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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> Hurewicz cofibration </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/10168/#Item_34" 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" 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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> <h4 id="homotopy_theory">Homotopy theory</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopy theory</a>, <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category+theory">(∞,1)-category theory</a>, <a class="existingWikiWord" href="/nlab/show/homotopy+type+theory">homotopy type theory</a></strong></p> <p>flavors: <a class="existingWikiWord" href="/nlab/show/stable+homotopy+theory">stable</a>, <a class="existingWikiWord" href="/nlab/show/equivariant+homotopy+theory">equivariant</a>, <a class="existingWikiWord" href="/nlab/show/rational+homotopy+theory">rational</a>, <a class="existingWikiWord" href="/nlab/show/p-adic+homotopy+theory">p-adic</a>, <a class="existingWikiWord" href="/nlab/show/proper+homotopy+theory">proper</a>, <a class="existingWikiWord" href="/nlab/show/geometric+homotopy+theory">geometric</a>, <a class="existingWikiWord" href="/nlab/show/cohesive+homotopy+theory">cohesive</a>, <a class="existingWikiWord" href="/nlab/show/directed+homotopy+theory">directed</a>…</p> <p>models: <a class="existingWikiWord" href="/nlab/show/topological+homotopy+theory">topological</a>, <a class="existingWikiWord" href="/nlab/show/simplicial+homotopy+theory">simplicial</a>, <a class="existingWikiWord" href="/nlab/show/localic+homotopy+theory">localic</a>, …</p> <p>see also <strong><a class="existingWikiWord" href="/nlab/show/algebraic+topology">algebraic topology</a></strong></p> <p><strong>Introductions</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Introduction+to+Topology+--+2">Introduction to Basic Homotopy Theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Introduction+to+Homotopy+Theory">Introduction to Abstract Homotopy Theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/geometry+of+physics+--+homotopy+types">geometry of physics – homotopy types</a></p> </li> </ul> <p><strong>Definitions</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy">homotopy</a>, <a class="existingWikiWord" href="/nlab/show/higher+homotopy">higher homotopy</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+type">homotopy type</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Pi-algebra">Pi-algebra</a>, <a class="existingWikiWord" href="/nlab/show/spherical+object+and+Pi%28A%29-algebra">spherical object and Pi(A)-algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+coherent+category+theory">homotopy coherent category theory</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopical+category">homotopical category</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/model+category">model category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/category+of+fibrant+objects">category of fibrant objects</a>, <a class="existingWikiWord" href="/nlab/show/cofibration+category">cofibration category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Waldhausen+category">Waldhausen category</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+category">homotopy category</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Ho%28Top%29">Ho(Top)</a></li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category">(∞,1)-category</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/homotopy+category+of+an+%28%E2%88%9E%2C1%29-category">homotopy category of an (∞,1)-category</a></li> </ul> </li> </ul> <p><strong>Paths and cylinders</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/left+homotopy">left homotopy</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/cylinder+object">cylinder object</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/mapping+cone">mapping cone</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/right+homotopy">right homotopy</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/path+object">path object</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/mapping+cocone">mapping cocone</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/generalized+universal+bundle">universal bundle</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/interval+object">interval object</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+localization">homotopy localization</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/infinitesimal+interval+object">infinitesimal interval object</a></p> </li> </ul> </li> </ul> <p><strong>Homotopy groups</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+group">homotopy group</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+group">fundamental group</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/fundamental+group+of+a+topos">fundamental group of a topos</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Brown-Grossman+homotopy+group">Brown-Grossman homotopy group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/categorical+homotopy+groups+in+an+%28%E2%88%9E%2C1%29-topos">categorical homotopy groups in an (∞,1)-topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/geometric+homotopy+groups+in+an+%28%E2%88%9E%2C1%29-topos">geometric homotopy groups in an (∞,1)-topos</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+%E2%88%9E-groupoid">fundamental ∞-groupoid</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+groupoid">fundamental groupoid</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/path+groupoid">path groupoid</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+%E2%88%9E-groupoid+in+a+locally+%E2%88%9E-connected+%28%E2%88%9E%2C1%29-topos">fundamental ∞-groupoid in a locally ∞-connected (∞,1)-topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+%E2%88%9E-groupoid+of+a+locally+%E2%88%9E-connected+%28%E2%88%9E%2C1%29-topos">fundamental ∞-groupoid of a locally ∞-connected (∞,1)-topos</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+%28%E2%88%9E%2C1%29-category">fundamental (∞,1)-category</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/fundamental+category">fundamental category</a></li> </ul> </li> </ul> <p><strong>Basic facts</strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/fundamental+group+of+the+circle+is+the+integers">fundamental group of the circle is the integers</a></li> </ul> <p><strong>Theorems</strong></p> <ul> <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/Freudenthal+suspension+theorem">Freudenthal suspension theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Blakers-Massey+theorem">Blakers-Massey theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/higher+homotopy+van+Kampen+theorem">higher homotopy van Kampen 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/Whitehead%27s+theorem">Whitehead's theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hurewicz+theorem">Hurewicz theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Galois+theory">Galois theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+hypothesis">homotopy hypothesis</a>-theorem</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='#properties'>Properties</a></li> <ul> <li><a href='#Closedness'>(Non-)Closed images</a></li> <li><a href='#Inclusions'>Characterization for closed subspace inclusions</a></li> <ul> <li><a href='#via_retracts_of_the_pushoutproduct_with_'>Via retracts of the pushout-product with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>0</mn><mo>↪</mo><mo stretchy="false">[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">0 \hookrightarrow [0,1]</annotation></semantics></math></a></li> <li><a href='#via_neighbourhood_deformations'>Via neighbourhood deformations</a></li> </ul> <li><a href='#AssortedFacts'>Assorted facts</a></li> <li><a href='#InteractionWithFiberProducts'>Interaction with (fiber) producs</a></li> <li><a href='#StromModelStructure'>Strøm’s model structure</a></li> </ul> <li><a href='#Examples'>Examples</a></li> <ul> <li><a href='#relative_cell_complex_inclusions'>Relative cell complex inclusions</a></li> <li><a href='#BasepointsInLocallyEuclideanSpaces'>Points in locally Euclidean Hausdorff spaces</a></li> <li><a href='#ClosedSubmanifoldsOfBanachManifolds'>Closed submanifolds of Banach manifolds</a></li> <li><a href='#further_examples'>Further examples</a></li> </ul> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> </ul> </div> <h2 id="idea">Idea</h2> <p>In <a class="existingWikiWord" href="/nlab/show/algebraic+topology">algebraic topology</a> and <a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopy theory</a>, <em>Hurewicz cofibrations</em> are a kind of <a class="existingWikiWord" href="/nlab/show/cofibration">cofibration</a> of <a class="existingWikiWord" href="/nlab/show/topological+spaces">topological spaces</a>, hence a kind of <a class="existingWikiWord" href="/nlab/show/continuous+function">continuous function</a> satisfying certain <a class="existingWikiWord" href="/nlab/show/extension">extension</a> properties.</p> <p>Specifically, a <a class="existingWikiWord" href="/nlab/show/continuous+function">continuous function</a> is a <em>Hurewicz cofibration</em> (<a href="#Strom66">Strøm 1966</a>) if it satisfies the <a class="existingWikiWord" href="/nlab/show/homotopy+extension+property">homotopy extension property</a> for all target spaces and with respect to the standard notion of <a class="existingWikiWord" href="/nlab/show/left+homotopy">left homotopy</a> of topological spaces given by the standard topological <a class="existingWikiWord" href="/nlab/show/interval+object">interval object</a>/<a class="existingWikiWord" href="/nlab/show/cylinder+object">cylinder object</a>.</p> <p>A <a class="existingWikiWord" href="/nlab/show/pointed+topological+space">pointed topological space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(X,x)</annotation></semantics></math> for which the base-point inclusion <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><mi>x</mi><mo stretchy="false">}</mo><mover><mo>↪</mo><mspace width="thickmathspace"></mspace></mover><mi>X</mi></mrow><annotation encoding="application/x-tex">\{x\} \xhookrightarrow{\;} X</annotation></semantics></math> is a Hurewicz cofibration is called a <em><a class="existingWikiWord" href="/nlab/show/well-pointed+topological+space">well-pointed topological space</a></em>. A <a class="existingWikiWord" href="/nlab/show/simplicial+topological+space">simplicial topological space</a> all whose degeneracy maps are Hurewicz cofibrations is called a <em><a class="existingWikiWord" href="/nlab/show/good+simplicial+topological+space">good simplicial topological space</a></em>.</p> <p>In <a class="existingWikiWord" href="/nlab/show/point-set+topology">point-set topology</a> Hurewicz cofibrations are often just called <em>cofibrations</em>,for short. If their <a class="existingWikiWord" href="/nlab/show/image">image</a> is a <a class="existingWikiWord" href="/nlab/show/closed+subspace">closed subspace</a> they are called <em><a class="existingWikiWord" href="/nlab/show/closed+cofibrations">closed cofibrations</a></em>.</p> <p>Beware that there are other relevant classes of cofibrations between topological spaces, notably the <em><a class="existingWikiWord" href="/nlab/show/Serre+cofibrations">Serre cofibrations</a></em>, and (in <a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopy theory</a> and <a class="existingWikiWord" href="/nlab/show/model+category">model category</a>-theory, at least) also often just called “cofibrations”. But “closed cofibration” always refers to closed Hurewicz cofibrations.</p> <p>In fact, the notion of <a class="existingWikiWord" href="/nlab/show/homotopy+extension+property">homotopy extension property</a> makes sense in any <a class="existingWikiWord" href="/nlab/show/category">category</a> with a chosen <a class="existingWikiWord" href="/nlab/show/cylinder+object">cylinder object</a>; and in this generality one also speaks of <em><a class="existingWikiWord" href="/nlab/show/h-cofibrations">h-cofibrations</a></em> (following <a href="#MandellMaySchwedeShipley01">Mandell, May, Schwede & Shipley 2001, p. 16</a>).</p> <h2 id="Definition">Definition</h2> <p> <div class='num_defn' id='HurewiczCofibration'> <h6>Definition</h6> <p><strong>(Hurewicz cofibration)</strong> <br /> A map (<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>i</mi><mo lspace="verythinmathspace">:</mo><mi>A</mi><mover><mo>→</mo><mspace width="thickmathspace"></mspace></mover><mi>X</mi></mrow><annotation encoding="application/x-tex">i \colon A \xrightarrow{\;} X</annotation></semantics></math> betwee <a class="existingWikiWord" href="/nlab/show/topological+spaces">topological spaces</a> is a <em>Hurewicz cofibration</em> if it satisfies the <a class="existingWikiWord" href="/nlab/show/homotopy+extension+property">homotopy extension property</a> for all spaces, hence if</p> <ul> <li>for any map <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 \longrightarrow Y</annotation></semantics></math> and any <a class="existingWikiWord" href="/nlab/show/left+homotopy">left homotopy</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>η</mi><mspace width="thinmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thinmathspace"></mspace><mi>A</mi><mo>×</mo><mo stretchy="false">[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy="false">]</mo><mo>⟶</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">\eta \,\colon\, A \times [0,1] \longrightarrow Y</annotation></semantics></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>η</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mn>0</mn><mo stretchy="false">)</mo><mo>=</mo><mi>f</mi><mo>∘</mo><mi>i</mi></mrow><annotation encoding="application/x-tex">\eta(-,0) = f \circ i</annotation></semantics></math>, there exists a <a class="existingWikiWord" href="/nlab/show/left+homotopy">left homotopy</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>η</mi><mo>^</mo></mover><mspace width="thinmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thinmathspace"></mspace><mi>X</mi><mo>×</mo><mo stretchy="false">[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy="false">]</mo><mo>⟶</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">\widehat{\eta} \,\colon\, X \times [0,1] \longrightarrow Y</annotation></semantics></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>η</mi><mo>^</mo></mover><mo>∘</mo><mo stretchy="false">(</mo><mi>i</mi><mo>×</mo><mi>id</mi><mo stretchy="false">)</mo><mo>=</mo><mi>η</mi></mrow><annotation encoding="application/x-tex">\widehat{\eta} \circ (i \times id) = \eta</annotation></semantics></math>;</li> </ul> <p>equivalently:</p> <ul> <li>given a <a class="existingWikiWord" href="/nlab/show/commuting+diagram">commuting diagram</a> in <a 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<p><strong>(re-formulation in terms of right homotopies)</strong> <br /> In a <a class="existingWikiWord" href="/nlab/show/convenient+category+of+topological+spaces">convenient</a> <a class="existingWikiWord" href="/nlab/show/cartesian+closed+category">cartesian closed category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>TopSp</mi></mrow><annotation encoding="application/x-tex">TopSp</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/topological+spaces">topological spaces</a> (such as that of <a class="existingWikiWord" href="/nlab/show/compactly+generated+topological+spaces">compactly generated topological spaces</a>) the <a class="existingWikiWord" href="/nlab/show/product">product</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⊣</mo></mrow><annotation encoding="application/x-tex">\dashv</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/internal+hom">internal hom</a>-<a class="existingWikiWord" href="/nlab/show/adjunction">adjunction</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>TopSp</mi><munderover><mrow><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></mrow><munder><mo>⟶</mo><mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><msup><mo stretchy="false">)</mo> <mrow><mo stretchy="false">[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy="false">]</mo></mrow></msup></mrow></munder><mover><mo>⟵</mo><mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>×</mo><mo stretchy="false">[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy="false">]</mo></mrow></mover></munderover><mi>TopSp</mi></mrow><annotation encoding="application/x-tex"> TopSp \underoverset {\underset{ (-)^{[0,1]} }{\longrightarrow}} {\overset{ (-) \times [0,1] }{\longleftarrow}} {\;\;\;\;\;\bot\;\;\;\;\;} TopSp </annotation></semantics></math></div> <p>allows to pass to <a class="existingWikiWord" href="/nlab/show/adjunct">adjunct</a> morphisms in Def. <a class="maruku-ref" href="#HurewiczCofibration"></a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>TopSp</mi><mo maxsize="1.2em" minsize="1.2em">(</mo><mi>A</mi><mo>×</mo><mo stretchy="false">[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy="false">]</mo><mo>,</mo><mspace width="thinmathspace"></mspace><mi>Y</mi><mo maxsize="1.2em" minsize="1.2em">)</mo></mtd> <mtd><mo>≃</mo></mtd> <mtd><mi>TopSp</mi><mo maxsize="1.2em" minsize="1.2em">(</mo><mi>A</mi><mo>,</mo><mspace width="thinmathspace"></mspace><msup><mi>Y</mi> <mrow><mo stretchy="false">[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy="false">]</mo></mrow></msup><mo maxsize="1.2em" minsize="1.2em">)</mo></mtd></mtr> <mtr><mtd><mi>η</mi></mtd> <mtd><mo>↔</mo></mtd> <mtd><mi>η</mi><mo>′</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ TopSp \big( A \times [0,1] ,\, Y \big) &\simeq& TopSp \big( A ,\, Y^{[0,1]} \big) \\ \eta &\leftrightarrow& \eta' } </annotation></semantics></math></div> <p>and equivalently express the above diagram of <a class="existingWikiWord" href="/nlab/show/left+homotopies">left homotopies</a> as the following (somewhat more transparent) diagram of <a class="existingWikiWord" href="/nlab/show/right+homotopies">right homotopies</a>, where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ev</mi> <mn>0</mn></msub><mspace width="thinmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thinmathspace"></mspace><msup><mi>Y</mi> <mrow><mo stretchy="false">[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy="false">]</mo></mrow></msup><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">ev_0 \,\colon\, Y^{[0,1]}\to Y</annotation></semantics></math> denotes <a class="existingWikiWord" href="/nlab/show/evaluation">evaluation</a> at 0 <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>γ</mi><mo>↦</mo><mi>γ</mi><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\gamma \mapsto 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stroke-linejoin="round" stroke="rgb(0%, 0%, 0%)" stroke-opacity="1" stroke-miterlimit="10" d="M -2.485291 2.867991 C -2.032166 1.149241 -1.020448 0.332835 -0.00091625 0.00080375 C -1.020448 -0.335134 -2.032166 -1.147634 -2.485291 -2.87029 " transform="matrix(1, 0, 0, -1, 91.90326, 79.62971)"></path> <g fill="rgb(0%, 0%, 0%)" fill-opacity="1"> <use xlink:href="#L0xfvgS-2F5hajWbbbQkR_byRLU=-glyph-2-4" x="56.197" y="90.064"></use> </g> <path fill="none" stroke-width="0.47818" stroke-linecap="butt" stroke-linejoin="miter" stroke="rgb(0%, 0%, 0%)" stroke-opacity="1" stroke-dasharray="3.34735 1.91277" stroke-miterlimit="10" d="M -37.0705 -25.138156 L 27.113094 19.2095 " transform="matrix(1, 0, 0, -1, 63.508, 45.272)"></path> <path fill="none" stroke-width="0.47818" stroke-linecap="round" stroke-linejoin="round" stroke="rgb(0%, 0%, 0%)" stroke-opacity="1" stroke-miterlimit="10" d="M -2.485075 2.868187 C -2.030318 1.148564 -1.019793 0.336443 0.00188906 0.000917719 C -1.017917 -0.334333 -2.031583 -1.148635 -2.486017 -2.871597 " transform="matrix(0.8227, -0.5684, -0.5684, -0.8227, 90.81928, 25.92761)"></path> <path fill-rule="nonzero" fill="rgb(100%, 100%, 100%)" fill-opacity="1" d="M 66.457031 40.082031 L 50.996094 40.082031 L 50.996094 56.121094 L 66.457031 56.121094 Z M 66.457031 40.082031 "></path> <g fill="rgb(0%, 0%, 0%)" fill-opacity="1"> <use xlink:href="#L0xfvgS-2F5hajWbbbQkR_byRLU=-glyph-5-1" x="54.6955" y="50.967"></use> </g> <g fill="rgb(0%, 0%, 0%)" fill-opacity="1"> <use xlink:href="#L0xfvgS-2F5hajWbbbQkR_byRLU=-glyph-2-3" x="54.213" y="50.967"></use> </g> <g fill="rgb(0%, 0%, 0%)" fill-opacity="1"> <use xlink:href="#L0xfvgS-2F5hajWbbbQkR_byRLU=-glyph-3-1" x="59.883" y="47.45075"></use> </g> </svg> <p>In this equivalent formulation, the homotopy extension property is simply the <a class="existingWikiWord" href="/nlab/show/right+lifting+property">right lifting property</a> against the <a class="existingWikiWord" href="/nlab/show/evaluation">evaluation</a> map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ev</mi> <mn>0</mn></msub><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><msup><mi>Y</mi> <mrow><mo stretchy="false">[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy="false">]</mo></mrow></msup><mover><mo>→</mo><mspace width="thickmathspace"></mspace></mover><mi>Y</mi></mrow><annotation encoding="application/x-tex">ev_0 \;\colon\; Y^{[0,1]} \xrightarrow{\;} Y</annotation></semantics></math> out of the <a class="existingWikiWord" href="/nlab/show/path+space">path space</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math>.</p> </div> </p> <p>(e.g. <a href="#May99">May 1999, p. 43 (51 of 251)</a>)</p> <p> <div class='num_defn' id='ClosedHurewiczCofibration'> <h6>Definition</h6> <p><strong>(closed cofibrations)</strong> <br /> A Hurewicz cofibration <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi><mo lspace="verythinmathspace">:</mo><mi>A</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">i \colon A\to X</annotation></semantics></math> (Def. <a class="maruku-ref" href="#HurewiczCofibration"></a>) is called a <em><a class="existingWikiWord" href="/nlab/show/closed+cofibration">closed cofibration</a></em> if the <a class="existingWikiWord" href="/nlab/show/image">image</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">i(A)</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/closed+subspace">closed subspace</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>. In this case the pair <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>A</mi><mo>,</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(A,X)</annotation></semantics></math> is also called an <em><a class="existingWikiWord" href="/nlab/show/NDR-pair">NDR-pair</a></em>.</p> </div> </p> <h2 id="properties">Properties</h2> <h3 id="Closedness">(Non-)Closed images</h3> <p> <div class='num_prop'> <h6>Proposition</h6> <p>Every Hurewicz cofibration <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi><mo lspace="verythinmathspace">:</mo><mi>A</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">i \colon A \to X</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/injective+function">injective</a> and a <a class="existingWikiWord" href="/nlab/show/homeomorphism">homeomorphism</a> onto its image. 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> is <a class="existingWikiWord" href="/nlab/show/Hausdorff+space">Hausdorff</a>, then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/closed+map">closed</a>.</p> </div> </p> <p>(<a href="#tDKP70">tDKP 1970</a> (1.17)).</p> <p> <div class='num_prop' id='HurewiczCofibrationsInCGWHSpacesAreClosed'> <h6>Proposition</h6> <p>In the category of <a class="existingWikiWord" href="/nlab/show/weakly+Hausdorff+space">weakly Hausdorff</a> <a class="existingWikiWord" href="/nlab/show/compactly+generated+spaces">compactly generated spaces</a>, the image <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">i(A)</annotation></semantics></math> of a Hurewicz cofibration is always closed. The same holds in the category of all <a class="existingWikiWord" href="/nlab/show/Hausdorff+spaces">Hausdorff spaces</a>.</p> </div> </p> <p>(e.g. <a href="#May99">May 1999, Sec. 6.2, p. 44 (52 of 251)</a>)</p> <p>But in the plain category <a class="existingWikiWord" href="/nlab/show/Top">Top</a> of all topological spaces there are pathological counterexamples:</p> <p> <div class='num_remark'> <h6>Example</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>=</mo><mo stretchy="false">{</mo><mi>a</mi><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">A =\{a\}</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>=</mo><mo stretchy="false">{</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">X=\{a,b\}</annotation></semantics></math> be the one and two element sets, both with the <a class="existingWikiWord" href="/nlab/show/codiscrete+topology">codiscrete topology</a> (only <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><mo>∅</mo></mrow><annotation encoding="application/x-tex">\varnothing</annotation></semantics></math> are <a class="existingWikiWord" href="/nlab/show/open+subsets">open subsets</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>), and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi><mo>:</mo><mi>A</mi><mo>↪</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">i:A\hookrightarrow X</annotation></semantics></math> is the inclusion <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><mo>↦</mo><mi>a</mi></mrow><annotation encoding="application/x-tex">a\mapsto a</annotation></semantics></math>. Then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math> is a non-closed cofibration.</p> </div> (<a href="#Strom66">Strøm 1966, p. 5</a>)</p> <h3 id="Inclusions">Characterization for closed subspace inclusions</h3> <p>When the <a class="existingWikiWord" href="/nlab/show/image">image</a> of a Hurewicz cofibration is a <a class="existingWikiWord" href="/nlab/show/closed+subspace">closed subspace</a> – which is automatically the case in <a class="existingWikiWord" href="/nlab/show/weakly+Hausdorff+topological+spaces">weakly Hausdorff topological spaces</a>, by Prop. <a class="maruku-ref" href="#"></a>HurewiczCofibrationsInCGWHSpacesAreClosed – then there are a number of equivalent reformulations of the defining homotopy extension property (Def. <a class="maruku-ref" href="#HurewiczCofibration"></a>).</p> <h4 id="via_retracts_of_the_pushoutproduct_with_">Via retracts of the pushout-product with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>0</mn><mo>↪</mo><mo stretchy="false">[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">0 \hookrightarrow [0,1]</annotation></semantics></math></h4> <p> <div class='num_prop' id='CharacterizationViaRetractionOfPushoutProduct'> <h6>Proposition</h6> <p>A <a class="existingWikiWord" href="/nlab/show/closed+subset">closed</a> <a class="existingWikiWord" href="/nlab/show/topological+subspace">topological subspace</a>-inclusion <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mover><mo>↪</mo><mrow><mspace width="thickmathspace"></mspace><mi>i</mi><mspace width="thickmathspace"></mspace></mrow></mover><mi>X</mi></mrow><annotation encoding="application/x-tex">A \xhookrightarrow{\;i\;} X</annotation></semantics></math> is a Hurewicz cofibration precisely if its <a class="existingWikiWord" href="/nlab/show/pushout+product">pushout product</a> with the endpoint inclusion <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>0</mn><mspace width="thinmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thinmathspace"></mspace><mo>*</mo><mover><mo>↪</mo><mspace width="thickmathspace"></mspace></mover><mo stretchy="false">[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">0 \,\colon\, \ast \xhookrightarrow{\;} [0,1]</annotation></semantics></math> into the <a class="existingWikiWord" href="/nlab/show/topological+interval">topological interval</a></p> <div class="maruku-equation" id="eq:PushoutProductMap"><span class="maruku-eq-number">(1)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>X</mi><mspace width="negativethinmathspace"></mspace><mo>×</mo><mspace width="negativethinmathspace"></mspace><mo stretchy="false">{</mo><mn>0</mn><mo stretchy="false">}</mo><mspace width="thinmathspace"></mspace><mo>∪</mo><mspace width="thinmathspace"></mspace><mi>A</mi><mspace width="negativethinmathspace"></mspace><mo>×</mo><mspace width="negativethinmathspace"></mspace><mo stretchy="false">[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy="false">]</mo><mspace width="thickmathspace"></mspace><mover><mo>→</mo><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></mrow></mover><mspace width="thickmathspace"></mspace><mi>X</mi><mo>×</mo><mo stretchy="false">[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex"> X \!\times\! \{0\} \,\cup\, A \!\times\! 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stroke-linecap="round" stroke-linejoin="round" stroke="rgb(0%, 0%, 0%)" stroke-opacity="1" stroke-miterlimit="10" d="M -2.484694 2.870172 C -2.031569 1.147516 -1.01985 0.335016 -0.00031875 -0.00092125 C -1.01985 -0.332953 -2.031569 -1.149359 -2.484694 -2.868109 " transform="matrix(0, -1, -1, 0, 398.79986, 26.8864)"></path> <g fill="rgb(0%, 0%, 0%)" fill-opacity="1"> <use xlink:href="#e6x9XnqiJRyn2a1HvH5LrwvRUnA=-glyph-6-1" x="229.596" y="48.054"></use> <use xlink:href="#e6x9XnqiJRyn2a1HvH5LrwvRUnA=-glyph-6-2" x="232.441338" y="48.054"></use> </g> <path fill="none" stroke-width="0.47818" stroke-linecap="butt" stroke-linejoin="miter" stroke="rgb(0%, 0%, 0%)" stroke-opacity="1" stroke-dasharray="3.34735 1.91277" stroke-miterlimit="10" d="M 40.066688 -0.00003125 L 95.281531 -0.00003125 " transform="matrix(1, 0, 0, -1, 233.863, 13.457)"></path> <path fill="none" stroke-width="0.47818" stroke-linecap="round" stroke-linejoin="round" stroke="rgb(0%, 0%, 0%)" stroke-opacity="1" stroke-miterlimit="10" d="M -2.487669 2.871063 C -2.030638 1.148406 -1.018919 0.335906 0.0006125 -0.00003125 C -1.018919 -0.335969 -2.030638 -1.148469 -2.487669 -2.867219 " transform="matrix(1, 0, 0, -1, 329.3822, 13.457)"></path> <g fill="rgb(0%, 0%, 0%)" fill-opacity="1"> <use xlink:href="#e6x9XnqiJRyn2a1HvH5LrwvRUnA=-glyph-3-2" x="299.24" y="9.941"></use> </g> </svg> <p></p> </div> (<a href="#Strom68">Strøm 1968 Thm. 2</a>; review in <a href="#May99">May 1999, Sec. 6.4, p. 45 (53 of 251)</a>; <a href="#AGP02">AGP 2002, Thm. 4.1.7</a>; <a href="#Gutierrez">Gutiérrez, Prop. 8.3</a>)</p> <p> <div class='num_remark' id='KifiedProductsOfCofibrationsWithCompactlyGeneratedSpaces'> <h6>Example</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mover><mo>↪</mo><mi>i</mi></mover><mi>X</mi></mrow><annotation encoding="application/x-tex">A \xhookrightarrow{i} X</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/closed+subset">closed</a> <a class="existingWikiWord" href="/nlab/show/topological+subspace">topological subspace</a>-inclusion of <a class="existingWikiWord" href="/nlab/show/compactly+generated+topological+spaces">compactly generated topological spaces</a> which is a Hurewicz cofibration. Then for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math> any <a class="existingWikiWord" href="/nlab/show/compactly+generated+topological+space">compactly generated topological space</a>, the <a class="existingWikiWord" href="/nlab/show/k-ification">k-ified</a> <a class="existingWikiWord" href="/nlab/show/product+space">product space</a>-construction <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi><mo>×</mo><mi>A</mi><mover><mo>↪</mo><mrow><msub><mi>id</mi> <mi>Y</mi></msub><mo>×</mo><mi>i</mi></mrow></mover><mi>Y</mi><mo>×</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">Y \times A \xhookrightarrow{ id_Y \times i } Y \times X</annotation></semantics></math> is itself a Hurewicz cofibration.</p> </div> (e.g. <a href="#AGP02">AGP 2002, Ex. 4.2.16</a>) <div class='proof'> <h6>Proof</h6> <p>Since <a class="existingWikiWord" href="/nlab/show/compactly+generated+spaces">compactly generated spaces</a> with the k-ified product space construction form a <a class="existingWikiWord" href="/nlab/show/cartesian+closed+category">cartesian closed category</a>, the operation <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi><mo>×</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Y \times (-)</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/left+adjoint">left adjoint</a> and <a class="existingWikiWord" href="/nlab/show/left+adjoints+preserve+colimits">hence</a> <a class="existingWikiWord" href="/nlab/show/preserved+colimit">preserves</a> the <a class="existingWikiWord" href="/nlab/show/pushout">pushout</a> in <a class="maruku-eqref" href="#eq:PushoutProductMap">(1)</a>. It follows that the retract which exhibits, according to Prop. <a class="maruku-ref" href="#CharacterizationViaRetractionOfPushoutProduct"></a>, the cofibration property of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math>, may be extended as a <a class="existingWikiWord" href="/nlab/show/constant+function">constant function</a> 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> to yield a retract that exhibits the cofibration property of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi><mo>×</mo><mi>i</mi></mrow><annotation encoding="application/x-tex">Y \times i</annotation></semantics></math>.</p> </div> </p> <h4 id="via_neighbourhood_deformations">Via neighbourhood deformations</h4> <p> <div class='num_prop' id='CharactrerizationViaNeighbourhoodDeformation'> <h6>Proposition</h6> <p>A <a class="existingWikiWord" href="/nlab/show/closed+subset">closed</a> <a class="existingWikiWord" href="/nlab/show/topological+subspace">topological subspace</a>-inclusion <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mover><mo>↪</mo><mrow><mspace width="thickmathspace"></mspace><mi>i</mi><mspace width="thickmathspace"></mspace></mrow></mover><mi>X</mi></mrow><annotation encoding="application/x-tex">A \xhookrightarrow{\;i\;} X</annotation></semantics></math> is a Hurewicz cofibration precisely if the following condition holds:</p> <p>There exists:</p> <p>(1) a <a class="existingWikiWord" href="/nlab/show/neighbourhood">neighbourhood</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi></mrow><annotation encoding="application/x-tex">U</annotation></semantics></math> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>A</mi><mover><mo>↪</mo><mspace width="thickmathspace"></mspace></mover><mi>U</mi><mover><mo>↪</mo><mspace width="thickmathspace"></mspace></mover><mi>X</mi></mrow><annotation encoding="application/x-tex"> A \xhookrightarrow{\;} U \xhookrightarrow{\;} X </annotation></semantics></math></div> <p>(2) a <a class="existingWikiWord" 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stroke-linejoin="round" stroke="rgb(0%, 0%, 0%)" stroke-opacity="1" stroke-miterlimit="10" d="M -2.488306 2.868126 C -2.031275 1.149376 -1.019556 0.33297 -0.000025 0.00093875 C -1.019556 -0.334999 -2.031275 -1.147499 -2.488306 -2.870155 " transform="matrix(0, -1, -1, 0, 165.02047, 73.0156)"></path> </svg> <p>(3) another <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>ϕ</mi><mo lspace="verythinmathspace">:</mo><mi>X</mi><mover><mo>→</mo><mspace width="thickmathspace"></mspace></mover><mo stretchy="false">[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">\phi \colon X \xrightarrow{\;} [0,1]</annotation></semantics></math> which makes the following <a class="existingWikiWord" href="/nlab/show/commuting+diagram">diagram commute</a>:</p> <svg xmlns="http://www.w3.org/2000/svg" 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-1, 0, 114.50279, 52.38535)"></path> </svg> <p>and in fact such that only <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 the <a class="existingWikiWord" href="/nlab/show/preimage">preimage</a> of zero: <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mspace width="thinmathspace"></mspace><mo>=</mo><mspace width="thinmathspace"></mspace><msup><mi>ϕ</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mo stretchy="false">{</mo><mn>0</mn><mo stretchy="false">}</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">A \,=\, \phi^{-1}(\{0\})</annotation></semantics></math>.</p> </div> </p> <p>(This is due to <a href="#Strom66">Strøm 1966, Thm. 2</a>; recalled, e.g., in <a href="#Bredon93">Bredon 1993, Thm. 1.5 on p. 431</a>)</p> <h3 id="AssortedFacts">Assorted facts</h3> <p> <div class='num_prop' id='hCofibrationIntoNormalSpaceIffSoIntoAnyOpenNeighbourhood'> <h6>Proposition</h6> <p>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> is a <a class="existingWikiWord" href="/nlab/show/normal+topological+space">normal topological space</a> then any inclusion <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mover><mo>↪</mo><mspace width="thickmathspace"></mspace></mover><mi>X</mi></mrow><annotation encoding="application/x-tex">A \xhookrightarrow{\;} X</annotation></semantics></math> is an h-cofibration iff its factorization <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>↪</mo><mspace width="thickmathspace"></mspace><msub><mi>U</mi> <mi>A</mi></msub></mrow><annotation encoding="application/x-tex">A \hookrightarrow{\;} U_A</annotation></semantics></math> through any <a class="existingWikiWord" href="/nlab/show/open+neighbourhood">open neighbourhood</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>U</mi> <mi>A</mi></msub><mo>⊂</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">U_A \subset X</annotation></semantics></math> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> is an h-cofibration.</p> </div> (<a href="#AGP02">AGP 2002, Prop. 4.1.9</a>)</p> <p> <div class='num_prop'> <h6>Proposition</h6> <p>The <a class="existingWikiWord" href="/nlab/show/composition">composition</a> of two h-cofibrations is itself an h-cofibration.</p> </div> (e.g. <a href="#AGP02">AGP 2002, Ex. 4.2.17</a>)</p> <h3 id="InteractionWithFiberProducts">Interaction with (fiber) producs</h3> <p> <div class='num_prop' id='CrossedUnionOfhCofsIshCof'> <h6>Proposition</h6> <p>If</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><msub><mi>A</mi> <mn>1</mn></msub></mtd> <mtd></mtd> <mtd><msub><mi>A</mi> <mn>2</mn></msub></mtd></mtr> <mtr><mtd><mo maxsize="1.2em" minsize="1.2em">↓</mo><msup><mrow></mrow> <mpadded width="0"><mrow><msub><mi>i</mi> <mn>1</mn></msub></mrow></mpadded></msup></mtd> <mtd><mo>,</mo></mtd> <mtd><mo maxsize="1.2em" minsize="1.2em">↓</mo><msup><mrow></mrow> <mpadded width="0"><mrow><msub><mi>i</mi> <mn>2</mn></msub></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><msub><mi>X</mi> <mn>1</mn></msub></mtd> <mtd></mtd> <mtd><msub><mi>X</mi> <mn>2</mn></msub></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ A_1 & & A_2 \\ \big\downarrow {}^{\mathrlap{i_1}} & , & \big\downarrow {}^{\mathrlap{i_2}} \\ X_1 & & X_2 } </annotation></semantics></math></div> <p>is a <a class="existingWikiWord" href="/nlab/show/pair">pair</a> of Hurewicz cofibrations, with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>i</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><msub><mi>A</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><munder><mo>⊂</mo><mi>clsd</mi></munder><msub><mi>X</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">i_1(A_1) \underset{clsd}{\subset} X_1</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/closed+subspace">closed</a>, then the inclusion</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><msub><mi>A</mi> <mn>1</mn></msub><mo>×</mo><msub><mi>X</mi> <mn>2</mn></msub><mspace width="thickmathspace"></mspace><mo>∪</mo><mspace width="thickmathspace"></mspace><msub><mi>X</mi> <mn>1</mn></msub><mo>×</mo><msub><mi>A</mi> <mn>2</mn></msub></mtd></mtr> <mtr><mtd><mo maxsize="1.2em" minsize="1.2em">↓</mo></mtd></mtr> <mtr><mtd><msub><mi>X</mi> <mn>1</mn></msub><mo>×</mo><msub><mi>X</mi> <mn>2</mn></msub></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ A_1 \times X_2 \;\cup\; X_1 \times A_2 \\ \big\downarrow \\ X_1 \times X_2 } </annotation></semantics></math></div> <p>into the <a class="existingWikiWord" href="/nlab/show/product+space">product space</a> of the <a class="existingWikiWord" href="/nlab/show/codomains">codomains</a> is itself a Hurewicz cofibration.</p> <p></p> </div> (<a href="#Strom68">Strøm 1968, Thm. 6</a>)</p> <p> <div class='num_prop' id='FiberProductPreserveshCofibrations'> <h6>Proposition</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/fiber+product">fiber product</a> of <a class="existingWikiWord" href="/nlab/show/Hurewicz+fibrations">Hurewicz fibrations</a> preserves <a class="existingWikiWord" href="/nlab/show/Hurewicz+cofibrations">Hurewicz cofibrations</a>)</strong> <br /> Let</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><msub><mi>X</mi> <mn>0</mn></msub></mtd> <mtd><mo>↪</mo></mtd> <mtd><mi>X</mi></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><msub><mi>p</mi> <mn>0</mn></msub></mrow></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>p</mi></mpadded></msup></mtd></mtr> <mtr><mtd><msub><mi>B</mi> <mn>0</mn></msub></mtd> <mtd><mo>↪</mo></mtd> <mtd><mi>B</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">↑</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↑</mo></mtd></mtr> <mtr><mtd><msub><mi>E</mi> <mn>0</mn></msub></mtd> <mtd><mo>↪</mo></mtd> <mtd><mi>E</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ X_0 &\hookrightarrow & X \\ {}^{\mathllap{p_0}}\downarrow && \downarrow^{\mathrlap{p}} \\ B_0 &\hookrightarrow& B \\ \uparrow && \uparrow \\ E_0 &\hookrightarrow& E } </annotation></semantics></math></div> <p>be a <a class="existingWikiWord" href="/nlab/show/commuting+diagram">commuting diagram</a> of <a class="existingWikiWord" href="/nlab/show/topological+spaces">topological spaces</a> such that</p> <ul> <li> <p>the horizontal morphisms are closed cofibrations;</p> </li> <li> <p>the morphisms <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>p</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">p_0</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math> are <a class="existingWikiWord" href="/nlab/show/Hurewicz+fibrations">Hurewicz fibrations</a>.</p> </li> </ul> <p>Then the induced morphism on <a class="existingWikiWord" href="/nlab/show/fiber+products">fiber products</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mn>0</mn></msub><msub><mo>×</mo> <mrow><msub><mi>B</mi> <mn>0</mn></msub></mrow></msub><msub><mi>E</mi> <mn>0</mn></msub><mo>↪</mo><mi>X</mi><msub><mo>×</mo> <mi>B</mi></msub><mi>E</mi></mrow><annotation encoding="application/x-tex"> X_0 \times_{B_0} E_0 \hookrightarrow X \times_B E </annotation></semantics></math></div> <p>is also a closed cofibration.</p> </div> (<a href="#Kieboom87">Kieboom 1987, Thm. 1</a>).</p> <p> <div class='num_cor'> <h6>Corollary</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/Cartesian+product">Cartesian product</a> preserves <a class="existingWikiWord" href="/nlab/show/Hurewicz+cofibrations">Hurewicz cofibrations</a>)</strong> <br /> If</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><msub><mi>A</mi> <mn>1</mn></msub></mtd> <mtd></mtd> <mtd><msub><mi>A</mi> <mn>2</mn></msub></mtd></mtr> <mtr><mtd><mo maxsize="1.2em" minsize="1.2em">↓</mo><msup><mrow></mrow> <mpadded width="0"><mrow><msub><mi>i</mi> <mn>1</mn></msub></mrow></mpadded></msup></mtd> <mtd><mo>,</mo></mtd> <mtd><mo maxsize="1.2em" minsize="1.2em">↓</mo><msup><mrow></mrow> <mpadded width="0"><mrow><msub><mi>i</mi> <mn>2</mn></msub></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><msub><mi>X</mi> <mn>1</mn></msub></mtd> <mtd></mtd> <mtd><msub><mi>X</mi> <mn>2</mn></msub></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ A_1 & & A_2 \\ \big\downarrow {}^{\mathrlap{i_1}} & , & \big\downarrow {}^{\mathrlap{i_2}} \\ X_1 & & X_2 } </annotation></semantics></math></div> <p>is a <a class="existingWikiWord" href="/nlab/show/pair">pair</a> of Hurewicz cofibrations, then their image under the <a class="existingWikiWord" href="/nlab/show/product">product</a> functor is, too:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><msub><mi>A</mi> <mn>1</mn></msub><mo>×</mo><msub><mi>A</mi> <mn>2</mn></msub></mtd></mtr> <mtr><mtd><mo maxsize="1.2em" minsize="1.2em">↓</mo></mtd></mtr> <mtr><mtd><msub><mi>X</mi> <mn>1</mn></msub><mo>×</mo><msub><mi>X</mi> <mn>2</mn></msub></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ A_1 \times A_2 \\ \big\downarrow \\ X_1 \times X_2 } </annotation></semantics></math></div> <p></p> </div> <div class='proof'> <h6>Proof</h6> <p>This is the special case of Prop. <a class="maruku-ref" href="#FiberProductPreserveshCofibrations"></a> for <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> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>B</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">B_0</annotation></semantics></math> being the <a class="existingWikiWord" href="/nlab/show/point+space">point space</a></p> </div> </p> <h3 id="StromModelStructure">Strøm’s model structure</h3> <p>The collections</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Hurewicz+fibrations">Hurewicz fibrations</a>,</p> </li> <li> <p>closed Hurewicz cofibrations (def. <a class="maruku-ref" href="#HurewiczCofibration"></a>, def. <a class="maruku-ref" href="#ClosedHurewiczCofibration"></a>),</p> </li> <li> <p>and <a class="existingWikiWord" href="/nlab/show/homotopy+equivalences">homotopy equivalences</a></p> </li> </ul> <p>make one of the standard Quillen <a class="existingWikiWord" href="/nlab/show/model+category">model category</a> structures on the category <a class="existingWikiWord" href="/nlab/show/Top">Top</a> of all topological spaces <em><a class="existingWikiWord" href="/nlab/show/Str%C3%B8m%27s+model+category">Strøm's model category</a>.</em></p> <p>The identity functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>id</mi><mo lspace="verythinmathspace">:</mo><mi>Top</mi><mo>→</mo><mi>Top</mi></mrow><annotation encoding="application/x-tex">id \colon Top \to Top</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/Quillen+adjunction">left Quillen</a> from the <a class="existingWikiWord" href="/nlab/show/classical+model+structure+on+topological+spaces">classical model structure on topological spaces</a> (or the mixed model structure) to the Strøm model structure, and of course right Quillen in the other direction.</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>Top</mi> <mi>Strom</mi></msub><mover><munder><mo>⟶</mo><mi>id</mi></munder><mover><mo>⟵</mo><mi>id</mi></mover></mover><msub><mi>Top</mi> <mi>Quillen</mi></msub><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> Top_{Strom} \stackrel{\overset{id}{\longleftarrow}}{\underset{id}{\longrightarrow}} Top_{Quillen} \,. </annotation></semantics></math></div> <p>This means in particular that any <a class="existingWikiWord" href="/nlab/show/retract">retract</a> of a <a class="existingWikiWord" href="/nlab/show/relative+cell+complex">relative cell complex</a> inclusion is a closed Hurewicz cofibration.</p> <h2 id="Examples">Examples</h2> <h3 id="relative_cell_complex_inclusions">Relative cell complex inclusions</h3> <p> <div class='num_remark' id='RelativeCWComplexes'> <h6>Example</h6> <p>Any <a class="existingWikiWord" href="/nlab/show/relative+CW+complex">relative CW complex</a>-inclusion is an h-cofibration.</p> </div> Proofs may be found spelled out in: <a href="#Bredon93">Bredon 1993, Cor. I.4 on p. 431</a>, <a href="rational+homotopy+theopry#FelixHalperinThomas00">Félix, Halperin & Thomas 2000, Prop. 1.9</a>, <a href="#Mathew10b">Mathew 2010b</a></p> <p>More generally, every <a class="existingWikiWord" href="/nlab/show/retract">retract</a> of a <a class="existingWikiWord" href="/nlab/show/relative+cell+complex">relative cell complex</a> inclusion is a closed Hurewicz cofibration.</p> <p>This is part of the statement of the <a class="existingWikiWord" href="/nlab/show/Quillen+adjunction">Quillen adjunction</a> between then <a class="existingWikiWord" href="/nlab/show/classical+model+structure+on+topological+spaces">classical model structure on topological spaces</a> and the <a class="existingWikiWord" href="/nlab/show/Str%C3%B8m+model+structure">Strøm model structure</a> (see <a href="#StromModelStructure">below</a>).</p> <h3 id="BasepointsInLocallyEuclideanSpaces">Points in locally Euclidean Hausdorff spaces</h3> <p> <div class='num_prop'> <h6>Proposition</h6> <p>Any <a class="existingWikiWord" href="/nlab/show/point">point</a>-inclusion into a (<a class="existingWikiWord" href="/nlab/show/finite+number">finite</a>-<a class="existingWikiWord" href="/nlab/show/dimension+of+a+manifold">dimensional</a>) <a class="existingWikiWord" href="/nlab/show/locally+Euclidean+space">locally Euclidean</a> <a class="existingWikiWord" href="/nlab/show/Hausdorff+space">Hausdorff space</a> (e.g. a <a class="existingWikiWord" href="/nlab/show/topological+manifold">topological manifold</a>) is an h-cofibration.</p> </div> This is a simple special case of the general Prop. <a class="maruku-ref" href="#ClosedSubmanifoldsOfParacompactBanachManifoldsArehCof"></a> below, but we give an explicit proof: <div class='proof' id='ProofThatPointInclusionsInLocallyEuclideanHausdorffSpacesArehCofibs'> <h6>Proof</h6> <p>By definition of <a class="existingWikiWord" href="/nlab/show/locally+Euclidean+spaces">locally Euclidean spaces</a>, any point has a <a class="existingWikiWord" href="/nlab/show/neighbourhood">neighbourhood</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi></mrow><annotation encoding="application/x-tex">U</annotation></semantics></math> which is <a class="existingWikiWord" href="/nlab/show/chart">chart</a>, being a <a class="existingWikiWord" href="/nlab/show/Euclidean+space">Euclidean space</a> that may be identified with a <a class="existingWikiWord" href="/nlab/show/vector+space">vector space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^n</annotation></semantics></math> with the given point being the origin <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>0</mn><mspace width="thinmathspace"></mspace><mo>∈</mo><mspace width="thinmathspace"></mspace><msup><mi>ℝ</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">0 \,\in\, \mathbb{R}^n</annotation></semantics></math>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><mn>0</mn><mo stretchy="false">}</mo><mspace width="thickmathspace"></mspace><mo>∈</mo><mspace width="thickmathspace"></mspace><msup><mi>ℝ</mi> <mi>n</mi></msup><mspace width="thickmathspace"></mspace><mo>≃</mo><mspace width="thickmathspace"></mspace><mi>U</mi><mspace width="thickmathspace"></mspace><mo>⊂</mo><mspace width="thickmathspace"></mspace><mi>X</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \{0\} \;\in\; \mathbb{R}^n \;\simeq\; U \;\subset\; X \,. </annotation></semantics></math></div> <p>Now let:</p> <ul> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>η</mi><mo lspace="verythinmathspace">:</mo><mi>U</mi><mo>×</mo><mo stretchy="false">[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy="false">]</mo><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">\eta \colon U \times [0,1]\to X</annotation></semantics></math> the homotopy given by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mover><mi>x</mi><mo stretchy="false">→</mo></mover><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo><mo>↦</mo><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><mi>t</mi><mo stretchy="false">)</mo><mo>⋅</mo><mover><mi>x</mi><mo stretchy="false">→</mo></mover></mrow><annotation encoding="application/x-tex">(\vec x, t) \mapsto (1-t)\cdot \vec x</annotation></semantics></math>;</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi><mspace width="thickmathspace"></mspace><mo>≔</mo><mspace width="thickmathspace"></mspace><msub><mi>B</mi> <mrow><mo>≤</mo><mn>1</mn></mrow></msub><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">K \;\coloneqq\; B_{\leq 1}(0)</annotation></semantics></math> the <a class="existingWikiWord" href="/nlab/show/closed+ball">closed ball</a> of unit <a class="existingWikiWord" href="/nlab/show/radius">radius</a> around the origin in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^n</annotation></semantics></math> (hence <a class="existingWikiWord" href="/nlab/show/compact+space">compact</a> by <a class="existingWikiWord" href="/nlab/show/Heine-Borel+theorem">Heine-Borel</a>);</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi><mo lspace="verythinmathspace">:</mo><mi>X</mi><mo>→</mo><mo stretchy="false">[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">\phi \colon X \to[0,1]</annotation></semantics></math> be given by:</p> <ol> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>↦</mo><mi>min</mi><mo maxsize="1.2em" minsize="1.2em">(</mo><mo stretchy="false">‖</mo><mi>x</mi><mo stretchy="false">‖</mo><mo>,</mo><mn>1</mn><mo maxsize="1.2em" minsize="1.2em">)</mo></mrow><annotation encoding="application/x-tex">x \mapsto min\big( \| x\|, 1 \big)</annotation></semantics></math> on <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> (the <a class="existingWikiWord" href="/nlab/show/distance">distance</a> from the origin cut off at 1),</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>↦</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">x \mapsto 1</annotation></semantics></math> on the <a class="existingWikiWord" href="/nlab/show/complement">complement</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>∖</mo><mi>K</mi></mrow><annotation encoding="application/x-tex">X \setminus K</annotation></semantics></math>.</p> </li> </ol> <p>It is manifest that this is a well-defined function and that the <a class="existingWikiWord" href="/nlab/show/restrictions">restrictions</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi><msub><mo stretchy="false">|</mo> <mi>U</mi></msub></mrow><annotation encoding="application/x-tex">\phi|_{U}</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi><msub><mo stretchy="false">|</mo> <mrow><mi>X</mi><mo>∖</mo><mi>K</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\phi|_{X\setminus K}</annotation></semantics></math> are <a class="existingWikiWord" href="/nlab/show/continuous+functions">continuous functions</a>. Moreover, since <a class="existingWikiWord" href="/nlab/show/compact+subspaces+of+Hausdorff+spaces+are+closed">compact subspaces of Hausdorff spaces are closed</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>∖</mo><mi>K</mi></mrow><annotation encoding="application/x-tex">X \setminus K</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/open+subset">open</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo maxsize="1.2em" minsize="1.2em">{</mo><mi>U</mi><mo>,</mo><mspace width="thinmathspace"></mspace><mi>X</mi><mo>∖</mo><mi>K</mi><mo maxsize="1.2em" minsize="1.2em">}</mo></mrow><annotation encoding="application/x-tex">\big\{ U ,\, X \setminus K \big\}</annotation></semantics></math> is an <a class="existingWikiWord" href="/nlab/show/open+cover">open cover</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>. Therefore (by the <a class="existingWikiWord" href="/nlab/show/sheaf">sheaf</a>-property of continuous functions), <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi></mrow><annotation encoding="application/x-tex">\phi</annotation></semantics></math> is continuous on all 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>.</p> </li> </ul> <p>It is immediate to see that this data satisfies the conditions discussed in Prop. <a class="maruku-ref" href="#CharactrerizationViaNeighbourhoodDeformation"></a>. Since <a class="existingWikiWord" href="/nlab/show/Hausdorff+spaces">Hausdorff spaces</a> are <a class="existingWikiWord" href="/nlab/show/T1"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"> <semantics> <mrow> <msub><mi>T</mi> <mn>1</mn></msub> </mrow> <annotation encoding="application/x-tex">T_1</annotation> </semantics> </math></a>, so that all of their points are <a class="existingWikiWord" href="/nlab/show/closed+point">closed</a>, that proposition applies and implies the claim.</p> </div> </p> <h3 id="ClosedSubmanifoldsOfBanachManifolds">Closed submanifolds of Banach manifolds</h3> <p> <div class='num_prop' id='ClosedInclusionIntoANRIsANRIffhCofibration'> <h6>Proposition</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> be an <a class="existingWikiWord" href="/nlab/show/absolute+neighbourhood+retract">absolute neighbourhood retract</a> (ANR) and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mover><mo>↪</mo><mi>i</mi></mover><mi>X</mi></mrow><annotation encoding="application/x-tex">A \xhookrightarrow{i} X</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/closed+subspace">closed subspace</a>-inclusion. Then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/Hurewicz+cofibration">Hurewicz cofibration</a> iff <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 itself an ANR.</p> </div> (<a href="#AGP02">Aguilar, Gitler & Prieto 2002, Thm. 4.2.15</a>)</p> <p> <div class='num_prop' id='ClosedSubmanifoldsOfParacompactBanachManifoldsArehCof'> <h6>Proposition</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/paracompact+topological+space">paracompact</a> <a class="existingWikiWord" href="/nlab/show/Banach+manifold">Banach manifold</a>. Then the inclusion <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>↪</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">A \hookrightarrow X</annotation></semantics></math> of any <a class="existingWikiWord" href="/nlab/show/closed+subspace">closed</a> sub-<a class="existingWikiWord" href="/nlab/show/Banach+manifold">Banach manifold</a> is a <a class="existingWikiWord" href="/nlab/show/Hurewicz+cofibration">Hurewicz cofibration</a>.</p> </div> <div class='proof'> <h6>Proof</h6> <p>Being a closed subspace of a paracompact space, <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 itself paracompact (by <a href="paracompact+topological+space#ClosedSubspacesOfParacompactsAreParacompact">this Prop.</a>). But paracompact Banach manifolds are <a class="existingWikiWord" href="/nlab/show/absolute+neighbourhood+retracts">absolute neighbourhood retracts</a> (<a href="absolute+retract#ParacompactBanachManifoldsAreANRs">this Prop.</a>) Therefore the statement follows with Prop. <a class="maruku-ref" href="#ClosedInclusionIntoANRIsANRIffhCofibration"></a>.</p> </div> </p> <h3 id="further_examples">Further examples</h3> <p> <div class='num_remark' id='PointInclusionIntoPUH'> <h6>Example</h6> <p><strong>(point inclusion into <a class="existingWikiWord" href="/nlab/show/PU%28%E2%84%8B%29">PU(ℋ)</a>)</strong> <br /> The <a class="existingWikiWord" href="/nlab/show/projective+unitary+group">projective unitary group</a> <a class="existingWikiWord" href="/nlab/show/PU%28%E2%84%8B%29">PU(ℋ)</a> on an infinite-dimensional <a class="existingWikiWord" href="/nlab/show/separable+Hilbert+space">separable</a> <a class="existingWikiWord" href="/nlab/show/Hilbert+space">Hilbert space</a> is</p> <ul> <li> <p>a <a class="existingWikiWord" href="/nlab/show/Banach+Lie+group">Banach Lie group</a> in its <a class="existingWikiWord" href="/nlab/show/norm+topology">norm topology</a>, and as such <a class="existingWikiWord" href="/nlab/show/well-pointed+topological+group">well-pointed</a> by Prop. <a class="maruku-ref" href="#ClosedSubmanifoldsOfParacompactBanachManifoldsArehCof"></a>;</p> </li> <li> <p>no longer a Banach space in its weak/strong <a class="existingWikiWord" href="/nlab/show/operator+topology">operator topology</a>, but nevertheless still <a class="existingWikiWord" href="/nlab/show/well-pointed+topological+group">well-pointed</a> in this case, by <a href="projective+unitary+group+on+a+Hilbert+space#WellPointedInOperatorTopology">this Prop.</a>.</p> </li> </ul> <p></p> </div> </p> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Hurewicz+fibration">Hurewicz fibration</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Str%C3%B8m+model+structure">Strøm model structure</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Serre+cofibration">Serre cofibration</a></p> </li> </ul> <h2 id="references">References</h2> <p>Named after:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Witold+Hurewicz">Witold Hurewicz</a>, <em>On the concept of fiber space</em>, Proc. Nat. Acad. Sci. USA <strong>41</strong> (1955) 956–961; (<a href="https://dx.doi.org/10.1073%2Fpnas.41.11.956">doi:10.1073%2Fpnas.41.11.956</a>, <a href="https://www.jstor.org/stable/89187">jstor:89187</a>, <a href="http://www.pnas.org/content/41/11/956.full.pdf">pdf</a>. MR0073987)</li> </ul> <p>Original articles:</p> <ul> <li id="Strom66"> <p><a class="existingWikiWord" href="/nlab/show/Arne+Str%C3%B8m">Arne Strøm</a>, <em>Note on cofibrations</em>, Math. Scand. <strong>19</strong> (1966) 11-14 (<a href="https://www.jstor.org/stable/24490229">jstor:24490229</a>, <a href="https://eudml.org/doc/165952">dml:165952</a>, MR0211403)</p> </li> <li id="Strom68"> <p><a class="existingWikiWord" href="/nlab/show/Arne+Str%C3%B8m">Arne Strøm</a>, <em>Note on cofibrations II</em>, Math. Scand. <strong>22</strong> (1968) 130–142 (<a href="https://www.jstor.org/stable/24489730">jstor:24489730</a>, <a href="https://eudml.org/doc/166037">dml:166037</a>, MR0243525)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Dieter+Puppe">Dieter Puppe</a>, <em>Bemerkungen über die Erweiterung von Homotopien</em>, Arch. Math. (Basel) 18 1967 81–88 (<a href="http://dx.doi.org/10.1007/BF01899475">doi:10.1007/BF01899475</a>, MR0206954)</p> </li> </ul> <p>Textbook accounts:</p> <ul> <li id="tDKP70"> <p><a class="existingWikiWord" href="/nlab/show/Tammo+tom+Dieck">Tammo tom Dieck</a>, <a class="existingWikiWord" href="/nlab/show/Klaus+Heiner+Kamps">Klaus Heiner Kamps</a>, <a class="existingWikiWord" href="/nlab/show/Dieter+Puppe">Dieter Puppe</a>, Chapter I of: <em>Homotopietheorie</em>, Lecture Notes in Mathematics <strong>157</strong> Springer 1970 (<a href="https://link.springer.com/book/10.1007/BFb0059721">doi:10.1007/BFb0059721</a>)</p> </li> <li id="Bredon93"> <p><a class="existingWikiWord" href="/nlab/show/Glen+Bredon">Glen Bredon</a>, Section VII.1 of: <em>Topology and Geometry</em>, Graduate texts in mathematics <strong>139</strong>, Springer 1993 (<a href="https://link.springer.com/book/10.1007/978-1-4757-6848-0">doi:10.1007/978-1-4757-6848-0</a>, <a href="http://virtualmath1.stanford.edu/~ralph/math215b/Bredon.pdf">pdf</a>)</p> </li> <li id="May99"> <p><a class="existingWikiWord" href="/nlab/show/Peter+May">Peter May</a>, Chapter 6 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="AGP02"> <p>Marcelo Aguilar, <a class="existingWikiWord" href="/nlab/show/Samuel+Gitler">Samuel Gitler</a>, Carlos Prieto, Section 4.1 in: <em>Algebraic topology from a homotopical viewpoint</em>, Springer (2002) (<a href="https://link.springer.com/book/10.1007/b97586">doi:10.1007/b97586</a>, <a href="http://tocs.ulb.tu-darmstadt.de/106999419.pdf">toc pdf</a>)</p> </li> </ul> <p>Lecture notes:</p> <ul> <li id="Gutierrez"><a class="existingWikiWord" href="/nlab/show/Javier+Guti%C3%A9rrez">Javier Gutiérrez</a>, <em>Cofibrations</em>, (<a href="https://www.math.ru.nl/~gutierrez/files/Lecture08.pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/Gutierrez_Cofibrations.pdf" title="pdf">pdf</a>)</li> </ul> <p>Exposition:</p> <ul> <li id="Mathew10a"> <p><a class="existingWikiWord" href="/nlab/show/Akhil+Mathew">Akhil Mathew</a>, <em>Cofibrations</em>, 2010 (<a href="https://amathew.wordpress.com/2010/10/07/cofibrations/">web</a>)</p> </li> <li id="Mathew10b"> <p><a class="existingWikiWord" href="/nlab/show/Akhil+Mathew">Akhil Mathew</a>, <em>Examples of cofibrations</em>, 2010 (<a href="https://amathew.wordpress.com/2010/10/08/examples-of-cofibrations/">web</a>)</p> </li> </ul> <p>The fact that morphisms of fibrant pullback diagrams along closed cofibrations induce closed cofibrations is in</p> <ul> <li id="Kieboom87"><a class="existingWikiWord" href="/nlab/show/Rudger+Kieboom">Rudger Kieboom</a>, <em>A pullback theorem for cofibrations</em>, Manuscripta Math <strong>58</strong> (1987) pp 381–384 (<a href="https://doi.org/10.1007/BF01165895">doi:10.1007/BF01165895</a>)</li> </ul> <p>The terminology “h-cofibration” is due to:</p> <ul> <li id="MandellMaySchwedeShipley01"><a class="existingWikiWord" href="/nlab/show/Michael+Mandell">Michael Mandell</a>, <a class="existingWikiWord" href="/nlab/show/Peter+May">Peter May</a>, <a class="existingWikiWord" href="/nlab/show/Stefan+Schwede">Stefan Schwede</a>, <a class="existingWikiWord" href="/nlab/show/Brooke+Shipley">Brooke Shipley</a>, p. 16 in: <em><a class="existingWikiWord" href="/nlab/show/Model+categories+of+diagram+spectra">Model categories of diagram spectra</a></em>, Proceedings of the London Mathematical Society, 82 (2001), 441-512 (<a href="http://www.math.uchicago.edu/~may/PAPERS/mmssLMSDec30.pdf">pdf</a>, <a href="https://doi.org/10.1112/S0024611501012692">doi:10.1112/S0024611501012692</a>)</li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on May 31, 2022 at 17:54:42. 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