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float: left;" onfocus="this.value == 'Search' ? this.value = '' : true" onblur="this.value == '' ? this.value = 'Search' : true" /></fieldset> </form> <span id='navEnd'></span> </div> <div id="revision"> <html xmlns="http://www.w3.org/1999/xhtml" xmlns:svg="http://www.w3.org/2000/svg" xml:lang="en" lang="en"> <head><meta http-equiv="Content-type" content="application/xhtml+xml;charset=utf-8" /><title>Contents</title></head> <body> <div class="rightHandSide"> <div class="toc clickDown" tabindex="0"> <h3 id="context">Context</h3> <h4 id="topology">Topology</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/topology">topology</a></strong> (<a class="existingWikiWord" href="/nlab/show/point-set+topology">point-set topology</a>, <a class="existingWikiWord" href="/nlab/show/point-free+topology">point-free topology</a>)</p> <p>see also <em><a class="existingWikiWord" href="/nlab/show/differential+topology">differential topology</a></em>, <em><a class="existingWikiWord" href="/nlab/show/algebraic+topology">algebraic topology</a></em>, <em><a class="existingWikiWord" href="/nlab/show/functional+analysis">functional analysis</a></em> and <em><a class="existingWikiWord" href="/nlab/show/topological+homotopy+theory">topological</a> <a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopy theory</a></em></p> <p><a class="existingWikiWord" href="/nlab/show/Introduction+to+Topology">Introduction</a></p> <p><strong>Basic concepts</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/open+subset">open subset</a>, <a class="existingWikiWord" href="/nlab/show/closed+subset">closed subset</a>, <a class="existingWikiWord" href="/nlab/show/neighbourhood">neighbourhood</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a>, <a class="existingWikiWord" href="/nlab/show/locale">locale</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/base+for+the+topology">base for the topology</a>, <a class="existingWikiWord" href="/nlab/show/neighbourhood+base">neighbourhood base</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/finer+topology">finer/coarser topology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+closure">closure</a>, <a class="existingWikiWord" href="/nlab/show/topological+interior">interior</a>, <a class="existingWikiWord" href="/nlab/show/topological+boundary">boundary</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/separation+axiom">separation</a>, <a class="existingWikiWord" href="/nlab/show/sober+topological+space">sobriety</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/continuous+function">continuous function</a>, <a class="existingWikiWord" href="/nlab/show/homeomorphism">homeomorphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/uniformly+continuous+function">uniformly continuous function</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+embedding">embedding</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/open+map">open map</a>, <a class="existingWikiWord" href="/nlab/show/closed+map">closed map</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/sequence">sequence</a>, <a class="existingWikiWord" href="/nlab/show/net">net</a>, <a class="existingWikiWord" href="/nlab/show/sub-net">sub-net</a>, <a class="existingWikiWord" href="/nlab/show/filter">filter</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/convergence">convergence</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/category">category</a><a class="existingWikiWord" href="/nlab/show/Top">Top</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/convenient+category+of+topological+spaces">convenient category of topological spaces</a></li> </ul> </li> </ul> <p><strong><a href="Top#UniversalConstructions">Universal constructions</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/initial+topology">initial topology</a>, <a class="existingWikiWord" href="/nlab/show/final+topology">final topology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/subspace">subspace</a>, <a class="existingWikiWord" href="/nlab/show/quotient+space">quotient space</a>,</p> </li> <li> <p>fiber space, <a class="existingWikiWord" href="/nlab/show/space+attachment">space attachment</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/product+space">product space</a>, <a class="existingWikiWord" href="/nlab/show/disjoint+union+space">disjoint union space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/mapping+cylinder">mapping cylinder</a>, <a class="existingWikiWord" href="/nlab/show/mapping+cocylinder">mapping cocylinder</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/mapping+cone">mapping cone</a>, <a class="existingWikiWord" href="/nlab/show/mapping+cocone">mapping cocone</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/mapping+telescope">mapping telescope</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/colimits+of+normal+spaces">colimits of normal spaces</a></p> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/stuff%2C+structure%2C+property">Extra stuff, structure, properties</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/nice+topological+space">nice topological space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/metric+space">metric space</a>, <a class="existingWikiWord" href="/nlab/show/metric+topology">metric topology</a>, <a class="existingWikiWord" href="/nlab/show/metrisable+space">metrisable space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Kolmogorov+space">Kolmogorov space</a>, <a class="existingWikiWord" href="/nlab/show/Hausdorff+space">Hausdorff space</a>, <a class="existingWikiWord" href="/nlab/show/regular+space">regular space</a>, <a class="existingWikiWord" href="/nlab/show/normal+space">normal space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/sober+space">sober space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/compact+space">compact space</a>, <a class="existingWikiWord" href="/nlab/show/proper+map">proper map</a></p> <p><a class="existingWikiWord" href="/nlab/show/sequentially+compact+topological+space">sequentially compact</a>, <a class="existingWikiWord" href="/nlab/show/countably+compact+topological+space">countably compact</a>, <a class="existingWikiWord" href="/nlab/show/locally+compact+topological+space">locally compact</a>, <a class="existingWikiWord" href="/nlab/show/sigma-compact+topological+space">sigma-compact</a>, <a class="existingWikiWord" href="/nlab/show/paracompact+space">paracompact</a>, <a class="existingWikiWord" href="/nlab/show/countably+paracompact+topological+space">countably paracompact</a>, <a class="existingWikiWord" href="/nlab/show/strongly+compact+topological+space">strongly compact</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/compactly+generated+space">compactly generated space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/second-countable+space">second-countable space</a>, <a class="existingWikiWord" href="/nlab/show/first-countable+space">first-countable space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/contractible+space">contractible space</a>, <a class="existingWikiWord" href="/nlab/show/locally+contractible+space">locally contractible space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/connected+space">connected space</a>, <a class="existingWikiWord" href="/nlab/show/locally+connected+space">locally connected space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/simply-connected+space">simply-connected space</a>, <a class="existingWikiWord" href="/nlab/show/locally+simply-connected+space">locally simply-connected space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cell+complex">cell complex</a>, <a class="existingWikiWord" href="/nlab/show/CW-complex">CW-complex</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/pointed+topological+space">pointed space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+vector+space">topological vector space</a>, <a class="existingWikiWord" href="/nlab/show/Banach+space">Banach space</a>, <a class="existingWikiWord" href="/nlab/show/Hilbert+space">Hilbert space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+group">topological group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+vector+bundle">topological vector bundle</a>, <a class="existingWikiWord" href="/nlab/show/topological+K-theory">topological K-theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+manifold">topological manifold</a></p> </li> </ul> <p><strong>Examples</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/empty+space">empty space</a>, <a class="existingWikiWord" href="/nlab/show/point+space">point space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/discrete+space">discrete space</a>, <a class="existingWikiWord" href="/nlab/show/codiscrete+space">codiscrete space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Sierpinski+space">Sierpinski space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/order+topology">order topology</a>, <a class="existingWikiWord" href="/nlab/show/specialization+topology">specialization topology</a>, <a class="existingWikiWord" href="/nlab/show/Scott+topology">Scott topology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Euclidean+space">Euclidean space</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/real+line">real line</a>, <a class="existingWikiWord" href="/nlab/show/plane">plane</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cylinder">cylinder</a>, <a class="existingWikiWord" href="/nlab/show/cone">cone</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/sphere">sphere</a>, <a class="existingWikiWord" href="/nlab/show/ball">ball</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/circle">circle</a>, <a class="existingWikiWord" href="/nlab/show/torus">torus</a>, <a class="existingWikiWord" href="/nlab/show/annulus">annulus</a>, <a class="existingWikiWord" href="/nlab/show/Moebius+strip">Moebius strip</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/polytope">polytope</a>, <a class="existingWikiWord" href="/nlab/show/polyhedron">polyhedron</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/projective+space">projective space</a> (<a class="existingWikiWord" href="/nlab/show/real+projective+space">real</a>, <a class="existingWikiWord" href="/nlab/show/complex+projective+space">complex</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/classifying+space">classifying space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/configuration+space+%28mathematics%29">configuration space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/path">path</a>, <a class="existingWikiWord" href="/nlab/show/loop">loop</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/mapping+spaces">mapping spaces</a>: <a class="existingWikiWord" href="/nlab/show/compact-open+topology">compact-open topology</a>, <a class="existingWikiWord" href="/nlab/show/topology+of+uniform+convergence">topology of uniform convergence</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/loop+space">loop space</a>, <a class="existingWikiWord" href="/nlab/show/path+space">path space</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Zariski+topology">Zariski topology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Cantor+space">Cantor space</a>, <a class="existingWikiWord" href="/nlab/show/Mandelbrot+space">Mandelbrot space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Peano+curve">Peano curve</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/line+with+two+origins">line with two origins</a>, <a class="existingWikiWord" href="/nlab/show/long+line">long line</a>, <a class="existingWikiWord" href="/nlab/show/Sorgenfrey+line">Sorgenfrey line</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/K-topology">K-topology</a>, <a class="existingWikiWord" href="/nlab/show/Dowker+space">Dowker space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Warsaw+circle">Warsaw circle</a>, <a class="existingWikiWord" href="/nlab/show/Hawaiian+earring+space">Hawaiian earring space</a></p> </li> </ul> <p><strong>Basic statements</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Hausdorff+spaces+are+sober">Hausdorff spaces are sober</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/schemes+are+sober">schemes are sober</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/continuous+images+of+compact+spaces+are+compact">continuous images of compact spaces are compact</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/closed+subspaces+of+compact+Hausdorff+spaces+are+equivalently+compact+subspaces">closed subspaces of compact Hausdorff spaces are equivalently compact subspaces</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/open+subspaces+of+compact+Hausdorff+spaces+are+locally+compact">open subspaces of compact Hausdorff spaces are locally compact</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quotient+projections+out+of+compact+Hausdorff+spaces+are+closed+precisely+if+the+codomain+is+Hausdorff">quotient projections out of compact Hausdorff spaces are closed precisely if the codomain is Hausdorff</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/compact+spaces+equivalently+have+converging+subnet+of+every+net">compact spaces equivalently have converging subnet of every net</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Lebesgue+number+lemma">Lebesgue number lemma</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/sequentially+compact+metric+spaces+are+equivalently+compact+metric+spaces">sequentially compact metric spaces are equivalently compact metric spaces</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/compact+spaces+equivalently+have+converging+subnet+of+every+net">compact spaces equivalently have converging subnet of every net</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/sequentially+compact+metric+spaces+are+totally+bounded">sequentially compact metric spaces are totally bounded</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/continuous+metric+space+valued+function+on+compact+metric+space+is+uniformly+continuous">continuous metric space valued function on compact metric space is uniformly continuous</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/paracompact+Hausdorff+spaces+are+normal">paracompact Hausdorff spaces are normal</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/paracompact+Hausdorff+spaces+equivalently+admit+subordinate+partitions+of+unity">paracompact Hausdorff spaces equivalently admit subordinate partitions of unity</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/closed+injections+are+embeddings">closed injections are embeddings</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/proper+maps+to+locally+compact+spaces+are+closed">proper maps to locally compact spaces are closed</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/injective+proper+maps+to+locally+compact+spaces+are+equivalently+the+closed+embeddings">injective proper maps to locally compact spaces are equivalently the closed embeddings</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/locally+compact+and+sigma-compact+spaces+are+paracompact">locally compact and sigma-compact spaces are paracompact</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/locally+compact+and+second-countable+spaces+are+sigma-compact">locally compact and second-countable spaces are sigma-compact</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/second-countable+regular+spaces+are+paracompact">second-countable regular spaces are paracompact</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/CW-complexes+are+paracompact+Hausdorff+spaces">CW-complexes are paracompact Hausdorff spaces</a></p> </li> </ul> <p><strong>Theorems</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Urysohn%27s+lemma">Urysohn's lemma</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Tietze+extension+theorem">Tietze extension theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Tychonoff+theorem">Tychonoff theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/tube+lemma">tube lemma</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Michael%27s+theorem">Michael's theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Brouwer%27s+fixed+point+theorem">Brouwer's fixed point theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+invariance+of+dimension">topological invariance of dimension</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Jordan+curve+theorem">Jordan curve theorem</a></p> </li> </ul> <p><strong>Analysis Theorems</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Heine-Borel+theorem">Heine-Borel theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/intermediate+value+theorem">intermediate value theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/extreme+value+theorem">extreme value theorem</a></p> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/topological+homotopy+theory">topological homotopy theory</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/left+homotopy">left homotopy</a>, <a class="existingWikiWord" href="/nlab/show/right+homotopy">right homotopy</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+equivalence">homotopy equivalence</a>, <a class="existingWikiWord" href="/nlab/show/deformation+retract">deformation retract</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+group">fundamental group</a>, <a class="existingWikiWord" href="/nlab/show/covering+space">covering space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+theorem+of+covering+spaces">fundamental theorem of covering spaces</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+group">homotopy group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/weak+homotopy+equivalence">weak homotopy equivalence</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Whitehead%27s+theorem">Whitehead's theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Freudenthal+suspension+theorem">Freudenthal suspension theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/nerve+theorem">nerve theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+extension+property">homotopy extension property</a>, <a class="existingWikiWord" href="/nlab/show/Hurewicz+cofibration">Hurewicz cofibration</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+cofiber+sequence">cofiber sequence</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Str%C3%B8m+model+category">Strøm model category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/classical+model+structure+on+topological+spaces">classical model structure on topological spaces</a></p> </li> </ul> </div></div> <h4 id="differential_geometry">Differential geometry</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/synthetic+differential+geometry">synthetic</a> <a class="existingWikiWord" href="/nlab/show/differential+geometry">differential geometry</a></strong></p> <p><strong>Introductions</strong></p> <p><a class="existingWikiWord" href="/nlab/show/Introduction+to+Topology+--+1">from point-set topology to differentiable manifolds</a></p> <p><a class="existingWikiWord" href="/nlab/show/geometry+of+physics">geometry of physics</a>: <a class="existingWikiWord" href="/nlab/show/geometry+of+physics+--+coordinate+systems">coordinate systems</a>, <a class="existingWikiWord" href="/nlab/show/geometry+of+physics+--+smooth+spaces">smooth spaces</a>, <a class="existingWikiWord" href="/nlab/show/geometry+of+physics+--+manifolds+and+orbifolds">manifolds</a>, <a class="existingWikiWord" href="/nlab/show/geometry+of+physics+--+smooth+homotopy+types">smooth homotopy types</a>, <a class="existingWikiWord" href="/nlab/show/geometry+of+physics+--+supergeometry">supergeometry</a></p> <p><strong>Differentials</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/differentiation">differentiation</a>, <a class="existingWikiWord" href="/nlab/show/chain+rule">chain rule</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/differentiable+function">differentiable function</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/infinitesimal+space">infinitesimal space</a>, <a class="existingWikiWord" href="/nlab/show/infinitesimally+thickened+point">infinitesimally thickened point</a>, <a class="existingWikiWord" href="/nlab/show/amazing+right+adjoint">amazing right adjoint</a></p> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/V-manifolds">V-manifolds</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/differentiable+manifold">differentiable manifold</a>, <a class="existingWikiWord" href="/nlab/show/coordinate+chart">coordinate chart</a>, <a class="existingWikiWord" href="/nlab/show/atlas">atlas</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/smooth+manifold">smooth manifold</a>, <a class="existingWikiWord" href="/nlab/show/smooth+structure">smooth structure</a>, <a class="existingWikiWord" href="/nlab/show/exotic+smooth+structure">exotic smooth structure</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/analytic+manifold">analytic manifold</a>, <a class="existingWikiWord" href="/nlab/show/complex+manifold">complex manifold</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/formal+smooth+manifold">formal smooth manifold</a>, <a class="existingWikiWord" href="/nlab/show/derived+smooth+manifold">derived smooth manifold</a></p> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/smooth+space">smooth space</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/diffeological+space">diffeological space</a>, <a class="existingWikiWord" href="/nlab/show/Fr%C3%B6licher+space">Frölicher space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/manifold+structure+of+mapping+spaces">manifold structure of mapping spaces</a></p> </li> </ul> <p><strong>Tangency</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/tangent+bundle">tangent bundle</a>, <a class="existingWikiWord" href="/nlab/show/frame+bundle">frame bundle</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/vector+field">vector field</a>, <a class="existingWikiWord" href="/nlab/show/multivector+field">multivector field</a>, <a class="existingWikiWord" href="/nlab/show/tangent+Lie+algebroid">tangent Lie algebroid</a>;</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+forms+in+synthetic+differential+geometry">differential forms</a>, <a class="existingWikiWord" href="/nlab/show/de+Rham+complex">de Rham complex</a>, <a class="existingWikiWord" href="/nlab/show/Dolbeault+complex">Dolbeault complex</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/pullback+of+differential+forms">pullback of differential forms</a>, <a class="existingWikiWord" href="/nlab/show/invariant+differential+form">invariant differential form</a>, <a class="existingWikiWord" href="/nlab/show/Maurer-Cartan+form">Maurer-Cartan form</a>, <a class="existingWikiWord" href="/nlab/show/horizontal+differential+form">horizontal differential form</a>,</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cogerm+differential+form">cogerm differential form</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/integration+of+differential+forms">integration of differential forms</a></p> </li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/local+diffeomorphism">local diffeomorphism</a>, <a class="existingWikiWord" href="/nlab/show/formally+%C3%A9tale+morphism">formally étale morphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/submersion">submersion</a>, <a class="existingWikiWord" href="/nlab/show/formally+smooth+morphism">formally smooth morphism</a>,</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/immersion">immersion</a>, <a class="existingWikiWord" href="/nlab/show/formally+unramified+morphism">formally unramified morphism</a>,</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/de+Rham+space">de Rham space</a>, <a class="existingWikiWord" href="/nlab/show/crystal">crystal</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/infinitesimal+disk+bundle">infinitesimal disk bundle</a></p> </li> </ul> <p><strong>The magic algebraic facts</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/embedding+of+smooth+manifolds+into+formal+duals+of+R-algebras">embedding of smooth manifolds into formal duals of R-algebras</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/smooth+Serre-Swan+theorem">smooth Serre-Swan theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/derivations+of+smooth+functions+are+vector+fields">derivations of smooth functions are vector fields</a></p> </li> </ul> <p><strong>Theorems</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Hadamard+lemma">Hadamard lemma</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Borel%27s+theorem">Borel's theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Boman%27s+theorem">Boman's theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Whitney+extension+theorem">Whitney extension theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Steenrod-Wockel+approximation+theorem">Steenrod-Wockel approximation theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Whitney+embedding+theorem">Whitney embedding theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Poincare+lemma">Poincare lemma</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Stokes+theorem">Stokes theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/de+Rham+theorem">de Rham theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hochschild-Kostant-Rosenberg+theorem">Hochschild-Kostant-Rosenberg theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+cohomology+hexagon">differential cohomology hexagon</a></p> </li> </ul> <p><strong>Axiomatics</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Kock-Lawvere+axiom">Kock-Lawvere axiom</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/smooth+topos">smooth topos</a>, <a class="existingWikiWord" href="/nlab/show/super+smooth+topos">super smooth topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/microlinear+space">microlinear space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/integration+axiom">integration axiom</a></p> </li> </ul> </li> </ul> <div> <p><strong><a class="existingWikiWord" href="/nlab/show/cohesion">cohesion</a></strong></p> <ul> <li> <p>(<a class="existingWikiWord" href="/nlab/show/shape+modality">shape modality</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/flat+modality">flat modality</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/sharp+modality">sharp modality</a>)</p> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mo lspace="0em" rspace="thinmathspace">ʃ</mo><mo>⊣</mo><mo>♭</mo><mo>⊣</mo><mo>♯</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\esh \dashv \flat \dashv \sharp )</annotation></semantics></math></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/discrete+object">discrete object</a>, <a class="existingWikiWord" href="/nlab/show/codiscrete+object">codiscrete object</a>, <a class="existingWikiWord" href="/nlab/show/concrete+object">concrete object</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/points-to-pieces+transform">points-to-pieces transform</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cohesive+%28infinity%2C1%29-topos+--+structures">structures in cohesion</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/dR-shape+modality">dR-shape modality</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/dR-flat+modality">dR-flat modality</a></p> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mo lspace="0em" rspace="thinmathspace">ʃ</mo> <mi>dR</mi></msub><mo>⊣</mo><msub><mo>♭</mo> <mi>dR</mi></msub></mrow><annotation encoding="application/x-tex">\esh_{dR} \dashv \flat_{dR}</annotation></semantics></math></p> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/infinitesimal+cohesion">infinitesimal cohesion</a></strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/classical+modality">classical modality</a></li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/tangent+cohesive+%28%E2%88%9E%2C1%29-topos">tangent cohesion</a></strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/differential+cohomology+diagram">differential cohomology diagram</a></li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/differential+cohesion">differential cohesion</a></strong></p> <ul> <li> <p>(<a class="existingWikiWord" href="/nlab/show/reduction+modality">reduction modality</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/infinitesimal+shape+modality">infinitesimal shape modality</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/infinitesimal+flat+modality">infinitesimal flat modality</a>)</p> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>ℜ</mi><mo>⊣</mo><mi>ℑ</mi><mo>⊣</mo><mi>&</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\Re \dashv \Im \dashv \&)</annotation></semantics></math></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/reduced+object">reduced object</a>, <a class="existingWikiWord" href="/nlab/show/coreduced+object">coreduced object</a>, <a class="existingWikiWord" href="/nlab/show/formally+smooth+object">formally smooth object</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/formally+%C3%A9tale+map">formally étale map</a></p> </li> <li> <p><a href="cohesive+%28infinity%2C1%29-topos+--+infinitesimal+cohesion#StructuresInDifferentialCohesion">structures in differential cohesion</a></p> </li> </ul> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/super+smooth+infinity-groupoid">graded differential cohesion</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/fermionic+modality">fermionic modality</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/bosonic+modality">bosonic modality</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/rheonomy+modality">rheonomy modality</a></p> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mo>⇉</mo><mo>⊣</mo><mo>⇝</mo><mo>⊣</mo><mi>Rh</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\rightrightarrows \dashv \rightsquigarrow \dashv Rh)</annotation></semantics></math></p> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/orbifold+cohomology">singular cohesion</a></strong></p> <div id="Diagram" class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd></mtd> <mtd></mtd> <mtd><mi>id</mi></mtd> <mtd><mo>⊣</mo></mtd> <mtd><mi>id</mi></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mo>∨</mo></mtd> <mtd></mtd> <mtd><mo>∨</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mover><mrow></mrow><mi>fermionic</mi></mover></mtd> <mtd><mo>⇉</mo></mtd> <mtd><mo>⊣</mo></mtd> <mtd><mo>⇝</mo></mtd> <mtd><mover><mrow></mrow><mi>bosonic</mi></mover></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mo>⊥</mo></mtd> <mtd></mtd> <mtd><mo>⊥</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mover><mrow></mrow><mi>bosonic</mi></mover></mtd> <mtd><mo>⇝</mo></mtd> <mtd><mo>⊣</mo></mtd> <mtd><mi mathvariant="normal">R</mi><mspace width="negativethinmathspace"></mspace><mspace width="negativethinmathspace"></mspace><mi mathvariant="normal">h</mi></mtd> <mtd><mover><mrow></mrow><mi>rheonomic</mi></mover></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mo>∨</mo></mtd> <mtd></mtd> <mtd><mo>∨</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mover><mrow></mrow><mi>reduced</mi></mover></mtd> <mtd><mi>ℜ</mi></mtd> <mtd><mo>⊣</mo></mtd> <mtd><mi>ℑ</mi></mtd> <mtd><mover><mrow></mrow><mi>infinitesimal</mi></mover></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mo>⊥</mo></mtd> <mtd></mtd> <mtd><mo>⊥</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mover><mrow></mrow><mi>infinitesimal</mi></mover></mtd> <mtd><mi>ℑ</mi></mtd> <mtd><mo>⊣</mo></mtd> <mtd><mi>&</mi></mtd> <mtd><mover><mrow></mrow><mtext>étale</mtext></mover></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mo>∨</mo></mtd> <mtd></mtd> <mtd><mo>∨</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mover><mrow></mrow><mi>cohesive</mi></mover></mtd> <mtd><mo lspace="0em" rspace="thinmathspace">ʃ</mo></mtd> <mtd><mo>⊣</mo></mtd> <mtd><mo>♭</mo></mtd> <mtd><mover><mrow></mrow><mi>discrete</mi></mover></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mo>⊥</mo></mtd> <mtd></mtd> <mtd><mo>⊥</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mover><mrow></mrow><mi>discrete</mi></mover></mtd> <mtd><mo>♭</mo></mtd> <mtd><mo>⊣</mo></mtd> <mtd><mo>♯</mo></mtd> <mtd><mover><mrow></mrow><mi>continuous</mi></mover></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mo>∨</mo></mtd> <mtd></mtd> <mtd><mo>∨</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mi>∅</mi></mtd> <mtd><mo>⊣</mo></mtd> <mtd><mo>*</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& \mathrm{R}\!\!\mathrm{h} & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{&#233;tale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& \esh &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast } </annotation></semantics></math></div></div> <p id="models_2"><strong>Models</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Models+for+Smooth+Infinitesimal+Analysis">Models for Smooth Infinitesimal Analysis</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/smooth+algebra">smooth algebra</a> (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mn>∞</mn></msup></mrow><annotation encoding="application/x-tex">C^\infty</annotation></semantics></math>-ring)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/smooth+locus">smooth locus</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Fermat+theory">Fermat theory</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Cahiers+topos">Cahiers topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/smooth+%E2%88%9E-groupoid">smooth ∞-groupoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/formal+smooth+%E2%88%9E-groupoid">formal smooth ∞-groupoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/super+formal+smooth+%E2%88%9E-groupoid">super formal smooth ∞-groupoid</a></p> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/Lie+theory">Lie theory</a>, <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Lie+theory">∞-Lie theory</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Lie+algebra">Lie algebra</a>, <a class="existingWikiWord" href="/nlab/show/Lie+n-algebra">Lie n-algebra</a>, <a class="existingWikiWord" href="/nlab/show/L-%E2%88%9E+algebra">L-∞ algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Lie+group">Lie group</a>, <a class="existingWikiWord" href="/nlab/show/Lie+2-group">Lie 2-group</a>, <a class="existingWikiWord" href="/nlab/show/smooth+%E2%88%9E-group">smooth ∞-group</a></p> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/differential+equations">differential equations</a>, <a class="existingWikiWord" href="/nlab/show/variational+calculus">variational calculus</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/D-geometry">D-geometry</a>, <a class="existingWikiWord" href="/nlab/show/D-module">D-module</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/jet+bundle">jet bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/variational+bicomplex">variational bicomplex</a>, <a class="existingWikiWord" href="/nlab/show/Euler-Lagrange+complex">Euler-Lagrange complex</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Euler-Lagrange+equation">Euler-Lagrange equation</a>, <a class="existingWikiWord" href="/nlab/show/de+Donder-Weyl+formalism">de Donder-Weyl formalism</a>,</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/phase+space">phase space</a></p> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/Chern-Weil+theory">Chern-Weil theory</a>, <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Chern-Weil+theory">∞-Chern-Weil theory</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/connection+on+a+bundle">connection on a bundle</a>, <a class="existingWikiWord" href="/nlab/show/connection+on+an+%E2%88%9E-bundle">connection on an ∞-bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+cohomology">differential cohomology</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/ordinary+differential+cohomology">ordinary differential cohomology</a>, <a class="existingWikiWord" href="/nlab/show/Deligne+complex">Deligne complex</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+K-theory">differential K-theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+cobordism+cohomology">differential cobordism cohomology</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/parallel+transport">parallel transport</a>, <a class="existingWikiWord" href="/nlab/show/higher+parallel+transport">higher parallel transport</a>, <a class="existingWikiWord" href="/nlab/show/fiber+integration+in+differential+cohomology">fiber integration in differential cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/holonomy">holonomy</a>, <a class="existingWikiWord" href="/nlab/show/higher+holonomy">higher holonomy</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/gauge+theory">gauge theory</a>, <a class="existingWikiWord" href="/nlab/show/higher+gauge+theory">higher gauge theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Wilson+line">Wilson line</a>, <a class="existingWikiWord" href="/nlab/show/Wilson+surface">Wilson surface</a></p> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/Cartan+geometry">Cartan geometry</a> (<a class="existingWikiWord" href="/nlab/show/super+Cartan+geometry">super</a>, <a class="existingWikiWord" href="/nlab/show/higher+Cartan+geometry">higher</a>)</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Klein+geometry">Klein geometry</a>, (<a class="existingWikiWord" href="/nlab/show/higher+Klein+geometry">higher</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/G-structure">G-structure</a>, <a class="existingWikiWord" href="/nlab/show/torsion+of+a+G-structure">torsion of a G-structure</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Euclidean+geometry">Euclidean geometry</a>, <a class="existingWikiWord" href="/nlab/show/hyperbolic+geometry">hyperbolic geometry</a>, <a class="existingWikiWord" href="/nlab/show/elliptic+geometry">elliptic geometry</a></p> </li> <li> <p>(<a class="existingWikiWord" href="/nlab/show/pseudo-Riemannian+geometry">pseudo</a>-)<a class="existingWikiWord" href="/nlab/show/Riemannian+geometry">Riemannian geometry</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/orthogonal+structure">orthogonal structure</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/isometry">isometry</a>, <a class="existingWikiWord" href="/nlab/show/Killing+vector+field">Killing vector field</a>, <a class="existingWikiWord" href="/nlab/show/Killing+spinor">Killing spinor</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/spacetime">spacetime</a>, <a class="existingWikiWord" href="/nlab/show/super-spacetime">super-spacetime</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/complex+geometry">complex geometry</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/symplectic+geometry">symplectic geometry</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/conformal+geometry">conformal geometry</a></p> </li> </ul> </div></div> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#definition'>Definition</a></li> <ul> <li><a href='#geometric'>Geometric</a></li> <li><a href='#combinatorial'>Combinatorial</a></li> </ul> <li><a href='#Properties'>Properties</a></li> <ul> <li><a href='#closed_balls'>Closed balls</a></li> <li><a href='#open_balls'>Open Balls</a></li> <ul> <li><a href='#observation'>Observation</a></li> </ul> <li><a href='#good_covers_by_balls'>Good covers by balls</a></li> </ul> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#References'>References</a></li> <ul> <li><a href='#geometric_2'>Geometric</a></li> <li><a href='#ReferencesStarShapedReasonDiffeomorphicToOpenBall'>Star-shaped regions diffeomorphic to open ball</a></li> <li><a href='#combinatorial_2'>Combinatorial</a></li> </ul> </ul> </div> <h2 id="definition">Definition</h2> <h3 id="geometric">Geometric</h3> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">n \in \mathbb{N}</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/natural+number">natural number</a>, the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/dimension">dimensional</a> <strong>ball</strong> or <strong><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>-disk</strong> 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> is the <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>D</mi> <mi>n</mi></msup><mo>:</mo><mo>=</mo><mo stretchy="false">{</mo><mover><mi>x</mi><mo stretchy="false">→</mo></mover><mo>∈</mo><msup><mi>ℝ</mi> <mi>n</mi></msup><mo stretchy="false">|</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mi>i</mi></munder><mo stretchy="false">(</mo><msup><mi>x</mi> <mi>i</mi></msup><msup><mo stretchy="false">)</mo> <mn>2</mn></msup><mo>≤</mo><mn>1</mn><mo stretchy="false">}</mo><mo>⊂</mo><msup><mi>ℝ</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex"> D^n := \{ \vec x \in \mathbb{R}^n | \sum_{i} (x^i)^2 \leq 1\} \subset \mathbb{R}^n </annotation></semantics></math></div> <p>equipped with the <a class="existingWikiWord" href="/nlab/show/induced+topology">induced topology</a> as a subspace of the <a class="existingWikiWord" href="/nlab/show/Cartesian+space">Cartesian space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^n</annotation></semantics></math>.</p> <p>Its <a class="existingWikiWord" href="/nlab/show/interior">interior</a> is the <strong>open <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>-ball</strong></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>𝔹</mi> <mi>n</mi></msup><mo>:</mo><mo>=</mo><mo stretchy="false">{</mo><mover><mi>x</mi><mo stretchy="false">→</mo></mover><mo>∈</mo><msup><mi>ℝ</mi> <mi>n</mi></msup><mo stretchy="false">|</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mi>i</mi></munder><mo stretchy="false">(</mo><msup><mi>x</mi> <mi>i</mi></msup><msup><mo stretchy="false">)</mo> <mn>2</mn></msup><mo><</mo><mn>1</mn><mo stretchy="false">}</mo><mo>⊂</mo><msup><mi>ℝ</mi> <mi>n</mi></msup><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \mathbb{B}^n := \{ \vec x \in \mathbb{R}^n | \sum_{i} (x^i)^2 \lt 1 \} \subset \mathbb{R}^n \,. </annotation></semantics></math></div> <p>Its <a class="existingWikiWord" href="/nlab/show/boundary">boundary</a> is the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(n-1)</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/sphere">sphere</a>.</p> <p>More generally, for <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>d</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(X,d)</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/metric+space">metric space</a> then an open ball 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> is a subset of the form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>B</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>r</mi><mo stretchy="false">)</mo><mo>≔</mo><mo stretchy="false">{</mo><mi>y</mi><mo>∈</mo><mi>X</mi><mspace width="thickmathspace"></mspace><mo stretchy="false">|</mo><mspace width="thickmathspace"></mspace><mi>d</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo><mo><</mo><mi>r</mi><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex"> B(x,r) \coloneqq \{y \in X \;|\; d(x,y) \lt r \} </annotation></semantics></math></div> <p>for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>∈</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">x \in X</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>r</mi><mo>∈</mo><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><mn>∞</mn><mo stretchy="false">)</mo><mo>⊂</mo><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">r \in (0,\infty) \subset \mathbb{R}</annotation></semantics></math>. (The collection of all open balls 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> form the <a class="existingWikiWord" href="/nlab/show/basis+of+a+topology">basis</a> of the <a class="existingWikiWord" href="/nlab/show/metric+topology">metric topology</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>.)</p> <h3 id="combinatorial">Combinatorial</h3> <p>There are also combinatorial notions of <em>disks</em>. For instance that due to (<a href="#Joyal">Joyal</a>), as entering the definition of the <a class="existingWikiWord" href="/nlab/show/Theta-category">Theta-category</a>. See for instance (<a href="#MakkaiZawadowski">Makkai-Zawadowski</a>).</p> <h2 id="Properties">Properties</h2> <h3 id="closed_balls">Closed balls</h3> <p>A simple result on the <em>homeomorphism</em> type of <em>closed</em> balls is the following:</p> <div class="num_theorem"> <h6 id="theorem">Theorem</h6> <p>A <a class="existingWikiWord" href="/nlab/show/compact+space">compact</a> <a class="existingWikiWord" href="/nlab/show/convex+subset">convex</a> <a class="existingWikiWord" href="/nlab/show/subset">subset</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math> 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> with <a class="existingWikiWord" href="/nlab/show/nonempty+set">nonempty</a> <a class="existingWikiWord" href="/nlab/show/interior">interior</a> is <a class="existingWikiWord" href="/nlab/show/homeomorphic">homeomorphic</a> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>D</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">D^n</annotation></semantics></math>.</p> </div> <div class="proof"> <h6 id="proof">Proof</h6> <p>Without loss of generality we may suppose the origin is an interior point of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math>. We claim that the map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi><mo>:</mo><mi>v</mi><mo>↦</mo><mi>v</mi><mo stretchy="false">/</mo><mrow><mo stretchy="false">‖</mo><mi>v</mi><mo stretchy="false">‖</mo></mrow></mrow><annotation encoding="application/x-tex">\phi: v \mapsto v/{\|v\|}</annotation></semantics></math> maps the boundary <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>∂</mo><mi>D</mi></mrow><annotation encoding="application/x-tex">\partial D</annotation></semantics></math> homeomorphically onto <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup></mrow><annotation encoding="application/x-tex">S^{n-1}</annotation></semantics></math>. By convexity, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math> is homeomorphic to the cone on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>∂</mo><mi>D</mi></mrow><annotation encoding="application/x-tex">\partial D</annotation></semantics></math>, and therefore to the cone on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup></mrow><annotation encoding="application/x-tex">S^{n-1}</annotation></semantics></math> which is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>D</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">D^n</annotation></semantics></math>.</p> <p>The claim reduces to the following three steps.</p> <ol> <li> <p>The restricted map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi><mo>:</mo><mo>∂</mo><mi>D</mi><mo>→</mo><msup><mi>S</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup></mrow><annotation encoding="application/x-tex">\phi: \partial D \to S^{n-1}</annotation></semantics></math> is continuous.</p> </li> <li> <p>It’s surjective: <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math> contains a ball <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi><mo>=</mo><msub><mi>B</mi> <mi>ε</mi></msub><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">B = B_{\varepsilon}(0)</annotation></semantics></math> in its interior, and for each <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>∈</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">x \in B</annotation></semantics></math>, the positive ray through <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> intersects <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math> in a bounded half-open line segment. For the extreme point <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> on this line segment, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi><mo stretchy="false">(</mo><mi>v</mi><mo stretchy="false">)</mo><mo>=</mo><mi>ϕ</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\phi(v) = \phi(x)</annotation></semantics></math>. Thus every unit vector <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>u</mi><mo>∈</mo><msup><mi>S</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup></mrow><annotation encoding="application/x-tex">u \in S^{n-1}</annotation></semantics></math> is of the form <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi><mo stretchy="false">(</mo><mi>v</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\phi(v)</annotation></semantics></math> for some extreme point <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>v</mi><mo>∈</mo><mi>D</mi></mrow><annotation encoding="application/x-tex">v \in D</annotation></semantics></math>, and such extreme points lie in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>∂</mo><mi>D</mi></mrow><annotation encoding="application/x-tex">\partial D</annotation></semantics></math>.</p> </li> <li> <p>It’s injective: for this we need to show that if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>v</mi><mo>,</mo><mi>w</mi><mo>∈</mo><mo>∂</mo><mi>D</mi></mrow><annotation encoding="application/x-tex">v, w \in \partial D</annotation></semantics></math> are distinct points, then neither is a positive multiple of the other. Supposing otherwise, we have <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>w</mi><mo>=</mo><mi>t</mi><mi>v</mi></mrow><annotation encoding="application/x-tex">w = 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><mn>1</mn></mrow><annotation encoding="application/x-tex">t \gt 1</annotation></semantics></math>, say. Let <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> be a ball inside <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math> containing <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>0</mn></mrow><annotation encoding="application/x-tex">0</annotation></semantics></math>; then the convex hull of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><mi>w</mi><mo stretchy="false">}</mo><mo>∪</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">\{w\} \cup B</annotation></semantics></math> is contained in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math> and contains <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> as an interior point, contradiction.</p> </li> </ol> <p>So the unit vector map, being a continuous bijection <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>∂</mo><mi>D</mi><mo>→</mo><msup><mi>S</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup></mrow><annotation encoding="application/x-tex">\partial D \to S^{n-1}</annotation></semantics></math> between <a class="existingWikiWord" href="/nlab/show/compact+Hausdorff+space">compact Hausdorff space</a>s, is a homeomorphism.</p> </div> <div class="num_cor"> <h6 id="corollary">Corollary</h6> <p>Any compact convex set <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math> of <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> is homeomorphic to a disk.</p> </div> <div class="proof"> <h6 id="proof_2">Proof</h6> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math> has nonempty interior relative to its affine span which is some <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math>-plane, and then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math> is homeomorphic to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>D</mi> <mi>k</mi></msup></mrow><annotation encoding="application/x-tex">D^k</annotation></semantics></math> by the theorem.</p> </div> <h3 id="open_balls">Open Balls</h3> <p>Open balls are a little less rigid than closed balls, in that one can more easily manipulate them within the <em>smooth</em> category:</p> <div class="num_lemma"> <h6 id="observation">Observation</h6> <p>The open <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>-ball is <a class="existingWikiWord" href="/nlab/show/homeomorphic">homeomorphic</a> and even <a class="existingWikiWord" href="/nlab/show/diffeomorphic">diffeomorphic</a> to the <a class="existingWikiWord" href="/nlab/show/Cartesian+space">Cartesian space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^n</annotation></semantics></math></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>𝔹</mi> <mi>n</mi></msup><mo>≃</mo><msup><mi>ℝ</mi> <mi>n</mi></msup><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \mathbb{B}^n \simeq \mathbb{R}^n \,. </annotation></semantics></math></div></div> <div class="proof"> <h6 id="proof_3">Proof</h6> <p>For instance, the smooth map</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>↦</mo><mfrac><mi>x</mi><msqrt><mrow><mn>1</mn><mo>+</mo><mo stretchy="false">|</mo><mi>x</mi><msup><mo stretchy="false">|</mo> <mn>2</mn></msup></mrow></msqrt></mfrac><mo>:</mo><msup><mi>ℝ</mi> <mi>n</mi></msup><mo>→</mo><msup><mi>𝔹</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex"> x\mapsto \frac{x}{\sqrt{1+|x|^2}} : \mathbb{R}^n \to \mathbb{B}^n </annotation></semantics></math></div> <p>has smooth inverse</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>y</mi><mo>↦</mo><mfrac><mi>y</mi><msqrt><mrow><mn>1</mn><mo>−</mo><mo stretchy="false">|</mo><mi>y</mi><msup><mo stretchy="false">|</mo> <mn>2</mn></msup></mrow></msqrt></mfrac><mo>:</mo><msup><mi>𝔹</mi> <mi>n</mi></msup><mo>→</mo><msup><mi>ℝ</mi> <mi>n</mi></msup><mo>.</mo></mrow><annotation encoding="application/x-tex"> y\mapsto \frac{y}{\sqrt{1-|y|^2}} : \mathbb{B}^n \to \mathbb{R}^n. </annotation></semantics></math></div></div> <p>This probe from <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> witnesses the property that the open <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>-ball is a (<a class="existingWikiWord" href="/nlab/show/smooth+manifold">smooth</a>) <a class="existingWikiWord" href="/nlab/show/manifold">manifold</a>. Hence, each (smooth) <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>-dimensional manifold is locally isomorphic to both <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> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>𝔹</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">\mathbb{B}^n</annotation></semantics></math>.</p> <p>From general existence results about <a class="existingWikiWord" href="/nlab/show/smooth+structure">smooth structure</a>s on <a class="existingWikiWord" href="/nlab/show/Cartesian+space">Cartesian space</a>s we have that</p> <div class="num_theorem"> <h6 id="theorem_2">Theorem</h6> <p>In <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>d</mi><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">d \in \mathbb{N}</annotation></semantics></math> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi><mo>≠</mo><mn>4</mn></mrow><annotation encoding="application/x-tex">d \neq 4</annotation></semantics></math> we have:</p> <p>every open subset of <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> which is <a class="existingWikiWord" href="/nlab/show/homeomorphic">homeomorphic</a> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>𝔹</mi> <mi>d</mi></msup></mrow><annotation encoding="application/x-tex">\mathbb{B}^d</annotation></semantics></math> is also <a class="existingWikiWord" href="/nlab/show/diffeomorphic">diffeomorphic</a> to it.</p> </div> <p>See the first page of (<a href="#Ozols">Ozols</a>) for a list of references.</p> <div class="num_remark"> <h6 id="remark">Remark</h6> <p>In dimension 4 the analog statement fails due to the existence of <a class="existingWikiWord" href="/nlab/show/exotic+smooth+structure">exotic smooth structure</a>s on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mn>4</mn></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^4</annotation></semantics></math>. See <a href="#DeMFreed">De Michelis-Freedman</a>.</p> </div> <div class="num_theorem" id="StarShapedOpenDiffeomorphicToOpenBall"> <h6 id="theorem_3">Theorem</h6> <p><strong>(star-shaped domains are diffeomorphic to open balls)</strong></p> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo>⊂</mo><msup><mi>ℝ</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">C \subset \mathbb{R}^n</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/star-shaped">star-shaped</a> <a class="existingWikiWord" href="/nlab/show/open+subset">open subset</a> of a <a class="existingWikiWord" href="/nlab/show/Cartesian+space">Cartesian space</a>. Then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/diffeomorphic">diffeomorphic</a> to <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> <div class="num_remark" id="LiteratureOnStarShapedOpenDiffeoToOpenBall"> <h6 id="remark_2">Remark</h6> <p>Theorem <a class="maruku-ref" href="#StarShapedOpenDiffeomorphicToOpenBall"></a> is a <a class="existingWikiWord" href="/nlab/show/folk+theorem">folk theorem</a>, but explicit <strong>proofs</strong> in the literature are hard to find. See the discussion in the References-section <em><a href="#ReferencesStarShapedReasonDiffeomorphicToOpenBall">here</a></em>. An explicit proof has been written out by Stefan Born, and this appears as the proof of <a href="http://www.math.tu-berlin.de/~ferus/ANA/Ana3.pdf#page=154">theorem 237</a> in (<a href="#Ferus07">Ferus 07</a>). A simpler proof is given in <a href="#GonnordTosel98">Gonnord-Tosel 98</a> reproduced <a href="http://mathoverflow.net/a/212595/381">here</a>.</p> </div> <p>Here is another proof:</p> <div class="proof"> <h6 id="proof_4">Proof</h6> <p>Suppose <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math> is a star-shaped open subset of <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> centered at the origin. Theorem 2.29 in <a href="#Lee09">Lee 2009</a> proves that there is a function <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> on <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> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>></mo><mn>0</mn></mrow><annotation encoding="application/x-tex">f\gt 0</annotation></semantics></math> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math> and <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> vanishes on the complement of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math>. By applying <a class="existingWikiWord" href="/nlab/show/bump+functions">bump functions</a> we can assume that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>≤</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">f\le 1</annotation></semantics></math> everywhere and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">f=1</annotation></semantics></math> in an open <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϵ</mi></mrow><annotation encoding="application/x-tex">\epsilon</annotation></semantics></math>-neighborhood of the origin; by rescaling the ambient space we can assume <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϵ</mi><mo>=</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">\epsilon=2</annotation></semantics></math>.</p> <p>The smooth vector field <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi><mo lspace="verythinmathspace">:</mo><mi>x</mi><mo>↦</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>⋅</mo><mi>x</mi><mo stretchy="false">/</mo><mrow><mo stretchy="false">‖</mo><mi>x</mi><mo stretchy="false">‖</mo></mrow></mrow><annotation encoding="application/x-tex">V\colon x\mapsto f(x)\cdot x/{\|x\|}</annotation></semantics></math> is defined on the complement of the origin in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math>. Multiply <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> by a smooth bump function <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>0</mn><mo>≤</mo><mi>b</mi><mo>≤</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">0\le b\le 1</annotation></semantics></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>b</mi><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">b=1</annotation></semantics></math> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mrow><mo stretchy="false">‖</mo><mi>x</mi><mo stretchy="false">‖</mo></mrow><mo>></mo><mn>1</mn><mo stretchy="false">/</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">{\|x\|} \gt 1/2</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>b</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">b=0</annotation></semantics></math> in a neighborhood of 0. The new vector field <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> extends smoothly to the origin and defines a smooth global flow <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo lspace="verythinmathspace">:</mo><mi>ℝ</mi><mo>×</mo><mi>T</mi><mo>→</mo><mi>T</mi></mrow><annotation encoding="application/x-tex">F\colon \mathbb{R} \times T\to T</annotation></semantics></math>. (The parameter of the flow is all of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">\mathbb{R}</annotation></semantics></math> and not just some interval <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>∞</mn><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(-\infty,A)</annotation></semantics></math> because the norm 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> is bounded by 1.) Observe that for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>1</mn><mo stretchy="false">/</mo><mn>2</mn><mo><</mo><mrow><mo stretchy="false">‖</mo><mi>x</mi><mo stretchy="false">‖</mo></mrow><mo><</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">1/2\lt {\|x\|} \lt 2</annotation></semantics></math> the vector field <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> equals <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>↦</mo><mi>x</mi><mo stretchy="false">/</mo><mrow><mo stretchy="false">‖</mo><mi>x</mi><mo stretchy="false">‖</mo></mrow></mrow><annotation encoding="application/x-tex">x\mapsto x/{\|x\|}</annotation></semantics></math>. Also, all flow lines 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> are radial rays.</p> <p>Now define the flow map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo lspace="verythinmathspace">:</mo><msubsup><mi>ℝ</mi> <mrow><mo>></mo><mn>1</mn><mo stretchy="false">/</mo><mn>2</mn></mrow> <mi>n</mi></msubsup><mo>→</mo><msub><mi>T</mi> <mrow><mo>></mo><mn>1</mn><mo stretchy="false">/</mo><mn>2</mn></mrow></msub></mrow><annotation encoding="application/x-tex">p\colon{\mathbb{R}}^n_{\gt 1/2}\to T_{\gt 1/2}</annotation></semantics></math> as <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>↦</mo><mi>F</mi><mo stretchy="false">(</mo><mrow><mo stretchy="false">‖</mo><mi>x</mi><mo stretchy="false">‖</mo></mrow><mo>−</mo><mn>1</mn><mo>,</mo><mfrac><mi>x</mi><mrow><mo stretchy="false">‖</mo><mi>x</mi><mo stretchy="false">‖</mo></mrow></mfrac><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">x\mapsto F({\|x\|}-1, \frac{x}{{\|x\|}})</annotation></semantics></math> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mrow><mo stretchy="false">‖</mo><mi>x</mi><mo stretchy="false">‖</mo></mrow><mo>></mo><mn>1</mn><mo stretchy="false">/</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">{\|x\|} \gt 1/2</annotation></semantics></math>. (The subscript <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>></mo><mn>1</mn><mo stretchy="false">/</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">\gt 1/2</annotation></semantics></math> removes the closed ball of radius <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>1</mn><mo stretchy="false">/</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">1/2</annotation></semantics></math>.) The flow map is the composition of two diffeomorphisms,</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msubsup><mi>ℝ</mi> <mrow><mo>></mo><mn>1</mn><mo stretchy="false">/</mo><mn>2</mn></mrow> <mi>n</mi></msubsup><mo>→</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn><mo stretchy="false">/</mo><mn>2</mn><mo>,</mo><mn>∞</mn><mo stretchy="false">)</mo><mo>×</mo><msup><mi>S</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup><mo>→</mo><msub><mi>T</mi> <mrow><mo>></mo><mn>1</mn><mo stretchy="false">/</mo><mn>2</mn></mrow></msub><mo>,</mo></mrow><annotation encoding="application/x-tex">{\mathbb{R}}^n_{\gt 1/2}\to(-1/2,\infty)\times S^{n-1} \to T_{\gt 1/2},</annotation></semantics></math></div> <p>hence itself is a diffeomorphism. (Note particularly that the latter map is surjective. In detail: a flow line is a smooth map of the form <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>L</mi><mo>:</mo><mo stretchy="false">(</mo><mi>A</mi><mo>,</mo><mi>B</mi><mo stretchy="false">)</mo><mo>→</mo><mi>T</mi></mrow><annotation encoding="application/x-tex">L: (A,B) \to T</annotation></semantics></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math> can be finite or infinite. If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math> is finite and the limit of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>L</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">L(t)</annotation></semantics></math> as <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>t</mi><mo>→</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">t \to B</annotation></semantics></math> exists, then the vector field <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> vanishes at <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>. In our case <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> can only vanish at the boundary of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math>, which is precisely what we want for surjectivity.)</p> <p>Finally, define the desired diffeomorphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi><mo lspace="verythinmathspace">:</mo><msup><mi>ℝ</mi> <mi>n</mi></msup><mo>→</mo><mi>T</mi></mrow><annotation encoding="application/x-tex">d\colon{\mathbb{R}}^n\to T</annotation></semantics></math> as the gluing of the identity map for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mrow><mo stretchy="false">‖</mo><mi>x</mi><mo stretchy="false">‖</mo></mrow><mo><</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">{\|x\|} \lt 2</annotation></semantics></math> and as <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> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mrow><mo stretchy="false">‖</mo><mi>x</mi><mo stretchy="false">‖</mo></mrow><mo>></mo><mn>1</mn><mo stretchy="false">/</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">{\|x\|}\gt 1/2</annotation></semantics></math>. The map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi></mrow><annotation encoding="application/x-tex">g</annotation></semantics></math> is smooth because for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>1</mn><mo stretchy="false">/</mo><mn>2</mn><mo><</mo><mrow><mo stretchy="false">‖</mo><mi>x</mi><mo stretchy="false">‖</mo></mrow><mo><</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">1/2\lt {\|x\|} \lt 2</annotation></semantics></math> both definitions give the same value.</p> </div> <p>And here is another proof, due to Gonnord and Tosel, translated into English by Erwann Aubry and available on MathOverflow:</p> <p> <div class='num_theorem'> <h6>Theorem</h6> <p>Every open star-shaped set <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ω</mi></mrow><annotation encoding="application/x-tex">\Omega</annotation></semantics></math> 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> is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mn>∞</mn></msup></mrow><annotation encoding="application/x-tex">C^\infty</annotation></semantics></math>-diffeomorphic to <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> <div class='proof'> <h6>Proof</h6> <p>For convenience assume that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ω</mi></mrow><annotation encoding="application/x-tex">\Omega</annotation></semantics></math> is star-shaped at <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>0</mn></mrow><annotation encoding="application/x-tex">0</annotation></semantics></math>.</p> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo>=</mo><msup><mstyle mathvariant="bold"><mi>R</mi></mstyle> <mi>n</mi></msup><mo>∖</mo><mi>Ω</mi></mrow><annotation encoding="application/x-tex">F=\mathbf{R}^n\setminus\Omega</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi><mo>:</mo><msup><mstyle mathvariant="bold"><mi>R</mi></mstyle> <mi>n</mi></msup><mo>→</mo><msub><mi>ℝ</mi> <mo>+</mo></msub></mrow><annotation encoding="application/x-tex">\phi:\mathbf{R}^n\rightarrow\mathbb{R}_+</annotation></semantics></math> (here <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mstyle mathvariant="bold"><mi>R</mi></mstyle> <mo>+</mo></msub><mo>=</mo><mo stretchy="false">[</mo><mn>0</mn><mo>,</mo><mn>∞</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbf{R}_+=[0,\infty)</annotation></semantics></math>) be a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mn>∞</mn></msup></mrow><annotation encoding="application/x-tex">C^\infty</annotation></semantics></math>-function such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo>=</mo><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">F=\phi^{-1}(\{0\})</annotation></semantics></math>. (Such <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> exists by the <a class="existingWikiWord" href="/nlab/show/Whitney+extension+theorem">Whitney extension theorem</a>.)</p> <p>Now we define <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><mi>Ω</mi><mo>→</mo><msup><mi>ℝ</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">f:\Omega\rightarrow\mathbb{R}^n</annotation></semantics></math> via the formula:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><msup><mover><mrow><mo>[</mo><mn>1</mn><mo>+</mo><msup><mrow><mo>(</mo><msubsup><mo>∫</mo> <mn>0</mn> <mn>1</mn></msubsup><mfrac><mi>dv</mi><mrow><mi>ϕ</mi><mo stretchy="false">(</mo><mi>vx</mi><mo stretchy="false">)</mo></mrow></mfrac><mo>)</mo></mrow> <mn>2</mn></msup><mo stretchy="false">‖</mo><mi>x</mi><msup><mo stretchy="false">‖</mo> <mn>2</mn></msup><mo>]</mo></mrow><mo>⏞</mo></mover> <mrow><mi>λ</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></msup><mo>⋅</mo><mi>x</mi><mo>=</mo><mrow><mo>[</mo><mn>1</mn><mo>+</mo><msup><mrow><mo>(</mo><msubsup><mo>∫</mo> <mn>0</mn> <mrow><mo stretchy="false">‖</mo><mi>x</mi><mo stretchy="false">‖</mo></mrow></msubsup><mfrac><mi>dt</mi><mrow><mi>ϕ</mi><mo stretchy="false">(</mo><mi>t</mi><mfrac><mi>x</mi><mrow><mo stretchy="false">‖</mo><mi>x</mi><mo stretchy="false">‖</mo></mrow></mfrac><mo stretchy="false">)</mo></mrow></mfrac><mo>)</mo></mrow> <mn>2</mn></msup><mo>]</mo></mrow><mo>⋅</mo><mi>x</mi><mo>.</mo></mrow><annotation encoding="application/x-tex">f(x)=\overbrace{\left[1+\left(\int_0^1\frac{dv}{\phi(vx)}\right)^2\|x\|^2\right]}^{\lambda(x)}\cdot x=\left[1+\left(\int_0^{\|x\|}\frac{dt}{\phi(t\frac{x}{\|x\|})}\right)^2\right]\cdot x.</annotation></semantics></math></div> <p>Clearly <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 smooth on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ω</mi></mrow><annotation encoding="application/x-tex">\Omega</annotation></semantics></math>.</p> <p>We set <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mi>sup</mi><mo stretchy="false">{</mo><mi>t</mi><mo>></mo><mn>0</mn><mo lspace="mediummathspace" rspace="mediummathspace">∣</mo><mi>t</mi><mfrac><mi>x</mi><mrow><mo stretchy="false">‖</mo><mi>x</mi><mo stretchy="false">‖</mo></mrow></mfrac><mo>∈</mo><mi>Ω</mi><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">A(x)=\sup\{t\gt0\mid t\frac{x}{\|x\|}\in\Omega\}</annotation></semantics></math>. <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> sends injectively the segment (or ray) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mn>0</mn><mo>,</mo><mi>A</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mfrac><mi>x</mi><mrow><mo stretchy="false">‖</mo><mi>x</mi><mo stretchy="false">‖</mo></mrow></mfrac></mrow><annotation encoding="application/x-tex">[0,A(x))\frac{x}{\|x\|}</annotation></semantics></math> to the ray <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mstyle mathvariant="bold"><mi>R</mi></mstyle> <mo>+</mo></msub><mfrac><mi>x</mi><mrow><mo stretchy="false">‖</mo><mi>x</mi><mo stretchy="false">‖</mo></mrow></mfrac></mrow><annotation encoding="application/x-tex">\mathbf{R}_+\frac{x}{\|x\|}</annotation></semantics></math>. Moreover, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mn>0</mn><mfrac><mi>x</mi><mrow><mo stretchy="false">‖</mo><mi>X</mi><mo stretchy="false">‖</mo></mrow></mfrac><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">f(0\frac{x}{\|X\|})=0</annotation></semantics></math> and</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><munder><mi>lim</mi> <mrow><mi>r</mi><mo>→</mo><mi>A</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></munder><mrow><mo>‖</mo><mi>f</mi><mo stretchy="false">(</mo><mi>r</mi><mfrac><mi>x</mi><mrow><mo stretchy="false">‖</mo><mi>x</mi><mo stretchy="false">‖</mo></mrow></mfrac><mo stretchy="false">)</mo><mo>‖</mo></mrow><mo>=</mo><munder><mi>lim</mi> <mrow><mi>r</mi><mo>→</mo><mi>A</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></munder><mrow><mo>[</mo><mn>1</mn><mo>+</mo><msup><mrow><mo>(</mo><msubsup><mo>∫</mo> <mn>0</mn> <mi>r</mi></msubsup><mfrac><mi>dt</mi><mrow><mi>ϕ</mi><mrow><mo>(</mo><mi>t</mi><mo>⋅</mo><mfrac><mi>rx</mi><mrow><mo stretchy="false">‖</mo><mi>x</mi><mo stretchy="false">‖</mo></mrow></mfrac><mo>⋅</mo><mrow><mo>‖</mo><mfrac><mrow><mo stretchy="false">‖</mo><mi>x</mi><mo stretchy="false">‖</mo></mrow><mi>rx</mi></mfrac><mo>‖</mo></mrow><mo>)</mo></mrow></mrow></mfrac><mo>)</mo></mrow> <mn>2</mn></msup><mo>]</mo></mrow><mo>⋅</mo><mi>r</mi><mo>=</mo><mrow><mo>[</mo><mn>1</mn><mo>+</mo><msup><mrow><mo>(</mo><msubsup><mo>∫</mo> <mn>0</mn> <mrow><mi>A</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></msubsup><mfrac><mi>dt</mi><mrow><mi>ϕ</mi><mo stretchy="false">(</mo><mi>t</mi><mfrac><mi>x</mi><mrow><mo stretchy="false">‖</mo><mi>x</mi><mo stretchy="false">‖</mo></mrow></mfrac><mo stretchy="false">)</mo></mrow></mfrac><mo>)</mo></mrow> <mn>2</mn></msup><mo>]</mo></mrow><mo>⋅</mo><mi>A</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mo lspace="verythinmathspace" rspace="0em">+</mo><mn>∞</mn><mo>.</mo></mrow><annotation encoding="application/x-tex">\lim_{r\rightarrow A(x)}\left\|f(r\frac{x}{\|x\|})\right\|=\lim_{r\to A(x)}\left[1+\left(\int_0^{r}\frac{dt}{\phi\left(t\cdot\frac{rx}{\|x\|}\cdot\left\|\frac{\|x\|}{rx}\right\|\right)}\right)^2\right]\cdot r= \left[1+\left(\int_0^{A(x)}\frac{dt}{\phi(t\frac{x}{\|x\|})}\right)^2\right]\cdot A(x)=+\infty.</annotation></semantics></math></div> <p>Indeed, if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mo lspace="verythinmathspace" rspace="0em">+</mo><mn>∞</mn></mrow><annotation encoding="application/x-tex">A(x)=+\infty</annotation></semantics></math>, then it holds for obvious reason. If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo><</mo><mo lspace="verythinmathspace" rspace="0em">+</mo><mn>∞</mn></mrow><annotation encoding="application/x-tex">A(x)\lt+\infty</annotation></semantics></math>, then by definitions of <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> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">A(x)</annotation></semantics></math> we get that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mfrac><mi>x</mi><mrow><mo stretchy="false">‖</mo><mi>x</mi><mo stretchy="false">‖</mo></mrow></mfrac><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\phi(A(x)\frac{x}{\|x\|})=0</annotation></semantics></math>. Hence by the <a class="existingWikiWord" href="/nlab/show/mean+value+theorem">mean value theorem</a> and the fact that <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 <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mn>1</mn></msup></mrow><annotation encoding="application/x-tex">C^1</annotation></semantics></math> due to</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi><mrow><mo>(</mo><mi>r</mi><mfrac><mi>x</mi><mrow><mo stretchy="false">‖</mo><mi>x</mi><mo stretchy="false">‖</mo></mrow></mfrac><mo>)</mo></mrow><mo>≤</mo><mi>M</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>−</mo><mi>r</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\phi\left(r\frac{x}{\|x\|}\right)\le M(A(x)-r)</annotation></semantics></math></div> <p>for some constant <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math> and every <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>r</mi></mrow><annotation encoding="application/x-tex">r</annotation></semantics></math>. As a result,</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msubsup><mo>∫</mo> <mn>0</mn> <mrow><mi>A</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></msubsup><mfrac><mi>dt</mi><mrow><mi>ϕ</mi><mrow><mo>(</mo><mi>t</mi><mfrac><mi>x</mi><mrow><mo stretchy="false">‖</mo><mi>x</mi><mo stretchy="false">‖</mo></mrow></mfrac><mo>)</mo></mrow></mrow></mfrac></mrow><annotation encoding="application/x-tex">\int_0^{A(x)}\frac{dt}{\phi\left(t\frac{x}{\|x\|}\right)}</annotation></semantics></math></div> <p>diverges. Hence we infer that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mo stretchy="false">[</mo><mn>0</mn><mo>,</mo><mi>A</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mfrac><mi>x</mi><mrow><mo stretchy="false">‖</mo><mi>x</mi><mo stretchy="false">‖</mo></mrow></mfrac><mo stretchy="false">)</mo><mo>=</mo><msub><mstyle mathvariant="bold"><mi>R</mi></mstyle> <mo>+</mo></msub><mfrac><mi>x</mi><mrow><mo stretchy="false">‖</mo><mi>x</mi><mo stretchy="false">‖</mo></mrow></mfrac></mrow><annotation encoding="application/x-tex">f([0,A(x))\frac{x}{\|x\|})=\mathbf{R}_+\frac{x}{\|x\|}</annotation></semantics></math> and so <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>Ω</mi><mo stretchy="false">)</mo><mo>=</mo><msup><mstyle mathvariant="bold"><mi>R</mi></mstyle> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">f(\Omega)=\mathbf{R}^n</annotation></semantics></math>.</p> <p>To end the proof we need to show that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> has a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mn>∞</mn></msup></mrow><annotation encoding="application/x-tex">C^\infty</annotation></semantics></math>-inverse. But as a corollary from the <a class="existingWikiWord" href="/nlab/show/inverse+function+theorem">inverse function theorem</a> we get that it is sufficient to show that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>df</mi></mrow><annotation encoding="application/x-tex">df</annotation></semantics></math> vanishes nowhere.</p> <p>Suppose that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>d</mi> <mi>x</mi></msub><mi>f</mi><mo stretchy="false">(</mo><mi>h</mi><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">d_x f(h)=0</annotation></semantics></math> for some <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>∈</mo><mi>Ω</mi></mrow><annotation encoding="application/x-tex">x\in\Omega</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>h</mi><mo>≠</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">h\neq 0</annotation></semantics></math>. From definition of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> we get that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>d</mi> <mi>x</mi></msub><mi>f</mi><mo stretchy="false">(</mo><mi>h</mi><mo stretchy="false">)</mo><mo>=</mo><mi>λ</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mi>h</mi><mo>+</mo><msub><mi>d</mi> <mi>x</mi></msub><mi>λ</mi><mo stretchy="false">(</mo><mi>h</mi><mo stretchy="false">)</mo><mi>x</mi><mo>.</mo></mrow><annotation encoding="application/x-tex">d_x f(h)=\lambda(x)h+d_x \lambda(h)x.</annotation></semantics></math></div> <p>Hence <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>h</mi><mo>=</mo><mi>μ</mi><mi>x</mi></mrow><annotation encoding="application/x-tex">h=\mu x</annotation></semantics></math> for some <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>μ</mi><mo>≠</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\mu\neq 0</annotation></semantics></math> and from that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>≠</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">x\neq 0</annotation></semantics></math>. As a result <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>λ</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>+</mo><msub><mi>d</mi> <mi>x</mi></msub><mi>λ</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\lambda(x)+d_x \lambda(x)=0</annotation></semantics></math>. But we have that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>λ</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>≥</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">\lambda(x)\ge1</annotation></semantics></math> and function <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>:</mo><mo>=</mo><mi>λ</mi><mo stretchy="false">(</mo><mi>tx</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">g(t):=\lambda(tx)</annotation></semantics></math> is increasing, so <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi><mo>′</mo><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo>=</mo><msub><mi>d</mi> <mi>x</mi></msub><mi>λ</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>></mo><mn>0</mn></mrow><annotation encoding="application/x-tex">g'(1)=d_x \lambda(x)\gt0</annotation></semantics></math>, which gives a contradiction.</p> </div> </p> <div class="num_example"> <h6 id="example">Example</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>I</mi><mo stretchy="false">(</mo><msup><mi>Δ</mi> <mi>n</mi></msup><mo stretchy="false">)</mo><mo>⊂</mo><msup><mi>ℝ</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">I(\Delta^n) \subset \mathbb{R}^n</annotation></semantics></math> be the <a class="existingWikiWord" href="/nlab/show/interior">interior</a> of the standard <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>-<a class="existingWikiWord" href="/nlab/show/simplex">simplex</a>. Then there is a diffeomorphism to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>𝔹</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">\mathbb{B}^n</annotation></semantics></math> defined as follows:</p> <p>Parameterize the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-simplex as</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>I</mi><mo stretchy="false">(</mo><msup><mi>Δ</mi> <mi>n</mi></msup><mo stretchy="false">)</mo><mo>=</mo><mrow><mo>{</mo><mo stretchy="false">(</mo><msup><mi>x</mi> <mn>1</mn></msup><mo>,</mo><mi>⋯</mi><mo>,</mo><msup><mi>x</mi> <mi>n</mi></msup><mo stretchy="false">)</mo><mo>∈</mo><mi>ℝ</mi><mo stretchy="false">|</mo><mo stretchy="false">(</mo><mo>∀</mo><mi>i</mi><mo>:</mo><msup><mi>x</mi> <mi>i</mi></msup><mo>></mo><mn>0</mn><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mi>and</mi><mspace width="thickmathspace"></mspace><mo stretchy="false">(</mo><munderover><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow> <mi>n</mi></munderover><msup><mi>x</mi> <mi>i</mi></msup><mo><</mo><mn>1</mn><mo stretchy="false">)</mo><mo>}</mo></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> I(\Delta^n) = \left\{ (x^1, \cdots, x^n) \in \mathbb{R} | (\forall i : x^i \gt 0)\; and \; ( \sum_{i=1}^n x^i \lt 1) \right\} \,. </annotation></semantics></math></div> <p>Then define the map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><mi>I</mi><mo stretchy="false">(</mo><msup><mi>Δ</mi> <mi>n</mi></msup><mo stretchy="false">)</mo><mo>→</mo><msup><mi>ℝ</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">f : I(\Delta^n) \to \mathbb{R}^n</annotation></semantics></math> by</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><msup><mi>x</mi> <mn>1</mn></msup><mo>,</mo><mi>…</mi><mo>,</mo><msup><mi>x</mi> <mi>n</mi></msup><mo stretchy="false">)</mo><mo>↦</mo><mo stretchy="false">(</mo><mi>log</mi><mo stretchy="false">(</mo><mfrac><mrow><msup><mi>x</mi> <mn>1</mn></msup></mrow><mrow><mn>1</mn><mo>−</mo><msup><mi>x</mi> <mn>1</mn></msup><mo>−</mo><mi>…</mi><mo>−</mo><msup><mi>x</mi> <mi>n</mi></msup></mrow></mfrac><mo stretchy="false">)</mo><mo>,</mo><mi>…</mi><mo>,</mo><mi>log</mi><mo stretchy="false">(</mo><mfrac><mrow><msup><mi>x</mi> <mi>n</mi></msup></mrow><mrow><mn>1</mn><mo>−</mo><msup><mi>x</mi> <mn>1</mn></msup><mo>−</mo><mi>…</mi><mo>−</mo><msup><mi>x</mi> <mi>n</mi></msup></mrow></mfrac><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> (x^1, \ldots, x^n) \mapsto (\log(\frac{x^1}{1 - x^1 - \ldots -x^n}), \ldots, \log(\frac{x^n}{1 - x^1 - \ldots - x^n})) \,. </annotation></semantics></math></div></div> <p>(Thanks to <a class="existingWikiWord" href="/nlab/show/Todd+Trimble">Todd Trimble</a>.) One way to think about it is that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>I</mi><mo stretchy="false">(</mo><msup><mi>Δ</mi> <mi>n</mi></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">I(\Delta^n)</annotation></semantics></math> is the positive orthant of an open <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>-ball in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>l</mi> <mn>1</mn></msup></mrow><annotation encoding="application/x-tex">l^1</annotation></semantics></math> norm, so that in the opposite direction we have a chain of invertible maps</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><msup><mi>ℝ</mi> <mi>n</mi></msup></mtd> <mtd><mover><mo>→</mo><mrow><msup><mi>exp</mi> <mi>n</mi></msup></mrow></mover></mtd> <mtd><msubsup><mi>ℝ</mi> <mo>+</mo> <mi>n</mi></msubsup></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>I</mi><mo stretchy="false">(</mo><msup><mi>Δ</mi> <mi>n</mi></msup><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mover><mi>x</mi><mo stretchy="false">→</mo></mover></mtd> <mtd><mo>↦</mo></mtd> <mtd><mover><mi>x</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">/</mo><mo stretchy="false">(</mo><mn>1</mn><mo>+</mo><msub><mrow><mo stretchy="false">‖</mo><mover><mi>x</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">‖</mo></mrow> <mn>1</mn></msub><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex">\array{ \mathbb{R}^n & \stackrel{\exp^n}{\to} & \mathbb{R}_+^n & \to & I(\Delta^n) \\ & & \vec{x} & \mapsto & \vec{x}/(1 + {\|\vec{x}\|}_1) } </annotation></semantics></math></div> <p>which we simply invert to get the map <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> above.</p> <h3 id="good_covers_by_balls">Good covers by balls</h3> <p>One central application of balls is as building blocks for <a class="existingWikiWord" href="/nlab/show/covering">covering</a>s. See <a class="existingWikiWord" href="/nlab/show/good+open+cover">good open cover</a> for some statements.</p> <h2 id="related_concepts">Related concepts</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/unit+ball">unit ball</a></li> </ul> <h2 id="References">References</h2> <h3 id="geometric_2">Geometric</h3> <ul> <li id="Ozols">V. Ozols, <em>Largest normal neighbourhoods</em> , Proceedings of the American Mathematical Society Vol. 61, No. 1 (Nov., 1976), pp. 99-101 (<a href="http://www.jstor.org/stable/2041672">jstor</a>)</li> </ul> <p>That an open subset <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo>⊆</mo><msup><mi>ℝ</mi> <mn>4</mn></msup></mrow><annotation encoding="application/x-tex">U \subseteq \mathbb{R}^4</annotation></semantics></math> homeomorphic to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mn>4</mn></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^4</annotation></semantics></math> equipped with the smooth structure inherited as an open submanifold of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mn>4</mn></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^4</annotation></semantics></math> might nevertheless be non-diffeomorphic to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mn>4</mn></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^4</annotation></semantics></math>, see</p> <ul id="DeMFreed"> <li>De Michelis, Stefano; Freedman, Michael H. (1992) “Uncountably many exotic <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mn>4</mn></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^4</annotation></semantics></math>‘s in standard 4-space”, J. Diff. Geom. 35, pp. 219-254.</li> </ul> <h3 id="ReferencesStarShapedReasonDiffeomorphicToOpenBall">Star-shaped regions diffeomorphic to open ball</h3> <p>The proof that open star-shaped regions are diffeomorphic to a ball appears as</p> <ul> <li id="Ferus07"><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>)</li> </ul> <p>It is a lengthy proof, due to Stefan Born.</p> <p>A simpler version of the proof appears in</p> <ul> <li id="GonnordTosel98">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>)</li> </ul> <p>These proofs had remained obscure (see also <a href="good+open+cover#LiteratureOnExistenceOfDifferentiablyGoodOpenCovers">this Remark</a> at <a class="existingWikiWord" href="/nlab/show/good+open+cover">good open cover</a>):</p> <p>For instance in a remark below lemma 10.5.5 of</p> <ul> <li id="Conlon08">Lawrence Conlon, <em>Differentiable manifolds</em>, Birkhäuser 2001/2008 (<a href="https://link.springer.com/book/10.1007/978-0-8176-4767-4">doi:10.1007/978-0-8176-4767-4</a>)</li> </ul> <p>it says:</p> <blockquote> <p>It seems that open star shaped sets <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo>⊂</mo><mi>M</mi></mrow><annotation encoding="application/x-tex">U \subset M</annotation></semantics></math> are always diffeomorphic to <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>, but this is extremely difficult to prove.</p> </blockquote> <p>And in</p> <ul> <li id="Lee09"><a class="existingWikiWord" href="/nlab/show/Jeffrey+Lee">Jeffrey Lee</a>, <em>Manifolds and differential geometry</em>, Graduate Studies in Mathematics <strong>107</strong> (2009) (<a href="https://bookstore.ams.org/gsm-107">ISBN: 978-0-8218-4815-9</a>, <a href="https://doi.org/10.1090/gsm/107">doi:10.1090/gsm/107</a>)</li> </ul> <p>one finds the statement:</p> <blockquote> <p>Actually, the assertion that an open geodesically convex set in a Riemannian manifold is diffeomorphic to <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> is common in literature, but it is a more subtle issue than it may seem, and references to a complete proof are hard to find (but see [Grom]).</p> </blockquote> <p>Here “Grom” refers to</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Mikhail+Gromov">Mikhail Gromov</a>, <em>Convex sets and Kähler manifolds</em>, in F. Tricerri (ed.) <em>Advances in differential geometry and topology</em>, World Scientific (1990) 1-38 [<a href="http://www.ihes.fr/~gromov/PDF/%5B68%5D.pdf">pdf</a>]</li> </ul> <p>where the relevant statement is 1.4.C1 on page 8. Note however that the diffeomorphism considered there is only of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mn>1</mn></msup></mrow><annotation encoding="application/x-tex">C^1</annotation></semantics></math> class, not <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mn>∞</mn></msup></mrow><annotation encoding="application/x-tex">C^\infty</annotation></semantics></math>, so this is not a proof either.</p> <p>A series of exercises covering the statement appears in</p> <ul> <li id="GuilleminHaine19"><a class="existingWikiWord" href="/nlab/show/Victor+Guillemin">Victor Guillemin</a>, <a class="existingWikiWord" href="/nlab/show/Peter+Haine">Peter Haine</a>, Exercises 5.3.i–5.3.iv in: <em>Differential Forms</em>, World Scientific (2019) [<a href="https://doi.org/10.1142/11058">doi:10.1142/11058</a>]</li> </ul> <p>See also the discussion at:</p> <ul> <li>Math Overflow, <em>explicit diffeomorphim between open simplex and open ball</em> [<a href="https://mathoverflow.net/q/41853/381">q/41853</a>]</li> </ul> <h3 id="combinatorial_2">Combinatorial</h3> <ul> <li id="Joyal"> <p><a class="existingWikiWord" href="/nlab/show/Andre+Joyal">Andre Joyal</a>, <em>Disks, duality and Theta-categories</em> (<a class="existingWikiWord" href="/nlab/files/JoyalThetaCategories.pdf" title="pdf">pdf</a>)</p> </li> <li id="MakkaiZawadowski"> <p><a class="existingWikiWord" href="/nlab/show/Mihaly+Makkai">Mihaly Makkai</a>, Marek Zawadowski, <em>Duality for Simple <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ω</mi></mrow><annotation encoding="application/x-tex">\omega</annotation></semantics></math>-Categories and Disks</em> (<a href="http://www.emis.de/journals/TAC/volumes/8/n7/8-07abs.html">TAC</a>)</p> </li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on June 2, 2024 at 07:03:58. 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