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value="Search" style="display:inline-block; float: left;" onfocus="this.value == 'Search' ? this.value = '' : true" onblur="this.value == '' ? this.value = 'Search' : true" /></fieldset> </form> <span id='navEnd'></span> </div> <div id="revision"> <html xmlns="http://www.w3.org/1999/xhtml" xmlns:svg="http://www.w3.org/2000/svg" xml:lang="en" lang="en"> <head><meta http-equiv="Content-type" content="application/xhtml+xml;charset=utf-8" /><title>Contents</title></head> <body> <div class="rightHandSide"> <div class="toc clickDown" tabindex="0"> <h3 id="context">Context</h3> <h4 id="topology">Topology</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/topology">topology</a></strong> (<a class="existingWikiWord" href="/nlab/show/point-set+topology">point-set topology</a>, <a class="existingWikiWord" href="/nlab/show/point-free+topology">point-free topology</a>)</p> <p>see also <em><a class="existingWikiWord" href="/nlab/show/differential+topology">differential topology</a></em>, <em><a class="existingWikiWord" href="/nlab/show/algebraic+topology">algebraic topology</a></em>, <em><a class="existingWikiWord" href="/nlab/show/functional+analysis">functional analysis</a></em> and <em><a class="existingWikiWord" href="/nlab/show/topological+homotopy+theory">topological</a> <a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopy theory</a></em></p> <p><a class="existingWikiWord" href="/nlab/show/Introduction+to+Topology">Introduction</a></p> <p><strong>Basic concepts</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/open+subset">open subset</a>, <a class="existingWikiWord" href="/nlab/show/closed+subset">closed subset</a>, <a class="existingWikiWord" href="/nlab/show/neighbourhood">neighbourhood</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a>, <a class="existingWikiWord" href="/nlab/show/locale">locale</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/base+for+the+topology">base for the topology</a>, <a class="existingWikiWord" href="/nlab/show/neighbourhood+base">neighbourhood base</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/finer+topology">finer/coarser topology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+closure">closure</a>, <a class="existingWikiWord" href="/nlab/show/topological+interior">interior</a>, <a class="existingWikiWord" href="/nlab/show/topological+boundary">boundary</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/separation+axiom">separation</a>, <a class="existingWikiWord" href="/nlab/show/sober+topological+space">sobriety</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/continuous+function">continuous function</a>, <a class="existingWikiWord" href="/nlab/show/homeomorphism">homeomorphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/uniformly+continuous+function">uniformly continuous function</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+embedding">embedding</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/open+map">open map</a>, <a class="existingWikiWord" href="/nlab/show/closed+map">closed map</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/sequence">sequence</a>, <a class="existingWikiWord" href="/nlab/show/net">net</a>, <a class="existingWikiWord" href="/nlab/show/sub-net">sub-net</a>, <a class="existingWikiWord" href="/nlab/show/filter">filter</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/convergence">convergence</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/category">category</a><a class="existingWikiWord" href="/nlab/show/Top">Top</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/convenient+category+of+topological+spaces">convenient category of topological spaces</a></li> </ul> </li> </ul> <p><strong><a href="Top#UniversalConstructions">Universal constructions</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/initial+topology">initial topology</a>, <a class="existingWikiWord" href="/nlab/show/final+topology">final topology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/subspace">subspace</a>, <a class="existingWikiWord" href="/nlab/show/quotient+space">quotient space</a>,</p> </li> <li> <p>fiber space, <a class="existingWikiWord" href="/nlab/show/space+attachment">space attachment</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/product+space">product space</a>, <a class="existingWikiWord" href="/nlab/show/disjoint+union+space">disjoint union space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/mapping+cylinder">mapping cylinder</a>, <a class="existingWikiWord" href="/nlab/show/mapping+cocylinder">mapping cocylinder</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/mapping+cone">mapping cone</a>, <a class="existingWikiWord" href="/nlab/show/mapping+cocone">mapping cocone</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/mapping+telescope">mapping telescope</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/colimits+of+normal+spaces">colimits of normal spaces</a></p> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/stuff%2C+structure%2C+property">Extra stuff, structure, properties</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/nice+topological+space">nice topological space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/metric+space">metric space</a>, <a class="existingWikiWord" href="/nlab/show/metric+topology">metric topology</a>, <a class="existingWikiWord" href="/nlab/show/metrisable+space">metrisable space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Kolmogorov+space">Kolmogorov space</a>, <a class="existingWikiWord" href="/nlab/show/Hausdorff+space">Hausdorff space</a>, <a class="existingWikiWord" href="/nlab/show/regular+space">regular space</a>, <a class="existingWikiWord" href="/nlab/show/normal+space">normal space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/sober+space">sober space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/compact+space">compact space</a>, <a class="existingWikiWord" href="/nlab/show/proper+map">proper map</a></p> <p><a class="existingWikiWord" href="/nlab/show/sequentially+compact+topological+space">sequentially compact</a>, <a class="existingWikiWord" href="/nlab/show/countably+compact+topological+space">countably compact</a>, <a class="existingWikiWord" href="/nlab/show/locally+compact+topological+space">locally compact</a>, <a class="existingWikiWord" href="/nlab/show/sigma-compact+topological+space">sigma-compact</a>, <a class="existingWikiWord" href="/nlab/show/paracompact+space">paracompact</a>, <a class="existingWikiWord" href="/nlab/show/countably+paracompact+topological+space">countably paracompact</a>, <a class="existingWikiWord" href="/nlab/show/strongly+compact+topological+space">strongly compact</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/compactly+generated+space">compactly generated space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/second-countable+space">second-countable space</a>, <a class="existingWikiWord" href="/nlab/show/first-countable+space">first-countable space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/contractible+space">contractible space</a>, <a class="existingWikiWord" href="/nlab/show/locally+contractible+space">locally contractible space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/connected+space">connected space</a>, <a class="existingWikiWord" href="/nlab/show/locally+connected+space">locally connected space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/simply-connected+space">simply-connected space</a>, <a class="existingWikiWord" href="/nlab/show/locally+simply-connected+space">locally simply-connected space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cell+complex">cell complex</a>, <a class="existingWikiWord" href="/nlab/show/CW-complex">CW-complex</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/pointed+topological+space">pointed space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+vector+space">topological vector space</a>, <a class="existingWikiWord" href="/nlab/show/Banach+space">Banach space</a>, <a class="existingWikiWord" href="/nlab/show/Hilbert+space">Hilbert space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+group">topological group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+vector+bundle">topological vector bundle</a>, <a class="existingWikiWord" href="/nlab/show/topological+K-theory">topological K-theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+manifold">topological manifold</a></p> </li> </ul> <p><strong>Examples</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/empty+space">empty space</a>, <a class="existingWikiWord" href="/nlab/show/point+space">point space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/discrete+space">discrete space</a>, <a class="existingWikiWord" href="/nlab/show/codiscrete+space">codiscrete space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Sierpinski+space">Sierpinski space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/order+topology">order topology</a>, <a class="existingWikiWord" href="/nlab/show/specialization+topology">specialization topology</a>, <a class="existingWikiWord" href="/nlab/show/Scott+topology">Scott topology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Euclidean+space">Euclidean space</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/real+line">real line</a>, <a class="existingWikiWord" href="/nlab/show/plane">plane</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cylinder">cylinder</a>, <a class="existingWikiWord" href="/nlab/show/cone">cone</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/sphere">sphere</a>, <a class="existingWikiWord" href="/nlab/show/ball">ball</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/circle">circle</a>, <a class="existingWikiWord" href="/nlab/show/torus">torus</a>, <a class="existingWikiWord" href="/nlab/show/annulus">annulus</a>, <a class="existingWikiWord" href="/nlab/show/Moebius+strip">Moebius strip</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/polytope">polytope</a>, <a class="existingWikiWord" href="/nlab/show/polyhedron">polyhedron</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/projective+space">projective space</a> (<a class="existingWikiWord" href="/nlab/show/real+projective+space">real</a>, <a class="existingWikiWord" href="/nlab/show/complex+projective+space">complex</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/classifying+space">classifying space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/configuration+space+%28mathematics%29">configuration space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/path">path</a>, <a class="existingWikiWord" href="/nlab/show/loop">loop</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/mapping+spaces">mapping spaces</a>: <a class="existingWikiWord" href="/nlab/show/compact-open+topology">compact-open topology</a>, <a class="existingWikiWord" href="/nlab/show/topology+of+uniform+convergence">topology of uniform convergence</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/loop+space">loop space</a>, <a class="existingWikiWord" href="/nlab/show/path+space">path space</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Zariski+topology">Zariski topology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Cantor+space">Cantor space</a>, <a class="existingWikiWord" href="/nlab/show/Mandelbrot+space">Mandelbrot space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Peano+curve">Peano curve</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/line+with+two+origins">line with two origins</a>, <a class="existingWikiWord" href="/nlab/show/long+line">long line</a>, <a class="existingWikiWord" href="/nlab/show/Sorgenfrey+line">Sorgenfrey line</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/K-topology">K-topology</a>, <a class="existingWikiWord" href="/nlab/show/Dowker+space">Dowker space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Warsaw+circle">Warsaw circle</a>, <a class="existingWikiWord" href="/nlab/show/Hawaiian+earring+space">Hawaiian earring space</a></p> </li> </ul> <p><strong>Basic statements</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Hausdorff+spaces+are+sober">Hausdorff spaces are sober</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/schemes+are+sober">schemes are sober</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/continuous+images+of+compact+spaces+are+compact">continuous images of compact spaces are compact</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/closed+subspaces+of+compact+Hausdorff+spaces+are+equivalently+compact+subspaces">closed subspaces of compact Hausdorff spaces are equivalently compact subspaces</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/open+subspaces+of+compact+Hausdorff+spaces+are+locally+compact">open subspaces of compact Hausdorff spaces are locally compact</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quotient+projections+out+of+compact+Hausdorff+spaces+are+closed+precisely+if+the+codomain+is+Hausdorff">quotient projections out of compact Hausdorff spaces are closed precisely if the codomain is Hausdorff</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/compact+spaces+equivalently+have+converging+subnet+of+every+net">compact spaces equivalently have converging subnet of every net</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Lebesgue+number+lemma">Lebesgue number lemma</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/sequentially+compact+metric+spaces+are+equivalently+compact+metric+spaces">sequentially compact metric spaces are equivalently compact metric spaces</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/compact+spaces+equivalently+have+converging+subnet+of+every+net">compact spaces equivalently have converging subnet of every net</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/sequentially+compact+metric+spaces+are+totally+bounded">sequentially compact metric spaces are totally bounded</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/continuous+metric+space+valued+function+on+compact+metric+space+is+uniformly+continuous">continuous metric space valued function on compact metric space is uniformly continuous</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/paracompact+Hausdorff+spaces+are+normal">paracompact Hausdorff spaces are normal</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/paracompact+Hausdorff+spaces+equivalently+admit+subordinate+partitions+of+unity">paracompact Hausdorff spaces equivalently admit subordinate partitions of unity</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/closed+injections+are+embeddings">closed injections are embeddings</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/proper+maps+to+locally+compact+spaces+are+closed">proper maps to locally compact spaces are closed</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/injective+proper+maps+to+locally+compact+spaces+are+equivalently+the+closed+embeddings">injective proper maps to locally compact spaces are equivalently the closed embeddings</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/locally+compact+and+sigma-compact+spaces+are+paracompact">locally compact and sigma-compact spaces are paracompact</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/locally+compact+and+second-countable+spaces+are+sigma-compact">locally compact and second-countable spaces are sigma-compact</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/second-countable+regular+spaces+are+paracompact">second-countable regular spaces are paracompact</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/CW-complexes+are+paracompact+Hausdorff+spaces">CW-complexes are paracompact Hausdorff spaces</a></p> </li> </ul> <p><strong>Theorems</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Urysohn%27s+lemma">Urysohn's lemma</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Tietze+extension+theorem">Tietze extension theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Tychonoff+theorem">Tychonoff theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/tube+lemma">tube lemma</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Michael%27s+theorem">Michael's theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Brouwer%27s+fixed+point+theorem">Brouwer's fixed point theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+invariance+of+dimension">topological invariance of dimension</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Jordan+curve+theorem">Jordan curve theorem</a></p> </li> </ul> <p><strong>Analysis Theorems</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Heine-Borel+theorem">Heine-Borel theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/intermediate+value+theorem">intermediate value theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/extreme+value+theorem">extreme value theorem</a></p> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/topological+homotopy+theory">topological homotopy theory</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/left+homotopy">left homotopy</a>, <a class="existingWikiWord" href="/nlab/show/right+homotopy">right homotopy</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+equivalence">homotopy equivalence</a>, <a class="existingWikiWord" href="/nlab/show/deformation+retract">deformation retract</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+group">fundamental group</a>, <a class="existingWikiWord" href="/nlab/show/covering+space">covering space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+theorem+of+covering+spaces">fundamental theorem of covering spaces</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+group">homotopy group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/weak+homotopy+equivalence">weak homotopy equivalence</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Whitehead%27s+theorem">Whitehead's theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Freudenthal+suspension+theorem">Freudenthal suspension theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/nerve+theorem">nerve theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+extension+property">homotopy extension property</a>, <a class="existingWikiWord" href="/nlab/show/Hurewicz+cofibration">Hurewicz cofibration</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+cofiber+sequence">cofiber sequence</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Str%C3%B8m+model+category">Strøm model category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/classical+model+structure+on+topological+spaces">classical model structure on topological spaces</a></p> </li> </ul> </div></div> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#exponentiable_spaces'>Exponentiable spaces</a></li> <ul> <li><a href='#theorem_exponentiability_i'>Theorem (Exponentiability, I)</a></li> </ul> <li><a href='#corecompactness'>Core-compactness</a></li> <ul> <li><a href='#theorem_exponentiability_ii'>Theorem (Exponentiability, II)</a></li> </ul> <li><a href='#ViaConvergence'>In terms of convergence</a></li> <li><a href='#general_exponential_laws'>General exponential laws</a></li> <ul> <li><a href='#theorem_exponential_law'>Theorem (Exponential law)</a></li> </ul> <li><a href='#based_exponential_laws'>Based exponential laws</a></li> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> <ul> <li><a href='#GeneralReferences'>General</a></li> <li><a href='#ParameterizedTopology'>Exponential law for parameterized topological spaces</a></li> </ul> </ul> </div> <h2 id="idea">Idea</h2> <p>When in a <a class="existingWikiWord" href="/nlab/show/convenient+category+of+topological+spaces">convenient category of topological spaces</a>, e.g. <a class="existingWikiWord" href="/nlab/show/compactly+generated+space">compactly generated space</a>s, the category is <a class="existingWikiWord" href="/nlab/show/cartesian+closed+category">cartesian closed</a>, so that there is an <a class="existingWikiWord" href="/nlab/show/adjunction">adjunction</a> between the mapping space and the cartesian product in that category. For general <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a>s there is no globally defined adjunction, but we can instead characterize exactly which spaces are exponentiable.</p> <h2 id="exponentiable_spaces">Exponentiable spaces</h2> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/finitely+complete+category">category with finite products</a>, recall that an object <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 <strong><a class="existingWikiWord" href="/nlab/show/exponentiable+object">exponentiable</a></strong> if the functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>c</mi><mo>×</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>:</mo><mi>C</mi><mo>→</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">c \times -: C \to C</annotation></semantics></math> has a <a class="existingWikiWord" href="/nlab/show/right+adjoint">right adjoint</a>, usually denoted <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><msup><mo stretchy="false">)</mo> <mi>c</mi></msup><mo>:</mo><mi>C</mi><mo>→</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">(-)^c: C \to C</annotation></semantics></math>.</p> <div class="un_theorem"> <h6 id="theorem_exponentiability_i">Theorem (Exponentiability, I)</h6> <p>Let <a class="existingWikiWord" href="/nlab/show/Top">Top</a> be the category of all topological spaces. An object <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> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Top</mi></mrow><annotation encoding="application/x-tex">Top</annotation></semantics></math> is exponentiable if and only if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>×</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>:</mo><mi>Top</mi><mo>→</mo><mi>Top</mi></mrow><annotation encoding="application/x-tex">X \times -: Top \to Top</annotation></semantics></math> preserves <a class="existingWikiWord" href="/nlab/show/coequalizer">coequalizer</a>s, or equivalently <a class="existingWikiWord" href="/nlab/show/quotient+space">quotient space</a>s.</p> </div> <p>This functor always preserves <a class="existingWikiWord" href="/nlab/show/coproducts">coproducts</a>, so this condition is equivalent to saying that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>×</mo><mo lspace="verythinmathspace" rspace="0em">−</mo></mrow><annotation encoding="application/x-tex">X \times -</annotation></semantics></math> preserves all small <a class="existingWikiWord" href="/nlab/show/colimits">colimits</a>. This is then equivalent to exponentiability by the <a class="existingWikiWord" href="/nlab/show/adjoint+functor+theorem">adjoint functor theorem</a>.</p> <p>This condition, however, is not really any more explicit. More interesting is to characterize the exponentiable spaces in terms of a point-set-topological condition.</p> <h3 id="corecompactness">Core-compactness</h3> <p>For open subsets <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi></mrow><annotation encoding="application/x-tex">U</annotation></semantics></math> and <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> of a topological space <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>, we write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi><mo>≪</mo><mi>U</mi></mrow><annotation encoding="application/x-tex">V\ll U</annotation></semantics></math> to mean that any open cover of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi></mrow><annotation encoding="application/x-tex">U</annotation></semantics></math> admits a finite subcover 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>; this is read as <strong><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 relatively compact under <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi></mrow><annotation encoding="application/x-tex">U</annotation></semantics></math></strong> or <strong><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 way below <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi></mrow><annotation encoding="application/x-tex">U</annotation></semantics></math></strong>. We say that <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 <strong>core-compact</strong> if for every open neighborhood <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi></mrow><annotation encoding="application/x-tex">U</annotation></semantics></math> of a point <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>, there exists an open neighborhood <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> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math> with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi><mo>≪</mo><mi>U</mi></mrow><annotation encoding="application/x-tex">V\ll U</annotation></semantics></math>. In other words, <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 core-compact iff for all open subsets <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>, we have <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi><mo>=</mo><mo lspace="thinmathspace" rspace="thinmathspace">⋃</mo><mo stretchy="false">{</mo><mi>U</mi><mo stretchy="false">|</mo><mi>U</mi><mo>≪</mo><mi>V</mi><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">V = \bigcup \{ U | U\ll V \}</annotation></semantics></math>. This says essentially the same thing as saying that the open-set lattice of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/continuous+lattice">continuous lattice</a>, which yields the corresponding definition for <a class="existingWikiWord" href="/nlab/show/locales">locales</a>.</p> <div class="un_theorem"> <h6 id="theorem_exponentiability_ii">Theorem (Exponentiability, II)</h6> <p>An object <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> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Top</mi></mrow><annotation encoding="application/x-tex">Top</annotation></semantics></math> is exponentiable if and only if it is core-compact.</p> </div> <p>If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/Hausdorff+space">Hausdorff</a>, then core-compactness is equivalent to <a class="existingWikiWord" href="/nlab/show/locally+compact+space">local compactness</a>; thus in particular all locally compact Hausdorff spaces are exponentiable. In fact, local compactness implies exponentiability even without the Hausdorff condition, if <a class="existingWikiWord" href="/nlab/show/locally+compact+space">local compactness</a> is defined appropriately (for every point the compact neighborhoods form a neighborhood basis). For this reason, that core-compactness is also called <strong>quasi local compactness</strong>.</p> <p>When <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> is core-compact, we can explicitly describe the exponential topology on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Y</mi> <mi>X</mi></msup></mrow><annotation encoding="application/x-tex">Y^X</annotation></semantics></math> (whose points are continuous maps <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><mi>X</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">f: X \to Y</annotation></semantics></math>). It is generated by <a class="existingWikiWord" href="/nlab/show/topological+basis">subbasis</a> elements <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>O</mi> <mrow><mi>U</mi><mo>,</mo><mi>V</mi></mrow></msub></mrow><annotation encoding="application/x-tex">O_{U,V}</annotation></semantics></math>, for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi></mrow><annotation encoding="application/x-tex">U</annotation></semantics></math> an open subset of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> and <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> an open subset of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math>, where a continuous map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo lspace="verythinmathspace">:</mo><mi>X</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">f\colon X \to Y</annotation></semantics></math> belongs to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>O</mi> <mrow><mi>U</mi><mo>,</mo><mi>V</mi></mrow></msub></mrow><annotation encoding="application/x-tex">O_{U,V}</annotation></semantics></math> iff <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo>≪</mo><msup><mi>f</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>V</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">U\ll f^{-1}(V)</annotation></semantics></math>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>O</mi> <mrow><mi>U</mi><mo>,</mo><mi>V</mi></mrow></msub><mo>=</mo><mo stretchy="false">{</mo><mi>f</mi><mo>∈</mo><msup><mi>Y</mi> <mi>X</mi></msup><mo>:</mo><mi>U</mi><mo>≪</mo><msup><mi>f</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>V</mi><mo stretchy="false">)</mo><mo stretchy="false">}</mo><mo>.</mo></mrow><annotation encoding="application/x-tex"> O_{U, V} = \{f \in Y^X: U \ll f^{-1}(V)\} .</annotation></semantics></math></div> <p>If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math> are Hausdorff, then this topology on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Y</mi> <mi>X</mi></msup></mrow><annotation encoding="application/x-tex">Y^X</annotation></semantics></math> coincides with the <a class="existingWikiWord" href="/nlab/show/compact-open+topology">compact-open topology</a>.</p> <h3 id="ViaConvergence">In terms of convergence</h3> <p>Exponentiable (i.e. core-compact) spaces can also be characterized in terms of <a class="existingWikiWord" href="/nlab/show/ultrafilter">ultrafilter</a> convergence. Recall that a topological space can equivalently be defined as a <a class="existingWikiWord" href="/nlab/show/relational+beta-module">lax algebra for the ultrafilter monad</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi></mrow><annotation encoding="application/x-tex">U</annotation></semantics></math> on the <a class="existingWikiWord" href="/nlab/show/%281%2C2%29-category">(1,2)-category</a> <a class="existingWikiWord" href="/nlab/show/Rel">Rel</a> of sets and relations. In other words, it consists of a set <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> and a relation <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi><mo lspace="verythinmathspace">:</mo><mi>U</mi><mi>X</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">R\colon U X \to X</annotation></semantics></math> called “convergence”, such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>id</mi> <mi>X</mi></msub><mo>⊆</mo><mi>R</mi><mo>∘</mo><mi>η</mi></mrow><annotation encoding="application/x-tex">id_X \subseteq R \circ \eta</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><mi>U</mi><mi>R</mi><mo>⊆</mo><mi>R</mi><mo>∘</mo><mi>μ</mi></mrow><annotation encoding="application/x-tex">R\circ U R \subseteq R\circ \mu</annotation></semantics></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>η</mi></mrow><annotation encoding="application/x-tex">\eta</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>μ</mi></mrow><annotation encoding="application/x-tex">\mu</annotation></semantics></math> are the unit and multiplication of the ultrafilter monad, regarded as relations. In the paper</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Claudio+Pisani">Claudio Pisani</a>, <em>Convergence in exponentiable spaces</em>, <a href="http://www.tac.mta.ca/tac/volumes/1999/n6/5-06abs.html">TAC</a></li> </ul> <p>it is shown that a space is exponentiable (i.e. core-compact) if and only if we have equality in the multiplication law <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi><mo>∘</mo><mi>U</mi><mi>R</mi><mo>=</mo><mi>R</mi><mo>∘</mo><mi>μ</mi></mrow><annotation encoding="application/x-tex">R\circ U R = R\circ \mu</annotation></semantics></math>.</p> <p>Some intuition for this characterization can be obtained as follows. Consider the standard non-locally-compact space, the rationals <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℚ</mi></mrow><annotation encoding="application/x-tex">\mathbb{Q}</annotation></semantics></math> as a subspace of the reals <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>. Suppose that <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 rational number and that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>y</mi> <mi>n</mi></msub></mrow><annotation encoding="application/x-tex">y_n</annotation></semantics></math> is a sequence of irrationals converging to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math>. Then for each <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> we can find a sequence <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>z</mi> <mi>m</mi> <mi>n</mi></msubsup></mrow><annotation encoding="application/x-tex">z^n_m</annotation></semantics></math> of rationals which converges to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>y</mi> <mi>n</mi></msub></mrow><annotation encoding="application/x-tex">y_n</annotation></semantics></math>; hence the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>z</mi> <mi>m</mi> <mi>n</mi></msubsup></mrow><annotation encoding="application/x-tex">z^n_m</annotation></semantics></math> form a “sequence of sequences” which “globally converges” to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℚ</mi></mrow><annotation encoding="application/x-tex">\mathbb{Q}</annotation></semantics></math>, i.e. which are related to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math> by the composite relation <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi><mo>∘</mo><mi>μ</mi></mrow><annotation encoding="application/x-tex">R\circ \mu</annotation></semantics></math>, but for which does not converge elementwise to an intermediate sequence which in turn converges to <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>, i.e. it is not related to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math> by the relation <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi><mo>∘</mo><mi>U</mi><mi>R</mi></mrow><annotation encoding="application/x-tex">R \circ U R</annotation></semantics></math>. It turns out that when generalized to ultrafilter convergence, this sort of behavior exactly characterizes what it means to fail to be (quasi) locally compact.</p> <h2 id="general_exponential_laws">General exponential laws</h2> <p>If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> is exponentiable, then the exponential law gives us an isomorphism of <em>sets</em> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Map</mi><mo stretchy="false">(</mo><mi>Y</mi><mo>,</mo><msup><mi>B</mi> <mi>X</mi></msup><mo stretchy="false">)</mo><mo>≅</mo><mi>Map</mi><mo stretchy="false">(</mo><mi>X</mi><mo>×</mo><mi>Y</mi><mo>,</mo><mi>B</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Map(Y,B^X) \cong Map(X\times Y,B)</annotation></semantics></math> for any other spaces <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math>. If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math> is also exponentiable, then the <a class="existingWikiWord" href="/nlab/show/Yoneda+lemma">Yoneda lemma</a> yields from this a homeomorphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>B</mi> <mrow><mi>X</mi><mo>×</mo><mi>Y</mi></mrow></msup><mo>≅</mo><mo stretchy="false">(</mo><msup><mi>B</mi> <mi>X</mi></msup><msup><mo stretchy="false">)</mo> <mi>Y</mi></msup></mrow><annotation encoding="application/x-tex">B^{X\times Y} \cong (B^X)^Y</annotation></semantics></math>. However, we can also say some things in general without all spaces involved being exponentiable.</p> <p>We now agree to denote by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Map</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>Y</mi><mo stretchy="false">)</mo><mo>=</mo><msup><mi>X</mi> <mi>Y</mi></msup></mrow><annotation encoding="application/x-tex">Map(X,Y)=X^Y</annotation></semantics></math> the space of continuous maps <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">X\to Y</annotation></semantics></math> in the <a class="existingWikiWord" href="/nlab/show/compact-open+topology">compact-open topology</a>.</p> <div class="un_theorem"> <h6 id="theorem_exponential_law">Theorem (Exponential law)</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>,</mo><mi>Y</mi><mo>,</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">X,Y,B</annotation></semantics></math> be <a class="existingWikiWord" href="/nlab/show/topological+spaces">topological spaces</a>. For any <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>∈</mo><msup><mi>B</mi> <mrow><mi>X</mi><mo>×</mo><mi>Y</mi></mrow></msup></mrow><annotation encoding="application/x-tex">f\in B^{X\times Y}</annotation></semantics></math>, the formula</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><mo stretchy="false">(</mo><mi>θ</mi><mi>f</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> [(\theta f)(y)](x) = f(x,y) </annotation></semantics></math></div> <p>defines a <a class="existingWikiWord" href="/nlab/show/continuous+map">continuous map</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>θ</mi><mi>f</mi><mo>:</mo><mi>Y</mi><mo>→</mo><msup><mi>B</mi> <mi>X</mi></msup></mrow><annotation encoding="application/x-tex">\theta f:Y\to B^X</annotation></semantics></math> which we call the map adjoint to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math>, or the <a class="existingWikiWord" href="/nlab/show/adjunct">adjunct</a> 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>.</p> <p>The adjunction map</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>θ</mi><mo>:</mo><mi>Map</mi><mo stretchy="false">(</mo><mi>X</mi><mo>×</mo><mi>Y</mi><mo>,</mo><mi>B</mi><mo stretchy="false">)</mo><mo>→</mo><mi>Map</mi><mo stretchy="false">(</mo><mi>Y</mi><mo>,</mo><msup><mi>B</mi> <mi>X</mi></msup><mo stretchy="false">)</mo><mo>,</mo><mspace width="thinmathspace"></mspace><mspace width="thinmathspace"></mspace><mspace width="thinmathspace"></mspace><mspace width="thinmathspace"></mspace><mspace width="thinmathspace"></mspace><mspace width="thinmathspace"></mspace><mspace width="thinmathspace"></mspace><mi>θ</mi><mo>:</mo><mi>f</mi><mo>↦</mo><mi>θ</mi><mi>f</mi></mrow><annotation encoding="application/x-tex"> \theta : Map(X\times Y,B)\to Map(Y,B^X), \,\,\,\,\,\,\,\theta:f\mapsto \theta f </annotation></semantics></math></div> <p>is a <a class="existingWikiWord" href="/nlab/show/one-to-one+function">one-to-one function</a>, and if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/locally+compact+space">locally compact and Hausdorff</a> then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>θ</mi></mrow><annotation encoding="application/x-tex">\theta</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/bijection">bijection</a>. Independently from that assumption 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>, if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/Hausdorff+space">Hausdorff</a>, then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>θ</mi></mrow><annotation encoding="application/x-tex">\theta</annotation></semantics></math> is continuous in the <a class="existingWikiWord" href="/nlab/show/compact-open+topology">compact-open topology</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>θ</mi><mo>:</mo><msup><mi>B</mi> <mrow><mi>X</mi><mo>×</mo><mi>Y</mi></mrow></msup><mo>→</mo><mo stretchy="false">(</mo><msup><mi>B</mi> <mi>X</mi></msup><msup><mo stretchy="false">)</mo> <mi>Y</mi></msup><mo>.</mo></mrow><annotation encoding="application/x-tex"> \theta : B^{X\times Y}\to (B^X)^Y. </annotation></semantics></math></div> <p>If both assumptions (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> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math>) are satisfied, then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>θ</mi></mrow><annotation encoding="application/x-tex">\theta</annotation></semantics></math> is not only a continuous bijection, but also open, hence a <a class="existingWikiWord" href="/nlab/show/homeomorphism">homeomorphism</a>.</p> </div> <h2 id="based_exponential_laws">Based exponential laws</h2> <p>There is also a version for based (= <a class="existingWikiWord" href="/nlab/show/pointed+space">pointed</a>) topological spaces. The <a class="existingWikiWord" href="/nlab/show/cartesian+product">cartesian product</a> then needs to be replace by the <a class="existingWikiWord" href="/nlab/show/smash+product">smash product</a> of the based spaces. Regarding that the maps preserve the base point, the adjunction map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>θ</mi></mrow><annotation encoding="application/x-tex">\theta</annotation></semantics></math> induces the adjunction map</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>θ</mi> <mo>*</mo></msub><mo>:</mo><msub><mi>Map</mi> <mo>*</mo></msub><mo stretchy="false">(</mo><mi>X</mi><mo>∧</mo><mi>Y</mi><mo>,</mo><mi>B</mi><mo stretchy="false">)</mo><mo>→</mo><msub><mi>Map</mi> <mo>*</mo></msub><mo stretchy="false">(</mo><mi>Y</mi><mo>,</mo><msub><mi>Map</mi> <mo>*</mo></msub><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>B</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\theta_*:Map_*(X\wedge Y,B)\to Map_*(Y,Map_*(X, B))</annotation></semantics></math></div> <p>where the mapping space <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Map</mi> <mo>*</mo></msub></mrow><annotation encoding="application/x-tex">Map_*</annotation></semantics></math> for based spaces is the subspace of the usual mapping space, in the compact-open topology, which consists of the mappings preserving the base point.</p> <p>It appears that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>θ</mi> <mo>*</mo></msub></mrow><annotation encoding="application/x-tex">\theta_*</annotation></semantics></math> is again one-to-one and continuous, and it is bijective if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> is locally compact Hausdorff. If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>,</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">X, Y</annotation></semantics></math> are compact Hausdorff then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>θ</mi> <mo>*</mo></msub></mrow><annotation encoding="application/x-tex">\theta_*</annotation></semantics></math> is a homeomorphism.</p> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/exponential+object">exponential object</a>, <a class="existingWikiWord" href="/nlab/show/mapping+space">mapping space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/compactly+generated+topological+space">compactly generated topological space</a></p> </li> </ul> <h2 id="references">References</h2> <h3 id="GeneralReferences">General</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/C.+R.+F.+Maunder">C. R. F. Maunder</a>, §6.2 in: <em>Algebraic Topology</em>, Cambridge University Press, Cambridge (1970, 1980) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo></mrow><annotation encoding="application/x-tex">[</annotation></semantics></math><a href="https://www.maths.ed.ac.uk/~v1ranick/papers/maunder.pdf">pdf</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">]</annotation></semantics></math></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Peter+Johnstone">Peter Johnstone</a>, §7.4 of: <em><a class="existingWikiWord" href="/nlab/show/Stone+Spaces">Stone Spaces</a></em>, Cambridge Studies in Advanced Mathematics <strong>3</strong>, Cambridge University Press (1982, 1986) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo></mrow><annotation encoding="application/x-tex">[</annotation></semantics></math><a href="https://www.cambridge.org/de/academic/subjects/mathematics/logic-categories-and-sets/stone-spaces?format=PB&isbn=9780521337793">ISBN:9780521337793</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">]</annotation></semantics></math></p> </li> <li> <p>Eva Lowen-Colebunders and Günther Richter, <em>An elementary approach to exponential spaces</em>, Applied Categorical Structures <strong>9</strong> (2001) 303–310 <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo></mrow><annotation encoding="application/x-tex">[</annotation></semantics></math><a href="https://doi.org/10.1023/A:1011268007097">doi:10.1023/A:1011268007097</a>, <a href="http://www.ams.org/mathscinet-getitem?mr=1836256">MR</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">]</annotation></semantics></math></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Mart%C3%ADn+Escard%C3%B3">Martín Escardó</a>, <a class="existingWikiWord" href="/nlab/show/Reinhold+Heckmann">Reinhold Heckmann</a>, <em>Topologies of spaces of continuous functions</em> (2001) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo></mrow><annotation encoding="application/x-tex">[</annotation></semantics></math><a href="http://www.cs.bham.ac.uk/~mhe/papers/newyork.pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/EscardoHeckmann-TopologiesOnFunctionSpaces.pdf" title="pdf">pdf</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">]</annotation></semantics></math></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Marcelo+Aguilar">Marcelo Aguilar</a>, <a class="existingWikiWord" href="/nlab/show/Samuel+Gitler">Samuel Gitler</a>, <a class="existingWikiWord" href="/nlab/show/Carlos+Prieto">Carlos Prieto</a>, §1.3 of: <em>Algebraic topology from a homotopical viewpoint</em>, Springer Science & Business Media, 2008 <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo></mrow><annotation encoding="application/x-tex">[</annotation></semantics></math><a href="https://link.springer.com/book/10.1007/b97586">doi:10.1007/b97586</a>, <a href="http://tocs.ulb.tu-darmstadt.de/106999419.pdf">toc pdf</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">]</annotation></semantics></math></p> </li> </ul> <div> <h3 id="ParameterizedTopology">Exponential law for parameterized topological spaces</h3> <p>On <a class="existingWikiWord" href="/nlab/show/exponential+law+for+spaces">exponential objects</a> (<a class="existingWikiWord" href="/nlab/show/internal+homs">internal homs</a>) in <a class="existingWikiWord" href="/nlab/show/slice+categories">slice categories</a> of (<a class="existingWikiWord" href="/nlab/show/compactly+generated+topological+spaces">compactly generated</a>) <a class="existingWikiWord" href="/nlab/show/topological+spaces">topological spaces</a> – see at <em><a class="existingWikiWord" href="/nlab/show/parameterized+homotopy+theory">parameterized homotopy theory</a></em>):</p> <ul> <li id="Booth70"> <p><a class="existingWikiWord" href="/nlab/show/Peter+I.+Booth">Peter I. Booth</a>, <em>The Exponential Law of Maps I</em>, Proceedings of the London Mathematical Society <strong>s3-20</strong> 1 (1970) 179-192 <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo></mrow><annotation encoding="application/x-tex">[</annotation></semantics></math><a href="https://doi.org/10.1112/plms/s3-20.1.179">doi:10.1112/plms/s3-20.1.179</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">]</annotation></semantics></math></p> </li> <li id="Booth71"> <p><a class="existingWikiWord" href="/nlab/show/Peter+I.+Booth">Peter I. Booth</a>, <em>The exponential law of maps. II</em>, Mathematische Zeitschrift <strong>121</strong> (1971) 311–319 <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo></mrow><annotation encoding="application/x-tex">[</annotation></semantics></math><a href="https://doi.org/10.1007/BF01109977">doi:10.1007/BF01109977</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">]</annotation></semantics></math></p> </li> <li id="BoothBrown78a"> <p><a class="existingWikiWord" href="/nlab/show/Peter+I.+Booth">Peter I. Booth</a>, <a class="existingWikiWord" href="/nlab/show/Ronnie+Brown">Ronnie Brown</a>, <em>Spaces of partial maps, fibred mapping spaces and the compact-open topology</em>, General Topology and its Applications <strong>8</strong> 2 (1978) 181-195 <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo></mrow><annotation encoding="application/x-tex">[</annotation></semantics></math><a href="https://doi.org/10.1016/0016-660X(78)90049-1">doi:10.1016/0016-660X(78)90049-1</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">]</annotation></semantics></math></p> </li> <li id="BoothBrown78b"> <p><a class="existingWikiWord" href="/nlab/show/Peter+I.+Booth">Peter I. Booth</a>, <a class="existingWikiWord" href="/nlab/show/Ronnie+Brown">Ronnie Brown</a>, <em>On the application of fibred mapping spaces to exponential laws for bundles, ex-spaces and other categories of maps</em>, General Topology and its Applications <strong>8</strong> 2 (1978) 165-179 <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo></mrow><annotation encoding="application/x-tex">[</annotation></semantics></math><a href="https://doi.org/10.1016/0016-660X(78)90048-X">doi:10.1016/0016-660X(78)90048-X</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">]</annotation></semantics></math></p> </li> <li id="Lewis85"> <p><a class="existingWikiWord" href="/nlab/show/L.+Gaunce+Lewis%2C+Jr.">L. Gaunce Lewis, Jr.</a>, §1 of: <em>Open maps, colimits, and a convenient category of fibre spaces</em>, Topology and its Applications <strong>19</strong> 1 (1985) 75-89 <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo></mrow><annotation encoding="application/x-tex">[</annotation></semantics></math><a href="https://doi.org/10.1016/0166-8641(85)90087-2">doi.org/10.1016/0166-8641(85)90087-2</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">]</annotation></semantics></math></p> </li> </ul> <p>And with an eye towards <a class="existingWikiWord" href="/nlab/show/parameterized+homotopy+theory">parameterized homotopy theory</a>:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Ioan+Mackenzie+James">Ioan Mackenzie James</a>: §II.9 in: <em>Fibrewise topology</em>, Cambridge Tracts in Mathematics, Cambridge University Press (1989) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo></mrow><annotation encoding="application/x-tex">[</annotation></semantics></math>ISBN:9780521360906<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">]</annotation></semantics></math></p> </li> <li id="MaySigurdsson06"> <p><a class="existingWikiWord" href="/nlab/show/Peter+May">Peter May</a>, <a class="existingWikiWord" href="/nlab/show/Johann+Sigurdsson">Johann Sigurdsson</a>, §1.3.7-§1.3.9 in: <em><a class="existingWikiWord" href="/nlab/show/Parametrized+Homotopy+Theory">Parametrized Homotopy Theory</a></em>, Mathematical Surveys and Monographs, vol. 132, AMS 2006 (<a href="https://bookstore.ams.org/surv-132">ISBN:978-0-8218-3922-5</a>, <a href="https://arxiv.org/abs/math/0411656">arXiv:math/0411656</a>, <a href="http://www.math.uchicago.edu/~may/EXTHEORY/MaySig.pdf">pdf</a>)</p> </li> </ul> </div></body></html> </div> <div class="revisedby"> <p> Last revised on June 20, 2022 at 09:39:05. 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