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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> <blockquote> <p>This page is about topology as a <strong>field of <a class="existingWikiWord" href="/nlab/show/mathematics">mathematics</a></strong>. For topology as a <strong><a class="existingWikiWord" href="/nlab/show/structured+set">structure</a></strong> on a <a class="existingWikiWord" href="/nlab/show/set">set</a>, see <em><a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a></em>.</p> <p>Parts of this page exists also in a German language version, see at <em><a class="existingWikiWord" href="/nlab/show/Topologie">Topologie</a></em>.</p> </blockquote> <hr /> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#Introduction'>Introduction</a></li> <ul> <li><a href='#Continuity'>Continuity</a></li> <li><a href='#TopologicalSpaces'>Topological spaces</a></li> <li><a href='#Homeomorphisms'>Homeomorphism</a></li> <li><a href='#Homotopy'>Homotopy</a></li> <li><a href='#ConnectedComponents'>Connected components</a></li> <li><a href='#FundamentalGroups'>Fundamental group</a></li> <li><a href='#CoveringSpaces'>Covering spaces</a></li> </ul> <li><a href='#BasicFacts'>Basic facts</a></li> <li><a href='#CentralTheorems'>Central theorems</a></li> <li><a href='#RelatedEntries'>Related entries</a></li> <ul> <li><a href='#topological_spaces_2'>Topological spaces</a></li> <li><a href='#manifolds_and_generalizations'>Manifolds and generalizations</a></li> <li><a href='#algebraic_topology_and_homotopy_theory'>Algebraic topology and homotopy theory</a></li> <ul> <li><a href='#topological_homotopy_theory'>Topological homotopy theory</a></li> <li><a href='#simplicial_homotopy_theory'>Simplicial homotopy theory</a></li> </ul> <li><a href='#sheaves_stacks_cohomology'>Sheaves, stacks, cohomology</a></li> <li><a href='#noncommutative_topology'>Non-commutative topology</a></li> <li><a href='#topological_physics'>Topological physics</a></li> <li><a href='#computational_topology'>Computational Topology</a></li> </ul> <li><a href='#References'>References</a></li> <ul> <li><a href='#ReferencesHistoricalOrigins'>Historical origins</a></li> <li><a href='#further'>Further</a></li> </ul> </ul> </div> <p><br /></p> <p><br /></p> <blockquote> <p>I believe that we lack another analysis properly geometric or linear which expresses location directly as algebra expresses magnitude.</p> <p><a class="existingWikiWord" href="/nlab/show/G.+W.+Leibniz">G. W. Leibniz</a> (letter to Huygens 1679, according to <a href="#Bredon93">Bredon 93, p. 430</a>)</p> </blockquote> <h2 id="idea">Idea</h2> <p><strong>Topology</strong> is one of the basic fields of <a class="existingWikiWord" href="/nlab/show/mathematics">mathematics</a>. The term is also used for a particular structure in a <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a>; see <a class="existingWikiWord" href="/nlab/show/topological+structure">topological structure</a> for that.</p> <p>The subject of topology deals with the expressions of <a class="existingWikiWord" href="/nlab/show/continuous+map">continuity</a> and <a class="existingWikiWord" href="/nlab/show/boundary">boundary</a>, and studying the geometric properties of (originally: <a class="existingWikiWord" href="/nlab/show/metric+space">metric</a>) <a class="existingWikiWord" href="/nlab/show/spaces">spaces</a> and relations of subspaces, which do not change under continuous deformations, regardless to other (such as in their metric properties).</p> <p>Topology as a structure enables one to model <a class="existingWikiWord" href="/nlab/show/continuity">continuity</a> and <a class="existingWikiWord" href="/nlab/show/convergence">convergence</a> locally. More recently, in metric spaces, topologists and geometric group theorists started looking at asymptotic properties at large, which are in some sense dual to the standard topological structure and are usually referred to as <em>coarse topology</em>.</p> <p>There are many cousins of the concept of <a class="existingWikiWord" href="/nlab/show/topological+spaces">topological spaces</a>, e.g. <a class="existingWikiWord" href="/nlab/show/sites">sites</a>, <a class="existingWikiWord" href="/nlab/show/locales">locales</a>, <a class="existingWikiWord" href="/nlab/show/topoi">topoi</a>, <a class="existingWikiWord" href="/nlab/show/higher+topoi">higher topoi</a>, <a class="existingWikiWord" href="/nlab/show/uniformity+spaces">uniformity spaces</a> and so on, which specialize or generalize some aspect or structure usually found in <a class="existingWikiWord" href="/nlab/show/Top">Top</a>.</p> <p>One of the tools of topology, <a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopy theory</a>, has long since crossed the boundaries of topology and applies to many other areas, thanks to many examples and motivations as well as of abstract categorical frameworks for homotopy like Quillen <a class="existingWikiWord" href="/nlab/show/model+categories">model categories</a>, Brown’s <a class="existingWikiWord" href="/nlab/show/category+of+fibrant+objects">categories of fibrant objects</a> and so on.</p> <h2 id="Introduction">Introduction</h2> <p>The following gives a quick introduction to some of the core concepts and tools of topology:</p> <ul> <li> <p><em><a href="#Continuity">Continuity</a></em></p> </li> <li> <p><em><a href="#TopologicalSpaces">Topological spaces</a></em></p> </li> <li> <p><em><a href="#Homeomorphisms">Homeomorphism</a></em></p> </li> <li> <p><em><a href="#Homotopy">Homotopy</a></em></p> </li> <li> <p><em><a href="#ConnectedComponents">Connected components</a></em></p> </li> <li> <p><em><a href="#FundamentalGroups">Fundamental group</a></em></p> </li> <li> <p><em><a href="#CoveringSpaces">Covering spaces</a></em></p> </li> </ul> <p>A detailed introduction is going to be at <em><a class="existingWikiWord" href="/nlab/show/Introduction+to+Topology">Introduction to Topology</a></em>.</p> <h3 id="Continuity">Continuity</h3> <p>The key idea of topology is to study <a class="existingWikiWord" href="/nlab/show/spaces">spaces</a> with “<a class="existingWikiWord" href="/nlab/show/continuous+maps">continuous maps</a>” between them. The concept of continuity was made precise first in <a class="existingWikiWord" href="/nlab/show/analysis">analysis</a>, in terms of <a class="existingWikiWord" href="/nlab/show/epsilontic+analysis">epsilontic analysis</a> of <a class="existingWikiWord" href="/nlab/show/open+balls">open balls</a>, recalled as def. <a class="maruku-ref" href="#EpsilonDeltaDefinitionOfContinuity"></a> below. Then it was realized that this has a more elegant formulation in terms of the more general concept of <em><a class="existingWikiWord" href="/nlab/show/open+sets">open sets</a></em>, this is prop. <a class="maruku-ref" href="#ContinuityBetweenMetricSpacesInTermsOfOpenSets"></a> below. Adopting the latter as the definition leads to the concept of <a class="existingWikiWord" href="/nlab/show/topological+spaces">topological spaces</a>, def. <a class="maruku-ref" href="#TopologicalSpace"></a> below.</p> <p>First recall the basic concepts from <a class="existingWikiWord" href="/nlab/show/analysis">analysis</a>:</p> <div class="num_defn" id="MetricSpace"> <h6 id="definition">Definition</h6> <p><strong>(metric space)</strong></p> <p>A <em><a class="existingWikiWord" href="/nlab/show/metric+space">metric space</a></em> is</p> <ol> <li> <p>a <a class="existingWikiWord" href="/nlab/show/set">set</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> (the “underlying set”);</p> </li> <li> <p>a <a class="existingWikiWord" href="/nlab/show/function">function</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>X</mi><mo>×</mo><mi>X</mi><mo>→</mo><mo stretchy="false">[</mo><mn>0</mn><mo>,</mo><mn>∞</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">d \;\colon\; X \times X \to [0,\infty)</annotation></semantics></math> (the “distance function”) from the <a class="existingWikiWord" href="/nlab/show/Cartesian+product">Cartesian product</a> of the set with itself to the <a class="existingWikiWord" href="/nlab/show/nonnegative+number">non-negative</a> <a class="existingWikiWord" href="/nlab/show/real+numbers">real numbers</a></p> </li> </ol> <p>such that for all <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>z</mi><mo>∈</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">x,y,z \in X</annotation></semantics></math>:</p> <ol> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo>⇔</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mi>x</mi><mo>=</mo><mi>y</mi></mrow><annotation encoding="application/x-tex">d(x,y) = 0 \;\;\Leftrightarrow\;\; x = y</annotation></semantics></math></p> </li> <li> <p>(symmetry) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo><mo>=</mo><mi>d</mi><mo stretchy="false">(</mo><mi>y</mi><mo>,</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">d(x,y) = d(y,x)</annotation></semantics></math></p> </li> <li> <p>(<a class="existingWikiWord" href="/nlab/show/triangle+inequality">triangle inequality</a>) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo><mo>+</mo><mi>d</mi><mo stretchy="false">(</mo><mi>y</mi><mo>,</mo><mi>z</mi><mo stretchy="false">)</mo><mo>≥</mo><mi>d</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>z</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">d(x,y)+ d(y,z) \geq d(x,z)</annotation></semantics></math>.</p> </li> </ol> </div> <div class="num_example" id="NormedVectorSpaceBecomesMetricSpace"> <h6 id="example">Example</h6> <p>Every <a class="existingWikiWord" href="/nlab/show/normed+vector+space">normed vector space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>V</mi><mo>,</mo><mrow><mo stretchy="false">|</mo><mo>−</mo><mo stretchy="false">|</mo></mrow><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(V, {\vert - \vert})</annotation></semantics></math> becomes a <a class="existingWikiWord" href="/nlab/show/metric+space">metric space</a> according to def. <a class="maruku-ref" href="#MetricSpace"></a> by setting</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>d</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo><mo>≔</mo><mrow><mo stretchy="false">|</mo><mi>x</mi><mo>−</mo><mi>y</mi><mo stretchy="false">|</mo></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> d(x,y) \coloneqq {\vert x-y \vert} \,. </annotation></semantics></math></div></div> <div class="num_defn" id="EpsilonDeltaDefinitionOfContinuity"> <h6 id="definition_2">Definition</h6> <p><strong>(epsilontic definition of continuity)</strong></p> <div style="float:right;margin:0 10px 10px 0;"> <img src="https://ncatlab.org/nlab/files/EpsilonDeltaBalls.png" width="250" /> </div> <p>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><msub><mi>d</mi> <mi>X</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(X,d_X)</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>Y</mi><mo>,</mo><msub><mi>d</mi> <mi>Y</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(Y,d_Y)</annotation></semantics></math> two <a class="existingWikiWord" href="/nlab/show/metric+spaces">metric spaces</a> (def. <a class="maruku-ref" href="#MetricSpace"></a>), then a <a class="existingWikiWord" href="/nlab/show/function">function</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>f</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>X</mi><mo>⟶</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex"> f \;\colon\; X \longrightarrow Y </annotation></semantics></math></div> <p>is said to be <em>continuous at a point <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></em> if for every <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϵ</mi><mo>></mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\epsilon \gt 0</annotation></semantics></math> there exists <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>δ</mi><mo>></mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\delta\gt 0</annotation></semantics></math> such 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><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo><mo><</mo><mi>δ</mi><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo>⇒</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><msub><mi>d</mi> <mi>Y</mi></msub><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>,</mo><mi>f</mi><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo><</mo><mi>ϵ</mi></mrow><annotation encoding="application/x-tex"> d_X(x,y) \lt \delta \;\;\Rightarrow\;\; d_Y(f(x), f(y)) \lt \epsilon </annotation></semantics></math></div> <p>or equivalently such that</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><mspace width="thickmathspace"></mspace><msubsup><mi>B</mi> <mi>x</mi> <mo>∘</mo></msubsup><mo stretchy="false">(</mo><mi>δ</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo>⊂</mo><mspace width="thickmathspace"></mspace><msubsup><mi>B</mi> <mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow> <mo>∘</mo></msubsup><mo stretchy="false">(</mo><mi>ϵ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> f(\;B_x^\circ(\delta)\;) \;\subset\; B^\circ_{f(x)}(\epsilon) </annotation></semantics></math></div> <p>where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>B</mi> <mo>∘</mo></msup></mrow><annotation encoding="application/x-tex">B^\circ</annotation></semantics></math> denotes the <a class="existingWikiWord" href="/nlab/show/open+ball">open ball</a> (definition <a class="maruku-ref" href="#OpenBalls"></a>).</p> <p>The 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> is called just <em>continuous</em> if it is continuous at every point <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>.</p> </div> <p>We now reformulate this analytic concept in terms of the simple but important concept of <em><a class="existingWikiWord" href="/nlab/show/open+sets">open sets</a></em>:</p> <div class="num_defn" id="OpenBalls"> <h6 id="definition_3">Definition</h6> <p><strong>(open ball)</strong></p> <p>Let <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>, be a <a class="existingWikiWord" href="/nlab/show/metric+space">metric space</a>. Then for every element <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 every <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϵ</mi><mo>∈</mo><msub><mi>ℝ</mi> <mo>+</mo></msub></mrow><annotation encoding="application/x-tex">\epsilon \in \mathbb{R}_+</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/positive+number">positive</a> <a class="existingWikiWord" href="/nlab/show/real+number">real number</a>, write</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msubsup><mi>B</mi> <mi>x</mi> <mo>∘</mo></msubsup><mo stretchy="false">(</mo><mi>ϵ</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo>≔</mo><mspace width="thickmathspace"></mspace><mrow><mo>{</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>ϵ</mi><mo>}</mo></mrow></mrow><annotation encoding="application/x-tex"> B^\circ_x(\epsilon) \;\coloneqq\; \left\{ y \in X \;\vert\; d(x,y) \lt \epsilon \right\} </annotation></semantics></math></div> <p>for the <a class="existingWikiWord" href="/nlab/show/open+ball">open ball</a> of <a class="existingWikiWord" href="/nlab/show/radius">radius</a> <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> around <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math>.</p> </div> <div class="num_defn" id="OpenSubsetsOfAMetricSpace"> <h6 id="definition_4">Definition</h6> <p><strong>(neighbourhood and open set)</strong></p> <p>Let <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> be a <a class="existingWikiWord" href="/nlab/show/metric+space">metric space</a> (def. <a class="maruku-ref" href="#MetricSpace"></a>). Say that</p> <ol> <li> <p>A <em><a class="existingWikiWord" href="/nlab/show/neighbourhood">neighbourhood</a></em> of a point <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> is 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>x</mi><mo>∈</mo><mi>U</mi><mo>⊂</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">x \in U \subset X</annotation></semantics></math> which contains some <a class="existingWikiWord" href="/nlab/show/open+ball">open ball</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>B</mi> <mi>x</mi> <mo>∘</mo></msubsup><mo stretchy="false">(</mo><mi>ϵ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">B_x^\circ(\epsilon)</annotation></semantics></math> around <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> (def. <a class="maruku-ref" href="#OpenBalls"></a>).</p> </li> <li> <p>An <em><a class="existingWikiWord" href="/nlab/show/open+subset">open subset</a></em> 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/subset">subset</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo>⊂</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">U \subset X</annotation></semantics></math> such that for every <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>∈</mo><mi>U</mi></mrow><annotation encoding="application/x-tex">x \in U</annotation></semantics></math> it also contains a <a class="existingWikiWord" href="/nlab/show/neighbourhood">neighbourhood</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math>.</p> </li> </ol> </div> <p>The following picture shows 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>, some <a class="existingWikiWord" href="/nlab/show/open+balls">open balls</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>B</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">B_i</annotation></semantics></math> containing it, and two of its <a class="existingWikiWord" href="/nlab/show/neighbourhoods">neighbourhoods</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>U</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">U_i</annotation></semantics></math>:</p> <p><img src="https://ncatlab.org/nlab/files/NeighbourhoodsAndOpenBalls.png" width="500" /></p> <blockquote> <p>graphics grabbed from <a href="#Munkres75">Munkres 75</a></p> </blockquote> <div class="num_prop" id="ContinuityBetweenMetricSpacesInTermsOfOpenSets"> <h6 id="proposition">Proposition</h6> <p><strong>(rephrasing continuity in terms of open sets)</strong></p> <p>A <a class="existingWikiWord" href="/nlab/show/function">function</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo lspace="verythinmathspace">:</mo><mi>X</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">f \colon X \to Y</annotation></semantics></math> between <a class="existingWikiWord" href="/nlab/show/metric+spaces">metric spaces</a> (def. <a class="maruku-ref" href="#MetricSpace"></a>) is continuous in the <a class="existingWikiWord" href="/nlab/show/epsilontic+analysis">epsilontic</a> sense of def. <a class="maruku-ref" href="#EpsilonDeltaDefinitionOfContinuity"></a> precisely if it has the property that its <a class="existingWikiWord" href="/nlab/show/pre-images">pre-images</a> of <a class="existingWikiWord" href="/nlab/show/open+subsets">open subsets</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math> (in the sense of def. <a class="maruku-ref" href="#OpenSubsetsOfAMetricSpace"></a>) are open subsets of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>.</p> </div> <div class="num_remark" style="border:solid #0000cc;background: #add8e6;border-width:2px 1px;padding:0 1em;margin:0 1em;"> <p><strong>principle of continuity</strong></p> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math> <em>Pre-Images of open subsets are open.</em></p> </div> <div class="proof"> <h6 id="proof">Proof</h6> <p>First assume 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> is continuous in the epsilontic sense. Then for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>O</mi> <mi>Y</mi></msub><mo>⊂</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">O_Y \subset Y</annotation></semantics></math> any <a class="existingWikiWord" href="/nlab/show/open+subset">open subset</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>∈</mo><msup><mi>f</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><msub><mi>O</mi> <mi>Y</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">x \in f^{-1}(O_Y)</annotation></semantics></math> any point in the pre-image, we need to show that there exists a <a class="existingWikiWord" href="/nlab/show/neighbourhood">neighbourhood</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>f</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><msub><mi>O</mi> <mi>Y</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f^{-1}(O_Y)</annotation></semantics></math>. But by assumption there exists an <a class="existingWikiWord" href="/nlab/show/open+ball">open ball</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>B</mi> <mi>x</mi> <mo>∘</mo></msubsup><mo stretchy="false">(</mo><mi>ϵ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">B_x^\circ(\epsilon)</annotation></semantics></math> with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><msubsup><mi>B</mi> <mi>x</mi> <mo>∘</mo></msubsup><mo stretchy="false">(</mo><mi>ϵ</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>⊂</mo><msub><mi>O</mi> <mi>Y</mi></msub></mrow><annotation encoding="application/x-tex">f(B_x^\circ(\epsilon)) \subset O_Y</annotation></semantics></math>. Since this is true for all <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 definition this means that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>f</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><msub><mi>O</mi> <mi>Y</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f^{-1}(O_Y)</annotation></semantics></math> is open in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>.</p> <p>Conversely, assume that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>f</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup></mrow><annotation encoding="application/x-tex">f^{-1}</annotation></semantics></math> takes open subsets to open subsets. Then for every <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><msubsup><mi>B</mi> <mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow> <mo>∘</mo></msubsup><mo stretchy="false">(</mo><mi>ϵ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">B_{f(x)}^\circ(\epsilon)</annotation></semantics></math> an <a class="existingWikiWord" href="/nlab/show/open+ball">open ball</a> around its image, we need to produce an open ball <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>B</mi> <mi>x</mi> <mo>∘</mo></msubsup><mo stretchy="false">(</mo><mi>δ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">B_x^\circ(\delta)</annotation></semantics></math> in its pre-image. But by assumption <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>f</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><msubsup><mi>B</mi> <mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow> <mo>∘</mo></msubsup><mo stretchy="false">(</mo><mi>ϵ</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f^{-1}(B_{f(x)}^\circ(\epsilon))</annotation></semantics></math> contains a <a class="existingWikiWord" href="/nlab/show/neighbourhood">neighbourhood</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math> which by definition means that it contains such an open ball around <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math>.</p> </div> <h3 id="TopologicalSpaces">Topological spaces</h3> <p>Therefore we should pay attention to <a class="existingWikiWord" href="/nlab/show/open+subsets">open subsets</a>. It turns out that the following closure property is what <em>characterizes</em> the concept:</p> <div class="num_prop"> <h6 id="proposition_2">Proposition</h6> <p><strong>(closure properties of open sets in a metric space)</strong></p> <p>The collection of <a class="existingWikiWord" href="/nlab/show/open+subsets">open subsets</a> of a <a class="existingWikiWord" href="/nlab/show/metric+space">metric space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>d</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(X,d)</annotation></semantics></math> as in def. <a class="maruku-ref" href="#OpenSubsetsOfAMetricSpace"></a> has the following properties:</p> <ol> <li> <p>The <a class="existingWikiWord" href="/nlab/show/intersection">intersection</a> of any <a class="existingWikiWord" href="/nlab/show/finite+number">finite number</a> of open subsets is again an open subset.</p> </li> <li> <p>The <a class="existingWikiWord" href="/nlab/show/union">union</a> of any <a class="existingWikiWord" href="/nlab/show/set">set</a> of open subsets is again an open subset.</p> </li> </ol> <p>In particular</p> <ul> <li>the <a class="existingWikiWord" href="/nlab/show/empty+set">empty set</a> is open (being the union of no subsets)</li> </ul> <p>and</p> <ul> <li>the whole 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> itself is open (being the intersection of no subsets).</li> </ul> </div> <p>This motivates the following generalized definition:</p> <div class="num_defn" id="TopologicalSpace"> <h6 id="definition_5">Definition</h6> <p><strong>(topological spaces)</strong></p> <p>Given a <a class="existingWikiWord" href="/nlab/show/set">set</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>, then a <em>topology</em> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> is a collection <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>τ</mi></mrow><annotation encoding="application/x-tex">\tau</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/subsets">subsets</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> called the <em><a class="existingWikiWord" href="/nlab/show/open+subsets">open subsets</a></em>, hence a <a class="existingWikiWord" href="/nlab/show/subset">subset</a> of the <a class="existingWikiWord" href="/nlab/show/power+set">power set</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>τ</mi><mo>⊂</mo><mi>P</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \tau \subset P(X) </annotation></semantics></math></div> <p>such that this is closed under forming</p> <ol> <li> <p>finite <a class="existingWikiWord" href="/nlab/show/intersections">intersections</a>;</p> </li> <li> <p>arbitrary <a class="existingWikiWord" href="/nlab/show/unions">unions</a>.</p> </li> </ol> <p>A <em><a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a></em> is 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> equipped with such a <a class="existingWikiWord" href="/nlab/show/topology">topology</a>.</p> </div> <p>The following shows all the topologies on the 3-element set (up to permutation of elements)</p> <p><img src="https://ncatlab.org/nlab/files/TopologiesOn3ElementSet.png" width="400" /></p> <blockquote> <p>graphics grabbed from <a href="#Munkres75">Munkres 75</a></p> </blockquote> <p>It is now immediate to formally implement the</p> <div class="num_remark" style="border:solid #0000cc;background: #add8e6;border-width:2px 1px;padding:0 1em;margin:0 1em;"> <p><strong>principle of continuity</strong></p> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math> <em>Pre-Images of open subsets are open.</em></p> </div> <div class="num_defn" id="ContinuousMaps"> <h6 id="definition_6">Definition</h6> <p><strong>(continuous maps)</strong></p> <p>A <em><a class="existingWikiWord" href="/nlab/show/continuous+function">continuous function</a></em> between <a class="existingWikiWord" href="/nlab/show/topological+spaces">topological spaces</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo lspace="verythinmathspace">:</mo><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><msub><mi>τ</mi> <mi>X</mi></msub><mo stretchy="false">)</mo><mo>→</mo><mo stretchy="false">(</mo><mi>Y</mi><mo>,</mo><msub><mi>τ</mi> <mi>Y</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> f \colon (X, \tau_X) \to (Y, \tau_Y) </annotation></semantics></math></div> <p>is a <a class="existingWikiWord" href="/nlab/show/function">function</a> between the underlying sets,</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 lspace="verythinmathspace">:</mo><mi>X</mi><mo>⟶</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex"> f \colon X \longrightarrow Y </annotation></semantics></math></div> <p>such that <a class="existingWikiWord" href="/nlab/show/pre-images">pre-images</a> under <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> of open subsets 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> are open subsets of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>.</p> </div> <p>The simple definition of <a class="existingWikiWord" href="/nlab/show/open+subsets">open subsets</a> and the simple <em>principle of continuity</em> gives topology its fundamental and universal flavor. The combinatorial nature of these definitions makes topology closely related to <a class="existingWikiWord" href="/nlab/show/formal+logic">formal logic</a> (for more on this see at <em><a class="existingWikiWord" href="/nlab/show/locale">locale</a></em>).</p> <p> <div class='num_remark'> <h6>Remark</h6> <p><strong>(the category of topological spaces)</strong> <br /> The <a class="existingWikiWord" href="/nlab/show/composition">composition</a> of <a class="existingWikiWord" href="/nlab/show/continuous+functions">continuous functions</a> is clearly <a class="existingWikiWord" href="/nlab/show/associativity">associative</a> and <a class="existingWikiWord" href="/nlab/show/unitality">unital</a>.</p> <p>One says that</p> <ol> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+spaces">topological spaces</a> constitute the <a class="existingWikiWord" href="/nlab/show/objects">objects</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/continuous+maps">continuous maps</a> constitute the <a class="existingWikiWord" href="/nlab/show/morphisms">morphisms</a> (<a class="existingWikiWord" href="/nlab/show/homomorphisms">homomorphisms</a>)</p> </li> </ol> <p>of a <em><a class="existingWikiWord" href="/nlab/show/category">category</a></em>. The <em><a class="existingWikiWord" href="/nlab/show/category+of+topological+spaces">category of topological spaces</a></em> (“<a class="existingWikiWord" href="/nlab/show/Top">Top</a>” for short).</p> <p>It is useful to depict collections of <a class="existingWikiWord" href="/nlab/show/objects">objects</a> with <a class="existingWikiWord" href="/nlab/show/morphisms">morphisms</a> between them by <a class="existingWikiWord" href="/nlab/show/diagrams">diagrams</a>, like this one:</p> <p><img src="https://ncatlab.org/nlab/files/AssociativityDiagram.png" width="400" /></p> <blockquote> <p>graphics grabbed from <a href="#category+theory#LawvereSchanuel09">Lawvere-Schanuel 09</a>.</p> </blockquote> <p></p> </div> </p> <p>Our motivating example now reads:</p> <div class="num_example" id="MetricTopology"> <h6 id="example_2">Example</h6> <p><strong>(metric topology)</strong></p> <p>Let <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> be a <a class="existingWikiWord" href="/nlab/show/metric+space">metric space</a>. Then the collection of open subsets in def. <a class="maruku-ref" href="#OpenSubsetsOfAMetricSpace"></a> constitutes a <em><a class="existingWikiWord" href="/nlab/show/topological+space">topology</a></em> on the 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>, making it a <em><a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a></em> in the sense of def. <a class="maruku-ref" href="#TopologicalSpace"></a>. This is called the <em><a class="existingWikiWord" href="/nlab/show/metric+topology">metric topology</a></em>.</p> <p>Stated more concisely: the <a class="existingWikiWord" href="/nlab/show/open+balls">open balls</a> in a metric space constitute a “<a class="existingWikiWord" href="/nlab/show/basis+of+a+topology">basis</a>” for the <a class="existingWikiWord" href="/nlab/show/metric+topology">metric topology</a>.</p> </div> <p>One point of the general definition of “<a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a>” is that it admits constructions which intuitively should exist on “continuous spaces”, but which do not in general exist, for instance, as metric spaces:</p> <div class="num_example" id="DiscreteTopologicalSpace"> <h6 id="example_3">Example</h6> <p><strong>(discrete topological space)</strong></p> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math> a bare <a class="existingWikiWord" href="/nlab/show/set">set</a>, then the <em><a class="existingWikiWord" href="/nlab/show/discrete+topology">discrete topology</a></em> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math> regards <em>every</em> subset of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math> as an <a class="existingWikiWord" href="/nlab/show/open+subset">open subset</a>.</p> </div> <div class="num_example" id="SubspaceTopology"> <h6 id="example_4">Example</h6> <p><strong>(subspace topology)</strong></p> <div style="float:right;margin:0 10px 10px 0;"> <img src="https://ncatlab.org/nlab/files/OpenSubsetsOfSquareInsidePlane.png" width="200" /> </div> <p>Let <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><msub><mi>τ</mi> <mi>X</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(X, \tau_X)</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a>, and let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mn>0</mn></msub><mo>↪</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">X_0 \hookrightarrow X</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/subset">subset</a> of the underlying set. Then the corresponding <em><a class="existingWikiWord" href="/nlab/show/topological+subspace">topological subspace</a></em> has <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">X_0</annotation></semantics></math> as its underlying set, and its open subsets are those subsets of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">X_0</annotation></semantics></math> which arise as restrictions of open subsets of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>.</p> <p>(Also called the <em><a class="existingWikiWord" href="/nlab/show/initial+topology">initial topology</a></em> of the inclusion map.)</p> <p>The picture on the right shows two open subsets inside the <a class="existingWikiWord" href="/nlab/show/square">square</a>, regarded as a <a class="existingWikiWord" href="/nlab/show/topological+subspace">topological subspace</a> of the <a class="existingWikiWord" href="/nlab/show/plane">plane</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mn>2</mn></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^2</annotation></semantics></math>:</p> <blockquote> <p>graphics grabbed from <a href="#Munkres75">Munkres 75</a></p> </blockquote> </div> <div class="num_example" id="QuotientTopologicalSpace"> <h6 id="example_5">Example</h6> <p><strong>(quotient topological space)</strong></p> <p>Let <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><msub><mi>τ</mi> <mi>X</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(X,\tau_X)</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a> (def. <a class="maruku-ref" href="#TopologicalSpace"></a>) and let</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>R</mi> <mo>∼</mo></msub><mo>⊂</mo><mi>X</mi><mo>×</mo><mi>X</mi></mrow><annotation encoding="application/x-tex"> R_\sim \subset X \times X </annotation></semantics></math></div> <p>be an <a class="existingWikiWord" href="/nlab/show/equivalence+relation">equivalence relation</a> on its underlying set. Then the <em><a class="existingWikiWord" href="/nlab/show/quotient+topological+space">quotient topological space</a></em> has</p> <ul> <li>as underlying set the <a class="existingWikiWord" href="/nlab/show/quotient+set">quotient set</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mo>∼</mo></msub></mrow><annotation encoding="application/x-tex">X_{\sim}</annotation></semantics></math>, hence the set of <a class="existingWikiWord" href="/nlab/show/equivalence+classes">equivalence classes</a>,</li> </ul> <p>and</p> <ul> <li> <p>a subset <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>O</mi><mo>⊂</mo><msub><mi>X</mi> <mo>∼</mo></msub></mrow><annotation encoding="application/x-tex">O \subset X_{\sim}</annotation></semantics></math> is declared to be an <a class="existingWikiWord" href="/nlab/show/open+subset">open subset</a> precisely if its <a class="existingWikiWord" href="/nlab/show/preimage">preimage</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>π</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>O</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\pi^{-1}(O)</annotation></semantics></math> under the canonical <a class="existingWikiWord" href="/nlab/show/projection+map">projection map</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>π</mi><mo lspace="verythinmathspace">:</mo><mi>X</mi><mo>→</mo><msub><mi>X</mi> <mo>∼</mo></msub></mrow><annotation encoding="application/x-tex"> \pi \colon X \to X_\sim </annotation></semantics></math></div> <p>is open in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>.</p> </li> </ul> <p>(This is also called the <em><a class="existingWikiWord" href="/nlab/show/final+topology">final topology</a></em> of the projection <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>π</mi></mrow><annotation encoding="application/x-tex">\pi</annotation></semantics></math>.)</p> </div> <p><img src="https://ncatlab.org/nlab/files/QuotientOfSquareIsCylinderAndTorus.png" width="660" /></p> <p>The above picture shows on the left the <a class="existingWikiWord" href="/nlab/show/square">square</a> (a <a class="existingWikiWord" href="/nlab/show/topological+subspace">topological subspace</a> of the <a class="existingWikiWord" href="/nlab/show/plane">plane</a>), then in the middle the resulting <a class="existingWikiWord" href="/nlab/show/quotient+topological+space">quotient topological space</a> obtained by identifying two opposite sides (the <em><a class="existingWikiWord" href="/nlab/show/cylinder">cylinder</a></em>), and on the right the further quotient obtained by identifying the remaining sides (the <em><a class="existingWikiWord" href="/nlab/show/torus">torus</a></em>).</p> <blockquote> <p>graphics grabbed from <a href="#Munkres75">Munkres 75</a></p> </blockquote> <div class="num_example" id="ProductTopologicalSpace"> <h6 id="example_6">Example</h6> <p><strong>(product topological space)</strong></p> <div style="float:right;margin:0 10px 10px 0;"> <img src="https://ncatlab.org/nlab/files/ProductTopology.png" width="300" /> </div> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> 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> two <a class="existingWikiWord" href="/nlab/show/topological+spaces">topological spaces</a>, then the <a class="existingWikiWord" href="/nlab/show/product+topological+space">product topological space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>×</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">X \times Y</annotation></semantics></math> has</p> <ul> <li>as underlying set the <a class="existingWikiWord" href="/nlab/show/Cartesian+product">Cartesian product</a> of the underlying sets 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>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math>,</li> </ul> <p>and</p> <ul> <li>its <a class="existingWikiWord" href="/nlab/show/open+sets">open sets</a> are those subsets <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>O</mi><mo>⊂</mo><mi>X</mi><mo>×</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">O \subset X \times Y</annotation></semantics></math> of the Cartesian product such that for all point <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>y</mi><mo stretchy="false">)</mo><mo>∈</mo><mi>O</mi></mrow><annotation encoding="application/x-tex">(x,y) \in O</annotation></semantics></math> there exists open sets <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>∈</mo><msub><mi>O</mi> <mi>x</mi></msub><mo>⊂</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">x \in O_x \subset X</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>y</mi><mo>∈</mo><msub><mi>O</mi> <mi>Y</mi></msub><mo>⊂</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">y \in O_Y \subset Y</annotation></semantics></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>O</mi> <mi>x</mi></msub><mo>×</mo><msub><mi>O</mi> <mi>y</mi></msub><mo>⊂</mo><mi>O</mi></mrow><annotation encoding="application/x-tex">O_x \times O_y \subset O</annotation></semantics></math>.</li> </ul> <blockquote> <p>graphics grabbed from <a href="#Munkres75">Munkres 75</a></p> </blockquote> </div> <p>These constructions of <a class="existingWikiWord" href="/nlab/show/discrete+topological+spaces">discrete topological spaces</a>, <a class="existingWikiWord" href="/nlab/show/quotient+topological+spaces">quotient topological spaces</a>, <a class="existingWikiWord" href="/nlab/show/topological+subspaces">topological subspaces</a> and of <a class="existingWikiWord" href="/nlab/show/product+topological+spaces">product topological spaces</a> are simple examples of <strong><a class="existingWikiWord" href="/nlab/show/limits">limits</a></strong> and of <strong><a class="existingWikiWord" href="/nlab/show/colimits">colimits</a></strong> of topological spaces. The <a class="existingWikiWord" href="/nlab/show/category">category</a> <a class="existingWikiWord" href="/nlab/show/Top">Top</a> of topological spaces has the convenient property that <em>all</em> <a class="existingWikiWord" href="/nlab/show/limits">limits</a> and <a class="existingWikiWord" href="/nlab/show/colimits">colimits</a> (over <a class="existingWikiWord" href="/nlab/show/small+diagrams">small diagrams</a>) exist in it. (For more on this see at <em><a href="Top#UniversalConstructions">Top – Universal constructions</a></em>.)</p> <h3 id="Homeomorphisms">Homeomorphism</h3> <p>With the <a class="existingWikiWord" href="/nlab/show/objects">objects</a> (<a class="existingWikiWord" href="/nlab/show/topological+spaces">topological spaces</a>) and the <a class="existingWikiWord" href="/nlab/show/morphisms">morphisms</a> (<a class="existingWikiWord" href="/nlab/show/continuous+maps">continuous maps</a>) of the <a class="existingWikiWord" href="/nlab/show/category">category</a> <a class="existingWikiWord" href="/nlab/show/Top">Top</a> of topology thus defined, we obtain the concept of “sameness” in topology.</p> <p>To make this precise, one says that a <a class="existingWikiWord" href="/nlab/show/morphism">morphism</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>X</mi><mover><mo>→</mo><mi>f</mi></mover><mi>Y</mi></mrow><annotation encoding="application/x-tex"> X \overset{f}{\to} Y </annotation></semantics></math></div> <p>in a <a class="existingWikiWord" href="/nlab/show/category">category</a> is an <em><a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphism</a></em> if there exists a morphism going the other way around</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>X</mi><mover><mo>⟵</mo><mrow><msup><mi>f</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup></mrow></mover><mi>Y</mi></mrow><annotation encoding="application/x-tex"> X \overset{f^{-1}}{\longleftarrow} Y </annotation></semantics></math></div> <p>which is an <a class="existingWikiWord" href="/nlab/show/inverse">inverse</a> in the sense that</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>∘</mo><msup><mi>f</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo>=</mo><msub><mi>id</mi> <mi>Y</mi></msub><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mi>and</mi><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><msup><mi>f</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo>∘</mo><mi>f</mi><mo>=</mo><msub><mi>id</mi> <mi>X</mi></msub><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> f \circ f^{-1} = id_Y \;\;\;\;\; and \;\;\;\;\; f^{-1} \circ f = id_X \,. </annotation></semantics></math></div> <div class="num_defn" id="Homeomorphism"> <h6 id="definition_7">Definition</h6> <p><strong>(homeomorphisms)</strong></p> <p>An <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphism</a> in the <a class="existingWikiWord" href="/nlab/show/category">category</a> <a class="existingWikiWord" href="/nlab/show/Top">Top</a> of <a class="existingWikiWord" href="/nlab/show/topological+spaces">topological spaces</a> with <a class="existingWikiWord" href="/nlab/show/continuous+functions">continuous functions</a> between them is called a <em><a class="existingWikiWord" href="/nlab/show/homeomorphism">homeomorphism</a></em>.</p> <p>Hence this is a <a class="existingWikiWord" href="/nlab/show/continuous+function">continuous function</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>f</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>X</mi><mo>⟶</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex"> f \;\colon\; X \longrightarrow Y </annotation></semantics></math></div> <p>such that there exists an <a class="existingWikiWord" href="/nlab/show/inverse">inverse</a> <a class="existingWikiWord" href="/nlab/show/morphism">morphism</a>, namely a <a class="existingWikiWord" href="/nlab/show/continuous+function">continuous function</a> the other way around</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><mi>Y</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><msup><mi>f</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup></mrow><annotation encoding="application/x-tex"> X \longleftarrow Y \;\colon\; f^{-1} </annotation></semantics></math></div> <p>such that</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>∘</mo><msup><mi>f</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo>=</mo><msub><mi>id</mi> <mi>Y</mi></msub><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mi>and</mi><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><msup><mi>f</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo>∘</mo><mi>f</mi><mo>=</mo><msub><mi>id</mi> <mi>X</mi></msub><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> f \circ f^{-1} = id_{Y} \;\;\;and\;\;\; f^{-1} \circ f = id_{X} \,. </annotation></semantics></math></div></div> <p><img src="https://ncatlab.org/nlab/files/Homeomorphism.png" width="560" /></p> <blockquote> <p>graphics grabbed from <a href="#Munkres75">Munkres 75</a></p> </blockquote> <div class="num_example" id="OpenBallsHomeomorphicToRn"> <h6 id="example_7">Example</h6> <p><strong>(open interval homeomorphic to the real line)</strong></p> <p>The open <a class="existingWikiWord" href="/nlab/show/interval">interval</a> <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>1</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(-1,1)</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/homeomorphic">homeomorphic</a> to all of the <a class="existingWikiWord" href="/nlab/show/real+line">real line</a></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 lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo><munder><mo>≃</mo><mi>homeo</mi></munder><msup><mi>ℝ</mi> <mn>1</mn></msup><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> (-1,1) \underset{homeo}{\simeq} \mathbb{R}^1 \,. </annotation></semantics></math></div> <p>An <a class="existingWikiWord" href="/nlab/show/inverse">inverse</a> pair of <a class="existingWikiWord" href="/nlab/show/continuous+functions">continuous functions</a> is for instance given by</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>f</mi></mtd> <mtd><mo lspace="verythinmathspace">:</mo></mtd> <mtd><msup><mi>ℝ</mi> <mn>1</mn></msup></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">+</mo><mn>1</mn><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mi>x</mi></mtd> <mtd><mo>↦</mo></mtd> <mtd><mfrac><mi>x</mi><msqrt><mrow><mn>1</mn><mo>+</mo><msup><mi>x</mi> <mn>2</mn></msup></mrow></msqrt></mfrac></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ f &\colon& \mathbb{R}^1 &\longrightarrow& (-1,+1) \\ && x &\mapsto& \frac{x}{\sqrt{1+ x^2}} } </annotation></semantics></math></div> <p>and</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>f</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup></mtd> <mtd><mo lspace="verythinmathspace">:</mo></mtd> <mtd><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">+</mo><mn>1</mn><mo stretchy="false">)</mo></mtd> <mtd><mo>⟶</mo></mtd> <mtd><msup><mi>ℝ</mi> <mn>1</mn></msup></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mi>x</mi></mtd> <mtd><mo>↦</mo></mtd> <mtd><mfrac><mi>x</mi><msqrt><mrow><mn>1</mn><mo>−</mo><msup><mi>x</mi> <mn>2</mn></msup></mrow></msqrt></mfrac></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ f^{-1} &\colon& (-1,+1) &\longrightarrow& \mathbb{R}^1 \\ && x &\mapsto& \frac{x}{\sqrt{1 - x^2}} } \,. </annotation></semantics></math></div> <p>Generally, every <a class="existingWikiWord" href="/nlab/show/open+ball">open ball</a> 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> (def. <a class="maruku-ref" href="#OpenBalls"></a>) is <a class="existingWikiWord" href="/nlab/show/homeomorphic">homeomorphic</a> to all 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>.</p> </div> <div class="num_example" id="HomeomorphismBetweenTopologicalAndCombinatorialCircle"> <h6 id="example_8">Example</h6> <p><strong>(interval glued at endpoints is homeomorphic to the circle)</strong></p> <p>As topological spaces, the <a class="existingWikiWord" href="/nlab/show/interval">interval</a> with its two endpoints identified is <a class="existingWikiWord" href="/nlab/show/homeomorphic">homeomorphic</a> (def. <a class="maruku-ref" href="#Homeomorphism"></a>) to the standard <a class="existingWikiWord" href="/nlab/show/circle">circle</a>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mn>0</mn><mo>,</mo><mn>1</mn><msub><mo stretchy="false">]</mo> <mrow><mo stretchy="false">/</mo><mo stretchy="false">(</mo><mn>0</mn><mo>∼</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msub><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><munder><mo>≃</mo><mi>homeo</mi></munder><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><msup><mi>S</mi> <mn>1</mn></msup><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> [0,1]_{/(0 \sim 1)} \;\; \underset{homeo}{\simeq} \;\; S^1 \,. </annotation></semantics></math></div> <p>More in detail: let</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mn>1</mn></msup><mo>↪</mo><msup><mi>ℝ</mi> <mn>2</mn></msup></mrow><annotation encoding="application/x-tex"> S^1 \hookrightarrow \mathbb{R}^2 </annotation></semantics></math></div> <p>be the unit <a class="existingWikiWord" href="/nlab/show/circle">circle</a> in the <a class="existingWikiWord" href="/nlab/show/plane">plane</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mn>1</mn></msup><mo>=</mo><mo stretchy="false">{</mo><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo><mo>∈</mo><msup><mi>ℝ</mi> <mn>2</mn></msup><mo>,</mo><msup><mi>x</mi> <mn>2</mn></msup><mo>+</mo><msup><mi>y</mi> <mn>2</mn></msup><mo>=</mo><mn>1</mn><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex"> S^1 = \{(x,y) \in \mathbb{R}^2, x^2 + y^2 = 1\} </annotation></semantics></math></div> <p>equipped with the <a class="existingWikiWord" href="/nlab/show/subspace+topology">subspace topology</a> (example <a class="maruku-ref" href="#SubspaceTopology"></a>) of the plane <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mn>2</mn></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^2</annotation></semantics></math>, which itself equipped with its standard <a class="existingWikiWord" href="/nlab/show/metric+topology">metric topology</a> (example <a class="maruku-ref" href="#MetricTopology"></a>).</p> <p>Moreover, let</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mn>0</mn><mo>,</mo><mn>1</mn><msub><mo stretchy="false">]</mo> <mrow><mo stretchy="false">/</mo><mo stretchy="false">(</mo><mn>0</mn><mo>∼</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msub></mrow><annotation encoding="application/x-tex"> [0,1]_{/(0 \sim 1)} </annotation></semantics></math></div> <p>be the <a class="existingWikiWord" href="/nlab/show/quotient+topological+space">quotient topological space</a> (example <a class="maruku-ref" href="#QuotientTopologicalSpace"></a>) obtained from the <a class="existingWikiWord" href="/nlab/show/interval">interval</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy="false">]</mo><mo>⊂</mo><msup><mi>ℝ</mi> <mn>1</mn></msup></mrow><annotation encoding="application/x-tex">[0,1] \subset \mathbb{R}^1</annotation></semantics></math> with its <a class="existingWikiWord" href="/nlab/show/subspace+topology">subspace topology</a> by applying the <a class="existingWikiWord" href="/nlab/show/equivalence+relation">equivalence relation</a> which identifies the two endpoints (and nothing else).</p> <p>Consider then the function</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>f</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mo stretchy="false">[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy="false">]</mo><mo>⟶</mo><msup><mi>S</mi> <mn>1</mn></msup></mrow><annotation encoding="application/x-tex"> f \;\colon\; [0,1] \longrightarrow S^1 </annotation></semantics></math></div> <p>given by</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>t</mi><mo>↦</mo><mo stretchy="false">(</mo><mi>cos</mi><mo stretchy="false">(</mo><mn>2</mn><mi>π</mi><mi>t</mi><mo stretchy="false">)</mo><mo>,</mo><mi>sin</mi><mo stretchy="false">(</mo><mn>2</mn><mi>π</mi><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> t \mapsto (cos(2\pi t), sin(2\pi t)) \,. </annotation></semantics></math></div> <p>This has the property that <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><mo stretchy="false">)</mo><mo>=</mo><mi>f</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(0) = f(1)</annotation></semantics></math>, so that it descends to the <a class="existingWikiWord" href="/nlab/show/quotient+topological+space">quotient topological space</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mo stretchy="false">[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy="false">]</mo></mtd> <mtd><mover><mo>⟶</mo><mrow></mrow></mover></mtd> <mtd><mo stretchy="false">[</mo><mn>0</mn><mo>,</mo><mn>1</mn><msub><mo stretchy="false">]</mo> <mrow><mo stretchy="false">/</mo><mo stretchy="false">(</mo><mn>0</mn><mo>∼</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msub></mtd></mtr> <mtr><mtd></mtd> <mtd><msub><mrow></mrow> <mpadded width="0" lspace="-100%width"><mi>f</mi></mpadded></msub><mo>↘</mo></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mover><mi>f</mi><mo stretchy="false">˜</mo></mover></mpadded></msup></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><msup><mi>S</mi> <mn>1</mn></msup></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ [0,1] &\overset{}{\longrightarrow}& [0,1]_{/(0 \sim 1)} \\ & {}_{\mathllap{f}}\searrow & \downarrow^{\mathrlap{\tilde f}} \\ && S^1 } \,. </annotation></semantics></math></div> <p>We claim that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>f</mi><mo stretchy="false">˜</mo></mover></mrow><annotation encoding="application/x-tex">\tilde f</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/homeomorphism">homeomorphism</a> (definition <a class="maruku-ref" href="#Homeomorphism"></a>).</p> <p>First of all it is immediate that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>f</mi><mo stretchy="false">˜</mo></mover></mrow><annotation encoding="application/x-tex">\tilde f</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/continuous+function">continuous function</a>. This follows immediately from the fact 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> is a <a class="existingWikiWord" href="/nlab/show/continuous+function">continuous function</a> and by definition of the <a class="existingWikiWord" href="/nlab/show/quotient+topology">quotient topology</a> (example <a class="maruku-ref" href="#QuotientTopologicalSpace"></a>).</p> <p>So we need to check that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>f</mi><mo stretchy="false">˜</mo></mover></mrow><annotation encoding="application/x-tex">\tilde f</annotation></semantics></math> has a continuous inverse function. Clearly the restriction 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> itself to the open interval <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(0,1)</annotation></semantics></math> has a continuous inverse. It fails to have a continuous inverse on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">[0,1)</annotation></semantics></math> and on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">(0,1]</annotation></semantics></math> and fails to have an inverse at all on [0,1], due to the fact that <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><mo stretchy="false">)</mo><mo>=</mo><mi>f</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(0) = f(1)</annotation></semantics></math>. But the relation quotiented out in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mn>0</mn><mo>,</mo><mn>1</mn><msub><mo stretchy="false">]</mo> <mrow><mo stretchy="false">/</mo><mo stretchy="false">(</mo><mn>0</mn><mo>∼</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msub></mrow><annotation encoding="application/x-tex">[0,1]_{/(0 \sim 1)}</annotation></semantics></math> is exactly such as to fix this failure.</p> </div> <p>Similarly:</p> <p>The <a class="existingWikiWord" href="/nlab/show/square">square</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mn>0</mn><mo>,</mo><mn>1</mn><msup><mo stretchy="false">]</mo> <mn>2</mn></msup></mrow><annotation encoding="application/x-tex">[0,1]^2</annotation></semantics></math> with two of its sides identified is the <a class="existingWikiWord" href="/nlab/show/cylinder">cylinder</a>, and with also the other two sides identified is the <a class="existingWikiWord" href="/nlab/show/torus">torus</a>:</p> <p><img src="https://ncatlab.org/nlab/files/TorusAsQuotientOfSquare.png" width="500" /></p> <p>If the sides are identified with opposite orientation, the result is the <em><a class="existingWikiWord" href="/nlab/show/M%C3%B6bius+strip">Möbius strip</a></em>:</p> <p><img src="https://ncatlab.org/nlab/files/MoebiusStripAsQuotientOfSquare.png" width="400" /></p> <blockquote> <p>graphics grabbed from <a href="#Lawson03">Lawson 03</a></p> </blockquote> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></p> <p>Important examples of pairs of spaces that are <em>not</em> homeomorphic include the following:</p> <p> <div class='num_theorem' id='TopologicalInvarianceOfDimension'> <h6>Theorem</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/topological+invariance+of+dimension">topological invariance of dimension</a>)</strong> <br /> For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>n</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>n</mi> <mn>2</mn></msub><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">n_1, n_2 \in \mathbb{N}</annotation></semantics></math> but <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>n</mi> <mn>1</mn></msub><mo>≠</mo><msub><mi>n</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">n_1 \neq n_2</annotation></semantics></math>, then the <a class="existingWikiWord" href="/nlab/show/Cartesian+spaces">Cartesian spaces</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mrow><msub><mi>n</mi> <mn>1</mn></msub></mrow></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^{n_1}</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mrow><msub><mi>n</mi> <mn>2</mn></msub></mrow></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^{n_2}</annotation></semantics></math> are <em>not</em> <a class="existingWikiWord" href="/nlab/show/homeomorphic">homeomorphic</a>.</p> <p>More generally, an <a class="existingWikiWord" href="/nlab/show/open+set">open set</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mrow><msub><mi>n</mi> <mn>1</mn></msub></mrow></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^{n_1}</annotation></semantics></math> is never homeomorphic to an open set in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mrow><msub><mi>n</mi> <mn>2</mn></msub></mrow></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^{n_2}</annotation></semantics></math> if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>n</mi> <mn>1</mn></msub><mo>≠</mo><msub><mi>n</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">n_1 \neq n_2</annotation></semantics></math>.</p> </div> </p> <p>The proof of theorem <a class="maruku-ref" href="#TopologicalInvarianceOfDimension"></a> is surprisingly hard, given how obvious the statement seems intuitively. It requires tools from a field called <em><a class="existingWikiWord" href="/nlab/show/algebraic+topology">algebraic topology</a></em> (notably <a class="existingWikiWord" href="/nlab/show/Brouwer%27s+fixed+point+theorem">Brouwer's fixed point theorem</a>).</p> <p>We showcase some basic tools of <a class="existingWikiWord" href="/nlab/show/algebraic+topology">algebraic topology</a> now and demonstrate the nature of their usage by proving two very simple special cases of the <a class="existingWikiWord" href="/nlab/show/topological+invariance+of+dimension">topological invariance of dimension</a> (prop. <a class="maruku-ref" href="#TopologicalInvarianceOfDimensionFirstSimpleCase"></a> and prop. <a class="maruku-ref" href="#topologicalInvarianceOfDimensionSecondSimpleCase"></a> below).</p> <div class="num_example"> <h6 id="example_9">Example</h6> <p><strong>(homeomorphism classes of surfaces)</strong></p> <p>The <a class="existingWikiWord" href="/nlab/show/2-sphere">2-sphere</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mn>2</mn></msup><mo>=</mo><mo stretchy="false">{</mo><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>z</mi><mo stretchy="false">)</mo><mo>∈</mo><msup><mi>ℝ</mi> <mn>3</mn></msup><mo stretchy="false">|</mo><msup><mi>x</mi> <mn>2</mn></msup><mo>+</mo><msup><mi>y</mi> <mn>2</mn></msup><mo>+</mo><msup><mi>z</mi> <mn>2</mn></msup><mo>=</mo><mn>1</mn><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">S^2 = \{(x,y,z) \in \mathbb{R}^3 \vert x^2 + y^2 + z^2 = 1\}</annotation></semantics></math> is <em>not</em> <a class="existingWikiWord" href="/nlab/show/homeomorphic">homeomorphic</a> to the <a class="existingWikiWord" href="/nlab/show/torus">torus</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>T</mi> <mn>2</mn></msup><mo>=</mo><msup><mi>S</mi> <mn>1</mn></msup><mo>×</mo><msup><mi>S</mi> <mn>1</mn></msup></mrow><annotation encoding="application/x-tex">T^2 = S^1 \times S^1</annotation></semantics></math>.</p> <p>Generally the <a class="existingWikiWord" href="/nlab/show/homeomorphism">homeomorphism</a> class of a <a class="existingWikiWord" href="/nlab/show/closed+manifold">closed</a> <a class="existingWikiWord" href="/nlab/show/orientable">orientable</a> <a class="existingWikiWord" href="/nlab/show/surface">surface</a> is determined by the number of “holes” it has, its <em><a class="existingWikiWord" href="/nlab/show/genus+of+a+surface">genus</a></em>.</p> </div> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></p> <h3 id="Homotopy">Homotopy</h3> <p>We have seen above that for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>≥</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">n \geq 1</annotation></semantics></math> then the <a class="existingWikiWord" href="/nlab/show/open+ball">open ball</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>B</mi> <mn>0</mn> <mo>∘</mo></msubsup><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">B_0^\circ(1)</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 <em>not</em> <a class="existingWikiWord" href="/nlab/show/homeomorphic">homeomorphic</a> to, notably, the point <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>*</mo><mo>=</mo><msup><mi>ℝ</mi> <mn>0</mn></msup></mrow><annotation encoding="application/x-tex">\ast = \mathbb{R}^0</annotation></semantics></math> (example <a class="maruku-ref" href="#OpenBallsHomeomorphicToRn"></a>, theorem <a class="maruku-ref" href="#TopologicalInvarianceOfDimension"></a>). Nevertheless, intuitively 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>-ball is a “continuous deformation” of the point, obtained as the radius of 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>-ball tends to zero.</p> <p>This intuition is made precise by observing that there is a <a class="existingWikiWord" href="/nlab/show/continuous+function">continuous function</a> out of the <a class="existingWikiWord" href="/nlab/show/product+topological+space">product topological space</a> (example <a class="maruku-ref" href="#ProductTopologicalSpace"></a>) of the open ball with the closed <a class="existingWikiWord" href="/nlab/show/interval">interval</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>η</mi><mo lspace="verythinmathspace">:</mo><mo stretchy="false">[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy="false">]</mo><mo>×</mo><msubsup><mi>B</mi> <mn>0</mn> <mo>∘</mo></msubsup><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo>⟶</mo><msup><mi>ℝ</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex"> \eta \colon [0,1] \times B_0^\circ(1) \longrightarrow \mathbb{R}^n </annotation></semantics></math></div> <p>which is given by rescaling:</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><mi>t</mi><mo>,</mo><mi>x</mi><mo stretchy="false">)</mo><mo>↦</mo><mi>t</mi><mo>⋅</mo><mi>x</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> (t,x) \mapsto t \cdot x \,. </annotation></semantics></math></div> <p>This continuously interpolates between the open ball and the point in that 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 = 1</annotation></semantics></math> then it restricts to the defining inclusion <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>B</mi> <mn>0</mn> <mo>∘</mo></msubsup><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">B_0^\circ(1)</annotation></semantics></math>, while for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>t</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">t = 0</annotation></semantics></math> then it restricts to the map constant on the origin.</p> <div style="float:right;margin:0 10px 10px 0;"> <img src="https://ncatlab.org/nlab/files/ShrinkingBalls.png" width="200" /> </div> <p>We may summarize this situation by saying that there is a <a class="existingWikiWord" href="/nlab/show/diagram">diagram</a> of <a class="existingWikiWord" href="/nlab/show/continuous+functions">continuous functions</a> of the form</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><msubsup><mi>B</mi> <mn>0</mn> <mo>∘</mo></msubsup><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo>×</mo><mo stretchy="false">{</mo><mn>0</mn><mo stretchy="false">}</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd><msup><mo>↘</mo> <mpadded width="0"><mrow><mi>x</mi><mo>↦</mo><mn>0</mn></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><mo stretchy="false">[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy="false">]</mo><mo>×</mo><msubsup><mi>B</mi> <mn>0</mn> <mo>∘</mo></msubsup><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>⟶</mo><mrow><mo stretchy="false">(</mo><mi>t</mi><mo>,</mo><mi>x</mi><mo stretchy="false">)</mo><mo>↦</mo><mi>t</mi><mo>⋅</mo><mi>x</mi></mrow></mover></mtd> <mtd><msup><mi>ℝ</mi> <mi>n</mi></msup></mtd></mtr> <mtr><mtd><mo stretchy="false">↑</mo></mtd> <mtd><msub><mo>↗</mo> <mpadded width="0"><mi>inclusion</mi></mpadded></msub></mtd></mtr> <mtr><mtd><msubsup><mi>B</mi> <mn>0</mn> <mo>∘</mo></msubsup><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo>×</mo><mo stretchy="false">{</mo><mn>1</mn><mo stretchy="false">}</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ B_0^\circ(1) \times \{0\} \\ \downarrow & \searrow^{\mathrlap{x \mapsto 0}} \\ [0,1] \times B_0^\circ(1) &\overset{(t,x) \mapsto t \cdot x}{\longrightarrow}& \mathbb{R}^n \\ \uparrow & \nearrow_{\mathrlap{inclusion}} \\ B_0^\circ(1) \times \{1\} } </annotation></semantics></math></div> <p>Such “continuous deformations” are called <em><a class="existingWikiWord" href="/nlab/show/homotopies">homotopies</a></em>:</p> <div class="num_defn" id="LeftHomotopy"> <h6 id="definition_8">Definition</h6> <p><strong>(homotopy)</strong></p> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>,</mo><mi>g</mi><mo lspace="verythinmathspace">:</mo><mi>X</mi><mo>⟶</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">f,g\colon X \longrightarrow Y</annotation></semantics></math> two <a class="existingWikiWord" href="/nlab/show/continuous+functions">continuous functions</a> between <a class="existingWikiWord" href="/nlab/show/topological+spaces">topological spaces</a> <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>, then a <em><a class="existingWikiWord" href="/nlab/show/left+homotopy">(left) homotopy</a></em></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>η</mi><mo lspace="verythinmathspace">:</mo><mi>f</mi><mspace width="thinmathspace"></mspace><msub><mo>⇒</mo> <mi>L</mi></msub><mspace width="thinmathspace"></mspace><mi>g</mi></mrow><annotation encoding="application/x-tex"> \eta \colon f \,\Rightarrow_L\, g </annotation></semantics></math></div> <p>is a <a class="existingWikiWord" href="/nlab/show/continuous+function">continuous function</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>η</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>X</mi><mo>×</mo><mi>I</mi><mo>⟶</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex"> \eta \;\colon\; X \times I \longrightarrow Y </annotation></semantics></math></div> <p>out of the <a class="existingWikiWord" href="/nlab/show/product+topological+space">product topological space</a> (example <a class="maruku-ref" href="#ProductTopologicalSpace"></a>) of the open ball with the standard interval, such that this fits into a <a class="existingWikiWord" href="/nlab/show/commuting+diagram">commuting diagram</a> of the form</p> <div style="float:right;margin:0 10px 10px 0;"> <img src="http://www.ncatlab.org/nlab/files/AHomotopy.jpg" width="400" /> </div><div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mn>0</mn><mo>×</mo><mi>X</mi></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><mo stretchy="false">(</mo><mi>id</mi><mo>,</mo><msub><mi>δ</mi> <mn>0</mn></msub><mo stretchy="false">)</mo></mrow></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd><msup><mo>↘</mo> <mpadded width="0"><mi>f</mi></mpadded></msup></mtd></mtr> <mtr><mtd><mo stretchy="false">[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy="false">]</mo><mo>×</mo><mi>X</mi></mtd> <mtd><mover><mo>⟶</mo><mi>η</mi></mover></mtd> <mtd><mi>Y</mi></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><mo stretchy="false">(</mo><mi>id</mi><mo>,</mo><msub><mi>δ</mi> <mn>1</mn></msub><mo stretchy="false">)</mo></mrow></mpadded></msup><mo stretchy="false">↑</mo></mtd> <mtd><msub><mo>↗</mo> <mpadded width="0"><mi>g</mi></mpadded></msub></mtd></mtr> <mtr><mtd><mo stretchy="false">{</mo><mn>1</mn><mo stretchy="false">}</mo><mo>×</mo><mi>X</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ {0} \times X \\ {}^{\mathllap{(id,\delta_0)}}\downarrow & \searrow^{\mathrlap{f}} \\ [0,1] \times X &\stackrel{\eta}{\longrightarrow}& Y \\ {}^{\mathllap{(id,\delta_1)}}\uparrow & \nearrow_{\mathrlap{g}} \\ \{1\} \times X } \,. </annotation></semantics></math></div> <blockquote> <p>graphics grabbed from J. Tauber <a href="http://jtauber.com/blog/2005/07/01/path_homotopy/">here</a></p> </blockquote> </div> <div class="num_defn" id="HomotopyEquivalence"> <h6 id="definition_9">Definition</h6> <p><strong>(homotopy equivalence)</strong></p> <p>A <a class="existingWikiWord" href="/nlab/show/continuous+function">continuous function</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>X</mi><mo>⟶</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">f \;\colon\; X \longrightarrow Y</annotation></semantics></math> is called a <em><a class="existingWikiWord" href="/nlab/show/homotopy+equivalence">homotopy equivalence</a></em> if</p> <ol> <li> <p>there exists a continuous function the other way around, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>Y</mi><mo>⟶</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">g \;\colon\; Y \longrightarrow X</annotation></semantics></math>, and</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/left+homotopies">left homotopies</a>, def. <a class="maruku-ref" href="#LeftHomotopy"></a>, from the two composites to the identity:</p> </li> </ol> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>η</mi> <mn>1</mn></msub><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>f</mi><mo>∘</mo><mi>g</mi><msub><mo>⇒</mo> <mi>L</mi></msub><msub><mi>id</mi> <mi>Y</mi></msub></mrow><annotation encoding="application/x-tex"> \eta_1 \;\colon\; f\circ g \Rightarrow_L id_Y </annotation></semantics></math></div> <p>and</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>η</mi> <mn>2</mn></msub><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>g</mi><mo>∘</mo><mi>f</mi><msub><mo>⇒</mo> <mi>L</mi></msub><msub><mi>id</mi> <mi>X</mi></msub><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \eta_2 \;\colon\; g\circ f \Rightarrow_L id_X \,. </annotation></semantics></math></div></div> <div class="num_example"> <h6 id="example_10">Example</h6> <p><strong>(open ball is contractible)</strong></p> <p>Any <a class="existingWikiWord" href="/nlab/show/open+ball">open ball</a> (or closed ball), hence (by example <a class="maruku-ref" href="#OpenBallsHomeomorphicToRn"></a>) any <a class="existingWikiWord" href="/nlab/show/Cartesian+space">Cartesian space</a> is <a class="existingWikiWord" href="/nlab/show/homotopy+equivalence">homotopy equivalent</a> to the point</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mi>n</mi></msup><munder><mo>≃</mo><mi>homotopy</mi></munder><mo>*</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \mathbb{R}^n \underset{homotopy}{\simeq} \ast \,. </annotation></semantics></math></div></div> <div class="num_example"> <h6 id="example_11">Example</h6> <p>The following three <a class="existingWikiWord" href="/nlab/show/graphs">graphs</a></p> <p><img src="https://ncatlab.org/nlab/files/ThreeNonHomeoButHomotopyEquivGraphs.png" width="400" /></p> <p>(i.e. the evident <a class="existingWikiWord" href="/nlab/show/topological+subspaces">topological subspaces</a> of the <a class="existingWikiWord" href="/nlab/show/plane">plane</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mn>2</mn></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^2</annotation></semantics></math> that these pictures indicate) are not <a class="existingWikiWord" href="/nlab/show/homeomorphic">homeomorphic</a>. But they are <a class="existingWikiWord" href="/nlab/show/homotopy+equivalence">homotopy equivalent</a>, in fact they are each homotopy equivalent to the <a class="existingWikiWord" href="/nlab/show/disk">disk</a> with two points removed, by the homotopies indicated by the following pictures:</p> <p><img src="https://ncatlab.org/nlab/files/HomotopyEquivalentsToBiAnnulus.png" width="400" /></p> <blockquote> <p>graphics grabbed from <a href="homotopy+equivalence#Hatcher">Hatcher</a></p> </blockquote> </div> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></p> <h3 id="ConnectedComponents">Connected components</h3> <p>Using the concept of <a class="existingWikiWord" href="/nlab/show/homotopy">homotopy</a> one obtains the basic tool of <a class="existingWikiWord" href="/nlab/show/algebraic+topology">algebraic topology</a>, namely the construction of algebraic <a class="existingWikiWord" href="/nlab/show/homotopy+invariants">homotopy invariants</a> of topological spaces. We introduce the simplest and indicate their use.</p> <div class="num_example"> <h6 id="example_12">Example</h6> <p>A <a class="existingWikiWord" href="/nlab/show/homotopy">homotopy</a> between two points</p> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>,</mo><mi>y</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mo>*</mo><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">x,y \;\colon\; \ast \to X</annotation></semantics></math></p> <p>is a continuous <a class="existingWikiWord" href="/nlab/show/path">path</a> between these points.</p> </div> <div class="num_defn" id="pi0"> <h6 id="definition_10">Definition</h6> <p><strong>(connected components)</strong></p> <p>The set of <a class="existingWikiWord" href="/nlab/show/homotopy+classes">homotopy classes</a> of points in a <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> is called its set of <em>path-<a class="existingWikiWord" href="/nlab/show/connected+components">connected components</a></em>, denoted.</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mn>0</mn></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>∈</mo><mi>Set</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \pi_0(X) \in Set \,. </annotation></semantics></math></div> <p>If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mn>0</mn></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>≃</mo><mo>*</mo></mrow><annotation encoding="application/x-tex">\pi_0(X) \simeq \ast</annotation></semantics></math> consists of a single element, 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 a <em><a class="existingWikiWord" href="/nlab/show/path-connected+topological+space">path-connected topological space</a></em>.</p> <p>If <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> is a <a class="existingWikiWord" href="/nlab/show/continuous+map">continuous map</a> between <a class="existingWikiWord" href="/nlab/show/topological+spaces">topological spaces</a>, then on <a class="existingWikiWord" href="/nlab/show/homotopy+classes">homotopy classes</a> of points it induces a <a class="existingWikiWord" href="/nlab/show/function">function</a> between the corresponding connected components, which we denote by:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mn>0</mn></msub><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><msub><mi>π</mi> <mn>0</mn></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>⟶</mo><msub><mi>π</mi> <mn>0</mn></msub><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \pi_0(f) \;\colon\; \pi_0(X) \longrightarrow \pi_0(Y) \,. </annotation></semantics></math></div></div> <p>This construction is evidently compatible with <a class="existingWikiWord" href="/nlab/show/composition">composition</a>, in that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mn>0</mn></msub><mo stretchy="false">(</mo><mi>g</mi><mo>∘</mo><mi>f</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mi>π</mi> <mn>0</mn></msub><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo><mo>∘</mo><msub><mi>π</mi> <mn>0</mn></msub><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \pi_0(g \circ f) = \pi_0(g) \circ \pi_0(f) </annotation></semantics></math></div> <p>and it evidently is <a class="existingWikiWord" href="/nlab/show/unitality">unital</a>, in that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mn>0</mn></msub><mo stretchy="false">(</mo><msub><mi>id</mi> <mi>X</mi></msub><mo stretchy="false">)</mo><mo>=</mo><msub><mi>id</mi> <mrow><msub><mi>π</mi> <mn>0</mn></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow></msub><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \pi_0(id_X) = id_{\pi_{0}(X)} \,. </annotation></semantics></math></div> <p>One summarizes this by saying that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">\pi_0</annotation></semantics></math> is a <em><a class="existingWikiWord" href="/nlab/show/functor">functor</a></em> from the <a class="existingWikiWord" href="/nlab/show/category">category</a> <a class="existingWikiWord" href="/nlab/show/Top">Top</a> of <a class="existingWikiWord" href="/nlab/show/topological+spaces">topological spaces</a> to the <a class="existingWikiWord" href="/nlab/show/category">category</a> <a class="existingWikiWord" href="/nlab/show/Set">Set</a> of <a class="existingWikiWord" href="/nlab/show/sets">sets</a>, denoted</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mn>0</mn></msub><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>Top</mi><mo>⟶</mo><mi>Set</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \pi_0 \;\colon\; Top \longrightarrow Set \,. </annotation></semantics></math></div> <p>An immediate but important consequence is this:</p> <div class="num_prop" id="ConnectedComponentsDistinctImpliesHeomeClassesDistinct"> <h6 id="proposition_3">Proposition</h6> <p>If two <a class="existingWikiWord" href="/nlab/show/topological+spaces">topological spaces</a> have sets of <a class="existingWikiWord" href="/nlab/show/connected+components">connected components</a> that are not in <a class="existingWikiWord" href="/nlab/show/bijection">bijection</a>, then those spaces are not <a class="existingWikiWord" href="/nlab/show/homeomorphic">homeomorphic</a> to each other:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mn>0</mn></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>≠</mo><msub><mi>π</mi> <mn>0</mn></msub><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo>⇒</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thinmathspace"></mspace><mi>X</mi><munder><mo>≠</mo><mi>homeo</mi></munder><mi>Y</mi></mrow><annotation encoding="application/x-tex"> \pi_0(X) \neq \pi_0(Y) \;\;\; \Rightarrow \;\;\, X \underset{homeo}{\neq} Y </annotation></semantics></math></div></div> <div class="proof"> <h6 id="proof_2">Proof</h6> <p>Since <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">\pi_0</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/functor">functorial</a>, it immediately follows that it sends <a class="existingWikiWord" href="/nlab/show/isomorphisms">isomorphisms</a> to <a class="existingWikiWord" href="/nlab/show/isomorphisms">isomorphisms</a>, hence <a class="existingWikiWord" href="/nlab/show/homeomorphisms">homeomorphisms</a> to <a class="existingWikiWord" href="/nlab/show/bijections">bijections</a>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd></mtd> <mtd><mi>f</mi><mo>∘</mo><mi>g</mi><mo>=</mo><mi>id</mi><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mi>and</mi><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mi>g</mi><mo>∘</mo><mi>f</mi><mo>=</mo><mi>id</mi></mtd></mtr> <mtr><mtd><mo>⇒</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace></mtd> <mtd><msub><mi>π</mi> <mn>0</mn></msub><mo stretchy="false">(</mo><mi>f</mi><mo>∘</mo><mi>g</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mi>π</mi> <mn>0</mn></msub><mo stretchy="false">(</mo><mi>id</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mi>and</mi><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><msub><mi>π</mi> <mn>0</mn></msub><mo stretchy="false">(</mo><mi>g</mi><mo>∘</mo><mi>f</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mi>π</mi> <mn>0</mn></msub><mo stretchy="false">(</mo><mi>id</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><mo>⇔</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace></mtd> <mtd><msub><mi>π</mi> <mn>0</mn></msub><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mo>∘</mo><msub><mi>π</mi> <mn>0</mn></msub><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo><mo>=</mo><mi>id</mi><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mi>and</mi><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><msub><mi>π</mi> <mn>0</mn></msub><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo><mo>∘</mo><msub><mi>π</mi> <mn>0</mn></msub><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mo>=</mo><mi>id</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \begin{aligned} & f \circ g = id \;\;and\;\; g \circ f = id \\ \Rightarrow \;\;\;\;\;\;& \pi_0(f \circ g) = \pi_0(id) \;\;and \;\; \pi_0(g \circ f) = \pi_0(id) \\ \Leftrightarrow \;\;\;\;\;\; & \pi_0(f) \circ \pi_0(g) = id \;\;and \;\; \pi_0(g) \circ \pi_0(f) = id \end{aligned} \,. </annotation></semantics></math></div></div> <p>This means that we may use path connected components as a first “topological invariant” that allows us to distinguish some topological spaces.</p> <div class="num_remark" style="border:solid #0000cc;background: #add8e6;border-width:2px 1px;padding:0 1em;margin:0 1em;"> <p><strong>principle of algebraic topology</strong></p> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,\,</annotation></semantics></math> <em>Use topological invariants to distinguish topological spaces.</em></p> </div> <p>As an example for how this is being used, we have the following proof of a simple special case of the <a class="existingWikiWord" href="/nlab/show/topological+invariance+of+dimension">topological invariance of dimension</a> (theorem <a class="maruku-ref" href="#TopologicalInvarianceOfDimension"></a>):</p> <div class="num_prop" id="TopologicalInvarianceOfDimensionFirstSimpleCase"> <h6 id="proposition_4">Proposition</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/topological+invariance+of+dimension">topological invariance of dimension</a> – first simple case)</strong></p> <p>The <a class="existingWikiWord" href="/nlab/show/Cartesian+spaces">Cartesian spaces</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mn>1</mn></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^1</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mn>2</mn></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^2</annotation></semantics></math> are not <a class="existingWikiWord" href="/nlab/show/homeomorphic">homeomorphic</a> (def. <a class="maruku-ref" href="#Homeomorphism"></a>).</p> </div> <div class="proof"> <h6 id="proof_3">Proof</h6> <p>Assume there were a <a class="existingWikiWord" href="/nlab/show/homeomorphism">homeomorphism</a></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 lspace="verythinmathspace">:</mo><msup><mi>ℝ</mi> <mn>1</mn></msup><mo>⟶</mo><msup><mi>ℝ</mi> <mn>2</mn></msup></mrow><annotation encoding="application/x-tex"> f \colon \mathbb{R}^1 \longrightarrow \mathbb{R}^2 </annotation></semantics></math></div> <p>we will derive a contradiction. If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> is a homeomorphism, then clearly so is its restriction to the <a class="existingWikiWord" href="/nlab/show/topological+subspaces">topological subspaces</a> (example <a class="maruku-ref" href="#SubspaceTopology"></a>) obtained by removing <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>0</mn><mo>∈</mo><msup><mi>ℝ</mi> <mn>1</mn></msup></mrow><annotation encoding="application/x-tex">0 \in \mathbb{R}^1</annotation></semantics></math> and <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><mo stretchy="false">)</mo><mo>∈</mo><msup><mi>ℝ</mi> <mn>2</mn></msup></mrow><annotation encoding="application/x-tex">f(0) \in \mathbb{R}^2</annotation></semantics></math>.</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>f</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mn>1</mn></msup><mo>−</mo><mo stretchy="false">{</mo><mn>0</mn><mo stretchy="false">}</mo><mo stretchy="false">)</mo><mo>⟶</mo><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mn>2</mn></msup><mo>−</mo><mo stretchy="false">{</mo><mi>f</mi><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo><mo stretchy="false">}</mo><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> f \;\colon\; (\mathbb{R}^1-\{0\}) \longrightarrow (\mathbb{R}^2 - \{f(0)\}) \,. </annotation></semantics></math></div> <p>It follows that we would get a bijection of <a class="existingWikiWord" href="/nlab/show/connected+components">connected components</a> between <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mn>0</mn></msub><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mn>1</mn></msup><mo>−</mo><mo stretchy="false">{</mo><mn>0</mn><mo stretchy="false">}</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\pi_0(\mathbb{R}^1 - \{0\})</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mn>0</mn></msub><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mn>2</mn></msup><mo>−</mo><mo stretchy="false">{</mo><mi>f</mi><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo><mo stretchy="false">}</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\pi_0(\mathbb{R}^2 - \{f(0)\})</annotation></semantics></math>. But clearly the first set has two elements, while the second has just one:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mn>0</mn></msub><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mn>1</mn></msup><mo>−</mo><mo stretchy="false">{</mo><mn>0</mn><mo stretchy="false">}</mo><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo>≠</mo><mspace width="thickmathspace"></mspace><msub><mi>π</mi> <mn>0</mn></msub><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mn>2</mn></msup><mo>−</mo><mo stretchy="false">{</mo><mi>f</mi><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo><mo stretchy="false">}</mo><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \pi_0(\mathbb{R}^1-\{0\}) \;\neq\; \pi_0(\mathbb{R}^2 - \{f(0)\}) \,. </annotation></semantics></math></div></div> <p>The key lesson of the proof of prop. <a class="maruku-ref" href="#TopologicalInvarianceOfDimensionFirstSimpleCase"></a> is its strategy:</p> <div class="num_remark" style="border:solid #0000cc;background: #add8e6;border-width:2px 1px;padding:0 1em;margin:0 1em;"> <p><strong>principle of algebraic topology</strong></p> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,\,</annotation></semantics></math> <em>Use topological invariants to distinguish topological spaces.</em></p> </div> <p>Of course in practice one uses more sophisticated invariants than just <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">\pi_0</annotation></semantics></math>.</p> <p>The next topological invariant after the <a class="existingWikiWord" href="/nlab/show/connected+components">connected components</a> is the <em><a class="existingWikiWord" href="/nlab/show/fundamental+group">fundamental group</a>:</em></p> <h3 id="FundamentalGroups">Fundamental group</h3> <div class="num_defn" id="FundamentalGroup"> <h6 id="definition_11">Definition</h6> <p><strong>(fundamental group)</strong></p> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a> and let <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> be a chosen point. Then write</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>x</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo>∈</mo><mspace width="thickmathspace"></mspace><mi>Grp</mi></mrow><annotation encoding="application/x-tex"> \pi_1(X,x) \;\in\; Grp </annotation></semantics></math></div> <p>for, to start with, the set of <a class="existingWikiWord" href="/nlab/show/homotopy+classes">homotopy classes</a> of <a class="existingWikiWord" href="/nlab/show/paths">paths</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> that start and end at <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>. Such paths are also called the continuous <a class="existingWikiWord" href="/nlab/show/loops">loops</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> based at <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> <ol> <li> <p>Under concatenation of loops, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\pi_1(X,x)</annotation></semantics></math> becomes a <a class="existingWikiWord" href="/nlab/show/semi-group">semi-group</a>.</p> </li> <li> <p>The constant loop is a <a class="existingWikiWord" href="/nlab/show/neutral+element">neutral element</a> under this composition (thus making <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\pi_1(X,x)</annotation></semantics></math> a “<a class="existingWikiWord" href="/nlab/show/monoid">monoid</a>”).</p> </li> <li> <p>The reverse of a loop is its <a class="existingWikiWord" href="/nlab/show/inverse">inverse</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\pi_1(X,x)</annotation></semantics></math>, making <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\pi_1(X,x)</annotation></semantics></math> indeed into a <a class="existingWikiWord" href="/nlab/show/group">group</a>.</p> </li> </ol> <p>This is called the <em><a class="existingWikiWord" href="/nlab/show/fundamental+group">fundamental group</a></em> 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> at <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math>.</p> </div> <p>The following picture indicates the four non-equivalent non-trivial generators of the <a class="existingWikiWord" href="/nlab/show/fundamental+group">fundamental group</a> of the oriented <a class="existingWikiWord" href="/nlab/show/surface">surface</a> of <a class="existingWikiWord" href="/nlab/show/genus+of+a+surface">genus</a> 2:</p> <p><img src="https://ncatlab.org/nlab/files/FundamentalGroupOfGenus2Surface.png" width="500" /></p> <blockquote> <p>graphics grabbed from <a href="#Lawson03">Lawson 03</a></p> </blockquote> <p>Again, this operation is <a class="existingWikiWord" href="/nlab/show/functor">functorial</a>, now on the <a class="existingWikiWord" href="/nlab/show/category">category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Top</mi> <mrow><mo>*</mo><mo stretchy="false">/</mo></mrow></msup></mrow><annotation encoding="application/x-tex">Top^{\ast/}</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/pointed+topological+spaces">pointed topological spaces</a>, whose <a class="existingWikiWord" href="/nlab/show/objects">objects</a> are topological spaces equipped with a chosen point, and whose <a class="existingWikiWord" href="/nlab/show/morphisms">morphisms</a> are <a class="existingWikiWord" href="/nlab/show/continuous+maps">continuous maps</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo lspace="verythinmathspace">:</mo><mi>X</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">f \colon X \to Y</annotation></semantics></math> that take the chosen basepoint 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> to that of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mn>1</mn></msub><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><msup><mi>Top</mi> <mrow><mo>*</mo><mo stretchy="false">/</mo></mrow></msup><mo>⟶</mo><mi>Grp</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \pi_1 \;\colon\; Top^{\ast/} \longrightarrow Grp \,. </annotation></semantics></math></div> <p>As <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">\pi_0</annotation></semantics></math>, so also <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">\pi_1</annotation></semantics></math> is a topological invariant. As before, we may use this to prove a simple case of the theorem of the <a class="existingWikiWord" href="/nlab/show/topological+invariance+of+dimension">topological invariance of dimension</a>:</p> <div class="num_defn" id="SimplyConnected"> <h6 id="definition_12">Definition</h6> <p>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> for which</p> <ol> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mn>0</mn></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>≃</mo><mo>*</mo></mrow><annotation encoding="application/x-tex">\pi_0(X) \simeq \ast</annotation></semantics></math> (<a class="existingWikiWord" href="/nlab/show/path-connected+topological+space">path connected</a>, def. <a class="maruku-ref" href="#pi0"></a>)</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>x</mi><mo stretchy="false">)</mo><mo>≃</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">\pi_1(X,x) \simeq 1</annotation></semantics></math> (the <a class="existingWikiWord" href="/nlab/show/fundamental+group">fundamental group</a> is <a class="existingWikiWord" href="/nlab/show/trivial+group">trivial</a>, def. <a class="maruku-ref" href="#FundamentalGroup"></a>),</p> </li> </ol> <p>is called <em><a class="existingWikiWord" href="/nlab/show/simply+connected+topological+space">simply connected</a></em>.</p> </div> <div class="num_prop" id="topologicalInvarianceOfDimensionSecondSimpleCase"> <h6 id="proposition_5">Proposition</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/topological+invariance+of+dimension">topological invariance of dimension</a> – second simple case)</strong></p> <p>There is <em>no</em> <a class="existingWikiWord" href="/nlab/show/homeomorphism">homeomorphism</a> between <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mn>2</mn></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^2</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mn>3</mn></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^3</annotation></semantics></math>.</p> </div> <div class="proof"> <h6 id="proof_4">Proof</h6> <p>Assume there were such a homeomorphism <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 will derive a contradiction.</p> <p>If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> is a homeomorphism, then so is its restriction to removing the origin from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mn>2</mn></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^2</annotation></semantics></math> and <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><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(0)</annotation></semantics></math> from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mn>3</mn></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^3</annotation></semantics></math>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mn>2</mn></msup><mo>−</mo><mo stretchy="false">{</mo><mn>0</mn><mo stretchy="false">}</mo><mo stretchy="false">)</mo><mo>⟶</mo><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mn>3</mn></msup><mo>−</mo><mo stretchy="false">{</mo><mi>f</mi><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo><mo stretchy="false">}</mo><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> (\mathbb{R}^2 - \{0\}) \longrightarrow (\mathbb{R}^3 - \{f(0)\}) \,. </annotation></semantics></math></div> <p>Thse two spaces are both path-connected, hence <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">\pi_0</annotation></semantics></math> does not distiguish them.</p> <p>But they do have different <a class="existingWikiWord" href="/nlab/show/fundamental+groups">fundamental groups</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">\pi_1</annotation></semantics></math>:</p> <ol> <li> <p>The fundamental group of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mn>2</mn></msup><mo>−</mo><mo stretchy="false">{</mo><mn>0</mn><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\mathbb{R}^{2} - \{0\}</annotation></semantics></math> is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">\mathbb{Z}</annotation></semantics></math> (counting the winding of loops around the removed point). We discuss this further below in example <a class="maruku-ref" href="#FundamentalGroupOfTheCircle"></a>.</p> </li> <li> <p>The fundamental group of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mn>3</mn></msup><mo>−</mo><mo stretchy="false">{</mo><mi>f</mi><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\mathbb{R}^3 - \{f(0)\}</annotation></semantics></math> is trivial: because the single removed point is no obstruction to sliding loops past it and contracting them.</p> </li> </ol> <p>But since passing to fundamental groups is <a class="existingWikiWord" href="/nlab/show/functor">functorial</a>, the same argument as in the proof of prop. <a class="maruku-ref" href="#ConnectedComponentsDistinctImpliesHeomeClassesDistinct"></a> shows 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> cannot be an <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphism</a>, hence not a <a class="existingWikiWord" href="/nlab/show/homeomorphism">homeomorphism</a>.</p> </div> <p>We now discuss a “dual incarnation” of fundamental groups, which often helps to compute them.</p> <h3 id="CoveringSpaces">Covering spaces</h3> <div class="num_defn" id="CoveringSpace"> <h6 id="definition_13">Definition</h6> <p><strong>(covering space)</strong></p> <p>A <em><a class="existingWikiWord" href="/nlab/show/covering+space">covering space</a></em> of a <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/continuous+map">continuous map</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo lspace="verythinmathspace">:</mo><mi>E</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex"> p \colon E \to X </annotation></semantics></math></div> <p>such that there exists an <a class="existingWikiWord" href="/nlab/show/open+cover">open cover</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><munder><mo>⊔</mo><mi>i</mi></munder><msub><mi>U</mi> <mi>i</mi></msub><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">\underset{i}{\sqcup}U_i \to X</annotation></semantics></math>, such that restricted to each <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>U</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">U_i</annotation></semantics></math> then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">E \to X</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/homeomorphic">homeomorphic</a> over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>U</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">U_i</annotation></semantics></math> to the <a class="existingWikiWord" href="/nlab/show/product+topological+space">product topological space</a> (example <a class="maruku-ref" href="#ProductTopologicalSpace"></a>) of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>U</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">U_i</annotation></semantics></math> with the <a class="existingWikiWord" href="/nlab/show/discrete+topological+space">discrete topological space</a> (example <a class="maruku-ref" href="#DiscreteTopologicalSpace"></a>) on a <a class="existingWikiWord" href="/nlab/show/set">set</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>F</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">F_i</annotation></semantics></math></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><munder><mo>⊔</mo><mi>i</mi></munder><msub><mi>U</mi> <mi>i</mi></msub><mo>×</mo><msub><mi>F</mi> <mi>i</mi></msub></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>E</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd><mo stretchy="false">(</mo><mi>pb</mi><mo stretchy="false">)</mo></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>p</mi></mpadded></msup></mtd></mtr> <mtr><mtd><munder><mo>⊔</mo><mi>i</mi></munder><msub><mi>U</mi> <mi>i</mi></msub></mtd> <mtd><munder><mo>⟶</mo><mrow></mrow></munder></mtd> <mtd><mi>X</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ \underset{i}{\sqcup} U_i \times F_i &\longrightarrow& E \\ \downarrow &(pb)& \downarrow^{\mathrlap{p}} \\ \underset{i}{\sqcup} U_i &\underset{}{\longrightarrow}& X } \,. </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><msub><mi>U</mi> <mi>i</mi></msub><mo>⊂</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">x \in U_i \subset X</annotation></semantics></math> a point, then the elements in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>F</mi> <mi>x</mi></msub><mo>=</mo><msub><mi>F</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">F_x = F_i</annotation></semantics></math> are called the <em>leaves</em> of the covering at <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math>.</p> </div> <div class="num_example" id="kForlCovringOfCircle"> <h6 id="example_13">Example</h6> <p><strong>(covering of circle by circle)</strong></p> <div style="float:left;margin:0 10px 10px 0;"> <img src="https://ncatlab.org/nlab/files/pFoldCoveringOfCircleB.png" width="180" /> </div> <p>Regard the <a class="existingWikiWord" href="/nlab/show/circle">circle</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mn>1</mn></msup><mo>=</mo><mo stretchy="false">{</mo><mi>z</mi><mo>∈</mo><mi>ℂ</mi><mspace width="thickmathspace"></mspace><mo stretchy="false">|</mo><mspace width="thickmathspace"></mspace><mrow><mo stretchy="false">|</mo><mi>z</mi><mo stretchy="false">|</mo></mrow><mo>=</mo><mn>1</mn><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">S^1 = \{ z \in \mathbb{C} \;\vert\; {\vert z\vert} = 1 \}</annotation></semantics></math> as the <a class="existingWikiWord" href="/nlab/show/topological+subspace">topological subspace</a> of elements of unit <a class="existingWikiWord" href="/nlab/show/absolute+value">absolute value</a> in the <a class="existingWikiWord" href="/nlab/show/complex+plane">complex plane</a>. For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">k \in \mathbb{N}</annotation></semantics></math>, consider the continuous function</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo>≔</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><msup><mo stretchy="false">)</mo> <mi>k</mi></msup><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><msup><mi>S</mi> <mn>1</mn></msup><mo>⟶</mo><msup><mi>S</mi> <mn>1</mn></msup></mrow><annotation encoding="application/x-tex"> p \coloneqq (-)^k \;\colon\; S^1 \longrightarrow S^1 </annotation></semantics></math></div> <p>given by taking a complex number to its <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>th power. This may be thought of as the result of “winding the circle <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> times around itself”. Precisely, for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi><mo>≥</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">k \geq 1</annotation></semantics></math> this is a <a class="existingWikiWord" href="/nlab/show/covering+space">covering space</a> (def. <a class="maruku-ref" href="#CoveringSpace"></a>) with <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> leaves at each point.</p> <blockquote> <p>graphics grabbed from <a href="homotopy+equivalence#Hatcher">Hatcher</a></p> </blockquote> </div> <div class="num_example" id="CoveringOfCircleByRealLine"> <h6 id="example_14">Example</h6> <p><strong>(covering of circle by real line)</strong></p> <div style="float:right;margin:0 10px 10px 0;"> <img src="https://ncatlab.org/nlab/files/UniversalCoveringOfCircle.png" width="150" /> </div> <p>Consider the <a class="existingWikiWord" href="/nlab/show/continuous+function">continuous function</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>exp</mi><mo stretchy="false">(</mo><mn>2</mn><mi>π</mi><mi>i</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><msup><mi>ℝ</mi> <mn>1</mn></msup><mo>⟶</mo><msup><mi>S</mi> <mn>1</mn></msup></mrow><annotation encoding="application/x-tex"> \exp(2 \pi i(-)) \;\colon\; \mathbb{R}^1 \longrightarrow S^1 </annotation></semantics></math></div> <p>from the <a class="existingWikiWord" href="/nlab/show/real+line">real line</a> to the <a class="existingWikiWord" href="/nlab/show/circle">circle</a>, which,</p> <ol> <li> <p>with the circle regarded as the unit circle in the <a class="existingWikiWord" href="/nlab/show/complex+plane">complex plane</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℂ</mi></mrow><annotation encoding="application/x-tex">\mathbb{C}</annotation></semantics></math>, is given by</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>t</mi><mo>↦</mo><mi>exp</mi><mo stretchy="false">(</mo><mn>2</mn><mi>π</mi><mi>i</mi><mi>t</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> t \mapsto \exp(2\pi i t) </annotation></semantics></math></div></li> <li> <p>with the circle regarded as the unit circle in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mn>2</mn></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^2</annotation></semantics></math>, is given by</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>t</mi><mo>↦</mo><mo stretchy="false">(</mo><mi>cos</mi><mo stretchy="false">(</mo><mn>2</mn><mi>π</mi><mi>t</mi><mo stretchy="false">)</mo><mo>,</mo><mi>sin</mi><mo stretchy="false">(</mo><mn>2</mn><mi>π</mi><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> t \mapsto ( cos(2\pi t), sin(2\pi t) ) \,. </annotation></semantics></math></div></li> </ol> <p>We may think of this as the result of “winding the line around the circle ad infinitum”. Precisely, this is a <a class="existingWikiWord" href="/nlab/show/covering+space">covering space</a> (def. <a class="maruku-ref" href="#CoveringSpace"></a>) with the <a class="existingWikiWord" href="/nlab/show/leaves">leaves</a> at each point forming the set <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">\mathbb{Z}</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/natural+numbers">natural numbers</a>.</p> </div> <div class="num_defn" id="ActionOfFundamentalGroupOnFibersOfCovering"> <h6 id="definition_14">Definition</h6> <p><strong>(action of fundamental group on fibers of covering)</strong></p> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi><mover><mo>⟶</mo><mi>π</mi></mover><mi>X</mi></mrow><annotation encoding="application/x-tex">E \overset{\pi}{\longrightarrow} X</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/covering+space">covering space</a> (def. <a class="maruku-ref" href="#CoveringSpace"></a>)</p> <p>Then 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> any point, and any choice of element <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>e</mi><mo>∈</mo><msub><mi>F</mi> <mi>x</mi></msub></mrow><annotation encoding="application/x-tex">e \in F_x</annotation></semantics></math> of the <a class="existingWikiWord" href="/nlab/show/leaf+space">leaf space</a> over <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 is, up to <a class="existingWikiWord" href="/nlab/show/homotopy">homotopy</a>, a unique way to lift a representative path 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> of an element <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>γ</mi></mrow><annotation encoding="application/x-tex">\gamma</annotation></semantics></math> of the the <a class="existingWikiWord" href="/nlab/show/fundamental+group">fundamental group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\pi_1(X,x)</annotation></semantics></math> (def. <a class="maruku-ref" href="#FundamentalGroup"></a>) to a continuous path in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math> that starts at <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>e</mi></mrow><annotation encoding="application/x-tex">e</annotation></semantics></math>. This path necessarily ends at some (other) point <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ρ</mi> <mi>γ</mi></msub><mo stretchy="false">(</mo><mi>e</mi><mo stretchy="false">)</mo><mo>∈</mo><msub><mi>F</mi> <mi>x</mi></msub></mrow><annotation encoding="application/x-tex">\rho_\gamma(e) \in F_x</annotation></semantics></math> in the same <a class="existingWikiWord" href="/nlab/show/fiber">fiber</a>. This construction provides a <a class="existingWikiWord" href="/nlab/show/function">function</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>ρ</mi></mtd> <mtd><mo lspace="verythinmathspace">:</mo></mtd> <mtd><msub><mi>F</mi> <mi>x</mi></msub><mo>×</mo><msub><mi>π</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>x</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>⟶</mo></mtd> <mtd><msub><mi>F</mi> <mi>x</mi></msub></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mo stretchy="false">(</mo><mi>e</mi><mo>,</mo><mi>γ</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>↦</mo></mtd> <mtd><msub><mi>ρ</mi> <mi>γ</mi></msub><mo stretchy="false">(</mo><mi>e</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ \rho &\colon& F_x \times \pi_1(X,x) &\longrightarrow& F_x \\ && (e,\gamma) &\mapsto& \rho_\gamma(e) } </annotation></semantics></math></div> <p>from the <a class="existingWikiWord" href="/nlab/show/Cartesian+product">Cartesian product</a> of the <a class="existingWikiWord" href="/nlab/show/leaf+space">leaf space</a> with the <a class="existingWikiWord" href="/nlab/show/fundamental+group">fundamental group</a>. This function is compatible with the <a class="existingWikiWord" href="/nlab/show/group">group</a>-structure on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\pi_1(X,x)</annotation></semantics></math>, in that the following <a class="existingWikiWord" href="/nlab/show/commuting+diagram">diagrams commute</a>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><msub><mi>F</mi> <mi>x</mi></msub><mo>×</mo><mo stretchy="false">{</mo><msub><mi>const</mi> <mi>x</mi></msub><mo stretchy="false">}</mo></mtd> <mtd></mtd> <mtd><mo>⟶</mo></mtd> <mtd></mtd> <mtd><msub><mi>F</mi> <mi>x</mi></msub><mo>×</mo><msub><mi>π</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>x</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><msub><mrow></mrow> <mpadded width="0" lspace="-100%width"><mi>id</mi></mpadded></msub><mo>↘</mo></mtd> <mtd></mtd> <mtd><msub><mo>↙</mo> <mpadded width="0"><mi>ρ</mi></mpadded></msub></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><msub><mi>F</mi> <mi>x</mi></msub></mtd></mtr></mtable></mrow><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mrow><mo>(</mo><mrow><mtable><mtr><mtd><mtext>the neutral element,</mtext></mtd></mtr> <mtr><mtd><mtext>i.e. the constant loop,</mtext></mtd></mtr> <mtr><mtd><mtext>acts trivially</mtext></mtd></mtr></mtable></mrow><mo>)</mo></mrow></mrow><annotation encoding="application/x-tex"> \array{ F_x \times \{const_x\} && \longrightarrow && F_x \times \pi_1(X,x) \\ & {}_{\mathllap{id}}\searrow && \swarrow_{\mathrlap{\rho}} \\ && F_x } \;\;\;\;\;\; \left( \array{ \text{the neutral element,} \\ \text{i.e. the constant loop,} \\ \text{acts trivially} } \right) </annotation></semantics></math></div> <p>and</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><msub><mi>F</mi> <mi>x</mi></msub><mo>×</mo><msub><mi>π</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>x</mi><mo stretchy="false">)</mo><mo>×</mo><msub><mi>π</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>x</mi><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>⟶</mo><mrow><mi>ρ</mi><mo>×</mo><mi>id</mi></mrow></mover></mtd> <mtd><msub><mi>F</mi> <mi>x</mi></msub><mo>×</mo><msub><mi>π</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>x</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><mi>id</mi><mo>×</mo><mo stretchy="false">(</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>⋅</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>ρ</mi></mpadded></msup></mtd></mtr> <mtr><mtd><msub><mi>F</mi> <mi>x</mi></msub><mo>×</mo><msub><mi>π</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>x</mi><mo stretchy="false">)</mo></mtd> <mtd><munder><mo>⟶</mo><mi>ρ</mi></munder></mtd> <mtd><msub><mi>F</mi> <mi>x</mi></msub></mtd></mtr></mtable></mrow><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mrow><mo>(</mo><mrow><mtable><mtr><mtd><mtext>acting with two group elements </mtext></mtd></mtr> <mtr><mtd><mtext>is the same as</mtext></mtd></mtr> <mtr><mtd><mtext>first multiplying them</mtext></mtd></mtr> <mtr><mtd><mtext> and then acting with their product element</mtext></mtd></mtr></mtable></mrow><mo>)</mo></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ F_x \times \pi_1(X,x) \times \pi_1(X,x) &\overset{\rho \times id}{\longrightarrow}& F_x \times \pi_1(X,x) \\ {}^{\mathllap{id \times ((-)\cdot(-))}}\downarrow && \downarrow^{\mathrlap{\rho}} \\ F_x \times \pi_1(X,x) &\underset{\rho}{\longrightarrow}& F_x } \;\;\;\;\;\; \left( \array{ \text{acting with two group elements } \\ \text{is the same as} \\ \text{first multiplying them} \\ \text{ and then acting with their product element} } \right) \,. </annotation></semantics></math></div> <p>One says that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ρ</mi></mrow><annotation encoding="application/x-tex">\rho</annotation></semantics></math> is an <em><a class="existingWikiWord" href="/nlab/show/action">action</a></em> or <em><a class="existingWikiWord" href="/nlab/show/permutation+representation">permutation representation</a></em> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\pi_1(X,x)</annotation></semantics></math> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>F</mi> <mi>x</mi></msub></mrow><annotation encoding="application/x-tex">F_x</annotation></semantics></math>.</p> <p>For <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> any <a class="existingWikiWord" href="/nlab/show/group">group</a>, then there is a <a class="existingWikiWord" href="/nlab/show/category">category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi><mi>Set</mi></mrow><annotation encoding="application/x-tex">G Set</annotation></semantics></math> whose <a class="existingWikiWord" href="/nlab/show/objects">objects</a> are <a class="existingWikiWord" href="/nlab/show/sets">sets</a> equipped with an <a class="existingWikiWord" href="/nlab/show/action">action</a> of <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>, and whose <a class="existingWikiWord" href="/nlab/show/morphisms">morphisms</a> are <a class="existingWikiWord" href="/nlab/show/function">functions</a> which respect these actions. The above construction yields a <a class="existingWikiWord" href="/nlab/show/functor">functor</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Cov</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>⟶</mo><msub><mi>π</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>x</mi><mo stretchy="false">)</mo><mi>Set</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> Cov(X) \longrightarrow \pi_1(X,x) Set \,. </annotation></semantics></math></div></div> <div class="num_example"> <h6 id="example_15">Example</h6> <p><strong>(three-sheeted covers of the circle)</strong></p> <div style="float:right;margin:0 10px 10px 0;"> <img src="https://ncatlab.org/nlab/files/The3SheetedCoveringsOfTheCircle.png" width="150" /> </div> <p>There are, up to <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphism</a>, three different 3-sheeted <a class="existingWikiWord" href="/nlab/show/covering+spaces">covering spaces</a> of the <a class="existingWikiWord" href="/nlab/show/circle">circle</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mn>1</mn></msup></mrow><annotation encoding="application/x-tex">S^1</annotation></semantics></math>.</p> <p>The one from example <a class="maruku-ref" href="#kForlCovringOfCircle"></a> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi><mo>=</mo><mn>3</mn></mrow><annotation encoding="application/x-tex">k = 3</annotation></semantics></math>. Another one. And the trivial one. Their corresponding <a class="existingWikiWord" href="/nlab/show/permutation+actions">permutation actions</a> may be seen from the pictures on the right.</p> <blockquote> <p>graphics grabbed from <a href="homotopy+equivalence#Hatcher">Hatcher</a></p> </blockquote> </div> <p>We are now ready to state the main theorem about the <a class="existingWikiWord" href="/nlab/show/fundamental+group">fundamental group</a>. Except that it does require the following slightly technical condition on the base topological space. This condition is satisfied for all “reasonable” topological spaces:</p> <div class="num_defn" id="SemiLocallySimplyConnected"> <h6 id="definition_15">Definition</h6> <p><strong>(semi-locally simply connected)</strong></p> <p>A <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> is called</p> <ol> <li> <p><em><a class="existingWikiWord" href="/nlab/show/locally+path-connected">locally path-connected</a></em> if for every point <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 for every <a class="existingWikiWord" href="/nlab/show/neighbourhood">neighbourhood</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>∈</mo><mi>U</mi><mo>⊂</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">x \in U \subset X</annotation></semantics></math> there exists a neighbourhood <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>∈</mo><mi>V</mi><mo>⊂</mo><mi>U</mi></mrow><annotation encoding="application/x-tex">x \in V \subset U</annotation></semantics></math> such that <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 <a class="existingWikiWord" href="/nlab/show/path-connected+topological+space">path-connected</a> (def. <a class="maruku-ref" href="#pi0"></a>);</p> </li> <li> <p><em><a class="existingWikiWord" href="/nlab/show/semi-locally+simply+connected+topological+space">semi-locally simply connected</a></em> if every point <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> has a <a class="existingWikiWord" href="/nlab/show/neighbourhood">neighbourhood</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>∈</mo><mi>U</mi><mo>⊂</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">x \in U \subset X</annotation></semantics></math> such that the induced morphism of <a class="existingWikiWord" href="/nlab/show/fundamental+groups">fundamental groups</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>U</mi><mo>,</mo><mi>x</mi><mo stretchy="false">)</mo><mo>→</mo><msub><mi>π</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\pi_1(U,x) \to \pi_1(X,x)</annotation></semantics></math> is trivial (i.e. sends everything to the <a class="existingWikiWord" href="/nlab/show/neutral+element">neutral element</a>).</p> </li> </ol> </div> <div class="num_theorem" id="FundamentalTheoremOfCoveringSpaces"> <h6 id="theorem">Theorem</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/fundamental+theorem+of+covering+spaces">fundamental theorem of covering spaces</a>)</strong></p> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a> which is <a class="existingWikiWord" href="/nlab/show/connected+topological+space">path-connected</a> (def. <a class="maruku-ref" href="#pi0"></a>), <a class="existingWikiWord" href="/nlab/show/locally+path-connected+topological+space">locally path connected</a> (def. <a class="maruku-ref" href="#SemiLocallySimplyConnected"></a>) and <a class="existingWikiWord" href="/nlab/show/semi-locally+simply+connected+topological+space">semi-locally simply connected</a> (def. <a class="maruku-ref" href="#SemiLocallySimplyConnected"></a>). Then for any <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> the functor</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>Fib</mi> <mi>x</mi></msub><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>Cov</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mover><mo>⟶</mo><mrow></mrow></mover><msub><mi>π</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>x</mi><mo stretchy="false">)</mo><mi>Set</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> Fib_x \;\colon\; Cov(X) \overset{}{\longrightarrow} \pi_1(X,x) Set \,. </annotation></semantics></math></div> <p>from def. <a class="maruku-ref" href="#ActionOfFundamentalGroupOnFibersOfCovering"></a> that describes the <a class="existingWikiWord" href="/nlab/show/action">action</a> of the <a class="existingWikiWord" href="/nlab/show/fundamental+group">fundamental group</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> on the set of <a class="existingWikiWord" href="/nlab/show/leaves">leaves</a> over <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> has the following property:</p> <ol> <li> <p>every <a class="existingWikiWord" href="/nlab/show/isomorphism+class">isomorphism class</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\pi_1(X,x)</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/actions">actions</a> is in the image of the functor (one says: the functor is <em><a class="existingWikiWord" href="/nlab/show/essentially+surjective+functor">essentially surjective</a></em>);</p> </li> <li> <p>for any two covering spaces <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>E</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>E</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">E_1, E_2</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> then the map on <a class="existingWikiWord" href="/nlab/show/hom-sets">morphism sets</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>Fib</mi> <mi>x</mi></msub><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><msub><mi>Hom</mi> <mrow><mi>Cov</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><msub><mi>E</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>E</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo>⟶</mo><mi>Hom</mi><mo stretchy="false">(</mo><msub><mi>Fib</mi> <mi>x</mi></msub><mo stretchy="false">(</mo><msub><mi>E</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mo>,</mo><msub><mi>Fib</mi> <mi>x</mi></msub><mo stretchy="false">(</mo><msub><mi>E</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> Fib_x \;\colon\; Hom_{Cov(X)}(E_1, E_2) \longrightarrow Hom( Fib_x(E_1), Fib_x(E_2) ) </annotation></semantics></math></div> <p>is a <a class="existingWikiWord" href="/nlab/show/bijection">bijection</a> (one says the functor is a <em><a class="existingWikiWord" href="/nlab/show/fully+faithful+functor">fully faithful functor</a></em> ).</p> </li> </ol> <p>A functor with these two properties one calls an <em><a class="existingWikiWord" href="/nlab/show/equivalence+of+categories">equivalence of categories</a></em>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Cov</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mover><mo>⟶</mo><mo>≃</mo></mover><msub><mi>π</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>x</mi><mo stretchy="false">)</mo><mi>Set</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> Cov(X) \overset{\simeq}{\longrightarrow} \pi_1(X,x) Set \,. </annotation></semantics></math></div></div> <p>This has some interesting implications:</p> <p>Every sufficiently nice 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> as above has a covering which is <a class="existingWikiWord" href="/nlab/show/simply+connected+topological+space">simply connected</a> (def. <a class="maruku-ref" href="#SimplyConnected"></a>). This is the covering corresponding, under the <a class="existingWikiWord" href="/nlab/show/fundamental+theorem+of+covering+spaces">fundamental theorem of covering spaces</a> (theorem <a class="maruku-ref" href="#FundamentalTheoremOfCoveringSpaces"></a>) to the action of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\pi_1(X)</annotation></semantics></math> on itself. This is called the <em><a class="existingWikiWord" href="/nlab/show/universal+covering+space">universal covering space</a></em> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>X</mi><mo stretchy="false">^</mo></mover><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">\hat X \to X</annotation></semantics></math>. The above theorem implies that the <a class="existingWikiWord" href="/nlab/show/fundamental+group">fundamental group</a> itself may be recovered as the <a class="existingWikiWord" href="/nlab/show/automorphisms">automorphisms</a> of the universal covering space:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>≃</mo><msub><mi>Aut</mi> <mrow><msub><mi>Cov</mi> <mrow><mo stretchy="false">/</mo><mi>X</mi></mrow></msub></mrow></msub><mo stretchy="false">(</mo><mover><mi>X</mi><mo stretchy="false">^</mo></mover><mo>,</mo><mover><mi>X</mi><mo stretchy="false">^</mo></mover><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \pi_1(X) \simeq Aut_{Cov_{/X}}(\hat X, \hat X) \,. </annotation></semantics></math></div> <div class="num_example" id="FundamentalGroupOfTheCircle"> <h6 id="example_16">Example</h6> <p><strong>(computing the fundamental group of the circle)</strong></p> <p>The covering <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>exp</mi><mo stretchy="false">(</mo><mn>2</mn><mi>π</mi><mi>i</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><msup><mi>ℝ</mi> <mn>1</mn></msup><mo>→</mo><msup><mi>S</mi> <mn>1</mn></msup></mrow><annotation encoding="application/x-tex">\exp(2\pi i(-)) \;\colon\; \mathbb{R}^1 \to S^1</annotation></semantics></math> from example <a class="maruku-ref" href="#CoveringOfCircleByRealLine"></a> is <a class="existingWikiWord" href="/nlab/show/simply+connected+topological+space">simply connected</a> (def. <a class="maruku-ref" href="#SimplyConnected"></a>), hence must be the <a class="existingWikiWord" href="/nlab/show/universal+covering+space">universal covering space</a>, up to <a class="existingWikiWord" href="/nlab/show/homeomorphism">homeomorphism</a>.</p> <p>It is fairly straightforward to see that the only <a class="existingWikiWord" href="/nlab/show/homeomorphisms">homeomorphisms</a> from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mn>1</mn></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^1</annotation></semantics></math> to itself over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mn>1</mn></msup></mrow><annotation encoding="application/x-tex">S^1</annotation></semantics></math> are given by <a class="existingWikiWord" href="/nlab/show/integer">integer</a> translations by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>∈</mo><mi>ℕ</mi><mo>↪</mo><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">n \in \mathbb{N} \hookrightarrow \mathbb{R}</annotation></semantics></math>:</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> <mn>1</mn></msup></mtd> <mtd></mtd> <mtd><munderover><mo>⟶</mo><mo>≃</mo><mrow><mi>t</mi><mo>↦</mo><mi>t</mi><mo>+</mo><mi>n</mi></mrow></munderover></mtd> <mtd></mtd> <mtd><msup><mi>ℝ</mi> <mn>1</mn></msup></mtd></mtr> <mtr><mtd></mtd> <mtd><msub><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><mi>exp</mi><mo stretchy="false">(</mo><mn>2</mn><mi>π</mi><mi>i</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow></mpadded></msub><mo>↘</mo></mtd> <mtd></mtd> <mtd><msub><mo>↙</mo> <mpadded width="0"><mrow><mi>exp</mi><mo stretchy="false">(</mo><mn>2</mn><mi>π</mi><mi>i</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow></mpadded></msub></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><msup><mi>S</mi> <mn>1</mn></msup></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ \mathbb{R}^1 && \underoverset{\simeq}{t \mapsto t + n}{\longrightarrow} && \mathbb{R}^1 \\ & {}_{\mathllap{\exp(2 \pi i(-))}}\searrow && \swarrow_{\mathrlap{\exp(2 \pi i(-))}} \\ && S^1 } \,. </annotation></semantics></math></div> <p>Hence</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>Aut</mi> <mrow><msub><mi>Cov</mi> <mrow><mo stretchy="false">/</mo><msup><mi>S</mi> <mn>1</mn></msup></mrow></msub></mrow></msub><mo stretchy="false">(</mo><msup><mover><mi>S</mi><mo stretchy="false">^</mo></mover> <mn>1</mn></msup><mo>,</mo><msup><mover><mi>S</mi><mo stretchy="false">^</mo></mover> <mn>1</mn></msup><mo stretchy="false">)</mo><mo>≃</mo><mi>ℤ</mi></mrow><annotation encoding="application/x-tex"> Aut_{Cov_{/S^1}}(\hat S^1, \hat S^1) \simeq \mathbb{Z} </annotation></semantics></math></div> <p>and hence the <a class="existingWikiWord" href="/nlab/show/fundamental+group">fundamental group</a> of the <a class="existingWikiWord" href="/nlab/show/circle">circle</a> is the additive group of <a class="existingWikiWord" href="/nlab/show/integers">integers</a>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><msup><mi>S</mi> <mn>1</mn></msup><mo stretchy="false">)</mo><mo>≃</mo><mi>ℤ</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \pi_1(S^1) \simeq \mathbb{Z} \,. </annotation></semantics></math></div></div> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></p> <hr /> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></p> <h2 id="BasicFacts">Basic facts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Hausdorff+spaces+are+sober">Hausdorff spaces are sober</a>, <a class="existingWikiWord" href="/nlab/show/schemes+are+sober">schemes are sober</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/CW-complexes+are+Hausdorff">CW-complexes are Hausdorff</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/compact+Hausdorff+spaces+are+normal">compact Hausdorff spaces are normal</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/continuous+image+of+a+compact+space+is+compact">continuous image of a compact space is 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/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> </ul> <h2 id="CentralTheorems">Central theorems</h2> <ul> <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/Heine-Borel+theorem">Heine-Borel 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/Brouwer%27s+fixed+point+theorem">Brouwer's fixed point theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Jordan+curve+theorem">Jordan curve theorem</a></p> </li> </ul> <h2 id="RelatedEntries">Related entries</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/algebraic+topology">algebraic topology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/computational+topology">computational topology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/applied+topology">applied topology</a></p> <p><a class="existingWikiWord" href="/nlab/show/topological+data+analysis">topological data analysis</a></p> </li> </ul> <p>The following is an (incomplete) list of available <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>Lab entries related to topology.</p> <h3 id="topological_spaces_2">Topological spaces</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Top">Top</a><a class="existingWikiWord" href="/nlab/show/CW-complex">CW-complex</a>, <a class="existingWikiWord" href="/nlab/show/general+topology">general topology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/induced+topology">induced topology</a>, <a class="existingWikiWord" href="/nlab/show/subspace">subspace</a>, <a class="existingWikiWord" href="/nlab/show/interior">interior</a>, <a class="existingWikiWord" href="/nlab/show/boundary">boundary</a>, <a class="existingWikiWord" href="/nlab/show/closure">closure</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/product+topological+space">product topological space</a>, <a class="existingWikiWord" href="/nlab/show/disjoint+union+topological+space">disjoint union topological space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/sphere">sphere</a>, <a class="existingWikiWord" href="/nlab/show/metric+space">metric space</a>, <a class="existingWikiWord" href="/nlab/show/metrizable+space">metrizable space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a>, <a class="existingWikiWord" href="/nlab/show/continuous+map">continuous map</a>, <a class="existingWikiWord" href="/nlab/show/homeomorphism">homeomorphism</a>, <a class="existingWikiWord" href="/nlab/show/neighborhood">neighborhood</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/pointed+space">pointed space</a>, <a class="existingWikiWord" href="/nlab/show/contractible+space">contractible space</a>, <a class="existingWikiWord" href="/nlab/show/connected+space">connected space</a>, <a class="existingWikiWord" href="/nlab/show/second+countable+space">second countable space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/convergence+space">convergence space</a>, <a class="existingWikiWord" href="/nlab/show/pretopological+space">pretopological space</a>, <a class="existingWikiWord" href="/nlab/show/pseudotopological+space">pseudotopological space</a>, <a class="existingWikiWord" href="/nlab/show/coarse+topology">coarse topology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/metric+space">metric space</a>, <a class="existingWikiWord" href="/nlab/show/filtered+space">filtered space</a>, <a class="existingWikiWord" href="/nlab/show/connected+filtered+space">connected filtered space</a>, <a class="existingWikiWord" href="/nlab/show/complete+space">complete space</a>, <a class="existingWikiWord" href="/nlab/show/net">net</a>, <a class="existingWikiWord" href="/nlab/show/Polish+space">Polish space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/separation+axioms">separation axioms</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/noetherian+topological+space">noetherian topological space</a>, <a class="existingWikiWord" href="/nlab/show/irreducible+topological+space">irreducible topological space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/compact+space">compact space</a>, <a class="existingWikiWord" href="/nlab/show/locally+compact+space">locally compact space</a>, <a class="existingWikiWord" href="/nlab/show/compactum">compactum</a>, <a class="existingWikiWord" href="/nlab/show/paracompact+space">paracompact space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Frechet-Uryson+space">Frechet-Uryson space</a>, <a class="existingWikiWord" href="/nlab/show/sequential+space">sequential space</a>, <a class="existingWikiWord" href="/nlab/show/uniform+space">uniform space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/convenient+category+of+topological+spaces">convenient category of topological spaces</a>, <a class="existingWikiWord" href="/nlab/show/compactly+generated+space">compactly generated space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/nice+topological+space">nice topological space</a>, <a class="existingWikiWord" href="/nlab/show/nice+category+of+spaces">nice category of spaces</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/pointless+topology">pointless topology</a>, <a class="existingWikiWord" href="/nlab/show/locale">locale</a>, <a class="existingWikiWord" href="/nlab/show/cover">cover</a>, <a class="existingWikiWord" href="/nlab/show/site">site</a>, <a class="existingWikiWord" href="/nlab/show/ringed+space">ringed space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Sierpinski+space">Sierpinski space</a>, <a class="existingWikiWord" href="/nlab/show/Warsaw+circle">Warsaw circle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cosmic+topology">cosmic topology</a></p> </li> </ul> <p>See also <a class="existingWikiWord" href="/nlab/show/examples+in+topology">examples in topology</a>.</p> <h3 id="manifolds_and_generalizations">Manifolds and generalizations</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/manifold">manifold</a>, <a class="existingWikiWord" href="/nlab/show/smooth+manifold">smooth manifold</a>, <a class="existingWikiWord" href="/nlab/show/generalized+smooth+space">generalized smooth space</a>, <a class="existingWikiWord" href="/nlab/show/diffeological+space">diffeological space</a>, <a class="existingWikiWord" href="/nlab/show/smooth+space">smooth space</a>, <a class="existingWikiWord" href="/nlab/show/Froelicher+space">Froelicher space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+topology">differential topology</a>, <a class="existingWikiWord" href="/nlab/show/microbundle">microbundle</a>, <a class="existingWikiWord" href="/nlab/show/cobordism">cobordism</a>, <a class="existingWikiWord" href="/nlab/show/derived+smooth+manifold">derived smooth manifold</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/low-dimensional+topology">low-dimensional topology</a>, <a class="existingWikiWord" href="/nlab/show/3-manifold">3-manifold</a></p> <p><a class="existingWikiWord" href="/nlab/show/quantum+topology">quantum topology</a></p> </li> </ul> <h3 id="algebraic_topology_and_homotopy_theory">Algebraic topology and homotopy theory</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy">homotopy</a>, <a class="existingWikiWord" href="/nlab/show/homotopy+inverse">homotopy inverse</a>, <a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopy theory</a>, <a class="existingWikiWord" href="/nlab/show/homotopy+equivalence">homotopy equivalence</a>, <a class="existingWikiWord" href="/nlab/show/rational+homotopy+theory">rational homotopy theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/shape+theory">shape theory</a>, <a class="existingWikiWord" href="/nlab/show/algebraic+topology">algebraic topology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+category">model category</a>, <a class="existingWikiWord" href="/nlab/show/category+of+fibrant+objects">category of fibrant objects</a>, <a class="existingWikiWord" href="/nlab/show/Quillen+adjunction">Quillen adjunction</a>, <a class="existingWikiWord" href="/nlab/show/Quillen+equivalence">Quillen equivalence</a>, <a class="existingWikiWord" href="/nlab/show/category+with+weak+equivalences">category with weak equivalences</a>,</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+2-category">model 2-category</a>, <a class="existingWikiWord" href="/nlab/show/model+stack">model stack</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Reedy+category">Reedy category</a>, <a class="existingWikiWord" href="/nlab/show/Reedy+model+structure">Reedy model structure</a>, <a class="existingWikiWord" href="/nlab/show/generalized+Reedy+category">generalized Reedy category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+chain+complexes">model structure on chain complexes</a>, <a class="existingWikiWord" href="/nlab/show/model+structure+on+crossed+complexes">model structure on crossed complexes</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+hypothesis">homotopy hypothesis</a>, <a class="existingWikiWord" href="/nlab/show/homotopy+theory+of+Grothendieck">homotopy theory of Grothendieck</a>, <a class="existingWikiWord" href="/nlab/show/test+category">test category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+limit">homotopy limit</a>, <a class="existingWikiWord" href="/nlab/show/homotopy+colimit">homotopy colimit</a>, <a class="existingWikiWord" href="/nlab/show/homotopy+pullback">homotopy pullback</a>, <a class="existingWikiWord" href="/nlab/show/homotopy+image">homotopy image</a>, <a class="existingWikiWord" href="/nlab/show/homotopy+coherent+category+theory">homotopy coherent category theory</a>, <a class="existingWikiWord" href="/nlab/show/homotopy+coherent+nerve">homotopy coherent nerve</a>, <a class="existingWikiWord" href="/nlab/show/Brown+representability+theorem">Brown representability theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+type">homotopy type</a>, <a class="existingWikiWord" href="/nlab/show/homotopy+1-type">homotopy 1-type</a>, <a class="existingWikiWord" href="/nlab/show/homotopy+2-type">homotopy 2-type</a>, <a class="existingWikiWord" href="/nlab/show/homotopy+3-type">homotopy 3-type</a>, <a class="existingWikiWord" href="/nlab/show/homotopy+n-type">homotopy n-type</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/deformation+retract">deformation retract</a>, <a class="existingWikiWord" href="/nlab/show/neighborhood+retract">neighborhood retract</a>, <a class="existingWikiWord" href="/nlab/show/Postnikov+system">Postnikov system</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+group">fundamental group</a>, <a class="existingWikiWord" href="/nlab/show/homotopy+group">homotopy group</a>, <a class="existingWikiWord" href="/nlab/show/suspension">suspension</a>, <a class="existingWikiWord" href="/nlab/show/Freudenthal+suspension+theorem">Freudenthal suspension theorem</a>, <a class="existingWikiWord" href="/nlab/show/delooping">delooping</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/loop+space">loop space</a>, <a class="existingWikiWord" href="/nlab/show/free+loop+space+object">free loop space object</a>,</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cup+product">cup product</a>, <a class="existingWikiWord" href="/nlab/show/Dold-Thom+theorem">Dold-Thom theorem</a></p> </li> </ul> <h4 id="topological_homotopy_theory">Topological homotopy theory</h4> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/classical+model+structure+on+topological+spaces">classical model structure on topological spaces</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+group">topological group</a>, <a class="existingWikiWord" href="/nlab/show/H-space">H-space</a>, <a class="existingWikiWord" href="/nlab/show/co-H-space">co-H-space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/principal+bundle">principal bundle</a>, <a class="existingWikiWord" href="/nlab/show/fiber+bundle">fiber bundle</a>, <a class="existingWikiWord" href="/nlab/show/vector+bundle">vector bundle</a>, <a class="existingWikiWord" href="/nlab/show/principal+2-bundle">principal 2-bundle</a>, <a class="existingWikiWord" href="/nlab/show/trivial+bundle">trivial bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fibration">fibration</a>, <a class="existingWikiWord" href="/nlab/show/Hurewicz+fibration">Hurewicz fibration</a>, <a class="existingWikiWord" href="/nlab/show/Hurewicz+connection">Hurewicz connection</a>, <a class="existingWikiWord" href="/nlab/show/homotopy+lifting+property">homotopy lifting property</a>, <a class="existingWikiWord" href="/nlab/show/Dold+fibration">Dold fibration</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>, <a class="existingWikiWord" href="/nlab/show/deformation+retract">deformation retract</a>, <a class="existingWikiWord" href="/nlab/show/model+structure+on+topological+spaces">model structure on topological spaces</a>, <a class="existingWikiWord" href="/nlab/show/Str%C3%B8m%27s+theorem">Strøm's theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cocylinder">cocylinder</a>, <a class="existingWikiWord" href="/nlab/show/cylinder+object">cylinder object</a>, <a class="existingWikiWord" href="/nlab/show/path+object">path object</a>, <a class="existingWikiWord" href="/nlab/show/mapping+cone">mapping cone</a>, <a class="existingWikiWord" href="/nlab/show/path+groupoid">path groupoid</a>, <a class="existingWikiWord" href="/nlab/show/path+n-groupoid">path n-groupoid</a>, <a class="existingWikiWord" href="/nlab/show/fundamental+groupoid">fundamental groupoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/classifying+space">classifying space</a>, <a class="existingWikiWord" href="/nlab/show/Eilenberg-MacLane+space">Eilenberg-MacLane space</a>,<a class="existingWikiWord" href="/nlab/show/Moore+space">Moore space</a>, <a class="existingWikiWord" href="/nlab/show/Moore+path+category">Moore path category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/spectrum">spectrum</a>, <a class="existingWikiWord" href="/nlab/show/smash+product+of+spectra">smash product of spectra</a>, <a class="existingWikiWord" href="/nlab/show/symmetric+spectrum">symmetric spectrum</a>, <a class="existingWikiWord" href="/nlab/show/stable+%28infinity%2C1%29-category">stable (infinity,1)-category</a>, <a class="existingWikiWord" href="/nlab/show/commutative+ring+spectrum">commutative ring spectrum</a></p> </li> </ul> <h4 id="simplicial_homotopy_theory">Simplicial homotopy theory</h4> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/classical+model+structure+on+simplicial+sets">classical model structure on simplicial sets</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/simplicial+set">simplicial set</a>, <a class="existingWikiWord" href="/nlab/show/simplex+category">simplex category</a>, <a class="existingWikiWord" href="/nlab/show/simplicial+identities">simplicial identities</a>, <a class="existingWikiWord" href="/nlab/show/simplicial+object">simplicial object</a>, <a class="existingWikiWord" href="/nlab/show/cosimplicial+object">cosimplicial object</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/simplicial+complex">simplicial complex</a>, <a class="existingWikiWord" href="/nlab/show/singular+simplicial+complex">singular simplicial complex</a>,</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/boundary+of+a+simplex">boundary of a simplex</a>, <a class="existingWikiWord" href="/nlab/show/simplicial+skeleton">simplicial skeleton</a>, <a class="existingWikiWord" href="/nlab/show/category+of+simplices">category of simplices</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/combinatorial+spectrum">combinatorial spectrum</a>, <a class="existingWikiWord" href="/nlab/show/simplicial+groupoid">simplicial groupoid</a>, <a class="existingWikiWord" href="/nlab/show/simplicial+model+category">simplicial model category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/simplicially+enriched+category">simplicially enriched category</a>, <a class="existingWikiWord" href="/nlab/show/quasicategory">quasicategory</a>, <a class="existingWikiWord" href="/nlab/show/Segal+category">Segal category</a>, <a class="existingWikiWord" href="/nlab/show/complete+Segal+space">complete Segal space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Kan+fibration">Kan fibration</a>, <a class="existingWikiWord" href="/nlab/show/Kan+complex">Kan complex</a>, <a class="existingWikiWord" href="/nlab/show/nerve">nerve</a>, <a class="existingWikiWord" href="/nlab/show/nerve+and+realization">nerve and realization</a>, <a class="existingWikiWord" href="/nlab/show/simplicial+homotopy">simplicial homotopy</a>, <a class="existingWikiWord" href="/nlab/show/simplicial+homotopy+group">simplicial homotopy group</a>, <a class="existingWikiWord" href="/nlab/show/simplicial+local+system">simplicial local system</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/simplicial+presheaf">simplicial presheaf</a>, <a class="existingWikiWord" href="/nlab/show/local+model+structure+on+simplicial+presheaves">local model structure on simplicial presheaves</a>, <a class="existingWikiWord" href="/nlab/show/local+model+structure+on+simplicial+sheaves">local model structure on simplicial sheaves</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/marked+simplicial+set">marked simplicial set</a>, <a class="existingWikiWord" href="/nlab/show/model+structure+on+marked+simplicial+over-sets">model structure on marked simplicial over-sets</a>, <a class="existingWikiWord" href="/nlab/show/infinity+topos">infinity topos</a>, <a class="existingWikiWord" href="/nlab/show/Higher+Topos+Theory">Higher Topos Theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/dendroidal+set">dendroidal set</a>, <a class="existingWikiWord" href="/nlab/show/model+structure+on+dendroidal+sets">model structure on dendroidal sets</a>, <a class="existingWikiWord" href="/nlab/show/cubical+set">cubical set</a>, <a class="existingWikiWord" href="/nlab/show/cellular+set">cellular set</a>, <a class="existingWikiWord" href="/nlab/show/cyclic+object">cyclic object</a>, <a class="existingWikiWord" href="/nlab/show/Theta+space">Theta space</a></p> </li> </ul> <h3 id="sheaves_stacks_cohomology">Sheaves, stacks, cohomology</h3> <ul> <li><a class="existingWikiWord" href="/nlab/show/site">site</a>, <a class="existingWikiWord" href="/nlab/show/sheaf">sheaf</a>, <a class="existingWikiWord" href="/nlab/show/flabby+sheaf">flabby sheaf</a>, <a class="existingWikiWord" href="/nlab/show/local+epimorphism">local epimorphism</a>, <a class="existingWikiWord" href="/nlab/show/etale+space">etale space</a>, <a class="existingWikiWord" href="/nlab/show/hypercover">hypercover</a></li> <li><a class="existingWikiWord" href="/nlab/show/topos">topos</a></li> <li><a class="existingWikiWord" href="/nlab/show/cohomology">cohomology</a>, <a class="existingWikiWord" href="/nlab/show/sheaf+cohomology">sheaf cohomology</a>, <a class="existingWikiWord" href="/nlab/show/abelian+sheaf+cohomology">abelian sheaf cohomology</a>, <a class="existingWikiWord" href="/nlab/show/local+system">local system</a>, <a class="existingWikiWord" href="/nlab/show/category+of+sheaves">category of sheaves</a></li> <li><a class="existingWikiWord" href="/nlab/show/%C4%8Cech+cohomology">Čech cohomology</a>, <a class="existingWikiWord" href="/nlab/show/Bredon+cohomology">Bredon cohomology</a>, <a class="existingWikiWord" href="/nlab/show/twisted+cohomology">twisted cohomology</a>, <a class="existingWikiWord" href="/nlab/show/twisting+cochain">twisting cochain</a></li> <li><a class="existingWikiWord" href="/nlab/show/nonabelian+algebraic+topology">nonabelian algebraic topology</a>, <a class="existingWikiWord" href="/nlab/show/nonabelian+cocycle">nonabelian cocycle</a>, <a class="existingWikiWord" href="/nlab/show/nonabelian+cohomology">nonabelian cohomology</a></li> <li><a class="existingWikiWord" href="/nlab/show/principal+infinity-bundle">principal infinity-bundle</a>, <a class="existingWikiWord" href="/nlab/show/BrownAHT">BrownAHT</a>, <a class="existingWikiWord" href="/nlab/show/category+of+fibrant+objects">category of fibrant objects</a></li> <li><a class="existingWikiWord" href="/nlab/show/topological+K-theory">topological K-theory</a>, <a class="existingWikiWord" href="/nlab/show/gerbe">gerbe</a>, <a class="existingWikiWord" href="/nlab/show/twisted+K-theory">twisted K-theory</a>, <a class="existingWikiWord" href="/nlab/show/Karoubi+K-theory">Karoubi K-theory</a>, <a class="existingWikiWord" href="/nlab/show/differential+K-theory">differential K-theory</a>, <a class="existingWikiWord" href="/nlab/show/K-theory+spectrum">K-theory spectrum</a></li> <li><a class="existingWikiWord" href="/nlab/show/de+Rham+cohomology">de Rham cohomology</a>, <a class="existingWikiWord" href="/nlab/show/string+topology">string topology</a></li> <li><a class="existingWikiWord" href="/nlab/show/Thom+space">Thom space</a>, <span class="newWikiWord">Thom bundle<a href="/nlab/new/Thom+bundle">?</a></span>, <a class="existingWikiWord" href="/nlab/show/fiber+integration">fiber integration</a>, <a class="existingWikiWord" href="/nlab/show/equivariant+cohomology">equivariant cohomology</a></li> <li><a class="existingWikiWord" href="/nlab/show/Poincar%C3%A9+duality">Poincaré duality</a>, <a class="existingWikiWord" href="/nlab/show/Spanier-Whitehead+duality">Spanier-Whitehead duality</a></li> <li><a class="existingWikiWord" href="/nlab/show/generalized+cohomology">generalized cohomology</a>, <a class="existingWikiWord" href="/nlab/show/generalized+%28Eilenberg-Steenrod%29+cohomology">generalized (Eilenberg-Steenrod) cohomology</a>, <a class="existingWikiWord" href="/nlab/show/generalized+%28Eilenberg-Steenrod%29+homotopy">generalized (Eilenberg-Steenrod) homotopy</a></li> <li><a class="existingWikiWord" href="/nlab/show/topological+stack">topological stack</a>, <a class="existingWikiWord" href="/nlab/show/orbispace">orbispace</a></li> <li><a class="existingWikiWord" href="/nlab/show/topological+quantum+field+theory">topological quantum field theory</a>, <a class="existingWikiWord" href="/nlab/show/tmf">tmf</a>, <a class="existingWikiWord" href="/nlab/show/cobordism+hypothesis">cobordism hypothesis</a>, <a class="existingWikiWord" href="/nlab/show/topological+T-duality">topological T-duality</a></li> <li><a class="existingWikiWord" href="/nlab/show/characteristic+class">characteristic class</a>, <a class="existingWikiWord" href="/nlab/show/Chern-Weil+theory">Chern-Weil theory</a></li> </ul> <h3 id="noncommutative_topology">Non-commutative topology</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/noncommutative+topology">noncommutative topology</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/C-star-algebra">C-star-algebra</a>, <a class="existingWikiWord" href="/nlab/show/KK-theory">KK-theory</a></li> </ul> </li> </ul> <h3 id="topological_physics">Topological physics</h3> <p>In <a class="existingWikiWord" href="/nlab/show/solid+state+physics">solid state physics</a>:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+phases+of+matter">topological phases of matter</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+insulators">topological insulators</a>, <a class="existingWikiWord" href="/nlab/show/topological+semi-metals">topological semi-metals</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/symmetry+protected+topological+phases">symmetry protected topological phases</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+order">topological order</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/K-theory+classification+of+topological+phases+of+matter">K-theory classification of topological phases of matter</a></p> </li> </ul> <p>In <a class="existingWikiWord" href="/nlab/show/string+theory">string theory</a>:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/K-theory+classification+of+D-brane+charge">D-brane charge quantization in topological K-theory</a></li> </ul> <h3 id="computational_topology">Computational Topology</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/computational+topology">computational topology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+data+analysis">topological data analysis</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+machine+learning">topological machine learning</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/sources+in+computational+topology">sources in computational topology</a></p> </li> </ul> <div> <h2 id="References">References</h2> <h3 id="ReferencesHistoricalOrigins">Historical origins</h3> <p>The general idea of <a class="existingWikiWord" href="/nlab/show/topology">topology</a> goes back to:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Henri+Poincar%C3%A9">Henri Poincaré</a>, <em><a class="existingWikiWord" href="/nlab/show/Analysis+Situs">Analysis Situs</a></em>, Journal de l’École Polytechnique <strong>2</strong> 1 (1895) 1–123 [<a href="https://gallica.bnf.fr/ark:/12148/bpt6k4337198/f7">gallica:12148/bpt6k4337198/f7</a>, Engl: <a href="https://www.maths.ed.ac.uk/~v1ranick/papers/poincare2009.pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/Stillwell_AnalysisSitus.pdf" title="pdf">pdf</a>]</li> </ul> <p>The notion of <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a> involving <a class="existingWikiWord" href="/nlab/show/neighbourhoods">neighbourhoods</a> was first developed, for the special case now known as <em><a class="existingWikiWord" href="/nlab/show/Hausdorff+spaces">Hausdorff spaces</a></em>, in:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Felix+Hausdorff">Felix Hausdorff</a>, <em><a class="existingWikiWord" href="/nlab/show/Grundz%C3%BCge+der+Mengenlehre">Grundzüge der Mengenlehre</a></em>, Leipzig: Veit (1914), Reprinted by Chelsea Publishing Company (1944, 1949, 1965) [ISBN:978-0-8284-0061-9, <a href="https://archive.org/details/grundzgedermen00hausuoft/page/n5/mode/2up">ark:/13960/t2891gn8g</a>]</li> </ul> <p>The more general definition – dropping Hausdorff’s <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>T</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">T_2</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/separation+axiom">separation axiom</a> and formulated in terms of <a class="existingWikiWord" href="/nlab/show/closure+operators">closure operators</a> that preserve finite <a class="existingWikiWord" href="/nlab/show/unions">unions</a> – is due to:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Kazimierz+Kuratowski">Kazimierz Kuratowski</a>, <em>Sur l’opération Ā de l’Analysis Situs</em>, Fundamenta Mathematicae <strong>3</strong> (1922) 182–199 [<a href="https://www.impan.pl/pl/wydawnictwa/czasopisma-i-serie-wydawnicze/fundamenta-mathematicae/all/3/0/92454/sur-l-operation-a-de-l-analysis-situs">doi:10.4064/fm-3-1-182-199</a>]</li> </ul> <p>The modern formulation via <a class="existingWikiWord" href="/nlab/show/open+set">open set</a> was widely popularized by:</p> <ul> <li id="Bourbak71"> <p><a class="existingWikiWord" href="/nlab/show/Nicolas+Bourbaki">Nicolas Bourbaki</a>], <em>Eléments de mathématique II. Première partie. Les structures fondamentales de l’analyse. Livre III. Topologie générale. Chapitre I. Structures topologiques.</em> Actualités scientifiques et industrielles, vol. 858. Hermann, Paris (1940)</p> <p><em>General topology</em>, Elements of Mathematics III, Springer (1971, 1990, 1995) [<a href="https://doi.org/10.1007/978-3-642-61701-0">doi:10.1007/978-3-642-61701-0</a>]</p> </li> </ul> <h3 id="further">Further</h3> <p>Further textbook accounts:</p> <ul> <li id="Kelley55"> <p><a class="existingWikiWord" href="/nlab/show/John+Kelley">John Kelley</a>, <em>General topology</em>, D. van Nostrand, New York (1955), reprinted as: Graduate Texts in Mathematics, Springer (1975) [<a href="https://www.springer.com/gp/book/9780387901251">ISBN:978-0-387-90125-1</a>]</p> </li> <li id="Dugundji66"> <p><a class="existingWikiWord" href="/nlab/show/James+Dugundji">James Dugundji</a>, <em>Topology</em>, Allyn and Bacon 1966 (<a href="https://www.southalabama.edu/mathstat/personal_pages/carter/Dugundji.pdf">pdf</a>)</p> </li> <li id="Munkres75"> <p><a class="existingWikiWord" href="/nlab/show/James+Munkres">James Munkres</a>, <em>Topology</em>, Prentice Hall (1975, 2000) [ISBN:0-13-181629-2, <a href="http://mathcenter.spb.ru/nikaan/2019/topology/4.pdf">pdf</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Richard+E.+Hodel">Richard E. Hodel</a> (ed.), <em>Set-Theoretic Topology</em>, Academic Press (1977) [<a href="https://doi.org/10.1016/C2013-0-11355-4">doi:10.1016/C2013-0-11355-4</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Klaus+J%C3%A4nich">Klaus Jänich</a>, <em>Topology</em>, Undergraduate Texts in Mathematics, Springer (1984, 1999) [<a href="https://link.springer.com/book/9780387908922">ISBN:9780387908922</a>, <a href="https://doi.org/10.1007/978-3-662-10574-0">doi:10.1007/978-3-662-10574-0</a>, Chapters 1-2: <a href="http://topologicalmedialab.net/xinwei/classes/readings/Janich/Janich_Topology_ch1-2.pdf">pdf</a>]</p> </li> <li id="Engelking89"> <p><a class="existingWikiWord" href="/nlab/show/Ryszard+Engelking">Ryszard Engelking</a>, <em>General Topology</em>, Sigma series in pure mathematics <strong>6</strong>, Heldermann 1989 (<a href="https://www.heldermann.de/SSPM/SSPM06/sspm06.htm">ISBN 388538-006-4</a>)</p> </li> <li id="Vickers89"> <p><a class="existingWikiWord" href="/nlab/show/Steven+Vickers">Steven Vickers</a>, <em>Topology via Logic</em>, Cambridge University Press (1989) (<a href="http://www.gbv.de/dms/ilmenau/toc/21309293X.PDF">toc pdf</a>)</p> </li> <li id="Bredon93"> <p><a class="existingWikiWord" href="/nlab/show/Glen+Bredon">Glen Bredon</a>, <em>Topology and Geometry</em>, Graduate Texts in Mathematics <strong>139</strong>, Springer (1993) [<a href="https://link.springer.com/book/10.1007/978-1-4757-6848-0">doi:10.1007/978-1-4757-6848-0</a>, <a href="http://virtualmath1.stanford.edu/~ralph/math215b/Bredon.pdf">pdf</a>]</p> </li> <li id="Lawson03"> <p>Terry Lawson, <em>Topology: A Geometric Approach</em>, Oxford University Press (2003) (<a href="http://users.metu.edu.tr/serge/courses/422-2014/supplementary/TGeometric.pdf">pdf</a>)</p> </li> <li> <p>Anatole Katok, Alexey Sossinsky, <em>Introduction to Modern Topology and Geometry</em> (2010) [<a href="http://akatok.s3-website-us-east-1.amazonaws.com/TOPOLOGY/Contents.pdf">toc pdf</a>, <a class="existingWikiWord" href="/nlab/files/KatokSossinsky-Topology-Ch1.pdf" title="pdf">pdf</a>]</p> </li> </ul> <p>and leading over to <a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopy theory</a>:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Ioan+Mackenzie+James">Ioan Mackenzie James</a>, <em>General Topology and Homotopy Theory</em>, Springer 1984 <p>(<a href="https://doi.org/10.1007/978-1-4613-8283-6">doi:10.1007/978-1-4613-8283-6</a>)</p> </li> </ul> <p>On <a class="existingWikiWord" href="/nlab/show/counterexamples">counterexamples</a> in topology:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Lynn+Arthur+Steen">Lynn Arthur Steen</a>, <a class="existingWikiWord" href="/nlab/show/J.+Arthur+Seebach">J. Arthur Seebach</a>, <em><a class="existingWikiWord" href="/nlab/show/Counterexamples+in+Topology">Counterexamples in Topology</a></em>, Springer 1971 (<a href="https://link.springer.com/book/10.1007/978-1-4612-6290-9">doi:10.1007/978-1-4612-6290-9</a>)</li> </ul> <p>With emphasis on <a class="existingWikiWord" href="/nlab/show/category+theory">category theoretic</a> aspects of <a class="existingWikiWord" href="/nlab/show/general+topology">general topology</a>, notably on <a href="separation+axioms#Reflection"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>T</mi> <mi>n</mi></msub></mrow><annotation encoding="application/x-tex">T_n</annotation></semantics></math>-reflections</a>:</p> <ul> <li id="HerrlichStrecker71"> <p><a class="existingWikiWord" href="/nlab/show/Horst+Herrlich">Horst Herrlich</a>, <a class="existingWikiWord" href="/nlab/show/George+Strecker">George Strecker</a>, <em>Categorical topology – Its origins as exemplified by the unfolding of the theory of topological reflections and coreflections before 1971</em> (<a href="https://link.springer.com/content/pdf/10.1007%2F978-94-017-0468-7_15.pdf">pdf</a>), pages 255-341 in: C. E. Aull, R Lowen (eds.), <em>Handbook of the History of General Topology. Vol. 1</em>, Kluwer 1997 (<a href="https://link.springer.com/book/10.1007/978-94-017-0468-7">doi:10.1007/978-94-017-0468-7</a>)</p> </li> <li id="BradleyBrysonTerilla20"> <p><a class="existingWikiWord" href="/nlab/show/Tai-Danae+Bradley">Tai-Danae Bradley</a>, <a class="existingWikiWord" href="/nlab/show/Tyler+Bryson">Tyler Bryson</a>, <a class="existingWikiWord" href="/nlab/show/John+Terilla">John Terilla</a>, <em>Topology – A categorical approach</em>, MIT Press 2020 (<a href="https://mitpress.mit.edu/books/topology">ISBN:9780262539357</a>, <a href="https://topology.pubpub.org/">web version</a>)</p> </li> </ul> <p>See also:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Alan+Hatcher">Alan Hatcher</a>, <em><a href="https://www.math.cornell.edu/~hatcher/AT/ATpage.html">Algebraic Topology</a></em></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Neil+Strickland">Neil Strickland</a>, <em>A Bestiary of Topological Objects</em> [<a href="https://strickland1.org/courses/bestiary/bestiary.pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/Strickland-BestiaryTopological.pdf" title="pdf">pdf</a>]</p> </li> </ul> <p>and see further references at <em><a class="existingWikiWord" href="/nlab/show/algebraic+topology">algebraic topology</a></em>.</p> <p>Lecture notes:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Friedhelm+Waldhausen">Friedhelm Waldhausen</a>, <em>Topologie</em> (<a href="https://www.math.uni-bielefeld.de/~fw/ein.pdf">pdf</a>)</p> </li> <li> <p>Alex Kuronya, <em>Introduction to topology</em>, 2010 (<a href="https://www.uni-
frankfurt.de/64271720/TopNotes_Spring10.pdf">pdf</a>)</p> </li> <li> <p>Anatole Katok, Alexey Sossinsky, <em>Introduction to modern topology and geometry</em> (<a href="http://www.personal.psu.edu/axk29/MASS-07/Background-forMASS.pdf">pdf</a>)</p> </li> <li id="Schreiber17"> <p><a class="existingWikiWord" href="/nlab/show/Urs+Schreiber">Urs Schreiber</a>, <em><a class="existingWikiWord" href="/nlab/show/Introduction+to+Topology">Introduction to Topology</a></em>, Bonn 2017</p> </li> <li id="Mueger18"> <p><a class="existingWikiWord" href="/nlab/show/Michael+M%C3%BCger">Michael Müger</a>, <em>Topology for the working mathematician</em>, Nijmegen 2018 (<a href="https://www.math.ru.nl/~mueger/topology.pdf">pdf</a>)</p> </li> </ul> <p>Basic topology set up in <a class="existingWikiWord" href="/nlab/show/intuitionistic+mathematics">intuitionistic mathematics</a> is discussed in</p> <ul> <li id="Waaldijk96"><a class="existingWikiWord" href="/nlab/show/Franka+Waaldijk">Franka Waaldijk</a>, <em>modern intuitionistic topology</em>, 1996 (<a href="http://www.fwaaldijk.nl/modern%20intuitionistic%20topology.pdf">pdf</a>)</li> </ul> <p>See also:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Topospaces">Topospaces</a>, a Wiki with basic material on topology.</li> </ul> </div></body></html> </div> <div class="revisedby"> <p> Last revised on June 7, 2024 at 08:50:43. See the <a href="/nlab/history/topology" style="color: #005c19">history</a> of this page for a list of all contributions to it. </p> </div> <div class="navigation navfoot"> <a href="/nlab/edit/topology" accesskey="E" class="navlink" id="edit" rel="nofollow">Edit</a><a href="https://nforum.ncatlab.org/discussion/10182/#Item_15">Discuss</a><span class="backintime"><a href="/nlab/revision/topology/111" accesskey="B" class="navlinkbackintime" id="to_previous_revision" rel="nofollow">Previous revision</a></span><a href="/nlab/show/diff/topology" accesskey="C" class="navlink" id="see_changes" rel="nofollow">Changes from previous revision</a><a href="/nlab/history/topology" accesskey="S" class="navlink" id="history" rel="nofollow">History (111 revisions)</a> <a href="/nlab/show/topology/cite" style="color: black">Cite</a> <a href="/nlab/print/topology" accesskey="p" id="view_print" rel="nofollow">Print</a> <a href="/nlab/source/topology" id="view_source" rel="nofollow">Source</a> </div> </div>  </div>  </body> </html>