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1.5 12.8 5.6 10 10.7-2.5 5.2-11.7 3-15.6 1.1-8.4-3.8-24.3-21.3-34.4-13.7-3.5 2.6-2.3 7.6-1.2 11.1 2.8 9 12.2 17.2 20.9 20.5 17.3 6.7 34.3-8 50.8-12.1z"/> <path fill="#a41e32" d="M145.9 121.3c-.2-7.5 0-19.6-4.5-26-5.4-7.5-12.9-1-14.1 5.8-1.4 7.8 2.7 14.1 4.8 21.3 3.4 12 5.8 29-.8 40.1-3.6-6.7-5.2-13-7-20.4-2.1-8.2-12.8-13.2-15.1-1.9-2 9.7 9 21.2 12 30.1 1.2 4 2 8.8 6.4 10.3 6.9 2.3 13.3-4.7 17.7-8.8 12.2-11.5 36.6-20.7 43.4-36.4 6.7-15.7-13.7-14-21.3-7.2-9.1 8-11.9 20.5-23.6 25.1 7.5-23.7 31.8-37.6 38.4-61.4 2-7.3-.8-29.6-13-19.8-14.5 11.6-6.6 37.6-23.3 49.2z"/> <path fill="#193c78" d="M86.3 47.5c0-13-10.2-27.6-5.8-40.4 2.8-8.4 14.1-10.1 17-1 3.8 11.6-.3 26.3-1.8 38 11.7-.7 10.5-16 14.8-24.3 2.1-4.2 5.7-9.1 11-6.7 6 2.7 7.4 9.2 6.6 15.1-2.2 14-12.2 18.8-22.4 27-3.4 2.7-8 6.6-5.9 11.6 2 4.4 7 4.5 10.7 2.8 7.4-3.3 13.4-16.5 21.7-16 14.6.7 12 21.9.9 26.2-5 1.9-10.2 2.3-15.2 3.9-5.8 1.8-9.4 8.7-15.7 8.9-6.1.1-9-6.9-14.3-9-14.4-6-33.3-2-44.7-14.7-3.7-4.2-9.6-12-4.9-17.4 9.3-10.7 28 7.2 35.7 12 2 1.1 11 6.9 11.4 1.1.4-5.2-10-8.2-13.5-10-11.1-5.2-30-15.3-35-27.3-2.5-6 2.8-13.8 9.4-13.6 6.9.2 13.4 7 17.5 12C70.9 34 75 43.8 86.3 47.4z"/> </svg> </span> <span class="webName">nLab</span> compactly generated topological space </h1> <div class="navigation"> <span class="skipNav"><a href='#navEnd'>Skip the Navigation Links</a> | </span> <span style="display:inline-block; width: 0.3em;"></span> <a href="/nlab/show/HomePage" accesskey="H" title="Home page">Home Page</a> | <a href="/nlab/all_pages" accesskey="A" title="List of all pages">All Pages</a> | <a href="/nlab/latest_revisions" accesskey="U" title="Latest edits and page creations">Latest Revisions</a> | <a href="https://nforum.ncatlab.org/discussion/8638/#Item_90" title="Discuss this page in its dedicated thread on the nForum" style="color: black">Discuss this page</a> | <form accept-charset="utf-8" action="/nlab/search" id="navigationSearchForm" method="get"> <fieldset class="search"><input type="text" id="searchField" name="query" value="Search" style="display:inline-block; float: left;" onfocus="this.value == 'Search' ? this.value = '' : true" onblur="this.value == '' ? this.value = 'Search' : true" /></fieldset> </form> <span id='navEnd'></span> </div> <div id="revision"> <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='#idea'>Idea</a></li> <li><a href='#Definition'>Definition</a></li> <li><a href='#Examples'>Examples</a></li> <li><a href='#properties'>Properties</a></li> <ul> <li><a href='#CoreflectionIntoTopologicalSpaces'>Coreflection into topological spaces</a></li> <li><a href='#ReflectionIntoWeakHausdorffSpaces'>Reflection into weak Hausdorff spaces</a></li> <li><a href='#cartesian_closure'>Cartesian closure</a></li> <li><a href='#RelationToLocallyCompactHausdorffSpaces'>Relation to locally compact Hausdorff spaces</a></li> <li><a href='#relation_to_compactly_generated_topological_space'>Relation to compactly generated topological space</a></li> <li><a href='#Regularity'>Regularity</a></li> <li><a href='#Homotopy'>Homotopy</a></li> </ul> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#References'>References</a></li> <ul> <li><a href='#ReferencesCGHausdorffSpaces'>k-Spaces and CG Hausdorff spaces</a></li> <li><a href='#ReferencesCGWeakHausdorffSpaces'>CG weak Hausdorff spaces</a></li> <li><a href='#ParameterizedTopology'>Exponential law for parameterized topological spaces</a></li> </ul> </ul> </div> <h2 id="idea">Idea</h2> <p id="IsCalled"> A <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a> is called <em>compactly generated</em> – also called a “k-space”<sup id="fnref:1"><a href="#fn:1" rel="footnote">1</a></sup> (<a href="#Gale50">Gale 1950, 1.</a>, following lectures by <a class="existingWikiWord" href="/nlab/show/Witold+Hurewicz">Hurewicz</a> in 1948), “Kelley space” (<a href="#GabrielZisman67">Gabriel & Zisman 1967, III.4</a>), or “kaonic space” (<a href="#Postnikov82">Postnikov 1982, p. 34</a>) – if its topology is detected by the <a class="existingWikiWord" href="/nlab/show/continuous+map">continuous</a> <a class="existingWikiWord" href="/nlab/show/images">images</a> of <a class="existingWikiWord" href="/nlab/show/compact+Hausdorff+spaces">compact Hausdorff spaces</a> inside it.</p> <p>As opposed to general topological spaces, compactly generated spaces form a <a class="existingWikiWord" href="/nlab/show/cartesian+closed+category">cartesian closed category</a> while still being general enough for most purposes of <a class="existingWikiWord" href="/nlab/show/general+topology">general topology</a>, hence form a <a class="existingWikiWord" href="/nlab/show/convenient+category+of+topological+spaces">convenient category of topological spaces</a> (<a href="#Steenrod67">Steenrod 1967</a>) and as such have come to be commonly used in the foundations of <a class="existingWikiWord" href="/nlab/show/algebraic+topology">algebraic topology</a> and <a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopy theory</a>, especially in their modern guise as compactly generated <em>weakly Hausdorff</em> spaces, due to <a href="#McCord69">McCord 1969, Sec. 2</a>.</p> <h2 id="Definition">Definition</h2> <p> <div class='num_defn' id='kContinuousFunction'> <h6>Definition</h6> <p><strong>(<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math>-continuous functions)</strong> <br /> A <a class="existingWikiWord" href="/nlab/show/function">function</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo lspace="verythinmathspace">:</mo><mi>X</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">f\colon X \to Y</annotation></semantics></math> between <a class="existingWikiWord" href="/nlab/show/underlying">underlying</a> <a class="existingWikiWord" href="/nlab/show/sets">sets</a> of <a class="existingWikiWord" href="/nlab/show/topological+spaces">topological spaces</a> is called <em><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math>-continuous</em> if for all <a class="existingWikiWord" href="/nlab/show/compact+Hausdorff+spaces">compact Hausdorff spaces</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> and <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>t</mi><mo lspace="verythinmathspace">:</mo><mi>C</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">t \colon C \to X</annotation></semantics></math> the <a class="existingWikiWord" href="/nlab/show/composition">composite</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>∘</mo><mi>t</mi><mo lspace="verythinmathspace">:</mo><mi>C</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">f \circ t \colon C \to Y</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/continuous+function">continuous</a>.</p> </div> </p> <p> <div class='num_prop' id='EquivalentConditionsForCompactGeneration'> <h6>Proposition</h6> <p><strong>(equivalent characterizations of compact generation)</strong> <br /> The following conditions on a <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a> <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> are equivalent:</p> <ol> <li> <p>For all spaces <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 all <a class="existingWikiWord" href="/nlab/show/functions">functions</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo lspace="verythinmathspace">:</mo><mi>X</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">f \colon X \to Y</annotation></semantics></math> (of <a class="existingWikiWord" href="/nlab/show/underlying">underlying</a> <a class="existingWikiWord" href="/nlab/show/sets">sets</a>), <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 <a class="existingWikiWord" href="/nlab/show/continuous+function">continuous</a> if and only 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> is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math>-continuous (Def. <a class="maruku-ref" href="#kContinuousFunction"></a>).</p> </li> <li> <p>There is 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> (instead of a <a class="existingWikiWord" href="/nlab/show/proper+class">proper class</a>) of <a class="existingWikiWord" href="/nlab/show/compact+Hausdorff+spaces">compact Hausdorff spaces</a> such that the previous condition holds for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo>∈</mo><mi>S</mi></mrow><annotation encoding="application/x-tex">C \in S</annotation></semantics></math>.</p> </li> <li> <p><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 an <a class="existingWikiWord" href="/nlab/show/identification+space">identification space</a> of a <a class="existingWikiWord" href="/nlab/show/disjoint+union">disjoint union</a> of compact Hausdorff spaces.</p> </li> <li> <p>A <a class="existingWikiWord" href="/nlab/show/subspace">subspace</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo>⊆</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">U \subseteq X</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/open+subspace">open</a> if and only if the <a class="existingWikiWord" href="/nlab/show/preimage">preimage</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>t</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>U</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">t^{-1}(U)</annotation></semantics></math> under 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>t</mi><mo lspace="verythinmathspace">:</mo><mi>C</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">t \colon C \to X</annotation></semantics></math> out of a <a class="existingWikiWord" href="/nlab/show/compact+Hausdorff+space">compact Hausdorff space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> is open.</p> </li> </ol> <p></p> </div> </p> <p> <div class='num_defn' id='kSpace'> <h6>Definition</h6> <p><strong>(k-spaces)</strong> <br /> A <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a> <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 a <strong><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math>-space</strong> if any (hence all) of the conditions in Prop. <a class="maruku-ref" href="#EquivalentConditionsForCompactGeneration"></a> hold.<br /></p> </div> <div class='num_remark'> <h6>Remark</h6> <p><strong>(terminology)</strong> <br /> Some authors say that a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math>-space (Def. <a class="maruku-ref" href="#kSpace"></a>) is <strong>compactly generated</strong>, while others reserve that term for a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math>-space which is also <em><a class="existingWikiWord" href="/nlab/show/weak+Hausdorff+space">weak Hausdorff</a></em>, meaning that the image of any <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>t</mi><mo lspace="verythinmathspace">:</mo><mi>C</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">t\colon C\to X</annotation></semantics></math> is closed (when <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> is compact Hausdorff). Some authors (especially the early authors on the subject) go on to require a <a class="existingWikiWord" href="/nlab/show/Hausdorff+space">Hausdorff</a> space, but this seems to be unnecessary.</p> </div> </p> <h2 id="Examples">Examples</h2> <p>Examples of compactly generated spaces include</p> <div class="num_example"> <h6 id="example">Example</h6> <p>Every <a class="existingWikiWord" href="/nlab/show/compact+Hausdorff+space">compact Hausdorff space</a> is compactly generated.</p> </div> <p>Note: it is not generally true that <a class="existingWikiWord" href="/nlab/show/compact+spaces">compact spaces</a> are compactly generated, even if they are <a class="existingWikiWord" href="/nlab/show/weakly+Hausdorff+space">weakly Hausdorff</a>. An example is the square of the <a class="existingWikiWord" href="/nlab/show/one-point+compactification">one-point compactification</a> of the rationals <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℚ</mi></mrow><annotation encoding="application/x-tex">\mathbb{Q}</annotation></semantics></math> with its standard topology. See for example this <a href="https://math.stackexchange.com/a/964546/43208">MathStackExchange post</a>.</p> <div class="num_example"> <h6 id="example_2">Example</h6> <p>Every <a class="existingWikiWord" href="/nlab/show/locally+compact+Hausdorff+space">locally compact Hausdorff space</a> is compactly generated.</p> </div> <div class="num_example"> <h6 id="example_3">Example</h6> <p>Every <a class="existingWikiWord" href="/nlab/show/topological+manifold">topological manifold</a> is compactly generated</p> </div> <div class="num_example" id="CWComplexIsCompactlyGenerated"> <h6 id="example_4">Example</h6> <p>Every <a class="existingWikiWord" href="/nlab/show/CW-complex">CW-complex</a> is a compactly generated topological space.</p> </div> <div class="proof"> <h6 id="proof">Proof</h6> <p>Since a CW-complex <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 a <a class="existingWikiWord" href="/nlab/show/colimit">colimit</a> in <a class="existingWikiWord" href="/nlab/show/Top">Top</a> over attachments of standard <a class="existingWikiWord" href="/nlab/show/n-disks">n-disks</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>D</mi> <mrow><msub><mi>n</mi> <mi>i</mi></msub></mrow></msup></mrow><annotation encoding="application/x-tex">D^{n_i}</annotation></semantics></math> (its cells), by the characterization of colimits 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> (<a href="Top#DescriptionOfLimitsAndColimitsInTop">prop.</a>) a 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> is open or closed precisely if its restriction to each cell is open or closed, respectively. Since the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-disks are compact, this implies one direction: if a subset <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> 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> intersected with all compact subsets is closed, then <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> is closed.</p> <p>For the converse direction, since <a class="existingWikiWord" href="/nlab/show/a+CW-complex+is+a+Hausdorff+space">a CW-complex is a Hausdorff space</a> and since <a class="existingWikiWord" href="/nlab/show/compact+subspaces+of+Hausdorff+spaces+are+closed">compact subspaces of Hausdorff spaces are closed</a>, the intersection of a closed subset with a compact subset is closed.</p> </div> <div class="num_example"> <h6 id="example_5">Example</h6> <p>Every <a class="existingWikiWord" href="/nlab/show/first+countable+space">first countable space</a> is a compactly generated space.</p> </div> <div class="proof"> <h6 id="proof_idea">Proof idea</h6> <p>Since the topology is determined by convergent sequences = maps from one-point <a class="existingWikiWord" href="/nlab/show/compactification">compactification</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℕ</mi><mo>∪</mo><mo stretchy="false">{</mo><mn>∞</mn><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\mathbb{N} \cup \{\infty\}</annotation></semantics></math>); these include all <a class="existingWikiWord" href="/nlab/show/Frechet%E2%80%93Uryson+spaces">Frechet–Uryson spaces</a>.</p> </div> <h2 id="properties">Properties</h2> <h3 id="CoreflectionIntoTopologicalSpaces">Coreflection into topological spaces</h3> <p> <div class='num_defn' id='CategoryOfKSpaces'> <h6>Definition</h6> <p><strong>(category of k-spaces)</strong> <br /> We write</p> <div class="maruku-equation" id="eq:kSpacesAsSubcategoryOfAllSpaces"><span class="maruku-eq-number">(1)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>k</mi><mo lspace="0em" rspace="thinmathspace">Top</mo><mover><mo>↪</mo><mrow><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace></mrow></mover><mi>Top</mi></mrow><annotation encoding="application/x-tex"> k\Top \xhookrightarrow{\;\;\;\;} Top </annotation></semantics></math></div> <p>for the category of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math>-spaces (Def. <a class="maruku-ref" href="#kSpace"></a>) with <a class="existingWikiWord" href="/nlab/show/continuous+functions">continuous functions</a> between them, hence for the <a class="existingWikiWord" href="/nlab/show/full+subcategory">full subcategory</a> of <a class="existingWikiWord" href="/nlab/show/Top">Top</a> on the k-spaces.</p> </div> </p> <p> <div class='num_prop' id='TheCoreflection'> <h6>Proposition</h6> <p>The inclusion <a class="maruku-eqref" href="#eq:kSpacesAsSubcategoryOfAllSpaces">(1)</a> is that of a <a class="existingWikiWord" href="/nlab/show/coreflective+subcategory">coreflective subcategory</a></p> <div class="maruku-equation" id="eq:kIficationReflection"><span class="maruku-eq-number">(2)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>k</mi><mi>Top</mi><munderover><mrow><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo>⊥</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace></mrow><munder><mo>⟵</mo><mi>k</mi></munder><mover><mo>↪</mo><mrow></mrow></mover></munderover><mi>Top</mi></mrow><annotation encoding="application/x-tex"> k Top \underoverset {\underset{k}{\longleftarrow}} {\overset{}{\hookrightarrow}} {\;\;\;\; \bot \;\;\;\;} Top </annotation></semantics></math></div> <p></p> </div> The <a class="existingWikiWord" href="/nlab/show/coreflection">coreflection</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math> is sometimes called <em><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math>-ification</em> (<a href="#May99">May 1999, p. 49</a>). <div class='proof'> <h6>Proof</h6> <p>The reflection functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math> is constructed as follows:</p> <p>We take <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>≔</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">k(X) \coloneqq X</annotation></semantics></math> on <a class="existingWikiWord" href="/nlab/show/underlying">underlying</a> <a class="existingWikiWord" href="/nlab/show/sets">sets</a>, and equip this with the <a class="existingWikiWord" href="/nlab/show/topological+space">topology</a> whose <a class="existingWikiWord" href="/nlab/show/closed+sets">closed sets</a> are those whose <a class="existingWikiWord" href="/nlab/show/intersection">intersection</a> with <a class="existingWikiWord" href="/nlab/show/compact+Hausdorff+space">compact Hausdorff</a> subsets of (the original topology 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 <a class="existingWikiWord" href="/nlab/show/closed+subset">closed</a> (in the original topology 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>). Then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">k(X)</annotation></semantics></math> has all the same closed sets and possibly more, hence all the same open sets and possibly more.</p> <p>In particular, the identity map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>id</mi><mo lspace="verythinmathspace">:</mo><mi>k</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">id \colon k(X)\to X</annotation></semantics></math> is continuous, and forms the <a class="existingWikiWord" href="/nlab/show/counit+of+an+adjunction">counit</a> of the <a class="existingWikiWord" href="/nlab/show/coreflective+subcategory">coreflection</a>. Thus this coreflection has a counit which is both a <a class="existingWikiWord" href="/nlab/show/monomorphism">monomorphism</a> as well as an <a class="existingWikiWord" href="/nlab/show/epimorphism">epimorphism</a>, i.e. a “bimorphism”—such a coreflection is sometimes called a “bicoreflection.”</p> </div> </p> <p> <div class='num_remark'> <h6>Remark</h6> <p>There is also the category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mo lspace="0em" rspace="thinmathspace">Top</mo> <mi>k</mi></msub></mrow><annotation encoding="application/x-tex">\Top_k</annotation></semantics></math> of all topological spaces and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math>-continuous maps (Def. <a class="maruku-ref" href="#kContinuousFunction"></a>). But the <a class="existingWikiWord" href="/nlab/show/composition">composite</a> sequence of inclusions</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>k</mi><mo lspace="0em" rspace="thinmathspace">Top</mo><mover><mo>↪</mo><mspace width="thickmathspace"></mspace></mover><mo lspace="0em" rspace="thinmathspace">Top</mo><mover><mo>→</mo><mspace width="thickmathspace"></mspace></mover><msub><mo lspace="0em" rspace="thinmathspace">Top</mo> <mi>k</mi></msub><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> k\Top \xhookrightarrow{\;} \Top \xrightarrow{\;} \Top_k \,, </annotation></semantics></math></div> <p>of which the first is the <a class="existingWikiWord" href="/nlab/show/full+subcategory">full</a> inclusion <a class="maruku-eqref" href="#eq:kSpacesAsSubcategoryOfAllSpaces">(1)</a> and the second is <a class="existingWikiWord" href="/nlab/show/bijective+on+objects+functor">bijective on objects</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi><mo lspace="0em" rspace="thinmathspace">Top</mo><mo>→</mo><msub><mi>Top</mi> <mi>k</mi></msub></mrow><annotation encoding="application/x-tex">k\Top \to Top_k</annotation></semantics></math>, is an <a class="existingWikiWord" href="/nlab/show/equivalence+of+categories">equivalence of categories</a>.</p> <p>Namely, the <a class="existingWikiWord" href="/nlab/show/identity+morphism">identity morphism</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>id</mi><mo lspace="verythinmathspace">:</mo><mi>X</mi><mo>→</mo><mi>k</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">id \colon X \to k(X)</annotation></semantics></math> is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math>-continuous, so that the <a class="existingWikiWord" href="/nlab/show/adjunction+counit">adjunction counit</a> from Prop. <a class="maruku-ref" href="#TheCoreflection"></a> becomes an <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphism</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mo lspace="0em" rspace="thinmathspace">Top</mo> <mi>k</mi></msub></mrow><annotation encoding="application/x-tex">\Top_k</annotation></semantics></math>. This shows that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi><mo lspace="0em" rspace="thinmathspace">Top</mo><mo>→</mo><msub><mo lspace="0em" rspace="thinmathspace">Top</mo> <mi>k</mi></msub></mrow><annotation encoding="application/x-tex">k\Top \to \Top_k</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/essentially+surjective+functor">essentially surjective</a>, and it is <a class="existingWikiWord" href="/nlab/show/fully+faithful+functor">fully faithful</a> since any <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math>-continuous function between <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math>-spaces is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math>-continuous; hence it is an equivalence.</p> </div> </p> <p> <div class='num_remark'> <h6>Remark</h6> <p>Since <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi><mo lspace="0em" rspace="thinmathspace">Top</mo><mo>↪</mo><mo lspace="0em" rspace="thinmathspace">Top</mo></mrow><annotation encoding="application/x-tex">k\Top \hookrightarrow \Top</annotation></semantics></math> is coreflective, it follows that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi><mo lspace="0em" rspace="thinmathspace">Top</mo></mrow><annotation encoding="application/x-tex">k\Top</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/complete+category">complete</a> and <a class="existingWikiWord" href="/nlab/show/cocomplete+category">cocomplete</a>. Its <a class="existingWikiWord" href="/nlab/show/colimits">colimits</a> are constructed as in <a class="existingWikiWord" href="/nlab/show/Top">Top</a>, but its <a class="existingWikiWord" href="/nlab/show/limits">limits</a> are the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math>-ification <a class="maruku-eqref" href="#eq:kIficationReflection">(2)</a> of <a href="Top#UniversalConstructions">limits in Top</a>.</p> <p>Notice that this is nontrivial already for <a class="existingWikiWord" href="/nlab/show/products">products</a>: the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math>-space product <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> is the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math>-ification of the usual <a class="existingWikiWord" href="/nlab/show/product+topology">product topology</a>. The <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math>-space product is better behaved in many ways; e.g. it enables <a class="existingWikiWord" href="/nlab/show/geometric+realization">geometric realization</a> to preserve products (and all <a class="existingWikiWord" href="/nlab/show/finite+limits">finite limits</a>), and the product of two <a class="existingWikiWord" href="/nlab/show/CW+complexes">CW complexes</a> to be another CW complex.</p> </div> </p> <h3 id="ReflectionIntoWeakHausdorffSpaces">Reflection into weak Hausdorff spaces</h3> <p> <div class='num_defn' id='FullSubcategoryOfCGWHSpacesInsideCGSpaces'> <h6>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>h</mi><mi>k</mi><mi>Top</mi><mover><mo>↪</mo><mrow><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace></mrow></mover><mi>k</mi><mi>Top</mi></mrow><annotation encoding="application/x-tex"> h k Top \xhookrightarrow{\;\;\;} k Top </annotation></semantics></math></div> <p>for the further <a class="existingWikiWord" href="/nlab/show/full+subcategory">full subcategory</a> inside that of k-spaces (Def. <a class="maruku-ref" href="#CategoryOfKSpaces"></a>) on those which in addition are <a class="existingWikiWord" href="/nlab/show/weak+Hausdorff+spaces">weak Hausdorff spaces</a>.</p> </div> </p> <p> <div class='num_prop' id='ReflectionOntoCGWHSpaces'> <h6>Proposition</h6> <p><strong>(cgwh spaces reflective in cg spaces)</strong> <br /> The <a class="existingWikiWord" href="/nlab/show/full+subcategory">full subcategory</a>-inclusion of <a class="existingWikiWord" href="/nlab/show/weakly+Hausdorff+space">weak Hausdorff spaces</a> in k-spaces (Def. <a class="maruku-ref" href="#FullSubcategoryOfCGWHSpacesInsideCGSpaces"></a>) is a <a class="existingWikiWord" href="/nlab/show/reflective+subcategory">reflective subcategory</a> inclusion:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>h</mi><mi>k</mi><mi>Top</mi><munderover><mrow><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo>⊥</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace></mrow><munder><mo>↪</mo><mrow></mrow></munder><mover><mo>⟵</mo><mi>h</mi></mover></munderover><mi>k</mi><mi>Top</mi></mrow><annotation encoding="application/x-tex"> h k Top \underoverset {\underset{}{\hookrightarrow}} {\overset{ h }{\longleftarrow}} {\;\;\;\; \bot \;\;\;\; } k Top </annotation></semantics></math></div> <p></p> </div> (e.g. <a href="#Strickland09">Strickland 2009, Prop. 2.22</a>)</p> <p> <div class='num_remark' id='SequencesOfCoReflections'> <h6>Remark</h6> <p><strong>(sequence of <a class="existingWikiWord" href="/nlab/show/coreflective+subcategory">(co-)</a><a class="existingWikiWord" href="/nlab/show/reflective+subcategory">reflections</a>)</strong> <br /> In summary, Prop. <a class="maruku-ref" href="#TheCoreflection"></a> and Prop. <a class="maruku-ref" href="#ReflectionOntoCGWHSpaces"></a> yield a sequence of <a class="existingWikiWord" href="/nlab/show/adjoint+functors">adjoint functors</a> of this form:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>h</mi><mi>k</mi><mi>Top</mi><munderover><mrow><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo>⊥</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace></mrow><munder><mo>↪</mo><mrow></mrow></munder><mover><mo>⟵</mo><mi>h</mi></mover></munderover><mi>k</mi><mi>Top</mi><munderover><mrow><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo>⊥</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace></mrow><munder><mo>⟵</mo><mi>k</mi></munder><mover><mo>↪</mo><mrow></mrow></mover></munderover><mi>Top</mi></mrow><annotation encoding="application/x-tex"> h k Top \underoverset {\underset{}{\hookrightarrow}} {\overset{ h }{\longleftarrow}} {\;\;\;\; \bot \;\;\;\; } k Top \underoverset {\underset{k}{\longleftarrow}} {\overset{}{\hookrightarrow}} {\;\;\;\; \bot \;\;\;\;} Top </annotation></semantics></math></div> <p>The <a class="existingWikiWord" href="/nlab/show/classical+model+structure+on+topological+spaces">classical model structure on topological spaces</a> restricts along these (co-)reflective embeddings to a <a class="existingWikiWord" href="/nlab/show/Quillen+equivalence">Quillen equivalent</a> <a class="existingWikiWord" href="/nlab/show/model+structure+on+compactly+generated+topological+spaces">model structure on compactly generated topological spaces</a>. See there for more.</p> </div> </p> <h3 id="cartesian_closure">Cartesian closure</h3> <p>The categories <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi><mo lspace="0em" rspace="thinmathspace">Top</mo><mo>≃</mo><msub><mo lspace="0em" rspace="thinmathspace">Top</mo> <mi>k</mi></msub></mrow><annotation encoding="application/x-tex">k\Top\simeq \Top_k</annotation></semantics></math> are <a class="existingWikiWord" href="/nlab/show/cartesian+closed+category">cartesian closed</a>. (While in <a class="existingWikiWord" href="/nlab/show/Top">Top</a> only some objects are exponentiable, see <a class="existingWikiWord" href="/nlab/show/exponential+law+for+spaces">exponential law for spaces</a>.) For arbitrary spaces <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>, define the <em>test-open</em> or <em>compact-open topology</em> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mo lspace="0em" rspace="thinmathspace">Top</mo> <mi>k</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>Y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Top_k(X,Y)</annotation></semantics></math> to have the <a class="existingWikiWord" href="/nlab/show/subbase">subbase</a> of sets <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi><mo stretchy="false">(</mo><mi>t</mi><mo>,</mo><mi>U</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">M(t,U)</annotation></semantics></math>, for a given compact Hausdorff space <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math>, a map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>t</mi><mo lspace="verythinmathspace">:</mo><mi>C</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">t\colon C \to X</annotation></semantics></math>, and an open set <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> 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>, where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi><mo stretchy="false">(</mo><mi>t</mi><mo>,</mo><mi>U</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">M(t,U)</annotation></semantics></math> consists of all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math>-continuous functions <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> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>⊆</mo><mi>U</mi></mrow><annotation encoding="application/x-tex">f(t(C))\subseteq U</annotation></semantics></math>.</p> <p>(This is slightly different from the usual <a class="existingWikiWord" href="/nlab/show/compact-open+topology">compact-open topology</a> if <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> happens to have non-Hausdorff compact subspaces; in that case the usual definition includes such subspaces as tests, while the above definition excludes them. Of course, if <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> itself is Hausdorff, then the two become identical.)</p> <p>With this topology, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mo lspace="0em" rspace="thinmathspace">Top</mo> <mi>k</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>Y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Top_k(X,Y)</annotation></semantics></math> becomes an <a class="existingWikiWord" href="/nlab/show/exponential+object">exponential object</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Top</mi> <mi>k</mi></msub></mrow><annotation encoding="application/x-tex">Top_k</annotation></semantics></math>. It follows, by <a class="existingWikiWord" href="/nlab/show/Yoneda+lemma">Yoneda lemma</a> arguments (<a href="closed+monoidal+category#TensorHomIsoInternalizes">prop.</a>), that the bijection</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>k</mi><mo lspace="0em" rspace="thinmathspace">Top</mo><mo stretchy="false">(</mo><mi>X</mi><mo>×</mo><mi>Y</mi><mo>,</mo><mi>Z</mi><mo stretchy="false">)</mo><mo>⟶</mo><mi>k</mi><mi>Top</mi><mo maxsize="1.2em" minsize="1.2em">(</mo><mi>X</mi><mo>,</mo><mi>k</mi><mo lspace="0em" rspace="thinmathspace">Top</mo><mo stretchy="false">(</mo><mi>Y</mi><mo>,</mo><mi>Z</mi><mo stretchy="false">)</mo><mo maxsize="1.2em" minsize="1.2em">)</mo></mrow><annotation encoding="application/x-tex"> k\Top(X \times Y, Z) \longrightarrow k Top\big(X,k\Top(Y,Z)\big) </annotation></semantics></math></div> <p>is actually an <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphism</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mo lspace="0em" rspace="thinmathspace">Top</mo> <mi>k</mi></msub></mrow><annotation encoding="application/x-tex">\Top_k</annotation></semantics></math>, which we may call a <em><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math>-homeomorphism</em> (e.g. <a href="#Strickland09">Strickland 09, prop. 2.12</a>). In fact, it is actually a homeomorphism, i.e. an isomorphism already 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>.</p> <p>It follows that the category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi><mo lspace="0em" rspace="thinmathspace">Top</mo></mrow><annotation encoding="application/x-tex">k\Top</annotation></semantics></math> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math>-spaces and continuous maps is also cartesian closed, since it is equivalent to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mo lspace="0em" rspace="thinmathspace">Top</mo> <mi>k</mi></msub></mrow><annotation encoding="application/x-tex">\Top_k</annotation></semantics></math>. Its exponential object is the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math>-ification of the one constructed above for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mo lspace="0em" rspace="thinmathspace">Top</mo> <mi>k</mi></msub></mrow><annotation encoding="application/x-tex">\Top_k</annotation></semantics></math>. Since for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math>-spaces, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math>-continuous implies continuous, the underlying set of this exponential space <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi><mo lspace="0em" rspace="thinmathspace">Top</mo><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>Y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">k\Top(X,Y)</annotation></semantics></math> is the set of all continuous maps from <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> to <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>. Thus, when <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 Hausdorff, we can identify this space with the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math>-ification of the usual <a class="existingWikiWord" href="/nlab/show/compact-open+topology">compact-open topology</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Top</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>Y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Top(X,Y)</annotation></semantics></math>.</p> <p>Finally, this all remains true if we also impose the weak Hausdorff, or Hausdorff, conditions.</p> <p> <div class='num_remark'> <h6>Remark</h6> <p><strong>(failure of local cartesian closure)</strong> <br /> Unfortunately neither of the above categories is <a class="existingWikiWord" href="/nlab/show/locally+cartesian+closed+category">locally cartesian closed</a> (<a href="#CagliariMatovaniVitale95">Cagliari-Matovani-Vitale 1995, p. 4</a>)</p> <p id="PartialLocalCartesianClosure"> However, if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi></mrow><annotation encoding="application/x-tex">K</annotation></semantics></math> is the category of not-necessarily-weak-Hausdorff k-spaces, and <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> are k-spaces that are weak Hausdorff, then the <a class="existingWikiWord" href="/nlab/show/pullback">pullback</a> (<a class="existingWikiWord" href="/nlab/show/base+change">base change</a>) <a class="existingWikiWord" href="/nlab/show/functor">functor</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>K</mi> <mrow><mo stretchy="false">/</mo><mi>B</mi></mrow></msub><mo>→</mo><msub><mi>K</mi> <mrow><mo stretchy="false">/</mo><mi>A</mi></mrow></msub></mrow><annotation encoding="application/x-tex">K_{/B} \to K_{/A}</annotation></semantics></math> has a <a class="existingWikiWord" href="/nlab/show/right+adjoint">right adjoint</a>. (see <a href="#BoothBrown78a">Booth & Brown 1978a, Thm. 3.5 & 7.3</a>; <a href="#MaySigurdsson06">May & Sigurdsson 2006, §1.3.7-§1.3.9</a>).</p> <p>There is still a lot of work on fibred exponential laws and their applications. One reason for the success and difficulties is that it is easy to give a topology on the space of closed subsets of a space <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> by regarding this as the space of maps to the <a class="existingWikiWord" href="/nlab/show/Sierpinski+space">Sierpinski space</a> (the set <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{0,1\}</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/truth+value">truth values</a> in which <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><mn>1</mn><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{1\}</annotation></semantics></math> is closed but not open). From this one can get an <a class="existingWikiWord" href="/nlab/show/exponential+law+for+spaces">exponential law for spaces</a> over <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> if <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> is <a class="existingWikiWord" href="/nlab/show/T0"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"> <semantics> <mrow> <msub><mi>T</mi> <mn>0</mn></msub> </mrow> <annotation encoding="application/x-tex">T_0</annotation> </semantics> </math></a>, so that all fibres of spaces over <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> are closed in their total space. Note that weak Hausdorff implies <a class="existingWikiWord" href="/nlab/show/T0"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"> <semantics> <mrow> <msub><mi>T</mi> <mn>0</mn></msub> </mrow> <annotation encoding="application/x-tex">T_0</annotation> </semantics> </math></a>.</p> </div> </p> <p><br /></p> <h3 id="RelationToLocallyCompactHausdorffSpaces">Relation to locally compact Hausdorff spaces</h3> <p> <div class='num_prop' id='LocallyCompactHausdorffSpacesAreCompactlyGeneratedWeaklyHausdorff'> <h6>Proposition</h6> <p>Every <a class="existingWikiWord" href="/nlab/show/locally+compact+Hausdorff+space">locally compact Hausdorff space</a> is a k-space and is <a class="existingWikiWord" href="/nlab/show/weakly+Hausdorff+topological+space">weakly Hausdorff</a>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>LCHausSp</mi><mover><mo>↪</mo><mspace width="thickmathspace"></mspace></mover><mi>h</mi><mi>k</mi><mi>TopSp</mi></mrow><annotation encoding="application/x-tex"> LCHausSp \xhookrightarrow{\;} h k TopSp </annotation></semantics></math></div> <p></p> </div> (<a href="#Dugundji66">Dugundji 1966, XI Thm. 9.3</a>; <a href="compactly+generated+topological+space#Strickland09">Strickland 2009, Prop. 1.7</a>)</p> <p> <div class='num_prop'> <h6>Proposition</h6> <p>The <a class="existingWikiWord" href="/nlab/show/product+topological+space">product topological space</a> of a <a class="existingWikiWord" href="/nlab/show/locally+compact+Hausdorff+space">locally compact Hausdorff space</a> with a k-space is already a k-space (i.e. without need of k-ification).</p> </div> </p> <p>(e.g. <a href="#Lewis78">Lewis 1978, Lem. 2.4</a>; <a href="#Piccinini92">Piccinini 1992, Thm. B.6</a>, <a href="compactly+generated+topological+space#Strickland09">Strickland 2009, Prop. 2.6</a>)</p> <p> <div class='num_prop' id='kSpacesAreTheQuotientSpacesOfLocallyCompactHausdorffSpaces'> <h6>Proposition</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/k-spaces">k-spaces</a> are the <a class="existingWikiWord" href="/nlab/show/quotient+topological+space">quotient spaces</a> of <a class="existingWikiWord" href="/nlab/show/locally+compact+Hausdorff+spaces">locally compact Hausdorff spaces</a>)</strong> <br /> A <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a> is a <a class="existingWikiWord" href="/nlab/show/k-space">k-space</a> (Def. <a class="maruku-ref" href="#kSpace"></a>) iff it is a <a class="existingWikiWord" href="/nlab/show/quotient+topological+space">quotient topological space</a> of a <a class="existingWikiWord" href="/nlab/show/locally+compact+Hausdorff+space">locally compact Hausdorff space</a>.</p> </div> This is proven in <a href="#Dugundji66">Dugundji 1966, XI Thm. 9.4</a> (also <a href="#Piccinini92">Piccinini 92, Thm. B.4</a>) assuming Hausdorffness, and without that assumption in <a href="#EscardoLawsonSimpson04">Escardo, Lawson & Simpson 2004, Cor. 3.4 (iii)</a>. Moreover: <div class='num_prop' id='kSpacesAreTheColimitsInTopOfCompactHausdorffSpaces'> <h6>Proposition</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/k-spaces">k-spaces</a> are the <a class="existingWikiWord" href="/nlab/show/Top#UniversalConstructions">colimits in Top</a> of <a class="existingWikiWord" href="/nlab/show/compact+Hausdorff+spaces">compact Hausdorff spaces</a>)</strong> <br /> A <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a> is a <a class="existingWikiWord" href="/nlab/show/k-space">k-space</a> (Def. <a class="maruku-ref" href="#kSpace"></a>) iff it is a <a class="existingWikiWord" href="/nlab/show/colimit">colimit</a> as formed in <a class="existingWikiWord" href="/nlab/show/Top">Top</a> (according to <a href="Top#DescriptionOfLimitsAndColimitsInTop">this Prop.</a>) of a <a class="existingWikiWord" href="/nlab/show/diagram">diagram</a> of <a class="existingWikiWord" href="/nlab/show/compact+Hausdorff+spaces">compact Hausdorff spaces</a>.</p> </div> (<a href="#EscardoLawsonSimpson04">Escardo, Lawson & Simpson 2004, Lem. 3.2 (v)</a>)</p> <h3 id="relation_to_compactly_generated_topological_space">Relation to compactly generated topological space</h3> <p>Insice <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math>-spaces there is the further <a class="existingWikiWord" href="/nlab/show/coreflective+subcategory">coreflective subcategory</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi><mi>Top</mi></mrow><annotation encoding="application/x-tex">D Top</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/Delta-generated+topological+spaces">Delta-generated topological spaces</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><munderover><mrow><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo>⊥</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace></mrow><munder><mo>⟶</mo><mi>k</mi></munder><mo>↩</mo></munderover><mi>k</mi><msub><mi>Top</mi> <mi>Qu</mi></msub><munderover><mrow><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo>⊥</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace></mrow><munder><mo>⟶</mo><mi>D</mi></munder><mo>↩</mo></munderover><mi>D</mi><msub><mi>Top</mi> <mi>Qu</mi></msub><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> Top \underoverset { \underset{ k }{\longrightarrow} } { {\hookleftarrow} } { \;\;\;\;\;\;\bot\;\;\;\;\;\; } k Top_{Qu} \underoverset { \underset{ D }{\longrightarrow} } { {\hookleftarrow} } { \;\;\;\;\;\;\bot\;\;\;\;\;\; } D Top_{Qu} \,. </annotation></semantics></math></div> <p>Both of these <a class="existingWikiWord" href="/nlab/show/adjoint+functor">coreflections</a> are <a class="existingWikiWord" href="/nlab/show/Quillen+equivalences">Quillen equivalences</a> with respect to the <a class="existingWikiWord" href="/nlab/show/classical+model+structure+on+topological+spaces">classical model structure on topological spaces</a> and the induced <a class="existingWikiWord" href="/nlab/show/model+structure+on+compactly+generated+topological+spaces">model structure on compactly generated topological spaces</a> and the <a class="existingWikiWord" href="/nlab/show/model+structure+on+Delta-generated+topological+spaces">model structure on Delta-generated topological spaces</a> (<a href="#Gaucher07">Gaucher 2009</a>, <a href="Delta-generated+topological+space#Haraguchi13">Haraguchi 2013</a>).</p> <h3 id="Regularity">Regularity</h3> <p> <div class='num_prop' id='kHausIsRegular'> <h6>Proposition</h6> <p>The categories of</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/compactly+generated+Hausdorff+spaces">compactly generated Hausdorff spaces</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/compactly+generated+weakly+Hausdorff+spaces">compactly generated weakly Hausdorff spaces</a></p> </li> </ul> <p>are both <a class="existingWikiWord" href="/nlab/show/regular+category">regular</a>.</p> </div> (<a href="#CagliariMatovaniVitale95">Cagliari-Matovani-Vitale 95, p. 3</a>).</p> <p>In particular this implies that in these categories <a class="existingWikiWord" href="/nlab/show/pullback">pullback</a> preserves <a class="existingWikiWord" href="/nlab/show/effective+epimorphisms">effective epimorphisms</a> (see <a href="regular+epimorphism#InRegularCategoryRegularEpisAreStableUnderPullback">there</a>).</p> <h3 id="Homotopy">Homotopy</h3> <p> <div class='num_prop' id='kIficationIsWeakHomotopyEquivalence'> <h6>Proposition</h6> <p>For every <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a> <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>, the canonical <a class="existingWikiWord" href="/nlab/show/continuous+function">continuous function</a> from the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math>-ification (the <a class="existingWikiWord" href="/nlab/show/adjunction+counit">adjunction counit</a>) is a <a class="existingWikiWord" href="/nlab/show/weak+homotopy+equivalence">weak homotopy equivalence</a>, hence induces an <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphism</a> on all <a class="existingWikiWord" href="/nlab/show/homotopy+groups">homotopy groups</a>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>k</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><munderover><mo>⟶</mo><mi>whe</mi><mrow><mspace width="thickmathspace"></mspace><msubsup><mi>ε</mi> <mi>X</mi> <mi>k</mi></msubsup><mspace width="thickmathspace"></mspace></mrow></munderover><mi>X</mi><mspace width="thinmathspace"></mspace><mo>,</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mtext>i.e.</mtext><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><msub><mi>π</mi> <mn>0</mn></msub><mo maxsize="1.2em" minsize="1.2em">(</mo><mi>k</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo maxsize="1.2em" minsize="1.2em">)</mo><mover><mo>→</mo><mrow><mspace width="thickmathspace"></mspace><mo>∼</mo><mspace width="thickmathspace"></mspace></mrow></mover><msub><mi>π</mi> <mn>0</mn></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>,</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mtext>and</mtext><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><munder><mo>∀</mo><mfrac linethickness="0"><mrow><mrow><mi>x</mi><mo>∈</mo><mi>X</mi></mrow></mrow><mrow><mrow><mi>n</mi><mo>∈</mo><msub><mi>ℕ</mi> <mo>+</mo></msub></mrow></mrow></mfrac></munder><mo maxsize="1.8em" minsize="1.8em">(</mo><msub><mi>π</mi> <mi>n</mi></msub><mo maxsize="1.2em" minsize="1.2em">(</mo><mi>k</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>,</mo><mspace width="thinmathspace"></mspace><mi>x</mi><mo maxsize="1.2em" minsize="1.2em">)</mo><mover><mo>→</mo><mo>∼</mo></mover><msub><mi>π</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mspace width="thinmathspace"></mspace><mi>x</mi><mo stretchy="false">)</mo><mo maxsize="1.8em" minsize="1.8em">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> k(X) \underoverset {whe} {\;\varepsilon^k_X\;} {\longrightarrow} X \,, \;\;\; \text{i.e.} \;\;\; \pi_0\big(k(X)\big) \xrightarrow{\;\sim\;} \pi_0(X) \,, \;\; \text{and} \;\; \underset{ {x \in X} \atop {n \in \mathbb{N}_+} }{\forall} \Big( \pi_n\big( k(X),\,x \big) \xrightarrow{\sim} \pi_n(X,\, x) \Big) \,. </annotation></semantics></math></div> <p></p> </div> E.g. <a href="#Vogt71">Vogt 1971, Prop. 1.2 (h)</a>. The proof is spelled out <a href="Introduction+to+Homotopy+Theory#kificationComparisonIsWeakHomotopyEquivalence">here</a> at <em><a class="existingWikiWord" href="/nlab/show/Introduction+to+Homotopy+Theory">Introduction to Homotopy Theory</a></em>.</p> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+compactly+generated+topological+spaces">model structure on compactly generated topological spaces</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Euclidean-generated+topological+spaces">Euclidean-generated topological spaces</a> (<a class="existingWikiWord" href="/nlab/show/Delta-generated+topological+spaces">Delta-generated topological spaces</a>)</p> </li> </ul> <h2 id="References">References</h2> <h3 id="ReferencesCGHausdorffSpaces">k-Spaces and CG Hausdorff spaces</h3> <p>The idea of compactly generated Hausdorff spaces first appears in print in:</p> <ul> <li id="Gale50"><a class="existingWikiWord" href="/nlab/show/David+Gale">David Gale</a>, Section 1 of: <em>Compact Sets of Functions and Function Rings</em>, Proc. AMS <strong>1</strong> (1950) pp.303-308. (<a href="http://www.ams.org/journals/proc/1950-001-03/S0002-9939-1950-0036503-X/S0002-9939-1950-0036503-X.pdf">pdf</a>, <a href="https://doi.org/10.2307/2032373">doi:10.2307/2032373</a>, <a href="https://www.jstor.org/stable/2032373">jstor:2032373</a>)</li> </ul> <p>where it is attributed to <a class="existingWikiWord" href="/nlab/show/Witold+Hurewicz">Witold Hurewicz</a>, who introduced the concept in a lecture series given in Princeton, 1948-49, which Gale attended.<sup id="fnref:2"><a href="#fn:2" rel="footnote">2</a></sup></p> <p>Early textbook accounts, assuming the Hausdorff condition:</p> <ul> <li id="Kelley55"> <p><a class="existingWikiWord" href="/nlab/show/John+Kelley">John Kelley</a>, p. 230 in: <em>General topology</em>, D. van Nostrand, New York (1955), reprinted as: Graduate Texts in Mathematics, Springer (1975) [<a href="https://www.springer.com/gp/book/9780387901251">ISBN:978-0-387-90125-1</a>]</p> </li> <li id="Dugundji66"> <p><a class="existingWikiWord" href="/nlab/show/James+Dugundji">James Dugundji</a>, Section XI.9 of: <em>Topology</em>, Allyn and Bacon 1966 (<a href="https://www.southalabama.edu/mathstat/personal_pages/carter/Dugundji.pdf">pdf</a>)</p> </li> <li id="GabrielZisman67"> <p><a class="existingWikiWord" href="/nlab/show/Pierre+Gabriel">Pierre Gabriel</a>, <a class="existingWikiWord" href="/nlab/show/Michel+Zisman">Michel Zisman</a>, sections I.1.5.3 and III.2 of <em><a class="existingWikiWord" href="/nlab/show/Calculus+of+fractions+and+homotopy+theory">Calculus of fractions and homotopy theory</a></em>, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 35, Springer (1967) (<a href="https://people.math.rochester.edu/faculty/doug/otherpapers/GZ.pdf">pdf</a>)</p> </li> </ul> <p>also:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Susan+Niefield">Susan Niefield</a>, Section 9 of: <em>Cartesianness</em>, PhD thesis, Rutgers 1978 (<a href="https://www.proquest.com/docview/302920643">proquest:302920643</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Francis+Borceux">Francis Borceux</a>, Section 7.2 of: <em>Categories and Structures</em>, Vol. 2 of: <em><a class="existingWikiWord" href="/nlab/show/Handbook+of+Categorical+Algebra">Handbook of Categorical Algebra</a></em>, Encyclopedia of Mathematics and its Applications <strong>50</strong> Cambridge University Press (1994) (<a href="https://doi.org/10.1017/CBO9780511525865">doi:10.1017/CBO9780511525865</a>)</p> </li> </ul> <p>Influential emphasis of the usefulness of the notion as providing a <a class="existingWikiWord" href="/nlab/show/convenient+category+of+topological+spaces">convenient category of topological spaces</a>:</p> <ul> <li id="Steenrod67"><a class="existingWikiWord" href="/nlab/show/Norman+Steenrod">Norman Steenrod</a>, <em>A convenient category of topological spaces</em>, Michigan Math. J. 14 (1967) 133–152 (<a href="http://projecteuclid.org/euclid.mmj/1028999711">euclid:mmj/1028999711</a>)</li> </ul> <p>Early discussion in the context of <a class="existingWikiWord" href="/nlab/show/geometric+realization+of+simplicial+topological+spaces">geometric realization of simplicial topological spaces</a>:</p> <ul> <li id="MacLane70"><a class="existingWikiWord" href="/nlab/show/Saunders+MacLane">Saunders MacLane</a>, Section 4 of: <em>The Milgram bar construction as a tensor product of functors</em>, In: F.P. Peterson (eds.) <em>The Steenrod Algebra and Its Applications: A Conference to Celebrate N.E. Steenrod’s Sixtieth Birthday</em>, Lecture Notes in Mathematics <em>168</em>, Springer 1970 (<a href="https://doi.org/10.1007/BFb0058523">doi:10.1007/BFb0058523</a>, <a href="https://link.springer.com/content/pdf/10.1007/BFb0058523.pdf">pdf</a>)</li> </ul> <p>and briefly in:</p> <ul> <li id="May72"><a class="existingWikiWord" href="/nlab/show/Peter+May">Peter May</a>, Section 1 of: <em>The geometry of iterated loop spaces</em>, Springer 1972 (<a href="https://www.math.uchicago.edu/~may/BOOKS/geom_iter.pdf">pdf</a>, <a href="https://link.springer.com/book/10.1007/BFb0067491">doi:10.1007/BFb0067491</a>)</li> </ul> <p>More history and early references, with emphasis on <a class="existingWikiWord" href="/nlab/show/category+theory">category-theoretic</a> aspects:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Horst+Herrlich">Horst Herrlich</a>, <a class="existingWikiWord" href="/nlab/show/George+Strecker">George Strecker</a>, Section 3.4 of: <em>Categorical topology – Its origins as exemplified by the unfolding of the theory of topological reflections and coreflections before 1971</em> (<a href="https://link.springer.com/content/pdf/10.1007%2F978-94-017-0468-7_15.pdf">pdf</a>), pages 255-341 in: C. E. Aull, R Lowen (eds.), <em>Handbook of the History of General Topology. Vol. 1</em> , Kluwer 1997 (<a href="https://link.springer.com/book/10.1007/978-94-017-0468-7">doi:10.1007/978-94-017-0468-7</a>)</li> </ul> <p>The terminology “kaonic spaces”, or rather the Russian version “каонные пространства” is used in</p> <ul> <li id="Postnikov71"> <p><a class="existingWikiWord" href="/nlab/show/M+M+Postnikov">M M Postnikov</a>, <em>Введение в теорию Морса</em>, Наука 1971 (<a href="http://libgen.is/book/index.php?md5=4BF450585846A0531FF485E34D062C0A">web</a>)</p> </li> <li id="Postnikov82"> <p><a class="existingWikiWord" href="/nlab/show/M+M+Postnikov">M M Postnikov</a>, p. 34 of: <em>Лекции по алгебраической топологии. Основы теории гомотопий</em>, Наука 1982 (<a href="http://libgen.is/book/index.php?md5=34A8C3C956EB80877F4E3CF5A297F514">web</a>)</p> </li> </ul> <p>Discussion of k-spaces in the generality of subcategory-generated spaces, including <a class="existingWikiWord" href="/nlab/show/Delta-generated+topological+spaces">Delta-generated topological spaces</a>:</p> <ul> <li id="Vogt71"> <p><a class="existingWikiWord" href="/nlab/show/Rainer+M.+Vogt">Rainer M. Vogt</a>, <em>Convenient categories of topological spaces for homotopy theory</em>, Arch. Math 22, 545–555 (1971) (<a href="https://doi.org/10.1007/BF01222616">doi:10.1007/BF01222616</a>)</p> </li> <li id="EscardoLawsonSimpson04"> <p><a class="existingWikiWord" href="/nlab/show/Mart%C3%ADn+Escard%C3%B3">Martín Escardó</a>, <a class="existingWikiWord" href="/nlab/show/Jimmie+Lawson">Jimmie Lawson</a>, <a class="existingWikiWord" href="/nlab/show/Alex+Simpson">Alex Simpson</a>, Section 3 of: <em>Comparing Cartesian closed categories of (core) compactly generated spaces</em>, Topology and its Applications Volume 143, Issues 1–3, 28 August 2004, Pages 105-145 (<a href="https://doi.org/10.1016/j.topol.2004.02.011">doi:10.1016/j.topol.2004.02.011</a>)</p> </li> <li id="Gaucher07"> <p><a class="existingWikiWord" href="/nlab/show/Philippe+Gaucher">Philippe Gaucher</a>, Section 2 of: <em>Homotopical interpretation of globular complex by multipointed d-space</em>, Theory and Applications of Categories, vol. 22, number 22, 588-621, 2009 (<a href="https://arxiv.org/abs/0710.3553">arXiv:0710.3553</a>)</p> </li> </ul> <p>Proof that k-spaces form a <a class="existingWikiWord" href="/nlab/show/regular+category">regular category</a>:</p> <ul> <li id="CagliariMatovaniVitale95">F. Cagliari, S. Mantovani, <a class="existingWikiWord" href="/nlab/show/Enrico+Vitale">Enrico Vitale</a>, <em>Regularity of the category of Kelley spaces</em>, Applied Categorical Structures volume 3, pages 357–361 (1995) (<a href="https://link.springer.com/article/10.1007/BF00872904">doi:10.1007/BF00872904</a>, <a href="http://www.dm.unibo.it/~cagliari/articoli/Regularkelley.pdf">pdf</a>)</li> </ul> <p>Further accounts:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/George+W.+Whitehead">George W. Whitehead</a>, Section I.4 of: <em>Elements of Homotopy Theory</em>, Springer 1978 (<a href="https://link.springer.com/book/10.1007/978-1-4612-6318-0">doi:10.1007/978-1-4612-6318-0</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Brian+Day">Brian J. Day</a>, <em>Relationship of Spanier’s Quasi-topological Spaces to k-Spaces</em> , M. Sc. thesis University of Sydney 1968. (<a href="http://www.math.mq.edu.au/~street/DayMasters.pdf">pdf</a>)</p> </li> <li> <p>Peter Booth, Philip R. Heath, <a class="existingWikiWord" href="/nlab/show/Renzo+A.+Piccinini">Renzo A. Piccinini</a>, <em>Fibre preserving maps and functional spaces</em>, Algebraic topology (Proc. Conf., Univ. British Columbia, Vancouver, B.C., 1977), pp. 158–167, Lecture Notes in Math., 673, Springer, Berlin, 1978.</p> </li> <li id="Piccinini92"> <p><a class="existingWikiWord" href="/nlab/show/Renzo+A.+Piccinini">Renzo A. Piccinini</a>, Appendix B in: <em>Lectures on Homotopy Theory</em>, Mathematics Studies <strong>171</strong>, North Holland 1992 (<a href="https://www.sciencedirect.com/bookseries/north-holland-mathematics-studies/vol/171/suppl/C">ISBN:978-0-444-89238-6</a>)</p> </li> <li id="FelixHalperinThomas00"> <p><a class="existingWikiWord" href="/nlab/show/Yves+F%C3%A9lix">Yves Félix</a>, <a class="existingWikiWord" href="/nlab/show/Stephen+Halperin">Stephen Halperin</a>, <a class="existingWikiWord" href="/nlab/show/Jean-Claude+Thomas">Jean-Claude Thomas</a>, Section 0 of: <em>Rational Homotopy Theory</em>, Graduate Texts in Mathematics, 205, Springer-Verlag, 2000 (<a href="https://link.springer.com/book/10.1007/978-1-4613-0105-9">doi:10.1007/978-1-4613-0105-9</a>)</p> <blockquote> <p>(in a context of <a class="existingWikiWord" href="/nlab/show/rational+homotopy+theory">rational homotopy theory</a>)</p> </blockquote> </li> <li id="AGP02"> <p>Marcelo Aguilar, <a class="existingWikiWord" href="/nlab/show/Samuel+Gitler">Samuel Gitler</a>, Carlos Prieto, around note 4.3.22 of <em>Algebraic topology from a homotopical viewpoint</em>, Springer (2002) (<a href="http://tocs.ulb.tu-darmstadt.de/106999419.pdf">toc pdf</a>)</p> </li> <li> <p>Samuel Smith, <em>The homotopy theory of function spaces: a survey</em> (<a href="http://arxiv.org/abs/1009.0804">arXiv:1009.0804</a>)</p> </li> </ul> <h3 id="ReferencesCGWeakHausdorffSpaces">CG weak Hausdorff spaces</h3> <p>The idea of generalizing compact generation to weakly Hausdorff spaces appears in:</p> <ul> <li id="McCord69"><a class="existingWikiWord" href="/nlab/show/Michael+C.+McCord">Michael C. McCord</a>, Section 2 of: <em>Classifying Spaces and Infinite Symmetric Products</em>, Transactions of the American Mathematical Society, Vol. 146 (Dec., 1969), pp. 273-298 (<a href="https://www.jstor.org/stable/1995173">jstor:1995173</a>, <a href="https://www.ams.org/journals/tran/1969-146-00/S0002-9947-1969-0251719-4/S0002-9947-1969-0251719-4.pdf">pdf</a>)</li> </ul> <p>where it is attributed to <a class="existingWikiWord" href="/nlab/show/John+C.+Moore">John C. Moore</a>.</p> <p>Review in this generality of CG weakly Hausdorff spaces:</p> <ul> <li id="Lewis78"> <p><a class="existingWikiWord" href="/nlab/show/Gaunce+Lewis">Gaunce Lewis</a>, <em>Compactly generated spaces</em> (<a href="http://www.math.uchicago.edu/~may/MISC/GaunceApp.pdf">pdf</a>), appendix A of <em>The Stable Category and Generalized Thom Spectra</em>, PhD thesis Chicago, 1978</p> </li> <li id="FritschPiccinini90"> <p><a class="existingWikiWord" href="/nlab/show/Rudolf+Fritsch">Rudolf Fritsch</a>, <a class="existingWikiWord" href="/nlab/show/Renzo+Piccinini">Renzo Piccinini</a>, Appendix A.1 of: <em>Cellular structures in topology</em>, Cambridge University Press (1990) (<a href="https://doi.org/10.1017/CBO9780511983948">doi:10.1017/CBO9780511983948</a>, <a href="https://epub.ub.uni-muenchen.de/4493/1/4493.pdf">pdf</a>)</p> </li> <li id="May99"> <p><a class="existingWikiWord" href="/nlab/show/Peter+May">Peter May</a>, Chapter 5 of: <em><a class="existingWikiWord" href="/nlab/show/A+concise+course+in+algebraic+topology">A concise course in algebraic topology</a></em>, University of Chicago Press 1999 (<a href="https://www.press.uchicago.edu/ucp/books/book/chicago/C/bo3777031.html">ISBN: 9780226511832</a>, <a href="http://www.math.uchicago.edu/~may/CONCISE/ConciseRevised.pdf">pdf</a>)</p> </li> <li id="Strickland09"> <p><a class="existingWikiWord" href="/nlab/show/Neil+Strickland">Neil Strickland</a>, <em>The category of CGWH spaces</em>, 2009 (<a href="http://neil-strickland.staff.shef.ac.uk/courses/homotopy/cgwh.pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/StricklandCGHWSpaces.pdf" title="pdf">pdf</a>)</p> </li> <li id="Schwede12"> <p><a class="existingWikiWord" href="/nlab/show/Stefan+Schwede">Stefan Schwede</a>, Section A.2 of: <em><a class="existingWikiWord" href="/nlab/show/Symmetric+spectra">Symmetric spectra</a></em> (2012)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Charles+Rezk">Charles Rezk</a>, <em>Compactly Generated Spaces</em>, 2018 (<a href="https://faculty.math.illinois.edu/~rezk/cg-spaces-better.pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/Rezk_CompactlyGeneratedSpaces.pdf" title="pdf">pdf</a>)</p> </li> </ul> <p>Brief review in preparation of the <a class="existingWikiWord" href="/nlab/show/model+structure+on+compactly+generated+topological+spaces">model structure on compactly generated topological spaces</a>:</p> <ul> <li id="Hovey99"><a class="existingWikiWord" href="/nlab/show/Mark+Hovey">Mark Hovey</a>, Def. 2.4.21 in: <em><a class="existingWikiWord" href="/nlab/show/Model+Categories">Model Categories</a></em>, Mathematical Surveys and Monographs, Volume 63, AMS (1999) (<a href="https://bookstore.ams.org/surv-63-s">ISBN:978-0-8218-4361-1</a>, <a href="https://doi.org/http://dx.doi.org/10.1090/surv/063">doi:10.1090/surv/063</a>, <a href="https://people.math.rochester.edu/faculty/doug/otherpapers/hovey-model-cats.pdf">pdf</a>, <a href="http://books.google.co.uk/books?id=Kfs4uuiTXN0C&printsec=frontcover">Google books</a>)</li> </ul> <p>Review with focus on compactly generated <a class="existingWikiWord" href="/nlab/show/topological+G-spaces">topological G-spaces</a> in <a class="existingWikiWord" href="/nlab/show/equivariant+homotopy+theory">equivariant homotopy theory</a> and specifically <a class="existingWikiWord" href="/nlab/show/equivariant+bundle">equivariant bundle</a>-theory:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Bernardo+Uribe">Bernardo Uribe</a>, <a class="existingWikiWord" href="/nlab/show/Wolfgang+L%C3%BCck">Wolfgang Lück</a>, Section 16 of: <em>Equivariant principal bundles and their classifying spaces</em>, Algebr. Geom. Topol. 14 (2014) 1925-1995 (<a href="https://arxiv.org/abs/1304.4862">arXiv:1304.4862</a>, <a href="http://dx.doi.org/10.2140/agt.2014.14.1925">doi:10.2140/agt.2014.14.1925</a>)</li> </ul> <div> <h3 id="ParameterizedTopology">Exponential law for parameterized topological spaces</h3> <p>On <a class="existingWikiWord" href="/nlab/show/exponential+law+for+spaces">exponential objects</a> (<a class="existingWikiWord" href="/nlab/show/internal+homs">internal homs</a>) in <a class="existingWikiWord" href="/nlab/show/slice+categories">slice categories</a> of (<a class="existingWikiWord" href="/nlab/show/compactly+generated+topological+spaces">compactly generated</a>) <a class="existingWikiWord" href="/nlab/show/topological+spaces">topological spaces</a> – see at <em><a class="existingWikiWord" href="/nlab/show/parameterized+homotopy+theory">parameterized homotopy theory</a></em>):</p> <ul> <li id="Booth70"> <p><a class="existingWikiWord" href="/nlab/show/Peter+I.+Booth">Peter I. Booth</a>, <em>The Exponential Law of Maps I</em>, Proceedings of the London Mathematical Society <strong>s3-20</strong> 1 (1970) 179-192 <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo></mrow><annotation encoding="application/x-tex">[</annotation></semantics></math><a href="https://doi.org/10.1112/plms/s3-20.1.179">doi:10.1112/plms/s3-20.1.179</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">]</annotation></semantics></math></p> </li> <li id="Booth71"> <p><a class="existingWikiWord" href="/nlab/show/Peter+I.+Booth">Peter I. Booth</a>, <em>The exponential law of maps. II</em>, Mathematische Zeitschrift <strong>121</strong> (1971) 311–319 <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo></mrow><annotation encoding="application/x-tex">[</annotation></semantics></math><a href="https://doi.org/10.1007/BF01109977">doi:10.1007/BF01109977</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">]</annotation></semantics></math></p> </li> <li id="BoothBrown78a"> <p><a class="existingWikiWord" href="/nlab/show/Peter+I.+Booth">Peter I. Booth</a>, <a class="existingWikiWord" href="/nlab/show/Ronnie+Brown">Ronnie Brown</a>, <em>Spaces of partial maps, fibred mapping spaces and the compact-open topology</em>, General Topology and its Applications <strong>8</strong> 2 (1978) 181-195 <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo></mrow><annotation encoding="application/x-tex">[</annotation></semantics></math><a href="https://doi.org/10.1016/0016-660X(78)90049-1">doi:10.1016/0016-660X(78)90049-1</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">]</annotation></semantics></math></p> </li> <li id="BoothBrown78b"> <p><a class="existingWikiWord" href="/nlab/show/Peter+I.+Booth">Peter I. Booth</a>, <a class="existingWikiWord" href="/nlab/show/Ronnie+Brown">Ronnie Brown</a>, <em>On the application of fibred mapping spaces to exponential laws for bundles, ex-spaces and other categories of maps</em>, General Topology and its Applications <strong>8</strong> 2 (1978) 165-179 <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo></mrow><annotation encoding="application/x-tex">[</annotation></semantics></math><a href="https://doi.org/10.1016/0016-660X(78)90048-X">doi:10.1016/0016-660X(78)90048-X</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">]</annotation></semantics></math></p> </li> <li id="Lewis85"> <p><a class="existingWikiWord" href="/nlab/show/L.+Gaunce+Lewis%2C+Jr.">L. Gaunce Lewis, Jr.</a>, §1 of: <em>Open maps, colimits, and a convenient category of fibre spaces</em>, Topology and its Applications <strong>19</strong> 1 (1985) 75-89 <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo></mrow><annotation encoding="application/x-tex">[</annotation></semantics></math><a href="https://doi.org/10.1016/0166-8641(85)90087-2">doi.org/10.1016/0166-8641(85)90087-2</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">]</annotation></semantics></math></p> </li> </ul> <p>And with an eye towards <a class="existingWikiWord" href="/nlab/show/parameterized+homotopy+theory">parameterized homotopy theory</a>:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Ioan+Mackenzie+James">Ioan Mackenzie James</a>: §II.9 in: <em>Fibrewise topology</em>, Cambridge Tracts in Mathematics, Cambridge University Press (1989) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo></mrow><annotation encoding="application/x-tex">[</annotation></semantics></math>ISBN:9780521360906<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">]</annotation></semantics></math></p> </li> <li id="MaySigurdsson06"> <p><a class="existingWikiWord" href="/nlab/show/Peter+May">Peter May</a>, <a class="existingWikiWord" href="/nlab/show/Johann+Sigurdsson">Johann Sigurdsson</a>, §1.3.7-§1.3.9 in: <em><a class="existingWikiWord" href="/nlab/show/Parametrized+Homotopy+Theory">Parametrized Homotopy Theory</a></em>, Mathematical Surveys and Monographs, vol. 132, AMS 2006 (<a href="https://bookstore.ams.org/surv-132">ISBN:978-0-8218-3922-5</a>, <a href="https://arxiv.org/abs/math/0411656">arXiv:math/0411656</a>, <a href="http://www.math.uchicago.edu/~may/EXTHEORY/MaySig.pdf">pdf</a>)</p> </li> </ul> </div><div class="footnotes"><hr /><ol><li id="fn:1"> <p>The reason for choosing the term “k-space” in <a href="#Gale50">Gale 1950</a> seems to be lost in history. The “k” is not for “Kelley”, as <a href="#Kelley55">Kelley 1955</a> came later. It might have been an allusion to the German word <em>kompakt</em>. <a href="#fnref:1" rev="footnote">↩</a></p> </li><li id="fn:2"> <p>This is according to personal communication by <a class="existingWikiWord" href="/nlab/show/David+Gale">David Gale</a> to <a class="existingWikiWord" href="/nlab/show/William+Lawvere">William Lawvere</a> in 2003, forwarded by Lawvere to <a class="existingWikiWord" href="/nlab/show/Martin+Escardo">Martin Escardo</a> at that time, and then kindly forwarded by Escardo to the nForum in 2021; see <a href="https://nforum.ncatlab.org/discussion/8638/compactly-generated-topological-space/?Focus=94755#Comment_94755">there</a>. <a href="#fnref:2" rev="footnote">↩</a></p> </li></ol></div></body></html> </div> <div class="revisedby"> <p> Last revised on March 28, 2024 at 21:15:11. 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