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1.1.4-5.2-10-8.2-13.5-10-11.1-5.2-30-15.3-35-27.3-2.5-6 2.8-13.8 9.4-13.6 6.9.2 13.4 7 17.5 12C70.9 34 75 43.8 86.3 47.4z"/> </svg> nLab well-pointed topological space (changes) </h1> <div class="navigation"> <a href='#navEnd'>Skip the Navigation Links</a> | <a href="/nlab/show/diff/HomePage" accesskey="H" title="Home page">Home Page</a> | <a href="/nlab/all_pages" accesskey="A" title="List of all pages">All Pages</a> | <a href="/nlab/latest_revisions" accesskey="U" title="Latest edits and page creations">Latest Revisions</a> | <a href="https://nforum.ncatlab.org/discussion/13487/#Item_3" title="Discuss this page in its dedicated thread on the nForum" style="color: black">Discuss this page</a> | <form accept-charset="utf-8" action="/nlab/search" id="navigationSearchForm" method="get"> <fieldset class="search"><input type="text" id="searchField" name="query" value="Search" style="display:inline-block; float: left;" onfocus="this.value == 'Search' ? this.value = '' : true" onblur="this.value == '' ? this.value = 'Search' : true" /></fieldset> </form> </div> <div id="revision"> Showing changes from revision #3 to #4: <ins class="diffins">Added</ins> | <del class="diffdel">Removed</del> | <del class="diffmod">Chan</del><ins class="diffmod">ged</ins> <div class='rightHandSide'> <div class='toc clickDown' tabindex='0'> <h3 id='context'>Context</h3> <h4 id='topology'>Topology</h4> <div class='hide'> <a class='existingWikiWord' href='/nlab/show/diff/topology'>topology</a> (<a class='existingWikiWord' href='/nlab/show/diff/general+topology'>point-set topology</a>, <a class='existingWikiWord' href='/nlab/show/diff/point-free+topology'>point-free topology</a>) see also <a class='existingWikiWord' href='/nlab/show/diff/differential+topology'>differential topology</a>, <a class='existingWikiWord' href='/nlab/show/diff/algebraic+topology'>algebraic topology</a>, <a class='existingWikiWord' href='/nlab/show/diff/functional+analysis'>functional analysis</a> and <a class='existingWikiWord' href='/nlab/show/diff/topological+homotopy+theory'>topological</a> <a class='existingWikiWord' href='/nlab/show/diff/homotopy+theory'>homotopy theory</a> <a class='existingWikiWord' href='/nlab/show/diff/Introduction+to+Topology'>Introduction</a> Basic concepts <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/open+subspace'>open subset</a>, <a class='existingWikiWord' href='/nlab/show/diff/closed+subspace'>closed subset</a>, <a class='existingWikiWord' href='/nlab/show/diff/neighborhood'>neighbourhood</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/topological+space'>topological space</a>, <a class='existingWikiWord' href='/nlab/show/diff/locale'>locale</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/topological+base'>base for the topology</a>, <a class='existingWikiWord' href='/nlab/show/diff/neighborhood+base'>neighbourhood base</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/finer+topology'>finer/coarser topology</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/closed+subspace'>closure</a>, <a class='existingWikiWord' href='/nlab/show/diff/interior'>interior</a>, <a class='existingWikiWord' href='/nlab/show/diff/boundary'>boundary</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/separation+axioms'>separation</a>, <a class='existingWikiWord' href='/nlab/show/diff/sober+topological+space'>sobriety</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/continuous+map'>continuous function</a>, <a class='existingWikiWord' href='/nlab/show/diff/homeomorphism'>homeomorphism</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/uniformly+continuous+map'>uniformly continuous function</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/embedding+of+topological+spaces'>embedding</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/open+map'>open map</a>, <a class='existingWikiWord' href='/nlab/show/diff/closed+map'>closed map</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/sequence'>sequence</a>, <a class='existingWikiWord' href='/nlab/show/diff/net'>net</a>, <a class='existingWikiWord' href='/nlab/show/diff/subnet'>sub-net</a>, <a class='existingWikiWord' href='/nlab/show/diff/filter'>filter</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/convergence'>convergence</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/category'>category</a> <a class='existingWikiWord' href='/nlab/show/diff/Top'>Top</a> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/convenient+category+of+topological+spaces'>convenient category of topological spaces</a></li> </ul> </li> </ul> <a href='Top#UniversalConstructions'>Universal constructions</a> <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/weak+topology'>initial topology</a>, <a class='existingWikiWord' href='/nlab/show/diff/weak+topology'>final topology</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/subspace'>subspace</a>, <a class='existingWikiWord' href='/nlab/show/diff/quotient+space'>quotient space</a>, </li> <li> fiber space, <a class='existingWikiWord' href='/nlab/show/diff/space+attachment'>space attachment</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/product+topological+space'>product space</a>, <a class='existingWikiWord' href='/nlab/show/diff/disjoint+union+topological+space'>disjoint union space</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/mapping+cylinder'>mapping cylinder</a>, <a class='existingWikiWord' href='/nlab/show/diff/cocylinder'>mapping cocylinder</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/mapping+cone'>mapping cone</a>, <a class='existingWikiWord' href='/nlab/show/diff/mapping+cocone'>mapping cocone</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/mapping+telescope'>mapping telescope</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/colimits+of+normal+spaces'>colimits of normal spaces</a> </li> </ul> <a class='existingWikiWord' href='/nlab/show/diff/stuff%2C+structure%2C+property'>Extra stuff, structure, properties</a> <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/nice+topological+space'>nice topological space</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/metric+space'>metric space</a>, <a class='existingWikiWord' href='/nlab/show/diff/metric+topology'>metric topology</a>, <a class='existingWikiWord' href='/nlab/show/diff/metrisable+topological+space'>metrisable space</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/Kolmogorov+topological+space'>Kolmogorov space</a>, <a class='existingWikiWord' href='/nlab/show/diff/Hausdorff+space'>Hausdorff space</a>, <a class='existingWikiWord' href='/nlab/show/diff/regular+space'>regular space</a>, <a class='existingWikiWord' href='/nlab/show/diff/normal+space'>normal space</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/sober+topological+space'>sober space</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/compact+space'>compact space</a>, <a class='existingWikiWord' href='/nlab/show/diff/proper+map'>proper map</a> <a class='existingWikiWord' href='/nlab/show/diff/sequentially+compact+topological+space'>sequentially compact</a>, <a class='existingWikiWord' href='/nlab/show/diff/countably+compact+topological+space'>countably compact</a>, <a class='existingWikiWord' href='/nlab/show/diff/locally+compact+topological+space'>locally compact</a>, <a class='existingWikiWord' href='/nlab/show/diff/sigma-compact+topological+space'>sigma-compact</a>, <a class='existingWikiWord' href='/nlab/show/diff/paracompact+topological+space'>paracompact</a>, <a class='existingWikiWord' href='/nlab/show/diff/countably+paracompact+topological+space'>countably paracompact</a>, <a class='existingWikiWord' href='/nlab/show/diff/strongly+compact+topological+space'>strongly compact</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/compactly+generated+topological+space'>compactly generated space</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/second-countable+space'>second-countable space</a>, <a class='existingWikiWord' href='/nlab/show/diff/first-countable+space'>first-countable space</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/contractible+space'>contractible space</a>, <a class='existingWikiWord' href='/nlab/show/diff/locally+contractible+space'>locally contractible space</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/connected+space'>connected space</a>, <a class='existingWikiWord' href='/nlab/show/diff/locally+connected+topological+space'>locally connected space</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/simply+connected+space'>simply-connected space</a>, <a class='existingWikiWord' href='/nlab/show/diff/semi-locally+simply-connected+topological+space'>locally simply-connected space</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/cell+complex'>cell complex</a>, <a class='existingWikiWord' href='/nlab/show/diff/CW+complex'>CW-complex</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/pointed+topological+space'>pointed space</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/topological+vector+space'>topological vector space</a>, <a class='existingWikiWord' href='/nlab/show/diff/Banach+space'>Banach space</a>, <a class='existingWikiWord' href='/nlab/show/diff/Hilbert+space'>Hilbert space</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/topological+group'>topological group</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/topological+vector+bundle'>topological vector bundle</a>, <a class='existingWikiWord' href='/nlab/show/diff/topological+K-theory'>topological K-theory</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/topological+manifold'>topological manifold</a> </li> </ul> Examples <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/empty+space'>empty space</a>, <a class='existingWikiWord' href='/nlab/show/diff/point+space'>point space</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/discrete+object'>discrete space</a>, <a class='existingWikiWord' href='/nlab/show/diff/codiscrete+space'>codiscrete space</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/Sierpinski+space'>Sierpinski space</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/order+topology'>order topology</a>, <a class='existingWikiWord' href='/nlab/show/diff/specialization+topology'>specialization topology</a>, <a class='existingWikiWord' href='/nlab/show/diff/Scott+topology'>Scott topology</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/Euclidean+space'>Euclidean space</a> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/real+number'>real line</a>, <a class='existingWikiWord' href='/nlab/show/diff/plane'>plane</a></li> </ul> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/cylinder+object'>cylinder</a>, <a class='existingWikiWord' href='/nlab/show/diff/cone'>cone</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/sphere'>sphere</a>, <a class='existingWikiWord' href='/nlab/show/diff/ball'>ball</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/circle'>circle</a>, <a class='existingWikiWord' href='/nlab/show/diff/torus'>torus</a>, <a class='existingWikiWord' href='/nlab/show/diff/annulus'>annulus</a>, <a class='existingWikiWord' href='/nlab/show/diff/M%C3%B6bius+strip'>Moebius strip</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/polytope'>polytope</a>, <a class='existingWikiWord' href='/nlab/show/diff/polyhedron'>polyhedron</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/projective+space'>projective space</a> (<a class='existingWikiWord' href='/nlab/show/diff/real+projective+space'>real</a>, <a class='existingWikiWord' href='/nlab/show/diff/complex+projective+space'>complex</a>) </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/classifying+space'>classifying space</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/configuration+space+of+points'>configuration space</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/path'>path</a>, <a class='existingWikiWord' href='/nlab/show/diff/loop'>loop</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/compact-open+topology'>mapping spaces</a>: <a class='existingWikiWord' href='/nlab/show/diff/compact-open+topology'>compact-open topology</a>, <a class='existingWikiWord' href='/nlab/show/diff/topology+of+uniform+convergence'>topology of uniform convergence</a> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/loop+space'>loop space</a>, <a class='existingWikiWord' href='/nlab/show/diff/path+space'>path space</a></li> </ul> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/Zariski+topology'>Zariski topology</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/Cantor+space'>Cantor space</a>, <a class='existingWikiWord' href='/nlab/show/diff/Mandelbrot+set'>Mandelbrot space</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/Peano+curve'>Peano curve</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/line+with+two+origins'>line with two origins</a>, <a class='existingWikiWord' href='/nlab/show/diff/long+line'>long line</a>, <a class='existingWikiWord' href='/nlab/show/diff/Sorgenfrey+line'>Sorgenfrey line</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/K-topology'>K-topology</a>, <a class='existingWikiWord' href='/nlab/show/diff/Dowker+space'>Dowker space</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/Warsaw+circle'>Warsaw circle</a>, <a class='existingWikiWord' href='/nlab/show/diff/Hawaiian+earring+space'>Hawaiian earring space</a> </li> </ul> Basic statements <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/Hausdorff+implies+sober'>Hausdorff spaces are sober</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/schemes+are+sober'>schemes are sober</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/continuous+images+of+compact+spaces+are+compact'>continuous images of compact spaces are compact</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/closed+subspaces+of+compact+Hausdorff+spaces+are+equivalently+compact+subspaces'>closed subspaces of compact Hausdorff spaces are equivalently compact subspaces</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/open+subspaces+of+compact+Hausdorff+spaces+are+locally+compact'>open subspaces of compact Hausdorff spaces are locally compact</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/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> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/compact+spaces+equivalently+have+converging+subnets'>compact spaces equivalently have converging subnet of every net</a> <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/Lebesgue+number+lemma'>Lebesgue number lemma</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/sequentially+compact+metric+spaces+are+equivalently+compact+metric+spaces'>sequentially compact metric spaces are equivalently compact metric spaces</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/compact+spaces+equivalently+have+converging+subnets'>compact spaces equivalently have converging subnet of every net</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/sequentially+compact+metric+spaces+are+totally+bounded'>sequentially compact metric spaces are totally bounded</a> </li> </ul> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/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> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/paracompact+Hausdorff+spaces+are+normal'>paracompact Hausdorff spaces are normal</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/paracompact+Hausdorff+spaces+equivalently+admit+subordinate+partitions+of+unity'>paracompact Hausdorff spaces equivalently admit subordinate partitions of unity</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/closed+injections+are+embeddings'>closed injections are embeddings</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/proper+maps+to+locally+compact+spaces+are+closed'>proper maps to locally compact spaces are closed</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/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> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/locally+compact+and+sigma-compact+spaces+are+paracompact'>locally compact and sigma-compact spaces are paracompact</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/locally+compact+and+second-countable+spaces+are+sigma-compact'>locally compact and second-countable spaces are sigma-compact</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/second-countable+regular+spaces+are+paracompact'>second-countable regular spaces are paracompact</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/CW-complexes+are+paracompact+Hausdorff+spaces'>CW-complexes are paracompact Hausdorff spaces</a> </li> </ul> Theorems <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/Urysohn%27s+lemma'>Urysohn's lemma</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/Tietze+extension+theorem'>Tietze extension theorem</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/Tychonoff+theorem'>Tychonoff theorem</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/tube+lemma'>tube lemma</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/Michael%27s+theorem'>Michael's theorem</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/Brouwer%27s+fixed+point+theorem'>Brouwer's fixed point theorem</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/topological+invariance+of+dimension'>topological invariance of dimension</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/Jordan+curve+theorem'>Jordan curve theorem</a> </li> </ul> Analysis Theorems <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/Heine-Borel+theorem'>Heine-Borel theorem</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/intermediate+value+theorem'>intermediate value theorem</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/extreme+value+theorem'>extreme value theorem</a> </li> </ul> <a class='existingWikiWord' href='/nlab/show/diff/topological+homotopy+theory'>topological homotopy theory</a> <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/homotopy'>left homotopy</a>, <a class='existingWikiWord' href='/nlab/show/diff/homotopy'>right homotopy</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/homotopy+equivalence'>homotopy equivalence</a>, <a class='existingWikiWord' href='/nlab/show/diff/deformation+retract'>deformation retract</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/fundamental+group'>fundamental group</a>, <a class='existingWikiWord' href='/nlab/show/diff/covering+space'>covering space</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/fundamental+theorem+of+covering+spaces'>fundamental theorem of covering spaces</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/homotopy+group'>homotopy group</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/weak+homotopy+equivalence'>weak homotopy equivalence</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/Whitehead+theorem'>Whitehead's theorem</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/Freudenthal+suspension+theorem'>Freudenthal suspension theorem</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/nerve+theorem'>nerve theorem</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/homotopy+extension+property'>homotopy extension property</a>, <a class='existingWikiWord' href='/nlab/show/diff/Hurewicz+cofibration'>Hurewicz cofibration</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/topological+cofiber+sequence'>cofiber sequence</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/Str%C3%B8m+model+structure'>Strøm model category</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/classical+model+structure+on+topological+spaces'>classical model structure on topological spaces</a> </li> </ul> </div> <h4 id='homotopy_theory'>Homotopy theory</h4> <div class='hide'> <a class='existingWikiWord' href='/nlab/show/diff/homotopy+theory'>homotopy theory</a>, <a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-category+theory'>(∞,1)-category theory</a>, <a class='existingWikiWord' href='/nlab/show/diff/homotopy+type+theory'>homotopy type theory</a> flavors: <a class='existingWikiWord' href='/nlab/show/diff/stable+homotopy+theory'>stable</a>, <a class='existingWikiWord' href='/nlab/show/diff/equivariant+homotopy+theory'>equivariant</a>, <a class='existingWikiWord' href='/nlab/show/diff/rational+homotopy+theory'>rational</a>, <a class='existingWikiWord' href='/nlab/show/diff/p-adic+homotopy+theory'>p-adic</a>, <a class='existingWikiWord' href='/nlab/show/diff/proper+homotopy+theory'>proper</a>, <a class='existingWikiWord' href='/nlab/show/diff/geometric+homotopy+type+theory'>geometric</a>, <a class='existingWikiWord' href='/nlab/show/diff/cohesive+homotopy+theory'>cohesive</a>, <a class='existingWikiWord' href='/nlab/show/diff/directed+homotopy+theory'>directed</a>… models: <a class='existingWikiWord' href='/nlab/show/diff/topological+homotopy+theory'>topological</a>, <a class='existingWikiWord' href='/nlab/show/diff/simplicial+homotopy+theory'>simplicial</a>, <a class='existingWikiWord' href='/nlab/show/diff/localic+homotopy+theory'>localic</a>, … see also <a class='existingWikiWord' href='/nlab/show/diff/algebraic+topology'>algebraic topology</a> Introductions <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/Introduction+to+Topology+--+2'>Introduction to Basic Homotopy Theory</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/Introduction+to+Homotopy+Theory'>Introduction to Abstract Homotopy Theory</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/geometry+of+physics+--+homotopy+types'>geometry of physics -- homotopy types</a> </li> </ul> Definitions <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/homotopy'>homotopy</a>, <a class='existingWikiWord' href='/nlab/show/diff/higher+homotopy'>higher homotopy</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/homotopy+type'>homotopy type</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/Pi-algebra'>Pi-algebra</a>, <a class='existingWikiWord' href='/nlab/show/diff/spherical+object'>spherical object and Pi(A)-algebra</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/homotopy+coherent+category+theory'>homotopy coherent category theory</a> <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/homotopical+category'>homotopical category</a> <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/model+category'>model category</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/category+of+fibrant+objects'>category of fibrant objects</a>, <a class='existingWikiWord' href='/nlab/show/diff/cofibration+category'>cofibration category</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/Waldhausen+category'>Waldhausen category</a> </li> </ul> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/homotopy+category'>homotopy category</a> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/Ho%28Top%29'>Ho(Top)</a></li> </ul> </li> </ul> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-category'>(∞,1)-category</a> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/homotopy+category+of+an+%28infinity%2C1%29-category'>homotopy category of an (∞,1)-category</a></li> </ul> </li> </ul> Paths and cylinders <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/homotopy'>left homotopy</a> <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/cylinder+object'>cylinder object</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/mapping+cone'>mapping cone</a> </li> </ul> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/homotopy'>right homotopy</a> <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/path+space+object'>path object</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/mapping+cocone'>mapping cocone</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/generalized+universal+bundle'>universal bundle</a> </li> </ul> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/interval+object'>interval object</a> <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/localization+at+geometric+homotopies'>homotopy localization</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/infinitesimal+interval+object'>infinitesimal interval object</a> </li> </ul> </li> </ul> Homotopy groups <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/homotopy+group'>homotopy group</a> <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/fundamental+group'>fundamental group</a> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/fundamental+group+of+a+topos'>fundamental group of a topos</a></li> </ul> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/Brown-Grossman+homotopy+group'>Brown-Grossman homotopy group</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/categorical+homotopy+groups+in+an+%28infinity%2C1%29-topos'>categorical homotopy groups in an (∞,1)-topos</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/geometric+homotopy+groups+in+an+%28infinity%2C1%29-topos'>geometric homotopy groups in an (∞,1)-topos</a> </li> </ul> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/fundamental+infinity-groupoid'>fundamental ∞-groupoid</a> <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/fundamental+groupoid'>fundamental groupoid</a> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/path+groupoid'>path groupoid</a></li> </ul> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/fundamental+infinity-groupoid+in+a+locally+infinity-connected+%28infinity%2C1%29-topos'>fundamental ∞-groupoid in a locally ∞-connected (∞,1)-topos</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/fundamental+infinity-groupoid+of+a+locally+infinity-connected+%28infinity%2C1%29-topos'>fundamental ∞-groupoid of a locally ∞-connected (∞,1)-topos</a> </li> </ul> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/fundamental+%28infinity%2C1%29-category'>fundamental (∞,1)-category</a> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/fundamental+category'>fundamental category</a></li> </ul> </li> </ul> Basic facts <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/fundamental+group+of+the+circle+is+the+integers'>fundamental group of the circle is the integers</a></li> </ul> Theorems <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/fundamental+theorem+of+covering+spaces'>fundamental theorem of covering spaces</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/Freudenthal+suspension+theorem'>Freudenthal suspension theorem</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/Blakers-Massey+theorem'>Blakers-Massey theorem</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/higher+homotopy+van+Kampen+theorem'>higher homotopy van Kampen theorem</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/nerve+theorem'>nerve theorem</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/Whitehead+theorem'>Whitehead's theorem</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/Hurewicz+theorem'>Hurewicz theorem</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/Galois+theory'>Galois theory</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/homotopy+hypothesis'>homotopy hypothesis</a>-theorem </li> </ul> </div> </div> </div> <h1 id='contents'>Contents</h1> <div class='maruku_toc'><ul><li><a href='#definition'>Definition</a></li><li><a href='#examples'>Examples</a></li><li><a href='#references'>References</a></li></ul></div> <h2 id='definition'>Definition</h2> A <a class='existingWikiWord' href='/nlab/show/diff/pointed+topological+space'>pointed topological space</a> <math class='maruku-mathml' display='inline' id='mathml_74655bebb96b0212c8398d62ac6fd054432c2945_1' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mi>X</mi><mo>,</mo><mi>x</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(X,x)</annotation></semantics></math> is called well-pointed if the base-point inclusion <math class='maruku-mathml' display='inline' id='mathml_74655bebb96b0212c8398d62ac6fd054432c2945_2' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>{</mo><mi>x</mi><mo stretchy='false'>}</mo><mover><mo>↪</mo><mspace width='thickmathspace' /></mover><mi>X</mi></mrow><annotation encoding='application/x-tex'>\{x\} \xhookrightarrow{\;} X</annotation></semantics></math> is a closed <a class='existingWikiWord' href='/nlab/show/diff/Hurewicz+cofibration'>Hurewicz cofibration</a> (e.g. <a href='#tomDieck2008'>tom Dieck 2008, p. 102</a><ins class='diffins'>). If the topological space is </ins><ins class='diffins'><a class='existingWikiWord' href='/nlab/show/diff/Hausdorff+space'>Hausdorff</a></ins><ins class='diffins'>, then closedness is implied (by </ins><ins class='diffins'><a href='Hurewicz+cofibration#HurewiczCofibrationsInCGWHSpacesAreClosed'>this Prop.</a></ins><ins class='diffins'>) and one may require just a Hurewicz cofibration (eg. </ins><ins class='diffins'><a href='#Bredon93'>Bredon 1993, VII, Def. 1.8</a></ins>). A <a class='existingWikiWord' href='/nlab/show/diff/topological+group'>topological group</a> is called well-pointed if it is so at its <a class='existingWikiWord' href='/nlab/show/diff/identity+element'>neutral element</a>, hence if <math class='maruku-mathml' display='inline' id='mathml_74655bebb96b0212c8398d62ac6fd054432c2945_3' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>{</mo><mi mathvariant='normal'>e</mi><mo stretchy='false'>}</mo><mover><mo>↪</mo><mspace width='thickmathspace' /></mover><mi>G</mi></mrow><annotation encoding='application/x-tex'>\{\mathrm{e}\} \xhookrightarrow{\;} G</annotation></semantics></math> is a closed <a class='existingWikiWord' href='/nlab/show/diff/Hurewicz+cofibration'>Hurewicz cofibration</a>. A <a class='existingWikiWord' href='/nlab/show/diff/simplicial+topological+group'>simplicial topological group</a> is well-pointed if all its component groups are. A key property of well-pointed topological groups (in the <a class='existingWikiWord' href='/nlab/show/diff/convenient+category+of+topological+spaces'>convenient</a> context of <a class='existingWikiWord' href='/nlab/show/diff/compactly+generated+topological+space'>compactly generated weak Hausdorff spaces</a>) is that the <a class='existingWikiWord' href='/nlab/show/diff/nerve'>nerves</a> of their <a class='existingWikiWord' href='/nlab/show/diff/delooping+groupoid'>delooping groupoids</a> are <a class='existingWikiWord' href='/nlab/show/diff/nice+simplicial+topological+space'>good simplicial topological spaces</a> (by <a href='Hurewicz+cofibration#KifiedProductsOfCofibrationsWithCompactlyGeneratedSpaces'>this Ex.</a>). Similarly, the underlying simplicial topological spaces of well-pointed simplicial topological groups are <a class='existingWikiWord' href='/nlab/show/diff/nice+simplicial+topological+space'>good</a> (<a href='simplicial+topological+group#RealizationOfWellPointedIsWellPointed'>this Prop.</a>). These facts explain the key role of well-pointedness in <a class='existingWikiWord' href='/nlab/show/diff/classifying+space'>classifying space</a>-theory, where it ensures that plain <a class='existingWikiWord' href='/nlab/show/diff/geometric+realization+of+simplicial+topological+spaces'>topological realization of simplicial topological spaces</a> coincides, up to <a class='existingWikiWord' href='/nlab/show/diff/weak+homotopy+equivalence'>weak homotopy equivalence</a>, with <a class='existingWikiWord' href='/nlab/show/diff/geometric+realization+of+simplicial+topological+spaces'>fat geometric realization</a> and hence with the <a class='existingWikiWord' href='/nlab/show/diff/homotopy+limit'>homotopy colimits</a>. <h2 id='examples'>Examples</h2> \begin{example}\label{ParacompactBanachManifoldsAreWellPointed} Every <a class='existingWikiWord' href='/nlab/show/diff/topological+manifold'>locally Euclidean</a> <a class='existingWikiWord' href='/nlab/show/diff/Hausdorff+space'>Hausdorff space</a> is well-pointed, in particular every <a class='existingWikiWord' href='/nlab/show/diff/topological+manifold'>topological manifold</a> is well pointed. In fact, every <a class='existingWikiWord' href='/nlab/show/diff/paracompact+topological+space'>paracompact</a> <a class='existingWikiWord' href='/nlab/show/diff/Banach+manifold'>Banach manifold</a> is well-pointed. \end{example} (Immediate by the discussion of <a href='Hurewicz+cofibration#Examples'>examples of Hurewicz cofibrations</a>). \begin{example}\label{PointInclusionIntoPUH} (the <a class='existingWikiWord' href='/nlab/show/diff/projective+unitary+group'>projective unitary group</a> <a class='existingWikiWord' href='/nlab/show/diff/PU%28%E2%84%8B%29'>PU(ℋ)</a>) \linebreak The <a class='existingWikiWord' href='/nlab/show/diff/projective+unitary+group'>projective unitary group</a> <a class='existingWikiWord' href='/nlab/show/diff/PU%28%E2%84%8B%29'>PU(ℋ)</a> on an infinite-dimensional <a class='existingWikiWord' href='/nlab/show/diff/separable+Hilbert+space'>separable</a> <a class='existingWikiWord' href='/nlab/show/diff/Hilbert+space'>Hilbert space</a> is: <ul> <li> a <a class='existingWikiWord' href='/nlab/show/diff/infinite-dimensional+Lie+group'>Banach Lie group</a> in its <a class='existingWikiWord' href='/nlab/show/diff/norm+topology'>norm topology</a>, and as such <a class='existingWikiWord' href='/nlab/show/diff/well-pointed+topological+space'>well-pointed</a> by Ex.\ref{ParacompactBanachManifoldsAreWellPointed}; </li> <li> no longer a Banach space in its weak/strong <a class='existingWikiWord' href='/nlab/show/diff/operator+topology'>operator topology</a>, but nevertheless still <a class='existingWikiWord' href='/nlab/show/diff/well-pointed+topological+space'>well-pointed</a> in this case, by <a href='projective+unitary+group+on+a+Hilbert+space#WellPointedInOperatorTopology'>this Prop.</a>. </li> </ul> \end{example} <h2 id='references'>References</h2> Textbook accounts: <ul> <li id='Bredon93'> <a class='existingWikiWord' href='/nlab/show/diff/Glen+Bredon'>Glen Bredon</a>, Section VII.1 of: Topology and Geometry, Graduate texts in mathematics 139, 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>) </li> <li id='tomDieck2008'> <a class='existingWikiWord' href='/nlab/show/diff/Tammo+tom+Dieck'>Tammo tom Dieck</a>, Algebraic topology. European Mathematical Society, Zürich (2008) (<a href='https://www.ems-ph.org/books/book.php?proj_nr=86'>doi:10.4171/048</a>) </li> </ul> </div> <div class="revisedby"> Last revised on May 6, 2022 at 12:49:22. 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