>τ</mi> <mi>final</mi></msub><mo stretchy="false">(</mo><mo stretchy="false">{</mo><msub><mi>f</mi> <mi>i</mi></msub><msub><mo stretchy="false">}</mo> <mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f_i \colon X_i \to (S,\tau_{final}(\{f_i\}_{i \in I}))</annotation></semantics></math> are <a class="existingWikiWord" href="/nlab/show/continuous+function">continuous</a>.</p> </li> </ul> </div> <div class="num_example" id="TopologicalSubspace"> <h6 id="example">Example</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> a single topological space, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ι</mi> <mi>S</mi></msub><mo lspace="verythinmathspace">:</mo><mi>S</mi><mo>↪</mo><mi>U</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\iota_S \colon S \hookrightarrow U(X)</annotation></semantics></math> a subset of its underlying set, then the initial topology <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>τ</mi> <mi>intial</mi></msub><mo stretchy="false">(</mo><msub><mi>ι</mi> <mi>S</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\tau_{intial}(\iota_S)</annotation></semantics></math>, def. <a class="maruku-ref" href="#InitialAndFinalTopologies"></a>, is the <a class="existingWikiWord" href="/nlab/show/subspace+topology">subspace topology</a>, making</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>ι</mi> <mi>S</mi></msub><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mo stretchy="false">(</mo><mi>S</mi><mo>,</mo><msub><mi>τ</mi> <mi>initial</mi></msub><mo stretchy="false">(</mo><msub><mi>ι</mi> <mi>S</mi></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>↪</mo><mi>X</mi></mrow><annotation encoding="application/x-tex"> \iota_S \;\colon\; (S, \tau_{initial}(\iota_S)) \hookrightarrow X </annotation></semantics></math></div> <p>a <a class="existingWikiWord" href="/nlab/show/topological+subspace">topological subspace</a> inclusion.</p> </div> <div class="num_example" id="QuotientTopology"> <h6 id="example_2">Example</h6> <p>Conversely, for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>p</mi> <mi>S</mi></msub><mo lspace="verythinmathspace">:</mo><mi>U</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>⟶</mo><mi>S</mi></mrow><annotation encoding="application/x-tex">p_S \colon U(X) \longrightarrow S</annotation></semantics></math> an <a class="existingWikiWord" href="/nlab/show/epimorphism">epimorphism</a>, then the final topology <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>τ</mi> <mi>final</mi></msub><mo stretchy="false">(</mo><msub><mi>p</mi> <mi>S</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\tau_{final}(p_S)</annotation></semantics></math> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math> is the <em><a class="existingWikiWord" href="/nlab/show/quotient+topology">quotient topology</a></em>.</p> </div> <div class="num_prop" id="DescriptionOfLimitsAndColimitsInTop"> <h6 id="proposition">Proposition</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>I</mi></mrow><annotation encoding="application/x-tex">I</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/small+category">small category</a> and let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mo>•</mo></msub><mo lspace="verythinmathspace">:</mo><mi>I</mi><mo>⟶</mo><mi>Top</mi></mrow><annotation encoding="application/x-tex">X_\bullet \colon I \longrightarrow Top</annotation></semantics></math> be an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>I</mi></mrow><annotation encoding="application/x-tex">I</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/diagram">diagram</a> in <a class="existingWikiWord" href="/nlab/show/Top">Top</a> (a <a class="existingWikiWord" href="/nlab/show/functor">functor</a> from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>I</mi></mrow><annotation encoding="application/x-tex">I</annotation></semantics></math> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Top</mi></mrow><annotation encoding="application/x-tex">Top</annotation></semantics></math>), with components denoted <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mi>i</mi></msub><mo>=</mo><mo stretchy="false">(</mo><msub><mi>S</mi> <mi>i</mi></msub><mo>,</mo><msub><mi>τ</mi> <mi>i</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">X_i = (S_i, \tau_i)</annotation></semantics></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>S</mi> <mi>i</mi></msub><mo>∈</mo><mi>Set</mi></mrow><annotation encoding="application/x-tex">S_i \in Set</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>τ</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">\tau_i</annotation></semantics></math> a topology on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>S</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">S_i</annotation></semantics></math>. Then:</p> <ol> <li> <p>The <a class="existingWikiWord" href="/nlab/show/limit">limit</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mo>•</mo></msub></mrow><annotation encoding="application/x-tex">X_\bullet</annotation></semantics></math> exists and is given by <a class="existingWikiWord" href="/nlab/show/generalized+the">the</a> topological space whose underlying set is <a class="existingWikiWord" href="/nlab/show/generalized+the">the</a> limit in <a class="existingWikiWord" href="/nlab/show/Set">Set</a> of the underlying sets in the diagram, and whose topology is the <a class="existingWikiWord" href="/nlab/show/initial+topology">initial topology</a>, def. <a class="maruku-ref" href="#InitialAndFinalTopologies"></a>, for the functions <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>p</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">p_i</annotation></semantics></math> which are the limiting <a class="existingWikiWord" href="/nlab/show/cone">cone</a> components:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd></mtd> <mtd></mtd> <mtd><msub><munder><mi>lim</mi><mo>⟵</mo></munder> <mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></msub><msub><mi>S</mi> <mi>i</mi></msub></mtd></mtr> <mtr><mtd></mtd> <mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><msub><mi>p</mi> <mi>i</mi></msub></mrow></mpadded></msup><mo>↙</mo></mtd> <mtd></mtd> <mtd><msup><mo>↘</mo> <mpadded width="0"><mrow><msub><mi>p</mi> <mi>j</mi></msub></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><msub><mi>S</mi> <mi>i</mi></msub></mtd> <mtd></mtd> <mtd><munder><mo>⟶</mo><mrow></mrow></munder></mtd> <mtd></mtd> <mtd><msub><mi>S</mi> <mi>j</mi></msub></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ && \underset{\longleftarrow}{\lim}_{i \in I} S_i \\ & {}^{\mathllap{p_i}}\swarrow && \searrow^{\mathrlap{p_j}} \\ S_i && \underset{}{\longrightarrow} && S_j } \,. </annotation></semantics></math></div> <p>Hence</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><munder><mi>lim</mi><mo>⟵</mo></munder> <mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></msub><msub><mi>X</mi> <mi>i</mi></msub><mo>≃</mo><mrow><mo>(</mo><msub><munder><mi>lim</mi><mo>⟵</mo></munder> <mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></msub><msub><mi>S</mi> <mi>i</mi></msub><mo>,</mo><mspace width="thickmathspace"></mspace><msub><mi>τ</mi> <mi>initial</mi></msub><mo stretchy="false">(</mo><mo stretchy="false">{</mo><msub><mi>p</mi> <mi>i</mi></msub><msub><mo stretchy="false">}</mo> <mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></msub><mo stretchy="false">)</mo><mo>)</mo></mrow></mrow><annotation encoding="application/x-tex"> \underset{\longleftarrow}{\lim}_{i \in I} X_i \simeq \left(\underset{\longleftarrow}{\lim}_{i \in I} S_i,\; \tau_{initial}(\{p_i\}_{i \in I})\right) </annotation></semantics></math></div></li> <li> <p>The <a class="existingWikiWord" href="/nlab/show/colimit">colimit</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mo>•</mo></msub></mrow><annotation encoding="application/x-tex">X_\bullet</annotation></semantics></math> exists and is the topological space whose underlying set is the colimit in <a class="existingWikiWord" href="/nlab/show/Set">Set</a> of the underlying diagram of sets, and whose topology is the <a class="existingWikiWord" href="/nlab/show/final+topology">final topology</a>, def. <a class="maruku-ref" href="#InitialAndFinalTopologies"></a> for the component maps <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ι</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">\iota_i</annotation></semantics></math> of the colimiting <a class="existingWikiWord" href="/nlab/show/cocone">cocone</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><msub><mi>S</mi> <mi>i</mi></msub></mtd> <mtd></mtd> <mtd><mo>⟶</mo></mtd> <mtd></mtd> <mtd><msub><mi>S</mi> <mi>j</mi></msub></mtd></mtr> <mtr><mtd></mtd> <mtd><msub><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><msub><mi>ι</mi> <mi>i</mi></msub></mrow></mpadded></msub><mo>↘</mo></mtd> <mtd></mtd> <mtd><msub><mo>↙</mo> <mpadded width="0"><mrow><msub><mi>ι</mi> <mi>j</mi></msub></mrow></mpadded></msub></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><msub><munder><mi>lim</mi><mo>⟶</mo></munder> <mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></msub><msub><mi>S</mi> <mi>i</mi></msub></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ S_i && \longrightarrow && S_j \\ & {}_{\mathllap{\iota_i}}\searrow && \swarrow_{\mathrlap{\iota_j}} \\ && \underset{\longrightarrow}{\lim}_{i \in I} S_i } \,. </annotation></semantics></math></div> <p>Hence</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><munder><mi>lim</mi><mo>⟶</mo></munder> <mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></msub><msub><mi>X</mi> <mi>i</mi></msub><mo>≃</mo><mrow><mo>(</mo><msub><munder><mi>lim</mi><mo>⟶</mo></munder> <mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></msub><msub><mi>S</mi> <mi>i</mi></msub><mo>,</mo><mspace width="thickmathspace"></mspace><msub><mi>τ</mi> <mi>final</mi></msub><mo stretchy="false">(</mo><mo stretchy="false">{</mo><msub><mi>ι</mi> <mi>i</mi></msub><msub><mo stretchy="false">}</mo> <mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></msub><mo stretchy="false">)</mo><mo>)</mo></mrow></mrow><annotation encoding="application/x-tex"> \underset{\longrightarrow}{\lim}_{i \in I} X_i \simeq \left(\underset{\longrightarrow}{\lim}_{i \in I} S_i,\; \tau_{final}(\{\iota_i\}_{i \in I})\right) </annotation></semantics></math></div></li> </ol> </div> <p>(e.g. <a href="#Bourbaki71">Bourbaki 71, section I.4</a>)</p> <div class="proof"> <h6 id="proof">Proof</h6> <p>The required <a class="existingWikiWord" href="/nlab/show/universal+property">universal property</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mrow><mo>(</mo><msub><munder><mi>lim</mi><mo>⟵</mo></munder> <mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></msub><msub><mi>S</mi> <mi>i</mi></msub><mo>,</mo><mspace width="thickmathspace"></mspace><msub><mi>τ</mi> <mi>initial</mi></msub><mo stretchy="false">(</mo><mo stretchy="false">{</mo><msub><mi>p</mi> <mi>i</mi></msub><msub><mo stretchy="false">}</mo> <mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></msub><mo stretchy="false">)</mo><mo>)</mo></mrow></mrow><annotation encoding="application/x-tex">\left(\underset{\longleftarrow}{\lim}_{i \in I} S_i,\; \tau_{initial}(\{p_i\}_{i \in I})\right)</annotation></semantics></math> is immediate: for</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd></mtd> <mtd></mtd> <mtd><mo stretchy="false">(</mo><mi>S</mi><mo>,</mo><mi>τ</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><msub><mi>f</mi> <mi>i</mi></msub></mrow></mpadded></msup><mo>↙</mo></mtd> <mtd></mtd> <mtd><msup><mo>↘</mo> <mpadded width="0"><mrow><msub><mi>f</mi> <mi>j</mi></msub></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><msub><mi>X</mi> <mi>i</mi></msub></mtd> <mtd></mtd> <mtd><munder><mo>⟶</mo><mrow></mrow></munder></mtd> <mtd></mtd> <mtd><msub><mi>X</mi> <mi>j</mi></msub></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ && (S,\tau) \\ & {}^{\mathllap{f_i}}\swarrow && \searrow^{\mathrlap{f_j}} \\ X_i && \underset{}{\longrightarrow} && X_j } </annotation></semantics></math></div> <p>any <a class="existingWikiWord" href="/nlab/show/cone">cone</a> over the diagram, then by construction there is a unique function of underlying sets <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi><mo>⟶</mo><msub><munder><mi>lim</mi><mo>⟵</mo></munder> <mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></msub><msub><mi>S</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">S \longrightarrow \underset{\longleftarrow}{\lim}_{i \in I} S_i</annotation></semantics></math> making the required diagrams commute, and so all that is required is that this unique function is always <a class="existingWikiWord" href="/nlab/show/continuous+function">continuous</a>. But this is precisely what the initial topology ensures.</p> <p>The case of the colimit is <a class="existingWikiWord" href="/nlab/show/formal+dual">formally dual</a>.</p> </div> <div class="num_example" id="PointTopologicalSpaceAsEmptyLimit"> <h6 id="example_3">Example</h6> <p>The limit over the empty diagram in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Top</mi></mrow><annotation encoding="application/x-tex">Top</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/point">point</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>*</mo></mrow><annotation encoding="application/x-tex">\ast</annotation></semantics></math> with its unique topology.</p> </div> <div class="num_example" id="DisjointUnionOfTopologicalSpacesIsCoproduct"> <h6 id="example_4">Example</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><msub><mi>X</mi> <mi>i</mi></msub><msub><mo stretchy="false">}</mo> <mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\{X_i\}_{i \in I}</annotation></semantics></math> a set of topological spaces, their <a class="existingWikiWord" href="/nlab/show/coproduct">coproduct</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><munder><mo>⊔</mo><mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></munder><msub><mi>X</mi> <mi>i</mi></msub><mo>∈</mo><mi>Top</mi></mrow><annotation encoding="application/x-tex">\underset{i \in I}{\sqcup} X_i \in Top</annotation></semantics></math> is their <em><a class="existingWikiWord" href="/nlab/show/disjoint+union">disjoint union</a></em>.</p> </div> <p>In particular:</p> <div class="num_example" id="DiscreteTopologicalSpaceAsCoproduct"> <h6 id="example_5">Example</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi><mo>∈</mo><mi>Set</mi></mrow><annotation encoding="application/x-tex">S \in Set</annotation></semantics></math>, the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math>-indexed <a class="existingWikiWord" href="/nlab/show/coproduct">coproduct</a> of the point, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><munder><mo lspace="thinmathspace" rspace="thinmathspace">∐</mo><mrow><mi>s</mi><mo>∈</mo><mi>S</mi></mrow></munder><mo>*</mo></mrow><annotation encoding="application/x-tex">\underset{s \in S}{\coprod}\ast </annotation></semantics></math>, is the set <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math> itself equipped with the <a class="existingWikiWord" href="/nlab/show/final+topology">final topology</a>, hence is the <a class="existingWikiWord" href="/nlab/show/discrete+topological+space">discrete topological space</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math>.</p> </div> <div class="num_example" id="ProductTopologicalSpace"> <h6 id="example_6">Example</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><msub><mi>X</mi> <mi>i</mi></msub><msub><mo stretchy="false">}</mo> <mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\{X_i\}_{i \in I}</annotation></semantics></math> a set of topological spaces, their <a class="existingWikiWord" href="/nlab/show/product">product</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo><mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></munder><msub><mi>X</mi> <mi>i</mi></msub><mo>∈</mo><mi>Top</mi></mrow><annotation encoding="application/x-tex">\underset{i \in I}{\prod} X_i \in Top</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/Cartesian+product">Cartesian product</a> of the underlying sets equipped with the <em><a class="existingWikiWord" href="/nlab/show/product+topology">product topology</a></em>, also called the <em><a class="existingWikiWord" href="/nlab/show/Tychonoff+product">Tychonoff product</a></em>.</p> <p>In the case that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/finite+set">finite set</a>, such as for binary product spaces <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>×</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">X \times Y</annotation></semantics></math>, then a <a class="existingWikiWord" href="/nlab/show/basis+for+a+topology">sub-basis</a> for the product topology is given by the <a class="existingWikiWord" href="/nlab/show/Cartesian+products">Cartesian products</a> of the open subsets of (a basis for) each factor space.</p> </div> <div class="num_example" id="EqualizerInTop"> <h6 id="example_7">Example</h6> <p>The <a class="existingWikiWord" href="/nlab/show/equalizer">equalizer</a> of two <a class="existingWikiWord" href="/nlab/show/continuous+functions">continuous functions</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>,</mo><mi>g</mi><mo lspace="verythinmathspace">:</mo><mi>X</mi><mover><mo>⟶</mo><mo>⟶</mo></mover><mi>Y</mi></mrow><annotation encoding="application/x-tex">f, g \colon X \stackrel{\longrightarrow}{\longrightarrow} Y</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Top</mi></mrow><annotation encoding="application/x-tex">Top</annotation></semantics></math> is the equalizer of the underlying functions of sets</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>eq</mi><mo stretchy="false">(</mo><mi>f</mi><mo>,</mo><mi>g</mi><mo stretchy="false">)</mo><mo>↪</mo><msub><mi>S</mi> <mi>X</mi></msub><mover><munder><mo>⟶</mo><mi>g</mi></munder><mover><mo>⟶</mo><mi>f</mi></mover></mover><msub><mi>S</mi> <mi>Y</mi></msub></mrow><annotation encoding="application/x-tex"> eq(f,g) \hookrightarrow S_X \stackrel{\overset{f}{\longrightarrow}}{\underset{g}{\longrightarrow}} S_Y </annotation></semantics></math></div> <p>(hence the largets subset of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>S</mi> <mi>X</mi></msub></mrow><annotation encoding="application/x-tex">S_X</annotation></semantics></math> on which both functions coincide) and equipped with the <a class="existingWikiWord" href="/nlab/show/subspace+topology">subspace topology</a>, example <a class="maruku-ref" href="#TopologicalSubspace"></a>.</p> </div> <div class="num_example" id="CoequalizerInTop"> <h6 id="example_8">Example</h6> <p>The <a class="existingWikiWord" href="/nlab/show/coequalizer">coequalizer</a> of two <a class="existingWikiWord" href="/nlab/show/continuous+functions">continuous functions</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>,</mo><mi>g</mi><mo lspace="verythinmathspace">:</mo><mi>X</mi><mover><mo>⟶</mo><mo>⟶</mo></mover><mi>Y</mi></mrow><annotation encoding="application/x-tex">f, g \colon X \stackrel{\longrightarrow}{\longrightarrow} Y</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Top</mi></mrow><annotation encoding="application/x-tex">Top</annotation></semantics></math> is the coequalizer of the underlying functions of sets</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>S</mi> <mi>X</mi></msub><mover><munder><mo>⟶</mo><mi>g</mi></munder><mover><mo>⟶</mo><mi>f</mi></mover></mover><msub><mi>S</mi> <mi>Y</mi></msub><mo>⟶</mo><mi>coeq</mi><mo stretchy="false">(</mo><mi>f</mi><mo>,</mo><mi>g</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> S_X \stackrel{\overset{f}{\longrightarrow}}{\underset{g}{\longrightarrow}} S_Y \longrightarrow coeq(f,g) </annotation></semantics></math></div> <p>(hence the <a class="existingWikiWord" href="/nlab/show/quotient+set">quotient set</a> by the <a class="existingWikiWord" href="/nlab/show/equivalence+relation">equivalence relation</a> generated by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>∼</mo><mi>g</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(x) \sim g(x)</annotation></semantics></math> for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>∈</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">x \in X</annotation></semantics></math>) and equipped with the <a class="existingWikiWord" href="/nlab/show/quotient+topology">quotient topology</a>, example <a class="maruku-ref" href="#QuotientTopology"></a>.</p> </div> <div class="num_example" id="PushoutInTop"> <h6 id="example_9">Example</h6> <p>For</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>A</mi></mtd> <mtd><mover><mo>⟶</mo><mi>g</mi></mover></mtd> <mtd><mi>Y</mi></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mi>f</mi></mpadded></msup><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi>X</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ A &\overset{g}{\longrightarrow}& Y \\ {}^{\mathllap{f}}\downarrow \\ X } </annotation></semantics></math></div> <p>two <a class="existingWikiWord" href="/nlab/show/continuous+functions">continuous functions</a> out of the same <a class="existingWikiWord" href="/nlab/show/domain">domain</a>, then the <a class="existingWikiWord" href="/nlab/show/colimit">colimit</a> under this diagram is also called the <em><a class="existingWikiWord" href="/nlab/show/pushout">pushout</a></em>, denoted</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>A</mi></mtd> <mtd><mover><mo>⟶</mo><mi>g</mi></mover></mtd> <mtd><mi>Y</mi></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mi>f</mi></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><msub><mi>g</mi> <mo>*</mo></msub><mi>f</mi></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><mi>X</mi></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>X</mi><msub><mo>⊔</mo> <mi>A</mi></msub><mi>Y</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ A &\overset{g}{\longrightarrow}& Y \\ {}^{\mathllap{f}}\downarrow && \downarrow^{\mathrlap{g_\ast f}} \\ X &\longrightarrow& X \sqcup_A Y \,. } \,. </annotation></semantics></math></div> <p>(Here <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>g</mi> <mo>*</mo></msub><mi>f</mi></mrow><annotation encoding="application/x-tex">g_\ast f</annotation></semantics></math> is also called the pushout of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math>, or the <em><a class="existingWikiWord" href="/nlab/show/base+change">cobase change</a></em> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> along <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi></mrow><annotation encoding="application/x-tex">g</annotation></semantics></math>.) If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi></mrow><annotation encoding="application/x-tex">g</annotation></semantics></math> is an inclusion, one also write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><msub><mo>∪</mo> <mi>f</mi></msub><mi>Y</mi></mrow><annotation encoding="application/x-tex">X \cup_f Y</annotation></semantics></math> and calls this the <em><a class="existingWikiWord" href="/nlab/show/attaching+space">attaching space</a></em>.</p> <div style="float:left;margin:0 10px 10px 0;"><img src="http://ncatlab.org/nlab/files/AttachingSpace.jpg" width="450" /></div> <p>By example <a class="maruku-ref" href="#CoequalizerInTop"></a> the <a class="existingWikiWord" href="/nlab/show/pushout">pushout</a>/<a class="existingWikiWord" href="/nlab/show/attaching+space">attaching space</a> is the <a class="existingWikiWord" href="/nlab/show/quotient+topological+space">quotient topological space</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>X</mi><msub><mo>⊔</mo> <mi>A</mi></msub><mi>Y</mi><mo>≃</mo><mo stretchy="false">(</mo><mi>X</mi><mo>⊔</mo><mi>Y</mi><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mo>∼</mo></mrow><annotation encoding="application/x-tex"> X \sqcup_A Y \simeq (X\sqcup Y)/\sim </annotation></semantics></math></div> <p>of the <a class="existingWikiWord" href="/nlab/show/disjoint+union">disjoint union</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math> subject to the <a class="existingWikiWord" href="/nlab/show/equivalence+relation">equivalence relation</a> which identifies a point in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> with a point in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math> if they have the same pre-image in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>.</p> <p>(graphics from <a href="#AguilarGitlerPrieto02">Aguilar-Gitler-Prieto 02</a>)</p> </div> <div class="num_example" id="TopologicalnSphereIsPushoutOfBoundaryOfnBallInclusionAlongItself"> <h6 id="example_10">Example</h6> <p>As an important special case of example <a class="maruku-ref" href="#PushoutInTop"></a>, let</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>i</mi> <mi>n</mi></msub><mo lspace="verythinmathspace">:</mo><msup><mi>S</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup><mo>⟶</mo><msup><mi>D</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex"> i_n \colon S^{n-1}\longrightarrow D^n </annotation></semantics></math></div> <p>be the canonical inclusion of the standard <a class="existingWikiWord" href="/nlab/show/n-sphere">(n-1)-sphere</a> as the <a class="existingWikiWord" href="/nlab/show/boundary">boundary</a> of the standard <a class="existingWikiWord" href="/nlab/show/n-disk">n-disk</a> (both regarded as <a class="existingWikiWord" href="/nlab/show/topological+spaces">topological spaces</a> with their <a class="existingWikiWord" href="/nlab/show/subspace+topology">subspace topology</a> as subspaces of the <a class="existingWikiWord" href="/nlab/show/Cartesian+space">Cartesian space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^n</annotation></semantics></math>).</p> <div style="float:left;margin:0 10px 10px 0;"> <img src="http://ncatlab.org/nlab/files/GluingHemispheres.jpg" width="400" /></div> <p>Then the colimit in <a class="existingWikiWord" href="/nlab/show/Top">Top</a> under the diagram, i.e. the <a class="existingWikiWord" href="/nlab/show/pushout">pushout</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>i</mi> <mi>n</mi></msub></mrow><annotation encoding="application/x-tex">i_n</annotation></semantics></math> along itself,</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mo>{</mo><msup><mi>D</mi> <mi>n</mi></msup><mover><mo>⟵</mo><mrow><msub><mi>i</mi> <mi>n</mi></msub></mrow></mover><msup><mi>S</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup><mover><mo>⟶</mo><mrow><msub><mi>i</mi> <mi>n</mi></msub></mrow></mover><msup><mi>D</mi> <mi>n</mi></msup><mo>}</mo></mrow><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \left\{ D^n \overset{i_n}{\longleftarrow} S^{n-1} \overset{i_n}{\longrightarrow} D^n \right\} \,, </annotation></semantics></math></div> <p>is the <a class="existingWikiWord" href="/nlab/show/n-sphere">n-sphere</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">S^n</annotation></semantics></math>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><msup><mi>S</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup></mtd> <mtd><mover><mo>⟶</mo><mrow><msub><mi>i</mi> <mi>n</mi></msub></mrow></mover></mtd> <mtd><msup><mi>D</mi> <mi>n</mi></msup></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><msub><mi>i</mi> <mi>n</mi></msub></mrow></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd><mo stretchy="false">(</mo><mi>po</mi><mo stretchy="false">)</mo></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><msup><mi>D</mi> <mi>n</mi></msup></mtd> <mtd><mo>⟶</mo></mtd> <mtd><msup><mi>S</mi> <mi>n</mi></msup></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ S^{n-1} &\overset{i_n}{\longrightarrow}& D^n \\ {}^{\mathllap{i_n}}\downarrow &(po)& \downarrow \\ D^n &\longrightarrow& S^n } \,. </annotation></semantics></math></div> <p>(graphics from Ueno-Shiga-Morita 95)</p> </div> <div class="num_example" id="ClosedSubspacesGluing"> <h6 id="example_11">Example</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/union">union</a> of two <a class="existingWikiWord" href="/nlab/show/open+subset">open</a> or two <a class="existingWikiWord" href="/nlab/show/closed+subset">closed</a> <a class="existingWikiWord" href="/nlab/show/subspaces">subspaces</a> is <a class="existingWikiWord" href="/nlab/show/pushout">pushout</a>)</strong></p> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a> and let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>,</mo><mi>B</mi><mo>⊂</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">A,B \subset X</annotation></semantics></math> be <a class="existingWikiWord" href="/nlab/show/subspaces">subspaces</a> such that</p> <ol> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>,</mo><mi>B</mi><mo>⊂</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">A,B \subset X</annotation></semantics></math> are both <a class="existingWikiWord" href="/nlab/show/open+subsets">open subsets</a> or are both <a class="existingWikiWord" href="/nlab/show/closed+subsets">closed subsets</a>;</p> </li> <li> <p>they constitute a <a class="existingWikiWord" href="/nlab/show/cover">cover</a>: <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>=</mo><mi>A</mi><mo>∪</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">X = A \cup B</annotation></semantics></math></p> </li> </ol> <p>Write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>i</mi> <mi>A</mi></msub><mo lspace="verythinmathspace">:</mo><mi>A</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">i_A \colon A \to X</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>i</mi> <mi>B</mi></msub><mo lspace="verythinmathspace">:</mo><mi>B</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">i_B \colon B \to X</annotation></semantics></math> for the corresponding inclusion <a class="existingWikiWord" href="/nlab/show/continuous+functions">continuous functions</a>.</p> <p>Then the <a class="existingWikiWord" href="/nlab/show/commuting+square">commuting square</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>A</mi><mo>∩</mo><mi>B</mi></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>A</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><msub><mi>i</mi> <mi>A</mi></msub></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><mi>B</mi></mtd> <mtd><munder><mo>⟶</mo><mrow><msub><mi>i</mi> <mi>B</mi></msub></mrow></munder></mtd> <mtd><mi>X</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ A \cap B &\longrightarrow& A \\ \downarrow && \downarrow^{\mathrlap{i_A}} \\ B &\underset{i_B}{\longrightarrow}& X } </annotation></semantics></math></div> <p>is a <a class="existingWikiWord" href="/nlab/show/pushout">pushout</a> square in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Top</mi></mrow><annotation encoding="application/x-tex">Top</annotation></semantics></math> (example <a class="maruku-ref" href="#PushoutInTop"></a>).</p> <p>By the <a class="existingWikiWord" href="/nlab/show/universal+property">universal property</a> of the <a class="existingWikiWord" href="/nlab/show/pushout">pushout</a> this means in particular that for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math> any <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a> then a function of underlying sets</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>f</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>X</mi><mo>⟶</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex"> f \;\colon\; X \longrightarrow Y </annotation></semantics></math></div> <p>is a <a class="existingWikiWord" href="/nlab/show/continuous+function">continuous function</a> as soon as its two restrictions</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>f</mi><msub><mo stretchy="false">|</mo> <mi>A</mi></msub><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>A</mi><mo>⟶</mo><mi>Y</mi><mphantom><mi>AAAA</mi></mphantom><mi>f</mi><msub><mo stretchy="false">|</mo> <mi>A</mi></msub><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>B</mi><mo>⟶</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex"> f\vert_A \;\colon\; A \longrightarrow Y \phantom{AAAA} f\vert_A \;\colon\; B \longrightarrow Y </annotation></semantics></math></div> <p>are continuous.</p> </div> <div class="proof"> <h6 id="proof_2">Proof</h6> <p>Clearly the underlying diagram of underlying <a class="existingWikiWord" href="/nlab/show/sets">sets</a> is a pushout in <a class="existingWikiWord" href="/nlab/show/Set">Set</a>. Therefore by prop. <a class="maruku-ref" href="#DescriptionOfLimitsAndColimitsInTop"></a> we need to show that the <a class="existingWikiWord" href="/nlab/show/topological+space">topology</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/final+topology">final topology</a> induced by the set of functions <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><msub><mi>i</mi> <mi>A</mi></msub><mo>,</mo><msub><mi>i</mi> <mi>B</mi></msub><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{i_A, i_B\}</annotation></semantics></math>, hence that a <a class="existingWikiWord" href="/nlab/show/subset">subset</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi><mo>⊂</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">S \subset X</annotation></semantics></math> is an <a class="existingWikiWord" href="/nlab/show/open+subset">open subset</a> precisely if the <a class="existingWikiWord" href="/nlab/show/pre-images">pre-images</a> (restrictions)</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msubsup><mi>i</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><mi>S</mi><mo stretchy="false">)</mo><mo>=</mo><mi>S</mi><mo>∩</mo><mi>A</mi><mphantom><mi>AAA</mi></mphantom><mtext>and</mtext><mphantom><mi>AAA</mi></mphantom><msubsup><mi>i</mi> <mi>B</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><mi>S</mi><mo stretchy="false">)</mo><mo>=</mo><mi>S</mi><mo>∩</mo><mi>B</mi></mrow><annotation encoding="application/x-tex"> i_A^{-1}(S) = S \cap A \phantom{AAA} \text{and} \phantom{AAA} i_B^{-1}(S) = S \cap B </annotation></semantics></math></div> <p>are open subsets of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math>, respectively.</p> <p>Now by definition of the <a class="existingWikiWord" href="/nlab/show/subspace+topology">subspace topology</a>, if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi><mo>⊂</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">S \subset X</annotation></semantics></math> is open, then the intersections <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>∩</mo><mi>S</mi><mo>⊂</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">A \cap S \subset A</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi><mo>∩</mo><mi>S</mi><mo>⊂</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">B \cap S \subset B</annotation></semantics></math> are open in these subspaces.</p> <p>Conversely, assume that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>∩</mo><mi>S</mi><mo>⊂</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">A \cap S \subset A</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi><mo>∩</mo><mi>S</mi><mo>⊂</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">B \cap S \subset B</annotation></semantics></math> are open. We need to show that then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi><mo>⊂</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">S \subset X</annotation></semantics></math> is open.</p> <p>Consider now first the case that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>;</mo><mi>B</mi><mo>⊂</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">A;B \subset X</annotation></semantics></math> are both open open. Then by the nature of the <a class="existingWikiWord" href="/nlab/show/subspace+topology">subspace topology</a>, that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>∩</mo><mi>S</mi></mrow><annotation encoding="application/x-tex">A \cap S</annotation></semantics></math> is open in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> means that there is an open subset <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>S</mi> <mi>A</mi></msub><mo>⊂</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">S_A \subset X</annotation></semantics></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>∩</mo><mi>S</mi><mo>=</mo><mi>A</mi><mo>∩</mo><msub><mi>S</mi> <mi>A</mi></msub></mrow><annotation encoding="application/x-tex">A \cap S = A \cap S_A</annotation></semantics></math>. Since the intersection of two open subsets is open, this implies that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>∩</mo><msub><mi>S</mi> <mi>A</mi></msub></mrow><annotation encoding="application/x-tex">A \cap S_A</annotation></semantics></math> and hence <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>∩</mo><mi>S</mi></mrow><annotation encoding="application/x-tex">A \cap S</annotation></semantics></math> is open. Similarly <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi><mo>∩</mo><mi>S</mi></mrow><annotation encoding="application/x-tex">B \cap S</annotation></semantics></math>. Therefore</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd><mi>S</mi></mtd> <mtd><mo>=</mo><mi>S</mi><mo>∩</mo><mi>X</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mi>S</mi><mo>∩</mo><mo stretchy="false">(</mo><mi>A</mi><mo>∪</mo><mi>B</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mo stretchy="false">(</mo><mi>S</mi><mo>∩</mo><mi>A</mi><mo stretchy="false">)</mo><mo>∪</mo><mo stretchy="false">(</mo><mi>S</mi><mo>∩</mo><mi>B</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \begin{aligned} S & = S \cap X \\ & = S \cap (A \cup B) \\ & = (S \cap A) \cup (S \cap B) \end{aligned} </annotation></semantics></math></div> <p>is the union of two open subsets and therefore open.</p> <p>Now consider the case that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>,</mo><mi>B</mi><mo>⊂</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">A,B \subset X</annotation></semantics></math> are both closed subsets.</p> <p>Again by the nature of the subspace topology, that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>∩</mo><mi>S</mi><mo>⊂</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">A \cap S \subset A</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi><mo>∩</mo><mi>S</mi><mo>⊂</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">B \cap S \subset B</annotation></semantics></math> are open means that there exist open subsets <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>S</mi> <mi>A</mi></msub><mo>,</mo><msub><mi>S</mi> <mi>B</mi></msub><mo>⊂</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">S_A, S_B \subset X</annotation></semantics></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>∩</mo><mi>S</mi><mo>=</mo><mi>A</mi><mo>∩</mo><msub><mi>S</mi> <mi>A</mi></msub></mrow><annotation encoding="application/x-tex">A \cap S = A \cap S_A</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi><mo>∩</mo><mi>S</mi><mo>=</mo><mi>B</mi><mo>∩</mo><msub><mi>S</mi> <mi>B</mi></msub></mrow><annotation encoding="application/x-tex">B \cap S = B \cap S_B</annotation></semantics></math>. Since <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>,</mo><mi>B</mi><mo>⊂</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">A,B \subset X</annotation></semantics></math> are closed by assumption, this means that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>∖</mo><mi>S</mi><mo>,</mo><mi>B</mi><mo>∖</mo><mi>S</mi><mo>⊂</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">A \setminus S, B \setminus S \subset X</annotation></semantics></math> are still closed, hence that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>∖</mo><mo stretchy="false">(</mo><mi>A</mi><mo>∖</mo><mi>S</mi><mo stretchy="false">)</mo><mo>,</mo><mi>X</mi><mo>∖</mo><mo stretchy="false">(</mo><mi>B</mi><mo>∖</mo><mi>S</mi><mo stretchy="false">)</mo><mo>⊂</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">X \setminus (A \setminus S), X \setminus (B \setminus S) \subset X</annotation></semantics></math> are open.</p> <p>Now observe that (by <a class="existingWikiWord" href="/nlab/show/de+Morgan+duality">de Morgan duality</a>)</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd><mi>S</mi></mtd> <mtd><mo>=</mo><mi>X</mi><mo>∖</mo><mo stretchy="false">(</mo><mi>X</mi><mo>∖</mo><mi>S</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mi>X</mi><mo>∖</mo><mo stretchy="false">(</mo><mo stretchy="false">(</mo><mi>A</mi><mo>∪</mo><mi>B</mi><mo stretchy="false">)</mo><mo>∖</mo><mi>S</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mi>X</mi><mo>∖</mo><mo stretchy="false">(</mo><mo stretchy="false">(</mo><mi>A</mi><mo>∖</mo><mi>S</mi><mo stretchy="false">)</mo><mo>∪</mo><mo stretchy="false">(</mo><mi>B</mi><mo>∖</mo><mi>S</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mo stretchy="false">(</mo><mi>X</mi><mo>∖</mo><mo stretchy="false">(</mo><mi>A</mi><mo>∖</mo><mi>S</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>∩</mo><mo stretchy="false">(</mo><mi>X</mi><mo>∖</mo><mo stretchy="false">(</mo><mi>B</mi><mo>∖</mo><mi>S</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \begin{aligned} S & = X \setminus (X \setminus S) \\ & = X \setminus ( (A \cup B) \setminus S ) \\ & = X \setminus ( (A \setminus S) \cup (B \setminus S) ) \\ & = (X \setminus (A \setminus S)) \cap (X \setminus (B \setminus S)) \,. \end{aligned} </annotation></semantics></math></div> <p>This exhibits <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math> as the intersection of two open subsets, hence as open.</p> </div> <div class="num_example" id="attach"> <h6 id="example_12">Example</h6> <p>If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>,</mo><mi>Y</mi><mo>,</mo><mi>Z</mi></mrow><annotation encoding="application/x-tex">X, Y, Z</annotation></semantics></math> are <a class="existingWikiWord" href="/nlab/show/normal+topological+spaces">normal topological spaces</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>h</mi><mo>:</mo><mi>X</mi><mo>→</mo><mi>Z</mi></mrow><annotation encoding="application/x-tex">h: X \to Z</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/closed+embedding+of+topological+spaces">closed embedding of topological spaces</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><mi>X</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">f: X \to Y</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/continuous+function">continuous function</a>, then in the <a class="existingWikiWord" href="/nlab/show/pushout">pushout</a> diagram in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Top</mi></mrow><annotation encoding="application/x-tex">Top</annotation></semantics></math> (example <a class="maruku-ref" href="#PushoutInTop"></a>)</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>X</mi></mtd> <mtd><mover><mo>→</mo><mi>h</mi></mover></mtd> <mtd><mi>Z</mi></mtd></mtr> <mtr><mtd><mpadded width="0" lspace="-100%width"><mi>f</mi></mpadded><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo><mpadded width="0"><mi>g</mi></mpadded></mtd></mtr> <mtr><mtd><mi>Y</mi></mtd> <mtd><munder><mo>→</mo><mi>k</mi></munder></mtd> <mtd><mi>W</mi><mo>,</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex">\array{ X & \stackrel{h}{\to} & Z \\ \mathllap{f} \downarrow & & \downarrow \mathrlap{g} \\ Y & \underset{k}{\to} & W, } </annotation></semantics></math></div> <p>the space <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>W</mi></mrow><annotation encoding="application/x-tex">W</annotation></semantics></math> is normal and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi><mo>:</mo><mi>Y</mi><mo>→</mo><mi>W</mi></mrow><annotation encoding="application/x-tex">k: Y \to W</annotation></semantics></math> is a closed embedding.</p> </div> <p>For <strong>proof</strong> of this and related statements see at <em><a class="existingWikiWord" href="/nlab/show/colimits+of+normal+spaces">colimits of normal spaces</a></em>.</p> <h3 id="relation_with_">Relation with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Set</mi></mrow><annotation encoding="application/x-tex">Set</annotation></semantics></math></h3> <p>Write <a class="existingWikiWord" href="/nlab/show/Set">Set</a> for the <a class="existingWikiWord" href="/nlab/show/category">category</a> of <a class="existingWikiWord" href="/nlab/show/sets">sets</a>.</p> <div class="num_defn" id="ForgetfulFunctorFromTopToSet"> <h6 id="definition_3">Definition</h6> <p>Write</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo lspace="verythinmathspace">:</mo><mi>Top</mi><mo>⟶</mo><mi>Set</mi></mrow><annotation encoding="application/x-tex"> U \colon Top \longrightarrow Set </annotation></semantics></math></div> <p>for the <a class="existingWikiWord" href="/nlab/show/forgetful+functor">forgetful functor</a> that sends a topological space <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>=</mo><mo stretchy="false">(</mo><mi>S</mi><mo>,</mo><mi>τ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">X = (S,\tau)</annotation></semantics></math> to its underlying set <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>=</mo><mi>S</mi><mo>∈</mo><mi>Set</mi></mrow><annotation encoding="application/x-tex">U(X) = S \in Set</annotation></semantics></math> and which regards a <a class="existingWikiWord" href="/nlab/show/continuous+function">continuous function</a> as a plain <a class="existingWikiWord" href="/nlab/show/function">function</a> on the underlying sets.</p> </div> <p>Prop. <a class="maruku-ref" href="#DescriptionOfLimitsAndColimitsInTop"></a> means in particular that:</p> <div class="num_prop"> <h6 id="proposition_2">Proposition</h6> <p>The category <a class="existingWikiWord" href="/nlab/show/Top">Top</a> has all small <a class="existingWikiWord" href="/nlab/show/limits">limits</a> and <a class="existingWikiWord" href="/nlab/show/colimits">colimits</a>. The <a class="existingWikiWord" href="/nlab/show/forgetful+functor">forgetful functor</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo lspace="verythinmathspace">:</mo><mi>Top</mi><mo>→</mo><mi>Set</mi></mrow><annotation encoding="application/x-tex">U \colon Top \to Set</annotation></semantics></math> from def. <a class="maruku-ref" href="#ForgetfulFunctorFromTopToSet"></a> <a class="existingWikiWord" href="/nlab/show/preserved+limit">preserves</a> and <a class="existingWikiWord" href="/nlab/show/lifted+limit">lifts</a> limits and colimits.</p> </div> <p>(But it does not <a class="existingWikiWord" href="/nlab/show/created+limit">create</a> or <a class="existingWikiWord" href="/nlab/show/reflected+limit">reflect</a> them.)</p> <div class="num_prop"> <h6 id="proposition_3">Proposition</h6> <p>The <a class="existingWikiWord" href="/nlab/show/forgetful+functor">forgetful functor</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi></mrow><annotation encoding="application/x-tex">U</annotation></semantics></math> from def. <a class="maruku-ref" href="#ForgetfulFunctorFromTopToSet"></a> has a <a class="existingWikiWord" href="/nlab/show/left+adjoint">left adjoint</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>disc</mi></mrow><annotation encoding="application/x-tex">disc</annotation></semantics></math>, given by sending a <a class="existingWikiWord" href="/nlab/show/set">set</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math> to the corresponding <a class="existingWikiWord" href="/nlab/show/discrete+topological+space">discrete topological space</a>, example <a class="maruku-ref" href="#DiscreteTopologicalSpaceAsCoproduct"></a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Top</mi><mover><munder><mo>⟶</mo><mi>U</mi></munder><mover><mo>⟵</mo><mi>disc</mi></mover></mover><mi>Set</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> Top \stackrel{\overset{disc}{\longleftarrow}}{\underset{U}{\longrightarrow}} Set \,. </annotation></semantics></math></div></div> <div class="num_prop" id="topcat"> <h6 id="proposition_4">Proposition</h6> <p>The <a class="existingWikiWord" href="/nlab/show/forgetful+functor">forgetful functor</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi></mrow><annotation encoding="application/x-tex">U</annotation></semantics></math> from def. <a class="maruku-ref" href="#ForgetfulFunctorFromTopToSet"></a> exhibits <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Top</mi></mrow><annotation encoding="application/x-tex">Top</annotation></semantics></math> as</p> <ul> <li> <p>a <a class="existingWikiWord" href="/nlab/show/concrete+category">concrete category</a></p> </li> <li> <p>a <a class="existingWikiWord" href="/nlab/show/topological+concrete+category">topological concrete category</a>.</p> </li> </ul> </div> <h3 id="MonoEpiMorphisms">Mono-/Epimorphisms</h3> <div class="num_prop" id="SubspaceInclusionsAreRegularMonos"> <h6 id="proposition_5">Proposition</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/regular+monomorphisms">regular monomorphisms</a> of <a class="existingWikiWord" href="/nlab/show/topological+spaces">topological spaces</a>)</strong></p> <p>In the <a class="existingWikiWord" href="/nlab/show/category">category</a> <a class="existingWikiWord" href="/nlab/show/Top">Top</a> of <a class="existingWikiWord" href="/nlab/show/topological+spaces">topological spaces</a>,</p> <ol> <li> <p>the <a class="existingWikiWord" href="/nlab/show/monomorphisms">monomorphisms</a> are those <a class="existingWikiWord" href="/nlab/show/continuous+functions">continuous functions</a> which are <a class="existingWikiWord" href="/nlab/show/injective+functions">injective functions</a>;</p> </li> <li> <p>the <a class="existingWikiWord" href="/nlab/show/regular+monomorphisms">regular monomorphisms</a> are the <a class="existingWikiWord" href="/nlab/show/topological+embeddings">topological embeddings</a> (i.e. those continuous functions which are <a class="existingWikiWord" href="/nlab/show/homeomorphisms">homeomorphisms</a> onto their <a class="existingWikiWord" href="/nlab/show/images">images</a> equipped with the <a class="existingWikiWord" href="/nlab/show/subspace+topology">subspace topology</a>).</p> </li> </ol> </div> <div class="proof"> <h6 id="proof_3">Proof</h6> <p>Regarding the first statement: An injective continuous function <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo lspace="verythinmathspace">:</mo><mi>X</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">f \colon X \to Y</annotation></semantics></math> clearly has the cancellation property that defines monomorphisms: for parallel continuous functions <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>g</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>g</mi> <mn>2</mn></msub><mo lspace="verythinmathspace">:</mo><mi>Z</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">g_1,g_2 \colon Z \to X</annotation></semantics></math>, if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>∘</mo><msub><mi>g</mi> <mn>1</mn></msub><mo>=</mo><mi>f</mi><mo>∘</mo><msub><mi>g</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">f \circ g_1 = f \circ g_2</annotation></semantics></math>, then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>g</mi> <mn>1</mn></msub><mo>=</mo><msub><mi>g</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">g_1 = g_2</annotation></semantics></math>, because continuous functions are equal precisely if their underlying functions of sets are equal. Conversely, if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> has the cancellation property, then testing on points <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>g</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>g</mi> <mn>2</mn></msub><mo lspace="verythinmathspace">:</mo><mo>*</mo><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">g_1, g_2 \colon \ast \to X</annotation></semantics></math> gives that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> is injective.</p> <p>Regarding the second statement: from the construction of <a class="existingWikiWord" href="/nlab/show/equalizers">equalizers</a> in <a class="existingWikiWord" href="/nlab/show/Top">Top</a> (example <a class="maruku-ref" href="#EqualizerInTop"></a>) we have that these are topological subspace inclusions.</p> <p>Conversely, let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi><mo lspace="verythinmathspace">:</mo><mi>X</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">i \colon X \to Y</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/topological+subspace+embedding">topological subspace embedding</a>. We need to show that this is the equalizer of some pair of parallel morphisms.</p> <p>To that end, form the <a class="existingWikiWord" href="/nlab/show/cokernel+pair">cokernel pair</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>i</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>i</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(i_1, i_2)</annotation></semantics></math> by taking the <a class="existingWikiWord" href="/nlab/show/pushout">pushout</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math> against itself (in the category of sets, and using the <a class="existingWikiWord" href="/nlab/show/quotient+topology">quotient topology</a> on a <a class="existingWikiWord" href="/nlab/show/disjoint+union+space">disjoint union space</a>). By <a href="regular+monomorphism#RegEquEff">this prop.</a>, the equalizer of that pair is the set-theoretic equalizer of that pair of functions endowed with the <a class="existingWikiWord" href="/nlab/show/subspace+topology">subspace topology</a>. Since monomorphisms in <a class="existingWikiWord" href="/nlab/show/Set">Set</a> are regular, we get the function <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math> back, and again by example <a class="maruku-ref" href="#EqualizerInTop"></a>, it gets equipped with the subspace topology. This completes the proof.</p> </div> <h3 id="intersections_and_quotients">Intersections and quotients</h3> <div class="num_lemma" id="pushout"> <h6 id="lemma">Lemma</h6> <p>The <a class="existingWikiWord" href="/nlab/show/pushout">pushout</a> in <a class="existingWikiWord" href="/nlab/show/Top">Top</a> of any (closed/open) <a class="existingWikiWord" href="/nlab/show/topological+subspace">topological subspace</a> inclusion <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi><mo lspace="verythinmathspace">:</mo><mi>A</mi><mo>↪</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">i \colon A \hookrightarrow B</annotation></semantics></math>, example <a class="maruku-ref" href="#TopologicalSubspace"></a>, along any <a class="existingWikiWord" href="/nlab/show/continuous+function">continuous function</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo lspace="verythinmathspace">:</mo><mi>A</mi><mo>→</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">f \colon A \to C</annotation></semantics></math> is itself an a (closed/open) subspace <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>j</mi><mo lspace="verythinmathspace">:</mo><mi>C</mi><mo>↪</mo><mi>D</mi></mrow><annotation encoding="application/x-tex">j \colon C \hookrightarrow D</annotation></semantics></math>.</p> </div> <p>For proof see <a href="subspace+topology#pushout">there</a>.</p> <h3 id="closed_monoidal_structure">Closed monoidal structure</h3> <p>It is well known that <a class="existingWikiWord" href="/nlab/show/Top">Top</a> is not <a class="existingWikiWord" href="/nlab/show/cartesian+closed">cartesian closed</a> (see for example at <a class="existingWikiWord" href="/nlab/show/convenient+category+of+topological+spaces">convenient category of topological spaces</a>).</p> <p>It is however <a class="existingWikiWord" href="/nlab/show/closed+monoidal+category">closed monoidal</a>.</p> <ul> <li> <p>The <a class="existingWikiWord" href="/nlab/show/tensor+product">tensor product</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>⊗</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">X\otimes Y</annotation></semantics></math> is given by the cartesian product of the underlying spaces, equipped with the <em>topology of separate continuity</em>, formed by the sets <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo>⊆</mo><mi>X</mi><mo>×</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">U\subseteq X\times Y</annotation></semantics></math> such that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>U</mi> <mi>x</mi></msub><mspace width="thickmathspace"></mspace><mo>≔</mo><mspace width="thickmathspace"></mspace><mo stretchy="false">{</mo><mi>y</mi><mo>∈</mo><mi>Y</mi><mo>:</mo><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>∈</mo><mi>U</mi><mo stretchy="false">)</mo><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex"> U_x \;\coloneqq\; \{y\in Y : (x,y\in U)\} </annotation></semantics></math></div> <p>is an open subset of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math> and</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>U</mi> <mi>x</mi></msub><mspace width="thickmathspace"></mspace><mo>≔</mo><mspace width="thickmathspace"></mspace><mo stretchy="false">{</mo><mi>x</mi><mo>∈</mo><mi>X</mi><mo>:</mo><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo><mo>∈</mo><mi>U</mi><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex"> U_x \;\coloneqq\; \{x\in X : (x,y)\in U\} </annotation></semantics></math></div> <p>is an open subset of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>. Equivalently, it is the topology such that for all spaces <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Z</mi></mrow><annotation encoding="application/x-tex">Z</annotation></semantics></math>, a function <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><mi>X</mi><mo>⊗</mo><mi>Y</mi><mo>→</mo><mi>Z</mi></mrow><annotation encoding="application/x-tex">f:X\otimes Y\to Z</annotation></semantics></math> is continuous if and only if: for every <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>∈</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">x\in X</annotation></semantics></math> the function <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>y</mi><mo>↦</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">y\mapsto f(x,y)</annotation></semantics></math> is continuous, and for every <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>y</mi><mo>∈</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">y\in Y</annotation></semantics></math>, the functions <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>↦</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">x\mapsto f(x,y)</annotation></semantics></math> is continuous.</p> </li> <li> <p>The <a class="existingWikiWord" href="/nlab/show/internal+hom">internal hom</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>X</mi><mo>,</mo><mi>Y</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[X,Y]</annotation></semantics></math> is given by the set of <a class="existingWikiWord" href="/nlab/show/continuous+functions">continuous functions</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">X\to Y</annotation></semantics></math>, together with the topology of pointwise convergence, generated by the (sub-basic) sets</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>S</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>V</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo>≔</mo><mspace width="thickmathspace"></mspace><mo stretchy="false">{</mo><mi>f</mi><mo>:</mo><mi>X</mi><mo>→</mo><mi>Y</mi><mo>:</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>∈</mo><mi>V</mi><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex"> S(x,V) \;\coloneqq\; \{f:X\to Y : f(x)\in V\} </annotation></semantics></math></div> <p>for each <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>∈</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">x\in X</annotation></semantics></math> and each open <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi><mo>⊆</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">V\subseteq Y</annotation></semantics></math>. Equivalently, a net <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>f</mi> <mi>α</mi></msub><mo>:</mo><mi>X</mi><mo>→</mo><mi>Y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(f_\alpha:X\to Y)</annotation></semantics></math> tends to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><mi>X</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">f:X\to Y</annotation></semantics></math> if and only if for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>∈</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">x\in X</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>f</mi> <mi>α</mi></msub><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></mrow><annotation encoding="application/x-tex">f_\alpha(x)\to f(x)</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math>.</p> </li> </ul> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+concrete+category">topological concrete category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Ho%28Top%29">Ho(Top)</a>, <a class="existingWikiWord" href="/nlab/show/%E2%88%9EGrpd">∞Grpd</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/convenient+category+of+topological+spaces">convenient category of topological spaces</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/TopGrp">TopGrp</a></p> </li> </ul> <h2 id="references">References</h2> <p>For general references see those listed at <em><a class="existingWikiWord" href="/nlab/show/topology">topology</a></em>, such as</p> <ul> <li id="Bourbaki71"><a class="existingWikiWord" href="/nlab/show/Nicolas+Bourbaki">Nicolas Bourbaki</a>, chapter 1 <em>Topological Structures</em> of <em>Elements of Mathematics III: General topology</em>, Springer 1971, 1990</li> </ul> <p>See also</p> <ul> <li id="AguilarGitlerPrieto02">Marcelo Aguilar, <a class="existingWikiWord" href="/nlab/show/Samuel+Gitler">Samuel Gitler</a>, Carlos Prieto, section 12 of <em>Algebraic topology from a homotopical viewpoint</em>, Springer (2002) (<a href="http://tocs.ulb.tu-darmstadt.de/106999419.pdf">toc pdf</a>)</li> </ul> <p>An axiomatic desciption of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Top</mi></mrow><annotation encoding="application/x-tex">Top</annotation></semantics></math> along the lines of <a class="existingWikiWord" href="/nlab/show/ETCS">ETCS</a> for <a class="existingWikiWord" href="/nlab/show/Set">Set</a> is discussed in</p> <ul> <li>Dana Schlomiuk, <em>An elementary theory of the category of topological space</em>, Transactions of the AMS, volume 149 (1970)</li> </ul> <p>For its <a href="#closed_monoidal_structure">closed monoidal structure</a>, see:</p> <ul> <li>Maria Cristina Pedicchio and Fabio Rossi, <em>Monoidal closed structures for topological spaces: counter-example to a question of Booth and Tillotson</em>, Cahiers de topologie et géométrie différentielle catégoriques, 24(4), 1983.</li> <li id="dagger_martingales">Appendix A of <a class="existingWikiWord" href="/nlab/show/Paolo+Perrone">Paolo Perrone</a> and Ruben Van Belle, <em>Convergence of martingales via enriched dagger categories</em>, 2024. (<a href="https://arxiv.org/abs/2404.15191">arXiv</a>)</li> </ul> <div class="property">category: <a class="category_link" href="/nlab/all_pages/category">category</a></div></body></html> </div> <div class="revisedby"> <p> Last revised on July 26, 2024 at 09:48:17. See the <a href="/nlab/history/Top" 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/Top" accesskey="E" class="navlink" id="edit" rel="nofollow">Edit</a><a href="https://nforum.ncatlab.org/discussion/18345/#Item_1">Discuss</a><span class="backintime"><a href="/nlab/revision/Top/45" accesskey="B" class="navlinkbackintime" id="to_previous_revision" rel="nofollow">Previous revision</a></span><a href="/nlab/show/diff/Top" accesskey="C" class="navlink" id="see_changes" rel="nofollow">Changes from previous revision</a><a href="/nlab/history/Top" accesskey="S" class="navlink" id="history" rel="nofollow">History (45 revisions)</a> <a href="/nlab/show/Top/cite" style="color: black">Cite</a> <a href="/nlab/print/Top" accesskey="p" id="view_print" rel="nofollow">Print</a> <a href="/nlab/source/Top" id="view_source" rel="nofollow">Source</a> </div> </div>  </div>  </body> </html>