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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> <span id='navEnd'></span> </div> <div id="revision"> <p class="show_diff"> Showing changes from revision #36 to #37: <ins class="diffins">Added</ins> | <del class="diffdel">Removed</del> | <del class="diffmod">Chan</del><ins class="diffmod">ged</ins> </p> <div class='rightHandSide'> <div class='toc clickDown' tabindex='0'> <h3 id='context'>Context</h3> <h4 id='topology'>Topology</h4> <div class='hide'> <p><strong><a class='existingWikiWord' href='/nlab/show/diff/topology'>topology</a></strong> (<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>)</p> <p>see also <em><a class='existingWikiWord' href='/nlab/show/diff/differential+topology'>differential topology</a></em>, <em><a class='existingWikiWord' href='/nlab/show/diff/algebraic+topology'>algebraic topology</a></em>, <em><a class='existingWikiWord' href='/nlab/show/diff/functional+analysis'>functional analysis</a></em> and <em><a class='existingWikiWord' href='/nlab/show/diff/topological+homotopy+theory'>topological</a> <a class='existingWikiWord' href='/nlab/show/diff/homotopy+theory'>homotopy theory</a></em></p> <p><a class='existingWikiWord' href='/nlab/show/diff/Introduction+to+Topology'>Introduction</a></p> <p><strong>Basic concepts</strong></p> <ul> <li> <p><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></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/topological+space'>topological space</a>, <a class='existingWikiWord' href='/nlab/show/diff/locale'>locale</a></p> </li> <li> <p><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></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/finer+topology'>finer/coarser topology</a></p> </li> <li> <p><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></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/separation+axioms'>separation</a>, <a class='existingWikiWord' href='/nlab/show/diff/sober+topological+space'>sobriety</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/continuous+map'>continuous function</a>, <a class='existingWikiWord' href='/nlab/show/diff/homeomorphism'>homeomorphism</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/uniformly+continuous+map'>uniformly continuous function</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/embedding+of+topological+spaces'>embedding</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/open+map'>open map</a>, <a class='existingWikiWord' href='/nlab/show/diff/closed+map'>closed map</a></p> </li> <li> <p><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></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/convergence'>convergence</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/category'>category</a> <a class='existingWikiWord' href='/nlab/show/diff/Top'>Top</a></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/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/diff/weak+topology'>initial topology</a>, <a class='existingWikiWord' href='/nlab/show/diff/weak+topology'>final topology</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/subspace'>subspace</a>, <a class='existingWikiWord' href='/nlab/show/diff/quotient+space'>quotient space</a>,</p> </li> <li> <p>fiber space, <a class='existingWikiWord' href='/nlab/show/diff/space+attachment'>space attachment</a></p> </li> <li> <p><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></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/mapping+cylinder'>mapping cylinder</a>, <a class='existingWikiWord' href='/nlab/show/diff/cocylinder'>mapping cocylinder</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/mapping+cone'>mapping cone</a>, <a class='existingWikiWord' href='/nlab/show/diff/mapping+cocone'>mapping cocone</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/mapping+telescope'>mapping telescope</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/colimits+of+normal+spaces'>colimits of normal spaces</a></p> </li> </ul> <p><strong><a class='existingWikiWord' href='/nlab/show/diff/stuff%2C+structure%2C+property'>Extra stuff, structure, properties</a></strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/nice+topological+space'>nice topological space</a></p> </li> <li> <p><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></p> </li> <li> <p><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></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/sober+topological+space'>sober space</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/compact+space'>compact space</a>, <a class='existingWikiWord' href='/nlab/show/diff/proper+map'>proper map</a></p> <p><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></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/compactly+generated+topological+space'>compactly generated space</a></p> </li> <li> <p><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></p> </li> <li> <p><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></p> </li> <li> <p><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></p> </li> <li> <p><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></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/cell+complex'>cell complex</a>, <a class='existingWikiWord' href='/nlab/show/diff/CW+complex'>CW-complex</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/pointed+topological+space'>pointed space</a></p> </li> <li> <p><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></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/topological+group'>topological group</a></p> </li> <li> <p><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></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/topological+manifold'>topological manifold</a></p> </li> </ul> <p><strong>Examples</strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/empty+space'>empty space</a>, <a class='existingWikiWord' href='/nlab/show/diff/point+space'>point space</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/discrete+object'>discrete space</a>, <a class='existingWikiWord' href='/nlab/show/diff/codiscrete+space'>codiscrete space</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Sierpinski+space'>Sierpinski space</a></p> </li> <li> <p><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></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Euclidean+space'>Euclidean space</a></p> <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> <p><a class='existingWikiWord' href='/nlab/show/diff/cylinder+object'>cylinder</a>, <a class='existingWikiWord' href='/nlab/show/diff/cone'>cone</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/sphere'>sphere</a>, <a class='existingWikiWord' href='/nlab/show/diff/ball'>ball</a></p> </li> <li> <p><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></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/polytope'>polytope</a>, <a class='existingWikiWord' href='/nlab/show/diff/polyhedron'>polyhedron</a></p> </li> <li> <p><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>)</p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/classifying+space'>classifying space</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/configuration+space+of+points'>configuration space</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/path'>path</a>, <a class='existingWikiWord' href='/nlab/show/diff/loop'>loop</a></p> </li> <li> <p><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></p> <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> <p><a class='existingWikiWord' href='/nlab/show/diff/Zariski+topology'>Zariski topology</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Cantor+space'>Cantor space</a>, <a class='existingWikiWord' href='/nlab/show/diff/Mandelbrot+set'>Mandelbrot space</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Peano+curve'>Peano curve</a></p> </li> <li> <p><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></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/K-topology'>K-topology</a>, <a class='existingWikiWord' href='/nlab/show/diff/Dowker+space'>Dowker space</a></p> </li> <li> <p><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></p> </li> </ul> <p><strong>Basic statements</strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Hausdorff+implies+sober'>Hausdorff spaces are sober</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/schemes+are+sober'>schemes are sober</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/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/diff/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/diff/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/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></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/compact+spaces+equivalently+have+converging+subnets'>compact spaces equivalently have converging subnet of every net</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Lebesgue+number+lemma'>Lebesgue number lemma</a></p> </li> <li> <p><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></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/compact+spaces+equivalently+have+converging+subnets'>compact spaces equivalently have converging subnet of every net</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/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/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></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/paracompact+Hausdorff+spaces+are+normal'>paracompact Hausdorff spaces are normal</a></p> </li> <li> <p><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></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/closed+injections+are+embeddings'>closed injections are embeddings</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/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/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></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/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/diff/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/diff/second-countable+regular+spaces+are+paracompact'>second-countable regular spaces are paracompact</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/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/diff/Urysohn%27s+lemma'>Urysohn's lemma</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Tietze+extension+theorem'>Tietze extension theorem</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Tychonoff+theorem'>Tychonoff theorem</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/tube+lemma'>tube lemma</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Michael%27s+theorem'>Michael's theorem</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Brouwer%27s+fixed+point+theorem'>Brouwer's fixed point theorem</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/topological+invariance+of+dimension'>topological invariance of dimension</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/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/diff/Heine-Borel+theorem'>Heine-Borel theorem</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/intermediate+value+theorem'>intermediate value theorem</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/extreme+value+theorem'>extreme value theorem</a></p> </li> </ul> <p><strong><a class='existingWikiWord' href='/nlab/show/diff/topological+homotopy+theory'>topological homotopy theory</a></strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/homotopy'>left homotopy</a>, <a class='existingWikiWord' href='/nlab/show/diff/homotopy'>right homotopy</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/homotopy+equivalence'>homotopy equivalence</a>, <a class='existingWikiWord' href='/nlab/show/diff/deformation+retract'>deformation retract</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/fundamental+group'>fundamental group</a>, <a class='existingWikiWord' href='/nlab/show/diff/covering+space'>covering space</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/fundamental+theorem+of+covering+spaces'>fundamental theorem of covering spaces</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/homotopy+group'>homotopy group</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/weak+homotopy+equivalence'>weak homotopy equivalence</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Whitehead+theorem'>Whitehead's theorem</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Freudenthal+suspension+theorem'>Freudenthal suspension theorem</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/nerve+theorem'>nerve theorem</a></p> </li> <li> <p><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></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/topological+cofiber+sequence'>cofiber sequence</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Str%C3%B8m+model+structure'>Strøm model category</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/classical+model+structure+on+topological+spaces'>classical model structure on topological spaces</a></p> </li> </ul> </div> </div> </div> <p><a class='existingWikiWord' href='/nlab/show/diff/Serre+fibration'>Serre fibration</a> <math class='maruku-mathml' display='inline' id='mathml_ea5b1053c58be0298f81ed941c37562201f02d51_1' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo>⇐</mo></mrow><annotation encoding='application/x-tex'>\Leftarrow</annotation></semantics></math> <strong>Hurewicz fibration</strong> <math class='maruku-mathml' display='inline' id='mathml_ea5b1053c58be0298f81ed941c37562201f02d51_2' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo>⇒</mo></mrow><annotation encoding='application/x-tex'>\Rightarrow</annotation></semantics></math> <a class='existingWikiWord' href='/nlab/show/diff/Dold+fibration'>Dold fibration</a> <math class='maruku-mathml' display='inline' id='mathml_ea5b1053c58be0298f81ed941c37562201f02d51_3' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo>⇐</mo></mrow><annotation encoding='application/x-tex'>\Leftarrow</annotation></semantics></math> <a class='existingWikiWord' href='/nlab/show/diff/shrinkable+map'>shrinkable map</a></p> <hr /> <h1 id='contents'>Contents</h1> <div class='maruku_toc'><ul><li><a href='#definition'>Definition</a></li><li><a href='#properties'>Properties</a><ul><li><a href='#appearance_in_a_model_structure'>Appearance in a model structure</a></li><li><a href='#relation_to_serre_fibrations'>Relation to Serre fibrations</a></li><li><a href='#LocalRecognition'>Local recognition over numerable open covers</a></li></ul></li><li><a href='#Abstractly'>Abstract Hurewicz fibrations</a></li><li><a href='#Examples'>Examples</a><ul><li><a href='#empty_bundles'>Empty bundles</a></li><li><a href='#covering_spaces'>Covering spaces</a></li><li><a href='#FiberBundles'>Fiber bundles</a></li></ul></li><li><a href='#related_concept'>Related concept</a></li><li><a href='#references'>References</a></li></ul></div> <h2 id='definition'>Definition</h2> <div class='num_defn'> <h6 id='definition_2'>Definition</h6> <p>A <a class='existingWikiWord' href='/nlab/show/diff/continuous+map'>continuous map</a> <math class='maruku-mathml' display='inline' id='mathml_ea5b1053c58be0298f81ed941c37562201f02d51_4' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>p</mi><mspace width='thickmathspace' /><mo lspace='verythinmathspace'>:</mo><mspace width='thickmathspace' /><mi>E</mi><mo>⟶</mo><mi>B</mi></mrow><annotation encoding='application/x-tex'>p \;\colon\; E\longrightarrow B</annotation></semantics></math> of <a class='existingWikiWord' href='/nlab/show/diff/topological+space'>topological space</a> is called a <strong>Hurewicz fibration</strong> if it satisfies the <a class='existingWikiWord' href='/nlab/show/diff/lifting+property'>right lifting property</a> against all <a class='existingWikiWord' href='/nlab/show/diff/continuous+map'>continuous functions</a></p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_ea5b1053c58be0298f81ed941c37562201f02d51_5' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>σ</mi> <mn>0</mn></msub><mspace width='thickmathspace' /><mo lspace='verythinmathspace'>:</mo><mspace width='thickmathspace' /><mi>X</mi><mo>≅</mo><mi>X</mi><mo>×</mo><mo stretchy='false'>{</mo><mn>0</mn><mo stretchy='false'>}</mo><mover><mo>↪</mo><mphantom><mrow><mo lspace='verythinmathspace' rspace='0em'>−</mo><mo>−</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo></mrow></mphantom></mover><mi>X</mi><mo>×</mo><mi>I</mi></mrow><annotation encoding='application/x-tex'> \sigma_0 \;\colon\; X \cong X \times\{0\} \xhookrightarrow{\phantom{---}} X \times I </annotation></semantics></math></div> <p>including any <a class='existingWikiWord' href='/nlab/show/diff/topological+space'>topological space</a> <math class='maruku-mathml' display='inline' id='mathml_ea5b1053c58be0298f81ed941c37562201f02d51_6' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> as one end of the <a class='existingWikiWord' href='/nlab/show/diff/cylinder+object'>cylinder</a> over it (where <math class='maruku-mathml' display='inline' id='mathml_ea5b1053c58be0298f81ed941c37562201f02d51_7' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>I</mi><mo>≔</mo><mo stretchy='false'>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy='false'>]</mo></mrow><annotation encoding='application/x-tex'>I \coloneqq [0,1]</annotation></semantics></math> denotes the <a class='existingWikiWord' href='/nlab/show/diff/topological+interval'>topological interval</a>).</p> </div> <div class='num_remark'> <h6 id='remark'>Remark</h6> <p>In this context, the defining <a class='existingWikiWord' href='/nlab/show/diff/lifting+property'>right lifting property</a> is called the <em><a class='existingWikiWord' href='/nlab/show/diff/homotopy+lifting+property'>homotopy lifting property</a></em>, because the maps from <math class='maruku-mathml' display='inline' id='mathml_ea5b1053c58be0298f81ed941c37562201f02d51_8' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi><mo>×</mo><mi>I</mi></mrow><annotation encoding='application/x-tex'>X\times I</annotation></semantics></math> are understood as <a class='existingWikiWord' href='/nlab/show/diff/homotopy'>homotopies</a>. In more detail, for every space <math class='maruku-mathml' display='inline' id='mathml_ea5b1053c58be0298f81ed941c37562201f02d51_9' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math>, any homotopy <math class='maruku-mathml' display='inline' id='mathml_ea5b1053c58be0298f81ed941c37562201f02d51_10' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>F</mi><mo>:</mo><mi>X</mi><mo>×</mo><mi>I</mi><mo>→</mo><mi>B</mi></mrow><annotation encoding='application/x-tex'>F:X\times I\to B</annotation></semantics></math>, and a continuous map <math class='maruku-mathml' display='inline' id='mathml_ea5b1053c58be0298f81ed941c37562201f02d51_11' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>f</mi><mo>:</mo><mi>X</mi><mo>→</mo><mi>E</mi></mrow><annotation encoding='application/x-tex'>f:X\to E</annotation></semantics></math>, there is a homotopy <math class='maruku-mathml' display='inline' id='mathml_ea5b1053c58be0298f81ed941c37562201f02d51_12' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mover><mi>F</mi><mo stretchy='false'>˜</mo></mover><mo>:</mo><mi>X</mi><mo>×</mo><mi>I</mi><mo>→</mo><mi>E</mi></mrow><annotation encoding='application/x-tex'>\tilde{F}:X\times I\to E</annotation></semantics></math> such that <math class='maruku-mathml' display='inline' id='mathml_ea5b1053c58be0298f81ed941c37562201f02d51_13' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>f</mi><mo>=</mo><mover><mi>F</mi><mo stretchy='false'>˜</mo></mover><mo>∘</mo><msub><mi>σ</mi> <mn>0</mn></msub><mo>:</mo><mo>=</mo><msub><mover><mi>F</mi><mo stretchy='false'>˜</mo></mover> <mn>0</mn></msub></mrow><annotation encoding='application/x-tex'>f =\tilde{F} \circ\sigma_0 :=\tilde{F}_0</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_ea5b1053c58be0298f81ed941c37562201f02d51_14' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>F</mi><mo>=</mo><mi>p</mi><mo>∘</mo><mover><mi>F</mi><mo stretchy='false'>˜</mo></mover></mrow><annotation encoding='application/x-tex'>F=p\circ\tilde{F}</annotation></semantics></math>:</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_ea5b1053c58be0298f81ed941c37562201f02d51_15' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mrow><mtable><mtr><mtd><mi>X</mi></mtd> <mtd><mover><mo>⟶</mo><mi>f</mi></mover></mtd> <mtd><mi>E</mi></mtd></mtr> <mtr><mtd><msup><mrow /> <mpadded lspace='-100%width' width='0'><mrow><msub><mi>σ</mi> <mn>0</mn></msub></mrow></mpadded></msup><mo maxsize='1.2em' minsize='1.2em'>↓</mo></mtd> <mtd><msup><mrow /> <mover><mi>F</mi><mo stretchy='false'>˜</mo></mover></msup><mo>↗</mo></mtd> <mtd><mo maxsize='1.2em' minsize='1.2em'>↓</mo><msup><mrow /> <mpadded width='0'><mi>p</mi></mpadded></msup></mtd></mtr> <mtr><mtd><mi>X</mi><mo>×</mo><mi>I</mi></mtd> <mtd><mover><mo>→</mo><mi>F</mi></mover></mtd> <mtd><mi>B</mi></mtd></mtr></mtable></mrow><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> \array{ X &\stackrel{f}\longrightarrow& E \\ {}^{\mathllap{\sigma_0}} \big\downarrow &{}^{\tilde{F}}\nearrow& \big\downarrow {}^{\mathrlap{p}} \\ X\times I &\stackrel{F}{\to}& B } \,. </annotation></semantics></math></div></div> <div class='num_remark'> <h6 id='remark_2'>Remark</h6> <p>Strictly speaking, the “all” in this context should be interpreted to refer to all spaces in whatever ambient <a class='existingWikiWord' href='/nlab/show/diff/category'>category</a> of <a class='existingWikiWord' href='/nlab/show/diff/Top'>TopologicalSpaces</a> one is working in, since frequently this is a <a class='existingWikiWord' href='/nlab/show/diff/convenient+category+of+topological+spaces'>convenient category of spaces</a>. In theory, therefore, a map in such a category could be a Hurewicz fibration in that category without necessarily being a Hurewicz fibration in the category of <em>all</em> topological spaces, but in practice this usually doesn’t make a whole lot of difference.</p> </div> <p>Instead of checking the homotopy lifting property, one can instead solve a universal problem:</p> <div class='num_theorem'> <h6 id='theorem'>Theorem</h6> <p>A map is a Hurewicz fibration precisely if it admits a <a class='existingWikiWord' href='/nlab/show/diff/Hurewicz+connection'>Hurewicz connection</a>. (See there for details.)</p> </div> <h2 id='properties'>Properties</h2> <h3 id='appearance_in_a_model_structure'>Appearance in a model structure</h3> <p>There is a <a class='existingWikiWord' href='/nlab/show/diff/model+category'>Quillen model category</a> structure on <a class='existingWikiWord' href='/nlab/show/diff/Top'>Top</a> where fibrations are Hurewicz fibrations, cofibrations are closed <a class='existingWikiWord' href='/nlab/show/diff/Hurewicz+cofibration'>Hurewicz cofibrations</a> and weak equivalences are <a class='existingWikiWord' href='/nlab/show/diff/homotopy+equivalence'>homotopy equivalences</a> – see at <em><a class='existingWikiWord' href='/nlab/show/diff/model+structure+on+topological+spaces'>model structure on topological spaces</a></em> and specifically at <em><a class='existingWikiWord' href='/nlab/show/diff/Str%C3%B8m+model+structure'>Strøm's model category</a></em>.</p> <p>There is a version of Hurewicz fibrations for <a class='existingWikiWord' href='/nlab/show/diff/pointed+space'>pointed spaces</a>, as well as in the <a class='existingWikiWord' href='/nlab/show/diff/over+category'>slice category</a> <math class='maruku-mathml' display='inline' id='mathml_ea5b1053c58be0298f81ed941c37562201f02d51_16' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Top</mi><mo stretchy='false'>/</mo><msub><mi>B</mi> <mn>0</mn></msub></mrow><annotation encoding='application/x-tex'>Top/B_0</annotation></semantics></math> where <math class='maruku-mathml' display='inline' id='mathml_ea5b1053c58be0298f81ed941c37562201f02d51_17' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>B</mi> <mn>0</mn></msub></mrow><annotation encoding='application/x-tex'>B_0</annotation></semantics></math> is a fixed base (the <a class='existingWikiWord' href='/nlab/show/diff/slice+model+structure'>slice model structure</a>).</p> <p>There is also a model category whose fibrations are the Hurewicz fibrations and whose weak equivalences are the <a class='existingWikiWord' href='/nlab/show/diff/weak+homotopy+equivalence'>weak homotopy equivalences</a>, obtained by <a class='existingWikiWord' href='/nlab/show/diff/mixed+model+structure'>mixing</a> the above model structure with the <a class='existingWikiWord' href='/nlab/show/diff/classical+model+structure+on+topological+spaces'>classical model structure on topological spaces</a>.</p> <h3 id='relation_to_serre_fibrations'>Relation to Serre fibrations</h3> <p>Every Hurewicz fibration is a <a class='existingWikiWord' href='/nlab/show/diff/Serre+fibration'>Serre fibration</a>. Conversely, <a class='existingWikiWord' href='/nlab/show/diff/a+Serre+fibration+between+CW-complexes+is+a+Hurewicz+fibration'>a Serre fibration between CW-complexes is a Hurewicz fibration</a>.</p> <h3 id='LocalRecognition'>Local recognition over numerable open covers</h3> <p>The following proposition says that being a Hurewicz fibration is a <em>local property</em> with respect to the <a class='existingWikiWord' href='/nlab/show/diff/target'>codomain</a> space, at least as long as the local <a class='existingWikiWord' href='/nlab/show/diff/open+cover'>open cover</a> used is <a class='existingWikiWord' href='/nlab/show/diff/numerable+open+cover'>numerable</a>.</p> <p>\begin{prop}\label{MapIsHurewiczFibrationIfPullbackToNumerableCoverIsSo} <strong>(Local recognition of Hurewicz fibrations over numerable open covers)</strong>\linebreak</p> <p>Let <math class='maruku-mathml' display='inline' id='mathml_ea5b1053c58be0298f81ed941c37562201f02d51_18' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>p</mi><mo lspace='verythinmathspace'>:</mo><mi>E</mi><mo>→</mo><mi>B</mi></mrow><annotation encoding='application/x-tex'>p \colon E \to B</annotation></semantics></math> be a continuous function such that there exists a <a class='existingWikiWord' href='/nlab/show/diff/numerable+open+cover'>numerable open cover</a> <math class='maruku-mathml' display='inline' id='mathml_ea5b1053c58be0298f81ed941c37562201f02d51_19' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>U</mi><mover><mo>⟶</mo><mi>ϕ</mi></mover><mi>B</mi></mrow><annotation encoding='application/x-tex'>U \overset{\phi}{\longrightarrow} B</annotation></semantics></math> over which <math class='maruku-mathml' display='inline' id='mathml_ea5b1053c58be0298f81ed941c37562201f02d51_20' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>E</mi></mrow><annotation encoding='application/x-tex'>E</annotation></semantics></math> is a Hurewicz fibration, hence such that the <a class='existingWikiWord' href='/nlab/show/diff/pullback'>pullback</a> <math class='maruku-mathml' display='inline' id='mathml_ea5b1053c58be0298f81ed941c37562201f02d51_21' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>ϕ</mi> <mo>*</mo></msup><mi>p</mi></mrow><annotation encoding='application/x-tex'>\phi^\ast p</annotation></semantics></math> is a Hurewicz fibration.</p> <p>Then <math class='maruku-mathml' display='inline' id='mathml_ea5b1053c58be0298f81ed941c37562201f02d51_22' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>p</mi></mrow><annotation encoding='application/x-tex'>p</annotation></semantics></math> is a Hurewicz fibration</p> <p>\end{prop}</p> <p>A proof may be found spelled out in e.g. <a href='#May99'>May 99, Sec. 7.4</a></p> <h2 id='Abstractly'>Abstract Hurewicz fibrations</h2> <p>The concept of Hurewicz fibrations makes sense also more generally in the presence of a (well behaved) <a class='existingWikiWord' href='/nlab/show/diff/interval+object'>interval object</a>, see for instance the early example of such in (<a href='#Kamps72'>Kamps 72</a>) and see (<a href='#Williamson13'>Williamson 13</a>) for review and further developments. Discussion with a view towards <a class='existingWikiWord' href='/nlab/show/diff/homotopy+type+theory'>homotopy type theory</a> is in (<a href='#Warren08'>Warren 08</a>).</p> <h2 id='Examples'>Examples</h2> <h3 id='empty_bundles'>Empty bundles</h3> <p>\begin{example}\label{EmptyBundlesAreHurewiczFibrations} <strong>(<a class='existingWikiWord' href='/nlab/show/diff/empty+bundle'>empty bundles</a> are <a class='existingWikiWord' href='/nlab/show/diff/Hurewicz+fibration'>Hurewicz fibrations</a>)</strong> \linebreak All <a class='existingWikiWord' href='/nlab/show/diff/empty+bundle'>empty bundles</a> <math class='maruku-mathml' display='inline' id='mathml_ea5b1053c58be0298f81ed941c37562201f02d51_23' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo>∅</mo><mo>⟶</mo><mi>B</mi></mrow><annotation encoding='application/x-tex'>\varnothing \longrightarrow B</annotation></semantics></math> are Hurewicz fibrations, because none of the <a class='existingWikiWord' href='/nlab/show/diff/commutative+square'>commuting squares</a> that one would have to non-trivially <a class='existingWikiWord' href='/nlab/show/diff/lifting+property'>lift in</a> actually exist:</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_ea5b1053c58be0298f81ed941c37562201f02d51_24' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mrow><mtable><mtr><mtd><mi>X</mi></mtd> <mtd><mover><mo>⟶</mo><mrow><mo>¬</mo><mo>∃</mo></mrow></mover></mtd> <mtd><mo>∅</mo></mtd></mtr> <mtr><mtd><mo maxsize='1.2em' minsize='1.2em'>↓</mo></mtd> <mtd /> <mtd><mo maxsize='1.2em' minsize='1.2em'>↓</mo></mtd></mtr> <mtr><mtd><mi>X</mi><mo>×</mo><mi>I</mi></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>B</mi><mpadded width='0'><mrow><mspace width='thinmathspace' /><mo>,</mo></mrow></mpadded></mtd></mtr></mtable></mrow></mrow><annotation encoding='application/x-tex'> \array{ X &\overset{ \not \exists }{\longrightarrow}& \varnothing \\ \big\downarrow && \big\downarrow \\ X \times I &\longrightarrow& B \mathrlap{\,,} } </annotation></semantics></math></div> <p>since the <a class='existingWikiWord' href='/nlab/show/diff/empty+space'>empty topological space</a> is a <a class='existingWikiWord' href='/nlab/show/diff/strict+initial+object'>strict initial object</a>: There is <em>no</em> morphism to it from any <a class='existingWikiWord' href='/nlab/show/diff/inhabited+set'>inhabited space</a> <math class='maruku-mathml' display='inline' id='mathml_ea5b1053c58be0298f81ed941c37562201f02d51_25' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math></p> <p>(If <math class='maruku-mathml' display='inline' id='mathml_ea5b1053c58be0298f81ed941c37562201f02d51_26' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> itself is <a class='existingWikiWord' href='/nlab/show/diff/empty+space'>empty</a>, then so is <math class='maruku-mathml' display='inline' id='mathml_ea5b1053c58be0298f81ed941c37562201f02d51_27' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi><mo>×</mo><mi>I</mi></mrow><annotation encoding='application/x-tex'>X \times I</annotation></semantics></math> and then a square only exists if <math class='maruku-mathml' display='inline' id='mathml_ea5b1053c58be0298f81ed941c37562201f02d51_28' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>B</mi></mrow><annotation encoding='application/x-tex'>B</annotation></semantics></math> in turn is empty. This square, consisting entirely of empty spaces, does have a unique lift, trivially, and hence also the empty bundle over the empty space is a Hurewicz fibration – for what it’s worth.)</p> <p>\end{example}</p> <h3 id='covering_spaces'>Covering spaces</h3> <div class='num_example' id='CoveringSpaceIsHurewiczFibration'> <h6 id='example'>Example</h6> <p><strong>(<a class='existingWikiWord' href='/nlab/show/diff/covering+space'>covering spaces</a> are Hurewicz fibrations)</strong></p> <p>Every <a class='existingWikiWord' href='/nlab/show/diff/covering+space'>covering space</a> projection is a Hurewicz fibration, by <a href='covering#space#HomotopyLiftingPropertyOfCoveringSpaces'>this prop.</a>.</p> </div> <h3 id='FiberBundles'>Fiber bundles</h3> <p>\begin{prop}\label{NumerableFiberBundlesAreSerreFibrations} <strong>(<a class='existingWikiWord' href='/nlab/show/diff/numerable+fiber+bundle'>numerable fiber bundles</a> are Hurewicz fibrations)</strong> \linebreak</p> <p>Every <a class='existingWikiWord' href='/nlab/show/diff/numerable+fiber+bundle'>numerable fiber bundle</a>, hence in particular every <a class='existingWikiWord' href='/nlab/show/diff/fiber+bundle'>fiber bundle</a> over a <a class='existingWikiWord' href='/nlab/show/diff/paracompact+topological+space'>paracompact topological space</a>, is a Serre fibration.</p> <p>\end{prop} \begin{proof} By definition of local triviality, for a <a class='existingWikiWord' href='/nlab/show/diff/numerable+fiber+bundle'>numerable fiber bundle</a> <math class='maruku-mathml' display='inline' id='mathml_ea5b1053c58be0298f81ed941c37562201f02d51_29' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>p</mi><mo lspace='verythinmathspace'>:</mo><mi>E</mi><mo>→</mo><mi>B</mi></mrow><annotation encoding='application/x-tex'>p \colon E \to B</annotation></semantics></math> – such as a <a class='existingWikiWord' href='/nlab/show/diff/fiber+bundle'>fiber bundle</a> over a <a class='existingWikiWord' href='/nlab/show/diff/paracompact+topological+space'>paracompact topological space</a> – there exists a <a class='existingWikiWord' href='/nlab/show/diff/numerable+open+cover'>numerable open cover</a> <math class='maruku-mathml' display='inline' id='mathml_ea5b1053c58be0298f81ed941c37562201f02d51_30' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>ϕ</mi><mo lspace='verythinmathspace'>:</mo><mi>U</mi><mo>→</mo><mi>B</mi></mrow><annotation encoding='application/x-tex'>\phi \colon U \to B</annotation></semantics></math> such that the <a class='existingWikiWord' href='/nlab/show/diff/pullback+bundle'>pullback bundle</a> <math class='maruku-mathml' display='inline' id='mathml_ea5b1053c58be0298f81ed941c37562201f02d51_31' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>ϕ</mi> <mo>*</mo></msup><mi>p</mi></mrow><annotation encoding='application/x-tex'>\phi^\ast p</annotation></semantics></math> is trivial, i.e. is the <a class='existingWikiWord' href='/nlab/show/diff/cartesian+product'>Cartesian product</a> <a class='existingWikiWord' href='/nlab/show/diff/projection'>projection</a></p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_ea5b1053c58be0298f81ed941c37562201f02d51_32' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>ϕ</mi> <mo>*</mo></msup><mi>p</mi><mspace width='thickmathspace' /><mo lspace='verythinmathspace'>:</mo><mspace width='thickmathspace' /><mi>U</mi><mo>×</mo><mi>F</mi><mo>⟶</mo><mi>U</mi></mrow><annotation encoding='application/x-tex'> \phi^\ast p \;\colon\; U \times F \longrightarrow U </annotation></semantics></math></div> <p>(for <math class='maruku-mathml' display='inline' id='mathml_ea5b1053c58be0298f81ed941c37562201f02d51_33' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>F</mi></mrow><annotation encoding='application/x-tex'>F</annotation></semantics></math> the <a class='existingWikiWord' href='/nlab/show/diff/fiber+bundle'>typical fiber</a>). Since such a projection is clearly a Hurewicz fibration the claim follows by the local recognition of Hurewicz fibrations over numerable open covers, from Prop. \ref{MapIsHurewiczFibrationIfPullbackToNumerableCoverIsSo}. \end{proof}</p> <p>Notice that the example of <a class='existingWikiWord' href='/nlab/show/diff/empty+bundle'>empty bundles</a> (Example \ref{EmptyBundlesAreHurewiczFibrations}) is a special case of a <a class='existingWikiWord' href='/nlab/show/diff/fiber+bundle'>fiber bundle</a>: With <a class='existingWikiWord' href='/nlab/show/diff/fiber+bundle'>typical fiber</a> the <a class='existingWikiWord' href='/nlab/show/diff/empty+space'>empty topological space</a>.</p> <h2 id='related_concept'>Related concept</h2> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Hurewicz+cofibration'>Hurewicz cofibration</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Str%C3%B8m+model+structure'>Strøm model structure</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Serre+fibration'>Serre fibration</a></p> </li> </ul> <h2 id='references'>References</h2> <p>The original article:</p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/Witold+Hurewicz'>Witold Hurewicz</a>, <em>On the concept of fiber space</em>, Proc. Nat. Acad. Sci. USA <strong>41</strong> (1955) 956–961; (<a href='https://dx.doi.org/10.1073%2Fpnas.41.11.956'>doi:10.1073%2Fpnas.41.11.956</a>, <a href='https://www.jstor.org/stable/89187'>jstor:89187</a>, <a href='http://www.pnas.org/content/41/11/956.full.pdf'>pdf</a>. MR0073987)</li> </ul> <p><span><del class='diffmod'> A</del><ins class='diffmod'> Review</ins><del class='diffdel'> decent</del><del class='diffdel'> review</del> of Hurewicz fibrations,</span><a class='existingWikiWord' href='/nlab/show/diff/Hurewicz+connection'>Hurewicz connections</a><span> and related<del class='diffmod'> issues</del><ins class='diffmod'> issues:</ins><del class='diffdel'> is</del><del class='diffdel'> in</del></span></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/James+Eells'>James Eells</a><span> ,<del class='diffmod'> Jr.,</del><ins class='diffmod'> Jr.:</ins></span><em>Fibring spaces of maps</em><span> ,<del class='diffmod'> in</del><ins class='diffmod'> in:</ins> Richard Anderson (ed.)</span><em>Symposium on infinite-dimensional topology</em><ins class='diffins'>, Annals of Mathematics Studies </ins><ins class='diffins'><strong>69</strong></ins><ins class='diffins'>, Princeton University Press (1972, 2016) 43-57 [[ISBN:9780691080871](https://press.princeton.edu/books/paperback/9780691080871/symposium-on-infinite-dimensional-topology-am-69-volume-69), </ins><ins class='diffins'><a class='existingWikiWord' href='/nlab/files/Eells-FibringSpaces.pdf' title='pdf'>pdf</a></ins><ins class='diffins'>]</ins></li> </ul> <p>Textbook accounts:</p> <ul> <li id='tDKP70'> <p><a class='existingWikiWord' href='/nlab/show/diff/Tammo+tom+Dieck'>Tammo tom Dieck</a>, <a class='existingWikiWord' href='/nlab/show/diff/Klaus+Heiner+Kamps'>Klaus Heiner Kamps</a>, <a class='existingWikiWord' href='/nlab/show/diff/Dieter+Puppe'>Dieter Puppe</a>, Chapter II of: <em>Homotopietheorie</em>, Lecture Notes in Mathematics <strong>157</strong> Springer 1970 (<a href='https://link.springer.com/book/10.1007/BFb0059721'>doi:10.1007/BFb0059721</a>)</p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Allen+Hatcher'>Alan Hatcher</a>, from p. 61 on in <em>Algebraic Topology</em> (<a href='http://www.math.cornell.edu/~hatcher/AT/ATpage.html'>web</a>)</p> </li> </ul> <p>See also</p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Roland+Schw%C3%A4nzl'>R. Schwänzl</a>, <a class='existingWikiWord' href='/nlab/show/diff/Rainer+Vogt'>R. Vogt</a>, <em>Strong cofibrations and fibrations in enriched categories</em>, 2002.</p> </li> <li> <p>the textbooks on algebraic topology by Whitehead and Spanier.</p> </li> <li id='May99'> <p><a class='existingWikiWord' href='/nlab/show/diff/Peter+May'>Peter May</a>, <em><a class='existingWikiWord' href='/nlab/show/diff/A+Concise+Course+in+Algebraic+Topology'>A concise course in algebraic topology</a></em>, University of Chicago Press 1999 (<a href='https://www.press.uchicago.edu/ucp/books/book/chicago/C/bo3777031.html'>ISBN: 9780226511832</a>, <a href='http://www.math.uchicago.edu/~may/CONCISE/ConciseRevised.pdf'>pdf</a>)</p> </li> <li id='Cutler20'> <p><a class='existingWikiWord' href='/nlab/show/diff/Tyrone+Cutler'>Tyrone Cutler</a>, <em>Fibrations III – Locally trivial maps and bundles</em>, 2020 (<a href='https://www.math.uni-bielefeld.de/~tcutler/pdf/Fibrations%20III.pdf'>pdf</a>, <a class='existingWikiWord' href='/nlab/files/CutlerFibrationsIII.pdf' title='pdf'>pdf</a>)</p> </li> </ul> <p>Abstract analogues of Hurewicz fibrations can be found in</p> <ul> <li id='Kamps72'><a class='existingWikiWord' href='/nlab/show/diff/Klaus+Heiner+Kamps'>K.H.Kamps</a>, <em>Kan-Bedingungen und abstrakte Homotopietheorie</em>, Math. Z. 124,1972, 215 -236</li> </ul> <p>summarised in</p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/Klaus+Heiner+Kamps'>K. H. Kamps</a>, <a class='existingWikiWord' href='/nlab/show/diff/Tim+Porter'>Tim Porter</a>, <em>Abstract Homotopy and Simple Homotopy Theory</em> (<a href='http://books.google.de/books?id=7JYKxInRMdAC&dq=Porter+Kamps&printsec=frontcover&source=bl&ots=uuyl_tIjs4&sig=Lt8I92xQBZ4DNKVXD0x76WkcxCE&hl=de&sa=X&oi=book_result&resnum=3&ct=result#PPP1,M1'>GoogleBooks</a>)</li> </ul> <p>and further developed in</p> <ul> <li id='Williamson13'><a class='existingWikiWord' href='/nlab/show/diff/Richard+Williamson'>Richard Williamson</a>, <em>Cylindrical model structures</em> (<a href='http://arxiv.org/abs/1304.0867'>arXiv:1304.0867</a>, <a href='http://rwilliamson-mathematics.info/cylindrical_model_structures.html'>web</a>)</li> </ul> <p>Discussion with an eye towards <a class='existingWikiWord' href='/nlab/show/diff/homotopy+type+theory'>homotopy type theory</a> is in</p> <ul> <li id='Warren08'><a class='existingWikiWord' href='/nlab/show/diff/Michael+Warren'>Michael Warren</a>, <em>Homotopy theoretic aspects of constructive type theory</em>, 2008 (<a href='http://mawarren.net/papers/phd.pdf'>PDF</a>)</li> </ul> <p> </p> </div> <div class="revisedby"> <p> Last revised on June 20, 2024 at 15:37:44. 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