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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="topos_theory">Topos theory</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/topos+theory">topos theory</a></strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Toposes">Toposes</a></li> </ul> <h2 id="background">Background</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/category+theory">category theory</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/category">category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/functor">functor</a></p> </li> </ul> </li> </ul> <h2 id="toposes">Toposes</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%280%2C1%29-topos">(0,1)-topos</a>, <a class="existingWikiWord" href="/nlab/show/Heyting+algebra">Heyting algebra</a>, <a class="existingWikiWord" href="/nlab/show/locale">locale</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/pretopos">pretopos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topos">topos</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Grothendieck+topos">Grothendieck topos</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/category+of+presheaves">category of presheaves</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/presheaf">presheaf</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/representable+functor">representable presheaf</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/category+of+sheaves">category of sheaves</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/site">site</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/sieve">sieve</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/coverage">coverage</a>, <a class="existingWikiWord" href="/nlab/show/Grothendieck+pretopology">pretopology</a>, <a class="existingWikiWord" href="/nlab/show/Grothendieck+topology">topology</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/sheaf">sheaf</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/sheafification">sheafification</a></p> </li> </ul> </li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quasitopos">quasitopos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/base+topos">base topos</a>, <a class="existingWikiWord" href="/nlab/show/indexed+topos">indexed topos</a></p> </li> </ul> <h2 id="toc_internal_logic">Internal Logic</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/categorical+semantics">categorical semantics</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/internal+logic">internal logic</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/subobject+classifier">subobject classifier</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/natural+numbers+object">natural numbers object</a></p> </li> </ul> </li> </ul> <h2 id="topos_morphisms">Topos morphisms</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/logical+morphism">logical morphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/geometric+morphism">geometric morphism</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/direct+image">direct image</a>/<a class="existingWikiWord" href="/nlab/show/inverse+image">inverse image</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/global+section">global sections</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/geometric+embedding">geometric embedding</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/surjective+geometric+morphism">surjective geometric morphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/essential+geometric+morphism">essential geometric morphism</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/locally+connected+geometric+morphism">locally connected geometric morphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/connected+geometric+morphism">connected geometric morphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/totally+connected+geometric+morphism">totally connected geometric morphism</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%C3%A9tale+geometric+morphism">étale geometric morphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/open+geometric+morphism">open geometric morphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/proper+geometric+morphism">proper geometric morphism</a>, <a class="existingWikiWord" href="/nlab/show/compact+topos">compact topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/separated+geometric+morphism">separated geometric morphism</a>, <a class="existingWikiWord" href="/nlab/show/Hausdorff+topos">Hausdorff topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/local+geometric+morphism">local geometric morphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/bounded+geometric+morphism">bounded geometric morphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/base+change">base change</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/localic+geometric+morphism">localic geometric morphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/hyperconnected+geometric+morphism">hyperconnected geometric morphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/atomic+geometric+morphism">atomic geometric morphism</a></p> </li> </ul> </li> </ul> <h2 id="extra_stuff_structure_properties">Extra stuff, structure, properties</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+locale">topological locale</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/localic+topos">localic topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/petit+topos">petit topos/gros topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/locally+connected+topos">locally connected topos</a>, <a class="existingWikiWord" href="/nlab/show/connected+topos">connected topos</a>, <a class="existingWikiWord" href="/nlab/show/totally+connected+topos">totally connected topos</a>, <a class="existingWikiWord" href="/nlab/show/strongly+connected+topos">strongly connected topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/local+topos">local topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cohesive+topos">cohesive topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/classifying+topos">classifying topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/smooth+topos">smooth topos</a></p> </li> </ul> <h2 id="cohomology_and_homotopy">Cohomology and homotopy</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/cohomology">cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+groups+in+an+%28infinity%2C1%29-topos">homotopy</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/abelian+sheaf+cohomology">abelian sheaf cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+simplicial+presheaves">model structure on simplicial presheaves</a></p> </li> </ul> <h2 id="in_higher_category_theory">In higher category theory</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/higher+topos+theory">higher topos theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%280%2C1%29-topos">(0,1)-topos</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/%280%2C1%29-site">(0,1)-site</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/2-topos">2-topos</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/2-site">2-site</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/2-sheaf">2-sheaf</a>, <a class="existingWikiWord" href="/nlab/show/stack">stack</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos">(∞,1)-topos</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-site">(∞,1)-site</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-sheaf">(∞,1)-sheaf</a>, <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-stack">∞-stack</a>, <a class="existingWikiWord" href="/nlab/show/derived+stack">derived stack</a></p> </li> </ul> </li> </ul> <h2 id="theorems">Theorems</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Diaconescu%27s+theorem">Diaconescu's theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Barr%27s+theorem">Barr's theorem</a></p> </li> </ul> <div> <p> <a href="/nlab/edit/topos+theory+-+contents">Edit this sidebar</a> </p> </div></div></div> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#definition'>Definition</a></li> <li><a href='#Properties'>Properties</a></li> <ul> <li><a href='#ExistenceOnParacompactManifolds'>Existence on paracompact smooth manifolds</a></li> <li><a href='#coverages_of_good_open_covers'>Coverages of good open covers</a></li> <li><a href='#existence_on_cw_complexes'>Existence on CW complexes</a></li> <li><a href='#nonexistence_for_topological_manifolds'>(Non-)Existence for topological manifolds</a></li> <li><a href='#refining_covers'>Refining covers</a></li> </ul> <li><a href='#pov'><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>POV</a></li> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#References'>References</a></li> </ul> </div> <h2 id="definition">Definition</h2> <div class="num_defn" id="GoodOpenCover"> <h6 id="definition_2">Definition</h6> <p>A <a class="existingWikiWord" href="/nlab/show/cover">cover</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><msub><mi>U</mi> <mi>i</mi></msub><mo>→</mo><mi>X</mi><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{U_i \to X\}</annotation></semantics></math> of a <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> is called a <strong>good cover</strong> – or <strong>good open cover</strong> if it is</p> <ol> <li> <p>an <a class="existingWikiWord" href="/nlab/show/open+cover">open cover</a>;</p> </li> <li> <p>such that all the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>U</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">U_i</annotation></semantics></math> and all their inhabited finite intersections are <a class="existingWikiWord" href="/nlab/show/contractible+topological+spaces">contractible topological spaces</a>.</p> </li> </ol> </div> <div class="num_remark"> <h6 id="remark">Remark</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/topological+manifold">topological manifold</a> one often requires that the inhabited finite intersections are <a class="existingWikiWord" href="/nlab/show/homeomorphism">homeomorphic</a> to an <a class="existingWikiWord" href="/nlab/show/open+ball">open ball</a>. Similarly, for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/smooth+manifold">smooth manifold</a> one often requires that the finite inhabited intersections are <a class="existingWikiWord" href="/nlab/show/diffeomorphism">diffeomorphic</a> to an <a class="existingWikiWord" href="/nlab/show/open+ball">open ball</a>.</p> <p>In the literature this is traditionally glossed over, but this is in fact a subtle point, see the discussion at <a class="existingWikiWord" href="/nlab/show/open+ball">open ball</a> and see below at <em><a href="#ExistenceOnParacompactManifolds">Existence on paracompact smooth manifolds</a></em>.</p> </div> <p>Due to this subtlety, it is instructive to make explicit the following definition:</p> <div class="num_defn" id="DifferentiablyGoodOpenCover"> <h6 id="definition_3">Definition</h6> <p>Given a <a class="existingWikiWord" href="/nlab/show/smooth+manifold">smooth manifold</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> a <em>differentiably good open cover</em> (<a href="#FSS10">FSS 2010, Def. 6.3.9</a>) is a good open cover one all whose finite non-empty intersections are in fact <a class="existingWikiWord" href="/nlab/show/diffeomorphism">diffeomorphic</a> to an <a class="existingWikiWord" href="/nlab/show/open+ball">open ball</a>, hence to a <a class="existingWikiWord" href="/nlab/show/Cartesian+space">Cartesian space</a>.</p> </div> <h2 id="Properties">Properties</h2> <h3 id="ExistenceOnParacompactManifolds">Existence on paracompact smooth manifolds</h3> <p> <div class='num_remark' id='LiteratureOnExistenceOfDifferentiablyGoodOpenCovers'> <h6>Remark</h6> <p>The existence result (Prop. <a class="maruku-ref" href="#ExistenceOfDifferentiablyGoodOpenCovers"></a> below) of differentiably good open covers (Def. <a class="maruku-ref" href="#DifferentiablyGoodOpenCover"></a>) on smooth manifolds seems to have been a <a class="existingWikiWord" href="/nlab/show/folk+theorem">folk theorem</a>, based in turn on the <a class="existingWikiWord" href="/nlab/show/folk+theorem">folk theorem</a> that every <a class="existingWikiWord" href="/nlab/show/geodesic+convexity">geodesically convex</a> <a class="existingWikiWord" href="/nlab/show/open+neighbourhood">open neighbourhood</a> in a <a class="existingWikiWord" href="/nlab/show/Riemannian+manifold">Riemannian manifold</a> is <em><a class="existingWikiWord" href="/nlab/show/diffeomorphic">diffeomorphic</a></em> (as opposed to just <a class="existingWikiWord" href="/nlab/show/homeomorphic">homeomorphic</a>) to an <a class="existingWikiWord" href="/nlab/show/open+ball">open ball</a>, hence to a <a class="existingWikiWord" href="/nlab/show/Cartesian+space">Cartesian space</a>.</p> <p>For instance, the latter fact is asserted without proof or reference inside the proof of Theorem 5.1 of <a href="#BottTu82">Bott & Tu 1982</a>.</p> <p>Actual proofs that <a class="existingWikiWord" href="/nlab/show/star-shaped+subsets">star-shaped subsets</a> of <a class="existingWikiWord" href="/nlab/show/Euclidean+space">Euclidean space</a> are <a class="existingWikiWord" href="/nlab/show/diffeomorphism">diffeomorphic</a> to <a class="existingWikiWord" href="/nlab/show/open+balls">open balls</a>/<a class="existingWikiWord" href="/nlab/show/Cartesian+spaces">Cartesian spaces</a> have then been made explicit in <a href="#GonnordTosel98">Gonnord & Tosel 1998, p. 60</a> and <a href="#Ferus07">Ferus 2007, Thm. 237</a>.</p> <p>(These proofs had remained obscure: In the textbooks <a href="ball#Conlon08">Conlon 2008</a> and <a href="ball#Lee09">Lee 2009</a> the statement is still referred to as lacking a citable proof: Conlon writes that a proof is “extremely difficult”, while Lee writes that “references to a complete proof are hard to find”, see also <a href="ball#LiteratureOnStarShapedOpenDiffeoToOpenBall">this remark</a> at <em><a class="existingWikiWord" href="/nlab/show/open+ball">open ball</a></em>.)</p> <p>That this argument completes the existence proof of differentiably good open covers has made explicit in <a href="#FSS10">FSS 2010, Prop. A.1</a>. In <a href="GuilleminHaine19">Guillemin & Haine 19 this is Thm. 5.3.2</a>, where the crucial step is left as Exercise 5.3, also App. C. See also <a href="#Demailly12">Demailly 2012, Lem. IV 6.9</a>.</p> </div> </p> <p> <div class='num_prop' id='ExistenceOfDifferentiablyGoodOpenCovers'> <h6>Proposition</h6> <p><strong>(existence of differentiably good open covers)</strong> <br /> Every <a class="existingWikiWord" href="/nlab/show/paracompact+manifold">paracompact</a> <a class="existingWikiWord" href="/nlab/show/smooth+manifold">smooth manifold</a> admits a good open cover, def. <a class="maruku-ref" href="#GoodOpenCover"></a>, in fact a differentiable good open cover, def. <a class="maruku-ref" href="#DifferentiablyGoodOpenCover"></a></p> </div> </p> <p>The following proof of Prop. <a class="maruku-ref" href="#ExistenceOfDifferentiablyGoodOpenCovers"></a> follows that given in <a href="#FSS10">FSS 2010, Prop. A.1</a>:</p> <p> <div class='proof'> <h6>Proof</h6> <p>By (<a href="#Greene">Greene</a>) every paracompact smooth manifold admits a <a class="existingWikiWord" href="/nlab/show/Riemannian+metric">Riemannian metric</a> with positive <a class="existingWikiWord" href="/nlab/show/convexity+radius">convexity radius</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>r</mi> <mi mathvariant="normal">conv</mi></msub><mo>∈</mo><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">r_{\mathrm{conv}} \in \mathbb{R}</annotation></semantics></math>. Choose such a metric and choose an <a class="existingWikiWord" href="/nlab/show/open+cover">open cover</a> consisting for each point <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo>∈</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">p\in X</annotation></semantics></math> of the geodesically convex open subset <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>U</mi> <mi>p</mi></msub><mo>:</mo><mo>=</mo><msub><mi>B</mi> <mi>p</mi></msub><mo stretchy="false">(</mo><msub><mi>r</mi> <mi>conv</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">U_p := B_p(r_{conv})</annotation></semantics></math> given by the geodesic <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>r</mi> <mi>conv</mi></msub></mrow><annotation encoding="application/x-tex">r_{conv}</annotation></semantics></math>-ball at <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>. Since the <a class="existingWikiWord" href="/nlab/show/injectivity+radius">injectivity radius</a> of any metric is at least <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>2</mn><msub><mi>r</mi> <mi mathvariant="normal">conv</mi></msub></mrow><annotation encoding="application/x-tex">2r_{\mathrm{conv}}</annotation></semantics></math> it follows from the minimality of the geodesics in a geodesically convex region that inside every finite nonempty intersection <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>U</mi> <mrow><msub><mi>p</mi> <mn>1</mn></msub></mrow></msub><mo>∩</mo><mi>⋯</mi><mo>∩</mo><msub><mi>U</mi> <mrow><msub><mi>p</mi> <mi>n</mi></msub></mrow></msub></mrow><annotation encoding="application/x-tex">U_{p_1} \cap \cdots \cap U_{p_n}</annotation></semantics></math> the <a class="existingWikiWord" href="/nlab/show/geodesic+flow">geodesic flow</a> around any point <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> is of radius less than or equal the injectivity radius and is therefore a diffeomorphism onto its image.</p> <p>Moreover, the <a class="existingWikiWord" href="/nlab/show/preimage">preimage</a> of the intersection region under the geometric flow is a <a class="existingWikiWord" href="/nlab/show/star-shaped">star-shaped</a> region in the <a class="existingWikiWord" href="/nlab/show/tangent+space">tangent space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>T</mi> <mi>u</mi></msub><mi>X</mi></mrow><annotation encoding="application/x-tex">T_u X</annotation></semantics></math>: because the intersection of geodesically convex regions is itself geodesically convex, so that for any <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>v</mi><mo>∈</mo><msub><mi>T</mi> <mi>u</mi></msub><mi>X</mi></mrow><annotation encoding="application/x-tex">v \in T_u X</annotation></semantics></math> with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>exp</mi><mo stretchy="false">(</mo><mi>v</mi><mo stretchy="false">)</mo><mo>∈</mo><msub><mi>U</mi> <mrow><msub><mi>p</mi> <mn>1</mn></msub></mrow></msub><mo>∩</mo><mi>⋯</mi><mo>∩</mo><msub><mi>U</mi> <mrow><msub><mi>p</mi> <mi>n</mi></msub></mrow></msub></mrow><annotation encoding="application/x-tex">\exp(v) \in U_{p_1} \cap \cdots \cap U_{p_n}</annotation></semantics></math> the whole geodesic segment <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>t</mi><mo>↦</mo><mi>exp</mi><mo stretchy="false">(</mo><mi>t</mi><mi>v</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">t \mapsto \exp(t v)</annotation></semantics></math> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>t</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">t \in [0,1]</annotation></semantics></math> is also in the region.</p> <p>So we have that every finite non-empty intersection of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>U</mi> <mi>p</mi></msub></mrow><annotation encoding="application/x-tex">U_p</annotation></semantics></math> is diffeomorphic to a star-shaped region in a vector space. By the results cited at <a class="existingWikiWord" href="/nlab/show/ball">ball</a> (e.g. theorem 237 of (<a href="#Ferus07">Ferus 2007</a>)) this star-shaped region is diffeomorphic to an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^n</annotation></semantics></math>.</p> </div> </p> <p>Here is another proof of Prop. <a class="maruku-ref" href="#ExistenceOfDifferentiablyGoodOpenCovers"></a>:</p> <p> <div class='proof'> <h6>Proof</h6> <p>Every <a class="existingWikiWord" href="/nlab/show/paracompact+manifold">paracompact</a> <a class="existingWikiWord" href="/nlab/show/smooth+manifold">smooth manifold</a> admits a <a class="existingWikiWord" href="/nlab/show/Riemannian+metric">Riemannian metric</a>, and for any point in a <a class="existingWikiWord" href="/nlab/show/Riemannian+manifold">Riemannian manifold</a> there is a <a class="existingWikiWord" href="/nlab/show/geodesically+convex">geodesically convex</a> <a class="existingWikiWord" href="/nlab/show/neighborhood">neighborhood</a> (any two points in the neighborhood are connected by a unique geodesic in the neighborhood, one whose length is the distance between the points; see for example the remark after (<a href="#Milnor">Milnor, lemma 10.3 on page 59</a>), or (<a href="#doCarmo">do Carmo, Proposition 4.2</a>)). A nonempty intersection of finitely many such geodesically convex neighborhoods is also geodesically convex. The inverse of the <a class="existingWikiWord" href="/nlab/show/exponential+map">exponential map</a> based at any interior point of a geodesically convex open subset gives a <a class="existingWikiWord" href="/nlab/show/diffeomorphism">diffeomorphism</a> from this subset to a star-shaped open subset of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mstyle mathvariant="bold"><mi>R</mi></mstyle> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">\mathbf{R}^n</annotation></semantics></math>. Indeed, the <a class="existingWikiWord" href="/nlab/show/Gauss+lemma">Gauss lemma</a> shows that the tangent map of the exponential map is invertible. By definition of geodesic convexity the exponential map is injective, hence a diffeomorphism. By <a href="open+ball#StarShapedOpenDiffeomorphicToOpenBall">this theorem</a>, star-shaped open subsets of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mstyle mathvariant="bold"><mi>R</mi></mstyle> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">\mathbf{R}^n</annotation></semantics></math> are diffeomorphic to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mstyle mathvariant="bold"><mi>R</mi></mstyle> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">\mathbf{R}^n</annotation></semantics></math>, which completes the proof.</p> </div> </p> <h3 id="coverages_of_good_open_covers">Coverages of good open covers</h3> <div class="num_prop" id="GoodOpenCoversFormACoverageOnParacompactSmooothManifolds"> <h6 id="proposition">Proposition</h6> <p>The <a class="existingWikiWord" href="/nlab/show/category">category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ParaSmMfd</mi></mrow><annotation encoding="application/x-tex">ParaSmMfd</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/paracompact+topological+space">paracompact</a> <a class="existingWikiWord" href="/nlab/show/smooth+manifolds">smooth manifolds</a> admits a <a class="existingWikiWord" href="/nlab/show/coverage">coverage</a> whose covering families are good open covers.</p> <p>The same holds true for <a class="existingWikiWord" href="/nlab/show/full+subcategories">full subcategories</a> such as</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/CartSp">CartSp</a>– <a class="existingWikiWord" href="/nlab/show/Cartesian+spaces">Cartesian spaces</a>.</li> </ul> </div> <div class="proof"> <h6 id="proof">Proof</h6> <p>It is sufficient to check this in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ParaSmMfd</mi></mrow><annotation encoding="application/x-tex">ParaSmMfd</annotation></semantics></math>. We need to check that for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><msub><mi>U</mi> <mi>i</mi></msub><mo>→</mo><mi>U</mi><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{U_i \to U\}</annotation></semantics></math> a good open cover and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><mi>V</mi><mo>→</mo><mi>U</mi></mrow><annotation encoding="application/x-tex">f : V \to U</annotation></semantics></math> any morphism, we get commuting squares</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>V</mi> <mi>j</mi></msub></mtd> <mtd><mo>→</mo></mtd> <mtd><msub><mi>U</mi> <mrow><mi>i</mi><mo stretchy="false">(</mo><mi>j</mi><mo stretchy="false">)</mo></mrow></msub></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi>V</mi></mtd> <mtd><mover><mo>→</mo><mi>f</mi></mover></mtd> <mtd><mi>U</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ V_j &\to& U_{i(j)} \\ \downarrow && \downarrow \\ V &\stackrel{f}{\to}& U } </annotation></semantics></math></div> <p>such that the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><msub><mi>V</mi> <mi>i</mi></msub><mo>→</mo><mi>V</mi><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{V_i \to V\}</annotation></semantics></math> form a good open cover of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math>.</p> <p>Now, while <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ParaSmMfd</mi></mrow><annotation encoding="application/x-tex">ParaSmMfd</annotation></semantics></math> does not have all <a class="existingWikiWord" href="/nlab/show/pullbacks">pullbacks</a>, the pullback of an <a class="existingWikiWord" href="/nlab/show/open+cover">open cover</a> does exist, and since <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> is necessarily a <a class="existingWikiWord" href="/nlab/show/continuous+function">continuous function</a> this is an <a class="existingWikiWord" href="/nlab/show/open+cover">open cover</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><msup><mi>f</mi> <mo>*</mo></msup><msub><mi>U</mi> <mi>i</mi></msub><mo>→</mo><mi>V</mi><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{f^* U_i \to V\}</annotation></semantics></math>. The <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>f</mi> <mo>*</mo></msup><msub><mi>U</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">f^* U_i</annotation></semantics></math> need not be contractible, but being open subsets of a paracompact manifold, they are themselves paracompact manifolds and hence admit themselves good open covers <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><msub><mi>W</mi> <mrow><mi>i</mi><mo>,</mo><mi>j</mi></mrow></msub><mo>→</mo><msup><mi>f</mi> <mo>*</mo></msup><msub><mi>U</mi> <mi>i</mi></msub><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{W_{i,j} \to f^* U_i\}</annotation></semantics></math>.</p> <p>Then the family of composites <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><msub><mi>W</mi> <mrow><mi>i</mi><mo>,</mo><mi>j</mi></mrow></msub><mo>→</mo><msup><mi>f</mi> <mo>*</mo></msup><msub><mi>U</mi> <mi>i</mi></msub><mo>→</mo><mi>V</mi><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{W_{i,j} \to f^* U_i \to V\}</annotation></semantics></math> is clearly a good open cover of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math>.</p> </div> <h3 id="existence_on_cw_complexes">Existence on CW complexes</h3> <div class="num_theorem"> <h6 id="theorem">Theorem</h6> <p>Every finite <a class="existingWikiWord" href="/nlab/show/CW+complex">CW complex</a> admits a good open cover.</p> </div> <p>Hopefully someone can find a clear reference to a proof. The assertion for finite CW complexes is found for example <a href="https://books.google.com/books?id=dL8FCAAAQBAJ&pg=PA37&lpg=PA37&dq=%22good+cover%22+%22CW+complex%22&source=bl&ots=GrZW-ZGSpU&sig=eLwTeBNnBYJuEYBJ-6JCPTPvw1g&hl=en&sa=X&ved=0CDgQ6AEwBGoVChMIlJCJzIOGyAIVCmg-Ch2FQwlj#v=onepage&q=%22good%20cover%22%20%22CW%20complex%22&f=false">here</a> (Topology of Tiling Spaces by Sadun, p. 37). It is not immediately clear from the remarks there what obstructions would exist to generalizing the assertion to all CW complexes.</p> <p>As indicated at <a class="existingWikiWord" href="/nlab/show/CW+complex">CW complex</a>, every CW complex is <em>homotopy equivalent</em> to a simplicial complex, and simplicial complexes certainly admit good covers by taking open stars.</p> <h3 id="nonexistence_for_topological_manifolds">(Non-)Existence for topological manifolds</h3> <p>For a (paracompact) <a class="existingWikiWord" href="/nlab/show/topological+manifold">topological manifold</a> the construction via <a class="existingWikiWord" href="/nlab/show/Riemannian+metrics">Riemannian metrics</a> or similar smooth constructions in general does not work.</p> <p>In (<a href="#OsborneStern69">Osborne-Stern 69</a>) the following discussion for sufficient conditions getting “close” to good open covers is discussed:</p> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/n-connected+space">k-connected</a> <a class="existingWikiWord" href="/nlab/show/topological+manifold">topological manifold</a> of <a class="existingWikiWord" href="/nlab/show/dimension">dimension</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math> (without <a class="existingWikiWord" href="/nlab/show/boundary">boundary</a>), and define</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>q</mi><mo>≔</mo><mi>min</mi><mo stretchy="false">(</mo><mi>k</mi><mo>,</mo><mi>n</mi><mo>−</mo><mn>3</mn><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> q \coloneqq min(k,n-3) \,. </annotation></semantics></math></div> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">p \in \mathbb{N}</annotation></semantics></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo stretchy="false">(</mo><mi>q</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo><mo>></mo><mi>n</mi></mrow><annotation encoding="application/x-tex">p(q+1) \gt n</annotation></semantics></math> then <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> admits a cover by <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> open balls and such that all nonempty intersections of the covering cells are <a class="existingWikiWord" href="/nlab/show/n-connected+space">(q−1)-connected</a>.</p> <h3 id="refining_covers">Refining covers</h3> <p>A cover <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><msub><mi>U</mi> <mi>i</mi></msub><mo>→</mo><mi>X</mi><msub><mo stretchy="false">}</mo> <mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\{U_i\to X\}_{i\in I}</annotation></semantics></math> refines another cover <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><msub><mi>V</mi> <mi>j</mi></msub><mo>→</mo><mi>X</mi><msub><mo stretchy="false">}</mo> <mrow><mi>j</mi><mo>∈</mo><mi>J</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\{V_j\to X\}_{j\in J}</annotation></semantics></math> if each map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>V</mi> <mi>j</mi></msub><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">V_j\to X</annotation></semantics></math> is some <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>U</mi> <mi>i</mi></msub><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">U_i\to X</annotation></semantics></math>.</p> <p>Each differentially good cover has a unique smallest refinement to a differentially good cover that is closed under intersection.</p> <h2 id="pov"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>POV</h2> <p>The following <a class="existingWikiWord" href="/nlab/show/nPOV">nPOV</a> perspective on good open covers gives a useful general “explanation” for their relevance, which also explains the role of good covers in <a class="existingWikiWord" href="/nlab/show/Cech+cohomology">Cech cohomology</a> generally and <a class="existingWikiWord" href="/nlab/show/abelian+sheaf+cohomology">abelian sheaf cohomology</a> in particular.</p> <div class="num_prop"> <h6 id="proposition_2">Proposition</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>sPSh</mi><mo stretchy="false">(</mo><mi>CartSp</mi><msub><mo stretchy="false">)</mo> <mi>proj</mi></msub></mrow><annotation encoding="application/x-tex">sPSh(CartSp)_{proj}</annotation></semantics></math> be the category of <a class="existingWikiWord" href="/nlab/show/simplicial+presheaves">simplicial presheaves</a> on the category <a class="existingWikiWord" href="/nlab/show/CartSp">CartSp</a> equipped with the projective <a class="existingWikiWord" href="/nlab/show/model+structure+on+simplicial+presheaves">model structure on simplicial presheaves</a>.</p> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/smooth+manifold">smooth manifold</a>, regarded as a 0-<a class="existingWikiWord" href="/nlab/show/truncated">truncated</a> object of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>sPSh</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">sPSh(C)</annotation></semantics></math>.</p> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><msub><mi>U</mi> <mi>i</mi></msub><mo>→</mo><mi>X</mi><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{U_i \to X\}</annotation></semantics></math> be a good open cover by <a class="existingWikiWord" href="/nlab/show/open+ball">open ball</a>s in the strong sense: such that every finite non-empty intersection is diffeomorphic to an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mi>d</mi></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^d</annotation></semantics></math>.</p> <p>Then: the <a class="existingWikiWord" href="/nlab/show/Cech+nerve">Cech nerve</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo stretchy="false">(</mo><mo stretchy="false">{</mo><mi>U</mi><mo stretchy="false">}</mo><mo stretchy="false">)</mo><mo>∈</mo><mi>sPSh</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">C(\{U\}) \in sPSh(C)</annotation></semantics></math> is a cofibrant resolution 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> in the <a class="existingWikiWord" href="/nlab/show/local+model+structure+on+simplicial+presheaves">local model structure on simplicial presheaves</a>.</p> </div> <div class="proof"> <h6 id="proof_2">Proof</h6> <p>By assumption we have that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo stretchy="false">(</mo><mi>U</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">C(U)</annotation></semantics></math> is degreewise a <a class="existingWikiWord" href="/nlab/show/coproduct">coproduct</a> of <a class="existingWikiWord" href="/nlab/show/representable+functor">representables</a>. It is also evidently a <a class="existingWikiWord" href="/nlab/show/split+hypercover">split hypercover</a>.</p> <p>This implies the statement by the <a href="http://ncatlab.org/nlab/show/model+structure+on+simplicial+presheaves#CofibrantObjects">characterization of cofibrant objects in the projective structure</a>.</p> </div> <p>This has a useful application in the <a class="existingWikiWord" href="/nlab/show/nerve+theorem">nerve theorem</a>.</p> <p>Notice that the <a class="existingWikiWord" href="/nlab/show/descent">descent</a> condition for simplicial presheaves on <a class="existingWikiWord" href="/nlab/show/CartSp">CartSp</a> at (good) covers is very weak, since all <a class="existingWikiWord" href="/nlab/show/Cartesian+space">Cartesian space</a>s are topologically contractible, so it is easy to find the fibrant objects <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>∈</mo><mi>sPSh</mi><mo stretchy="false">(</mo><mi>C</mi><msub><mo stretchy="false">)</mo> <mrow><mi>proj</mi><mo>,</mo><mi>loc</mi></mrow></msub></mrow><annotation encoding="application/x-tex">A \in sPSh(C)_{proj, loc}</annotation></semantics></math> in the <a class="existingWikiWord" href="/nlab/show/topological+localization">topological localization</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>sPSh</mi><mo stretchy="false">(</mo><mi>C</mi><msub><mo stretchy="false">)</mo> <mi>proj</mi></msub></mrow><annotation encoding="application/x-tex">sPSh(C)_{proj}</annotation></semantics></math> for the canonical <a class="existingWikiWord" href="/nlab/show/coverage">coverage</a> of <a class="existingWikiWord" href="/nlab/show/CartSp">CartSp</a>. The above observation says that for computing the <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>-valued <a class="existingWikiWord" href="/nlab/show/cohomology">cohomology</a> of a <a class="existingWikiWord" href="/nlab/show/diffeological+space">diffeological space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>, it is sufficient to evaluate <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> on (the Cech nerve of) a good cover 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> <p>We can turn this around and speak for any <a class="existingWikiWord" href="/nlab/show/site">site</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> of a covering family <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><msub><mi>U</mi> <mi>i</mi></msub><mo>→</mo><mi>X</mi><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{U_i \to X\}</annotation></semantics></math> as being <em>good</em> if the corresponding <a class="existingWikiWord" href="/nlab/show/Cech+nerve">Cech nerve</a> is degreewise a coproduct of representables. In the projective model structure on simplicial presheaves on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> such good covers will enjoy the central properties of good covers of topological spaces.</p> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/equivariant+good+open+cover">equivariant good open cover</a></p> </li> <li> <p><a href="Stein+manifold#GoodCoversBySteinManifolds">good covers by Stein manifolds</a> - note that this is a different concept, with vanishing Dolbeault cohomology replacing contractibility.</p> </li> <li> <p><span class="newWikiWord">quasicompact quasiseparated scheme<a href="/nlab/new/quasicompact+quasiseparated+scheme">?</a></span>: schemes that admit a finite cover by affine opens such that the intersection of any two elements is itself covered by finitely many affine opens.</p> </li> </ul> <h2 id="References">References</h2> <p>Proof of existence of differentiably good open covers (Def. <a class="maruku-ref" href="#DifferentiablyGoodOpenCover"></a>) of smooth manifolds:</p> <ul> <li id="BottTu82"> <p><a class="existingWikiWord" href="/nlab/show/Raoul+Bott">Raoul Bott</a>, <a class="existingWikiWord" href="/nlab/show/Loring+Tu">Loring Tu</a>, Thm. 5.1 in: <em><a class="existingWikiWord" href="/nlab/show/Differential+Forms+in+Algebraic+Topology">Differential Forms in Algebraic Topology</a></em>, Graduate Texts in Mathematics 82 Springer 1982 (<a href="https://link.springer.com/book/10.1007/978-1-4757-3951-0">doi:10.1007/978-1-4757-3951-0</a>, <a href="http://www.maths.ed.ac.uk/~aar/papers/botttu.pdf">pdf</a>)</p> </li> <li id="FSS10"> <p><a class="existingWikiWord" href="/nlab/show/Domenico+Fiorenza">Domenico Fiorenza</a>, <a class="existingWikiWord" href="/nlab/show/Urs+Schreiber">Urs Schreiber</a>, <a class="existingWikiWord" href="/nlab/show/Jim+Stasheff">Jim Stasheff</a>, Prop. A.1 of: <em><a class="existingWikiWord" href="/schreiber/show/Cech+cocycles+for+differential+characteristic+classes">Cech cocycles for differential characteristic classes</a></em>, Advances in Theoretical and Mathematical Physics, <strong>16</strong> 1 (2012) 149-250 (<a href="http://arxiv.org/abs/1011.4735">arXiv:1011.4735</a>, <a href="http://projecteuclid.org/euclid.atmp/1358950853">euclid:1358950853</a>, <a href="https://doi.org/10.1007/BF02104916">doi:10.1007/BF02104916</a>)</p> </li> <li id="Demailly12"> <p><a class="existingWikiWord" href="/nlab/show/Jean-Pierre+Demailly">Jean-Pierre Demailly</a>, Lemma IV.6.9 of: <em>Complex Analytic and Differential Geometry</em>, 2012 (<a href="https://www-fourier.ujf-grenoble.fr/~demailly/manuscripts/agbook.pdf">pdf</a>)</p> </li> <li id="GuilleminHaine19"> <p><a class="existingWikiWord" href="/nlab/show/Victor+Guillemin">Victor Guillemin</a>, <a class="existingWikiWord" href="/nlab/show/Peter+Haine">Peter Haine</a>, Thm. 5.3.2 and Appendix C of: <em>Differential Forms</em>, World Scientific (2019) (<a href="https://doi.org/10.1142/11058">doi:10.1142/11058</a>)</p> </li> </ul> <p>These proofs require one to show that <a class="existingWikiWord" href="/nlab/show/star-shaped+subsets">star-shaped subsets</a> of <a class="existingWikiWord" href="/nlab/show/Cartesian+space">Cartesian space</a>/<a class="existingWikiWord" href="/nlab/show/Euclidean+space">Euclidean space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mstyle mathvariant="bold"><mi>R</mi></mstyle> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">\mathbf{R}^n</annotation></semantics></math> are <a class="existingWikiWord" href="/nlab/show/diffeomorphism">diffeomorphic</a> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mstyle mathvariant="bold"><mi>R</mi></mstyle> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">\mathbf{R}^n</annotation></semantics></math> (see the article <em><a class="existingWikiWord" href="/nlab/show/ball">ball</a></em> for details, and see the references <a href="ball#ReferencesStarShapedReasonDiffeomorphicToOpenBall">there</a>).</p> <p>Such proofs have been given in:</p> <ul> <li id="GonnordTosel98"> <p>Stéphane Gonnord, Nicolas Tosel, page 60 of: <em>Calcul Différentiel</em>, ellipses (1998) (English translation: <a href="http://mathoverflow.net/a/212595">MO:a/212595</a>, <a class="existingWikiWord" href="/nlab/files/Aubry_reproducing_GonnordAndTosel.pdf" title="pdf">pdf</a>)</p> </li> <li id="Ferus07"> <p><a class="existingWikiWord" href="/nlab/show/Dirk+Ferus">Dirk Ferus</a>, <a href="http://www.math.tu-berlin.de/~ferus/ANA/Ana3.pdf#page=154">theorem 237</a> in: <em>Analysis III</em> (2007) (<a href="http://www.math.tu-berlin.de/~ferus/ANA/Ana3.pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/Ferus_AnalysisIII.pdf" title="pdf">pdf</a>)</p> </li> </ul> <p>Other references on good open covers:</p> <ul> <li id="doCarmo"> <p><a class="existingWikiWord" href="/nlab/show/Manfredo+do+Carmo">Manfredo do Carmo</a>, <em>Riemannian geometry</em> (trans. Francis Flaherty), Birkhäuser (1992)</p> </li> <li id="Milnor"> <p><a class="existingWikiWord" href="/nlab/show/John+Milnor">John Milnor</a>, <em>Morse theory</em> , Princeton University Press (1963)</p> </li> <li id="Greene"> <p>R. Greene, <em>Complete metrics of bounded curvature on noncompact manifolds</em> Archiv der Mathematik Volume 31, Number 1 (1978)</p> </li> <li id="OsborneStern69"> <p>RP Osborne and JL Stern. <em>Covering Manifolds with Cells</em>, Pacific Journal of Mathematics, Vol 30, No. 1, 1969.</p> </li> <li> <p>MathOverflow, <em><a href="http://mathoverflow.net/q/102161/381">Proving the existence of good covers</a></em></p> </li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on December 29, 2023 at 17:36:17. 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