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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 #6 to #7: <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> <h4 id='homotopy_theory'>Homotopy theory</h4> <div class='hide'> <p><strong><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></strong></p> <p>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>…</p> <p>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>, …</p> <p>see also <strong><a class='existingWikiWord' href='/nlab/show/diff/algebraic+topology'>algebraic topology</a></strong></p> <p><strong>Introductions</strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Introduction+to+Topology+--+2'>Introduction to Basic Homotopy Theory</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Introduction+to+Homotopy+Theory'>Introduction to Abstract Homotopy Theory</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/geometry+of+physics+--+homotopy+types'>geometry of physics -- homotopy types</a></p> </li> </ul> <p><strong>Definitions</strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/homotopy'>homotopy</a>, <a class='existingWikiWord' href='/nlab/show/diff/higher+homotopy'>higher homotopy</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/homotopy+type'>homotopy type</a></p> </li> <li> <p><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></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/homotopy+coherent+category+theory'>homotopy coherent category theory</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/homotopical+category'>homotopical category</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/model+category'>model category</a></p> </li> <li> <p><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></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Waldhausen+category'>Waldhausen category</a></p> </li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/homotopy+category'>homotopy category</a></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/Ho%28Top%29'>Ho(Top)</a></li> </ul> </li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-category'>(∞,1)-category</a></p> <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> <p><strong>Paths and cylinders</strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/homotopy'>left homotopy</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/cylinder+object'>cylinder object</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/mapping+cone'>mapping cone</a></p> </li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/homotopy'>right homotopy</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/path+space+object'>path object</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/mapping+cocone'>mapping cocone</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/generalized+universal+bundle'>universal bundle</a></p> </li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/interval+object'>interval object</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/localization+at+geometric+homotopies'>homotopy localization</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/infinitesimal+interval+object'>infinitesimal interval object</a></p> </li> </ul> </li> </ul> <p><strong>Homotopy groups</strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/homotopy+group'>homotopy group</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/fundamental+group'>fundamental group</a></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/fundamental+group+of+a+topos'>fundamental group of a topos</a></li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Brown-Grossman+homotopy+group'>Brown-Grossman homotopy group</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/categorical+homotopy+groups+in+an+%28infinity%2C1%29-topos'>categorical homotopy groups in an (∞,1)-topos</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/geometric+homotopy+groups+in+an+%28infinity%2C1%29-topos'>geometric homotopy groups in an (∞,1)-topos</a></p> </li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/fundamental+infinity-groupoid'>fundamental ∞-groupoid</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/fundamental+groupoid'>fundamental groupoid</a></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/path+groupoid'>path groupoid</a></li> </ul> </li> <li> <p><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></p> </li> <li> <p><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></p> </li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/fundamental+%28infinity%2C1%29-category'>fundamental (∞,1)-category</a></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/fundamental+category'>fundamental category</a></li> </ul> </li> </ul> <p><strong>Basic facts</strong></p> <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> <p><strong>Theorems</strong></p> <ul> <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/Freudenthal+suspension+theorem'>Freudenthal suspension theorem</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Blakers-Massey+theorem'>Blakers-Massey theorem</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/higher+homotopy+van+Kampen+theorem'>higher homotopy van Kampen 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/Whitehead+theorem'>Whitehead's theorem</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Hurewicz+theorem'>Hurewicz theorem</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Galois+theory'>Galois theory</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/homotopy+hypothesis'>homotopy hypothesis</a>-theorem</p> </li> </ul> </div> </div> </div> <h1 id='contents'>Contents</h1> <div class='maruku_toc'><ul><li><a href='#idea'>Idea</a></li><li><a href='#induced_constructions'>Induced constructions</a><ul><li><a href='#topological_cylinder'>Topological cylinder</a></li><li><a href='#left_homotopy'>Left homotopy</a></li><li><a href='#path_space'>Path space</a></li></ul></li><li><a href='#properties'>Properties</a><ul><li><a href='#freyds_characterization'>Freyd’s characterization</a></li></ul></li></ul></div> <h2 id='idea'>Idea</h2> <p>Generally, a <em>topological interval</em> is a (<a class='existingWikiWord' href='/nlab/show/diff/bounded+set'>bounded</a>) <a class='existingWikiWord' href='/nlab/show/diff/interval'>interval</a> in the <a class='existingWikiWord' href='/nlab/show/diff/real+number'>real line</a> (an <a class='existingWikiWord' href='/nlab/show/diff/interval'>open interval</a> <math class='maruku-mathml' display='inline' id='mathml_959a5257e1bdaf21e8ec816c6597738862b6180a_1' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(a,b)</annotation></semantics></math> or a <a class='existingWikiWord' href='/nlab/show/diff/interval'>closed interval</a> <math class='maruku-mathml' display='inline' id='mathml_959a5257e1bdaf21e8ec816c6597738862b6180a_2' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>[</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo stretchy='false'>]</mo></mrow><annotation encoding='application/x-tex'>[a,b]</annotation></semantics></math> or a <a class='existingWikiWord' href='/nlab/show/diff/interval'>half-open interval</a> <math class='maruku-mathml' display='inline' id='mathml_959a5257e1bdaf21e8ec816c6597738862b6180a_3' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo stretchy='false'>]</mo></mrow><annotation encoding='application/x-tex'>(a,b]</annotation></semantics></math> or <math class='maruku-mathml' display='inline' id='mathml_959a5257e1bdaf21e8ec816c6597738862b6180a_4' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>[</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>[a,b)</annotation></semantics></math>) equipped with the <a class='existingWikiWord' href='/nlab/show/diff/subspace+topology'>subspace topology</a> of the <a class='existingWikiWord' href='/nlab/show/diff/Euclidean+space'>Euclidean</a> <a class='existingWikiWord' href='/nlab/show/diff/metric+topology'>metric topology</a>.</p> <p>Specifically in the context of <a class='existingWikiWord' href='/nlab/show/diff/topological+homotopy+theory'>topological homotopy theory</a>, the <em>standard topological <a class='existingWikiWord' href='/nlab/show/diff/interval+object'>interval object</a></em> is the <a class='existingWikiWord' href='/nlab/show/diff/interval'>closed interval</a> <math class='maruku-mathml' display='inline' id='mathml_959a5257e1bdaf21e8ec816c6597738862b6180a_5' xmlns='http://www.w3.org/1998/Math/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> equipped with the <a class='existingWikiWord' href='/nlab/show/diff/continuous+map'>continuous functions</a></p> <ol> <li> <p><math class='maruku-mathml' display='inline' id='mathml_959a5257e1bdaf21e8ec816c6597738862b6180a_6' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>const</mi> <mn>0</mn></msub><mspace width='thickmathspace' /><mo lspace='verythinmathspace'>:</mo><mspace width='thickmathspace' /><mo>*</mo><mo>→</mo><mo stretchy='false'>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy='false'>]</mo></mrow><annotation encoding='application/x-tex'>const_0 \;\colon\; \ast \to [0,1]</annotation></semantics></math></p> </li> <li> <p><math class='maruku-mathml' display='inline' id='mathml_959a5257e1bdaf21e8ec816c6597738862b6180a_7' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>const</mi> <mn>1</mn></msub><mspace width='thickmathspace' /><mo lspace='verythinmathspace'>:</mo><mspace width='thickmathspace' /><mo>*</mo><mo>→</mo><mo stretchy='false'>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy='false'>]</mo></mrow><annotation encoding='application/x-tex'>const_1 \;\colon\; \ast \to [0,1]</annotation></semantics></math></p> </li> </ol> <p>which include the <a class='existingWikiWord' href='/nlab/show/diff/point+space'>point space</a> as the two endpoints, respectively.</p> <p>Together with the unique function <math class='maruku-mathml' display='inline' id='mathml_959a5257e1bdaf21e8ec816c6597738862b6180a_8' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy='false'>]</mo><mo>→</mo><mo>*</mo></mrow><annotation encoding='application/x-tex'>[0,1] \to \ast</annotation></semantics></math> this yields the factorization of the <a class='existingWikiWord' href='/nlab/show/diff/codiagonal'>codiagonal</a> on the <a class='existingWikiWord' href='/nlab/show/diff/point+space'>point space</a></p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_959a5257e1bdaf21e8ec816c6597738862b6180a_9' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mo>∇</mo> <mo>*</mo></msub><mspace width='thickmathspace' /><mo lspace='verythinmathspace'>:</mo><mspace width='thickmathspace' /><mo>*</mo><mo>⊔</mo><mo>*</mo><mover><mo>⟶</mo><mrow><mo stretchy='false'>(</mo><msub><mi>const</mi> <mn>0</mn></msub><mo>,</mo><msub><mi>const</mi> <mn>1</mn></msub><mo stretchy='false'>)</mo></mrow></mover><mo stretchy='false'>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy='false'>]</mo><mover><mo>⟶</mo><mrow><mo>∃</mo><mo>!</mo></mrow></mover><mo>*</mo></mrow><annotation encoding='application/x-tex'> \nabla_{\ast} \;\colon\; \ast \sqcup \ast \overset{(const_0,const_1)}{\longrightarrow} [0,1] \overset{\exists!}{\longrightarrow} \ast </annotation></semantics></math></div> <p>which exhibits an example of an <em><a class='existingWikiWord' href='/nlab/show/diff/interval+object'>interval object</a></em> in the general sense of <a class='existingWikiWord' href='/nlab/show/diff/model+category'>model category</a> theory with respect to the <a class='existingWikiWord' href='/nlab/show/diff/classical+model+structure+on+topological+spaces'>classical model structure on topological spaces</a>.</p> <h2 id='induced_constructions'>Induced constructions</h2> <h3 id='topological_cylinder'>Topological cylinder</h3> <p>For <math class='maruku-mathml' display='inline' id='mathml_959a5257e1bdaf21e8ec816c6597738862b6180a_10' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> a <a class='existingWikiWord' href='/nlab/show/diff/topological+space'>topological space</a>, then the <a class='existingWikiWord' href='/nlab/show/diff/product+topological+space'>product topological space</a> <math class='maruku-mathml' display='inline' id='mathml_959a5257e1bdaf21e8ec816c6597738862b6180a_11' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</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'>X \times [0,1]</annotation></semantics></math> with the topological interval is the <em>standard topological <a class='existingWikiWord' href='/nlab/show/diff/cylinder+object'>cylinder</a></em> over <math class='maruku-mathml' display='inline' id='mathml_959a5257e1bdaf21e8ec816c6597738862b6180a_12' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math>. Via the above inclusion functions, this inherits a factorization of the <a class='existingWikiWord' href='/nlab/show/diff/codiagonal'>codiagonal</a> of <math class='maruku-mathml' display='inline' id='mathml_959a5257e1bdaf21e8ec816c6597738862b6180a_13' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> (the canonical continuous function out of the <a class='existingWikiWord' href='/nlab/show/diff/disjoint+union+topological+space'>disjoint union space</a> of <math class='maruku-mathml' display='inline' id='mathml_959a5257e1bdaf21e8ec816c6597738862b6180a_14' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> with itself to <math class='maruku-mathml' display='inline' id='mathml_959a5257e1bdaf21e8ec816c6597738862b6180a_15' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math>):</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_959a5257e1bdaf21e8ec816c6597738862b6180a_16' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mo>∇</mo> <mi>X</mi></msub><mspace width='thickmathspace' /><mo lspace='verythinmathspace'>:</mo><mspace width='thickmathspace' /><mi>X</mi><mo>⊔</mo><mi>X</mi><mover><mo>⟶</mo><mrow><mo stretchy='false'>(</mo><msub><mi>id</mi> <mi>X</mi></msub><mo>×</mo><msub><mi>const</mi> <mn>0</mn></msub><mo>,</mo><msub><mi>id</mi> <mi>X</mi></msub><mo>×</mo><msub><mi>const</mi> <mn>1</mn></msub><mo stretchy='false'>)</mo></mrow></mover><mi>X</mi><mo>×</mo><mo stretchy='false'>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy='false'>]</mo><mo lspace='0em' rspace='thinmathspace'>overet</mo><mrow><msub><mi>pr</mi> <mn>1</mn></msub></mrow><mo>⟶</mo><mi>X</mi><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> \nabla_X \;\colon\; X \sqcup X \overset{(id_X \times const_0, id_X \times const_1)}{\longrightarrow} X \times [0,1] \overet{pr_1}{\longrightarrow} X \,. </annotation></semantics></math></div> <p>Accordingly, with respect to the <a class='existingWikiWord' href='/nlab/show/diff/classical+model+structure+on+topological+spaces'>classical model structure on topological spaces</a> this is an example a of <em><a class='existingWikiWord' href='/nlab/show/diff/cylinder+object'>cylinder object</a></em>.</p> <h3 id='left_homotopy'>Left homotopy</h3> <p>Given <math class='maruku-mathml' display='inline' id='mathml_959a5257e1bdaf21e8ec816c6597738862b6180a_17' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi><mo>,</mo><mi>Y</mi></mrow><annotation encoding='application/x-tex'>X,Y</annotation></semantics></math> two <a class='existingWikiWord' href='/nlab/show/diff/topological+space'>topological spaces</a> and <math class='maruku-mathml' display='inline' id='mathml_959a5257e1bdaf21e8ec816c6597738862b6180a_18' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>f</mi><mo>,</mo><mi>g</mi><mspace width='thickmathspace' /><mo lspace='verythinmathspace'>:</mo><mspace width='thickmathspace' /><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/diff/continuous+map'>continuous functions</a>,then a <strong><a class='existingWikiWord' href='/nlab/show/diff/homotopy'>left homotopy</a></strong></p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_959a5257e1bdaf21e8ec816c6597738862b6180a_19' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>η</mi><mo lspace='verythinmathspace'>:</mo><mi>f</mi><mspace width='thinmathspace' /><msub><mo>⇒</mo> <mi>L</mi></msub><mspace width='thinmathspace' /><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/diff/continuous+map'>continuous function</a></p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_959a5257e1bdaf21e8ec816c6597738862b6180a_20' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>η</mi><mspace width='thickmathspace' /><mo lspace='verythinmathspace'>:</mo><mspace width='thickmathspace' /><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/diff/product+topological+space'>product topological space</a> of <math class='maruku-mathml' display='inline' id='mathml_959a5257e1bdaf21e8ec816c6597738862b6180a_21' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> with the topological interval, such that this fits into a <a class='existingWikiWord' href='/nlab/show/diff/commutative+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 class='maruku-mathml' display='block' id='mathml_959a5257e1bdaf21e8ec816c6597738862b6180a_22' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mrow><mtable><mtr><mtd><mi>X</mi></mtd></mtr> <mtr><mtd><msup><mrow /> <mpadded lspace='-100%width' width='0'><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><mi>X</mi><mo>×</mo><mi>I</mi></mtd> <mtd><mover><mo>⟶</mo><mi>η</mi></mover></mtd> <mtd><mi>Y</mi></mtd></mtr> <mtr><mtd><msup><mrow /> <mpadded lspace='-100%width' width='0'><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><mi>X</mi></mtd></mtr></mtable></mrow><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> \array{ X \\ {}^{\mathllap{(id,\delta_0)}}\downarrow & \searrow^{\mathrlap{f}} \\ X \times I &\stackrel{\eta}{\longrightarrow}& Y \\ {}^{\mathllap{(id,\delta_1)}}\uparrow & \nearrow_{\mathrlap{g}} \\ X } \,. </annotation></semantics></math></div> <p>(graphics grabbed from J. Tauber <a href='http://jtauber.com/blog/2005/07/01/path_homotopy/'>here</a>)</p> <h3 id='path_space'>Path space</h3> <p>For <math class='maruku-mathml' display='inline' id='mathml_959a5257e1bdaf21e8ec816c6597738862b6180a_23' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> a <a class='existingWikiWord' href='/nlab/show/diff/topological+space'>topological space</a>, then its <em>path space</em> is the <a class='existingWikiWord' href='/nlab/show/diff/compact-open+topology'>mapping space</a> <math class='maruku-mathml' display='inline' id='mathml_959a5257e1bdaf21e8ec816c6597738862b6180a_24' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>X</mi> <mrow><mo stretchy='false'>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy='false'>]</mo></mrow></msup></mrow><annotation encoding='application/x-tex'>X^{[0,1]}</annotation></semantics></math>, out of the topological interval into <math class='maruku-mathml' display='inline' id='mathml_959a5257e1bdaf21e8ec816c6597738862b6180a_25' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math>, i.e. the set of <a class='existingWikiWord' href='/nlab/show/diff/continuous+map'>continuous function</a> <math class='maruku-mathml' display='inline' id='mathml_959a5257e1bdaf21e8ec816c6597738862b6180a_26' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>γ</mi><mspace width='thickmathspace' /><mo lspace='verythinmathspace'>:</mo><mspace width='thickmathspace' /><mo stretchy='false'>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy='false'>]</mo><mo>→</mo><mi>X</mi></mrow><annotation encoding='application/x-tex'>\gamma \;\colon\; [0,1] \to X</annotation></semantics></math> equipped with the <a class='existingWikiWord' href='/nlab/show/diff/compact-open+topology'>compact-open topology</a>.</p> <p>The two endpoint inclusions <math class='maruku-mathml' display='inline' id='mathml_959a5257e1bdaf21e8ec816c6597738862b6180a_27' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo>*</mo><mo lspace='verythinmathspace'>:</mo><mo stretchy='false'>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy='false'>]</mo></mrow><annotation encoding='application/x-tex'>\ast \colon [0,1]</annotation></semantics></math> and the unique projection <math class='maruku-mathml' display='inline' id='mathml_959a5257e1bdaf21e8ec816c6597738862b6180a_28' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy='false'>]</mo><mo>→</mo><mo>*</mo></mrow><annotation encoding='application/x-tex'>[0,1] \to \ast</annotation></semantics></math> induce continuous functions</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_959a5257e1bdaf21e8ec816c6597738862b6180a_29' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi><mover><mo>⟶</mo><mrow /></mover><msup><mi>X</mi> <mrow><mo stretchy='false'>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy='false'>]</mo></mrow></msup><mover><mo>⟶</mo><mrow><msup><mi>X</mi> <mrow><mo stretchy='false'>(</mo><msub><mi>const</mi> <mn>0</mn></msub><mo>,</mo><msub><mi>const</mi> <mn>1</mn></msub><mo stretchy='false'>)</mo></mrow></msup></mrow></mover><mi>X</mi><mo>×</mo><mi>X</mi></mrow><annotation encoding='application/x-tex'> X \overset{}{\longrightarrow} X^{[0,1]} \overset{X^{(const_0,const_1)}}{\longrightarrow} X \times X </annotation></semantics></math></div> <p>(inclusion of constant paths and endpoint evaluation of paths).</p> <h2 id='properties'>Properties</h2> <h3 id='freyds_characterization'>Freyd’s characterization</h3> <p>The topological interval <math class='maruku-mathml' display='inline' id='mathml_959a5257e1bdaf21e8ec816c6597738862b6180a_30' xmlns='http://www.w3.org/1998/Math/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> may be characterized by a <a class='existingWikiWord' href='/nlab/show/diff/coalgebra+for+an+endofunctor'>coalgebraic</a> definition first identified by <a class='existingWikiWord' href='/nlab/show/diff/Peter+Freyd'>Freyd</a><span><del class='diffmod'> .</del><ins class='diffmod'> :</ins><del class='diffdel'> Let</del></span><del class='diffdel'><math class='maruku-mathml' display='inline' id='mathml_959a5257e1bdaf21e8ec816c6597738862b6180a_31' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>Top</mi> <mrow><mo>*</mo><mo>,</mo><mo>*</mo></mrow></msub></mrow><annotation encoding='application/x-tex'>Top_{\ast, \ast}</annotation></semantics></math></del><del class='diffdel'> be the category of topological spaces </del><del class='diffdel'><math class='maruku-mathml' display='inline' id='mathml_959a5257e1bdaf21e8ec816c6597738862b6180a_32' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math></del><del class='diffdel'> equipped with a pair </del><del class='diffdel'><math class='maruku-mathml' display='inline' id='mathml_959a5257e1bdaf21e8ec816c6597738862b6180a_33' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>x</mi> <mn>0</mn></msub><mo>,</mo><msub><mi>x</mi> <mn>1</mn></msub></mrow><annotation encoding='application/x-tex'>x_0, x_1</annotation></semantics></math></del><del class='diffdel'> of distinct points, for example </del><del class='diffdel'><math class='maruku-mathml' display='inline' id='mathml_959a5257e1bdaf21e8ec816c6597738862b6180a_34' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>I</mi><mo>=</mo><mo stretchy='false'>(</mo><mo stretchy='false'>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy='false'>]</mo><mo>;</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>I = ([0, 1]; 0, 1)</annotation></semantics></math></del><del class='diffdel'>. Let </del><del class='diffdel'><math class='maruku-mathml' display='inline' id='mathml_959a5257e1bdaf21e8ec816c6597738862b6180a_35' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>F</mi><mo>:</mo><msub><mi>Top</mi> <mrow><mo>*</mo><mo>,</mo><mo>*</mo></mrow></msub><mo>→</mo><msub><mi>Top</mi> <mrow><mo>*</mo><mo>,</mo><mo>*</mo></mrow></msub></mrow><annotation encoding='application/x-tex'>F: Top_{\ast, \ast} \to Top_{\ast, \ast}</annotation></semantics></math></del><del class='diffdel'> be the </del><del class='diffdel'><a class='existingWikiWord' href='/nlab/show/diff/functor'>functor</a></del><del class='diffdel'> defined on objects by </del><del class='diffdel'><math class='maruku-mathml' display='inline' id='mathml_959a5257e1bdaf21e8ec816c6597738862b6180a_36' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>F</mi><mo stretchy='false'>(</mo><mi>X</mi><mo>;</mo><msub><mi>x</mi> <mn>0</mn></msub><mo>,</mo><msub><mi>x</mi> <mn>1</mn></msub><mo stretchy='false'>)</mo><mo>=</mo><mo stretchy='false'>(</mo><mi>X</mi><mo>∨</mo><mi>X</mi><mo>,</mo><msub><mi>y</mi> <mn>0</mn></msub><mo>,</mo><msub><mi>y</mi> <mn>1</mn></msub><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>F(X; x_0, x_1) = (X \vee X, y_0, y_1)</annotation></semantics></math></del><del class='diffdel'>, where </del><del class='diffdel'><math class='maruku-mathml' display='inline' id='mathml_959a5257e1bdaf21e8ec816c6597738862b6180a_37' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi><mo>∨</mo><mi>X</mi></mrow><annotation encoding='application/x-tex'>X \vee X</annotation></semantics></math></del><del class='diffdel'> is the quotient of </del><del class='diffdel'><math class='maruku-mathml' display='inline' id='mathml_959a5257e1bdaf21e8ec816c6597738862b6180a_38' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi><mo>⊔</mo><mi>X</mi></mrow><annotation encoding='application/x-tex'>X \sqcup X</annotation></semantics></math></del><del class='diffdel'> formed by identifying the element </del><del class='diffdel'><math class='maruku-mathml' display='inline' id='mathml_959a5257e1bdaf21e8ec816c6597738862b6180a_39' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>x</mi> <mn>1</mn></msub></mrow><annotation encoding='application/x-tex'>x_1</annotation></semantics></math></del><del class='diffdel'> of the first copy of </del><del class='diffdel'><math class='maruku-mathml' display='inline' id='mathml_959a5257e1bdaf21e8ec816c6597738862b6180a_40' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math></del><del class='diffdel'> with </del><del class='diffdel'><math class='maruku-mathml' display='inline' id='mathml_959a5257e1bdaf21e8ec816c6597738862b6180a_41' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>x</mi> <mn>0</mn></msub></mrow><annotation encoding='application/x-tex'>x_0</annotation></semantics></math></del><del class='diffdel'> of the second copy of </del><del class='diffdel'><math class='maruku-mathml' display='inline' id='mathml_959a5257e1bdaf21e8ec816c6597738862b6180a_42' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math></del><del class='diffdel'>, where </del><del class='diffdel'><math class='maruku-mathml' display='inline' id='mathml_959a5257e1bdaf21e8ec816c6597738862b6180a_43' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>y</mi> <mn>0</mn></msub></mrow><annotation encoding='application/x-tex'>y_0</annotation></semantics></math></del><del class='diffdel'> is identified with </del><del class='diffdel'><math class='maruku-mathml' display='inline' id='mathml_959a5257e1bdaf21e8ec816c6597738862b6180a_44' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>x</mi> <mn>0</mn></msub></mrow><annotation encoding='application/x-tex'>x_0</annotation></semantics></math></del><del class='diffdel'> of the first copy, and </del><del class='diffdel'><math class='maruku-mathml' display='inline' id='mathml_959a5257e1bdaf21e8ec816c6597738862b6180a_45' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>y</mi> <mn>1</mn></msub></mrow><annotation encoding='application/x-tex'>y_1</annotation></semantics></math></del><del class='diffdel'> is identified with </del><del class='diffdel'><math class='maruku-mathml' display='inline' id='mathml_959a5257e1bdaf21e8ec816c6597738862b6180a_46' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>x</mi> <mn>1</mn></msub></mrow><annotation encoding='application/x-tex'>x_1</annotation></semantics></math></del><del class='diffdel'> of the second copy.</del></p> <p><span><del class='diffmod'> For</del><ins class='diffmod'> Let</ins></span><math class='maruku-mathml' display='inline' id='mathml_959a5257e1bdaf21e8ec816c6597738862b6180a_47' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><del class='diffmod'><mi>I</mi></del><ins class='diffmod'><msub><mi>Top</mi> <mrow><mo>*</mo><mo>,</mo><mo>*</mo></mrow></msub></ins><del class='diffdel'><mo>=</mo></del><del class='diffdel'><mo stretchy='false'>(</mo></del><del class='diffdel'><mo stretchy='false'>[</mo></del><del class='diffdel'><mn>0</mn></del><del class='diffdel'><mo>,</mo></del><del class='diffdel'><mn>1</mn></del><del class='diffdel'><mo stretchy='false'>]</mo></del><del class='diffdel'><mo>;</mo></del><del class='diffdel'><mn>0</mn></del><del class='diffdel'><mo>,</mo></del><del class='diffdel'><mn>1</mn></del><del class='diffdel'><mo stretchy='false'>)</mo></del></mrow><annotation encoding='application/x-tex'><span><del class='diffmod'> I</del><ins class='diffmod'> Top_{\ast,</ins><del class='diffmod'> =</del><ins class='diffmod'> \ast}</ins><del class='diffdel'> ([0,</del><del class='diffdel'> 1];</del><del class='diffdel'> 0,</del><del class='diffdel'> 1)</del></span></annotation></semantics></math><span> <del class='diffmod'> there</del><ins class='diffmod'> be</ins><del class='diffmod'> is</del><ins class='diffmod'> the</ins><del class='diffdel'> an</del><del class='diffdel'> evident</del><del class='diffdel'> identification</del></span><del class='diffmod'><math class='maruku-mathml' display='inline' id='mathml_959a5257e1bdaf21e8ec816c6597738862b6180a_48' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>F</mi><mo stretchy='false'>(</mo><mi>I</mi><mo stretchy='false'>)</mo><mo>=</mo><mo stretchy='false'>(</mo><mo stretchy='false'>[</mo><mn>0</mn><mo>,</mo><mn>2</mn><mo stretchy='false'>]</mo><mo>;</mo><mn>0</mn><mo>,</mo><mn>2</mn><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>F(I) = ([0, 2]; 0, 2)</annotation></semantics></math></del><ins class='diffmod'><a class='existingWikiWord' href='/nlab/show/diff/category'>category</a></ins><span><del class='diffmod'> ,</del><ins class='diffmod'> </ins><del class='diffmod'> and</del><ins class='diffmod'> of</ins><del class='diffdel'> moreover</del><del class='diffdel'> there</del><del class='diffdel'> is</del><del class='diffdel'> an</del></span><del class='diffmod'><math class='maruku-mathml' display='inline' id='mathml_959a5257e1bdaf21e8ec816c6597738862b6180a_49' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>F</mi></mrow><annotation encoding='application/x-tex'>F</annotation></semantics></math></del><ins class='diffmod'><a class='existingWikiWord' href='/nlab/show/diff/topological+space'>topological spaces</a></ins><span><del class='diffdel'> -coalgebra</del><del class='diffdel'> structure</del></span><math class='maruku-mathml' display='inline' id='mathml_959a5257e1bdaf21e8ec816c6597738862b6180a_50' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi><span><del class='diffmod'> I</del><ins class='diffmod'> X</ins></span></mi><del class='diffdel'><mo>→</mo></del><del class='diffdel'><mi>F</mi></del><del class='diffdel'><mo stretchy='false'>(</mo></del><del class='diffdel'><mi>I</mi></del><del class='diffdel'><mo stretchy='false'>)</mo></del></mrow><annotation encoding='application/x-tex'><span><del class='diffmod'> I</del><ins class='diffmod'> X</ins><del class='diffdel'> \to</del><del class='diffdel'> F(I)</del></span></annotation></semantics></math><span> <del class='diffmod'> given</del><ins class='diffmod'> equipped</ins><del class='diffmod'> by</del><ins class='diffmod'> with</ins><del class='diffmod'> “multiplication</del><ins class='diffmod'> a</ins><del class='diffdel'> by</del></span><del class='diffmod'><math class='maruku-mathml' display='inline' id='mathml_959a5257e1bdaf21e8ec816c6597738862b6180a_51' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>2</mn></mrow><annotation encoding='application/x-tex'>2</annotation></semantics></math></del><ins class='diffmod'><a class='existingWikiWord' href='/nlab/show/diff/pair'>pair</a></ins><span><del class='diffdel'> ”.</del></span><ins class='diffins'><math class='maruku-mathml' display='inline' id='mathml_5d694da69bac01d48a59559292f479be75bae55e_33' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>x</mi> <mn>0</mn></msub><mo>,</mo><msub><mi>x</mi> <mn>1</mn></msub></mrow><annotation encoding='application/x-tex'>x_0, x_1</annotation></semantics></math></ins><ins class='diffins'> of distinct points, for example </ins><ins class='diffins'><math class='maruku-mathml' display='inline' id='mathml_5d694da69bac01d48a59559292f479be75bae55e_34' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>I</mi><mo>=</mo><mo stretchy='false'>(</mo><mo stretchy='false'>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy='false'>]</mo><mo>;</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>I = ([0, 1]; 0, 1)</annotation></semantics></math></ins><ins class='diffins'>. Let </ins><ins class='diffins'><math class='maruku-mathml' display='inline' id='mathml_5d694da69bac01d48a59559292f479be75bae55e_35' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>F</mi><mo lspace='verythinmathspace'>:</mo><msub><mi>Top</mi> <mrow><mo>*</mo><mo>,</mo><mo>*</mo></mrow></msub><mo>→</mo><msub><mi>Top</mi> <mrow><mo>*</mo><mo>,</mo><mo>*</mo></mrow></msub></mrow><annotation encoding='application/x-tex'>F \colon Top_{\ast, \ast} \to Top_{\ast, \ast}</annotation></semantics></math></ins><ins class='diffins'> be the </ins><ins class='diffins'><a class='existingWikiWord' href='/nlab/show/diff/functor'>functor</a></ins><ins class='diffins'> defined on </ins><ins class='diffins'><a class='existingWikiWord' href='/nlab/show/diff/object'>objects</a></ins><ins class='diffins'> by</ins></p><span /><div class='num_theorem'><del class='diffmod'> </del><ins class='diffmod'><math class='maruku-mathml' display='block' id='mathml_5d694da69bac01d48a59559292f479be75bae55e_36' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>F</mi><mo stretchy='false'>(</mo><mi>X</mi><mo>;</mo><msub><mi>x</mi> <mn>0</mn></msub><mo>,</mo><msub><mi>x</mi> <mn>1</mn></msub><mo stretchy='false'>)</mo><mo>=</mo><mo stretchy='false'>(</mo><mi>X</mi><mo>∨</mo><mi>X</mi><mo>,</mo><msub><mi>y</mi> <mn>0</mn></msub><mo>,</mo><msub><mi>y</mi> <mn>1</mn></msub><mo stretchy='false'>)</mo><mspace width='thinmathspace' /><mo>,</mo></mrow><annotation encoding='application/x-tex'> F(X; x_0, x_1) = (X \vee X, y_0, y_1) \,, </annotation></semantics></math></ins><del class='diffdel'><h6 id='theorem'>Theorem</h6></del><del class='diffdel'> </del><del class='diffdel'><p><strong>(Freyd)</strong></p></del><del class='diffdel'> </del><del class='diffdel'><p><math class='maruku-mathml' display='inline' id='mathml_959a5257e1bdaf21e8ec816c6597738862b6180a_52' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>I</mi><mo>=</mo><mo stretchy='false'>(</mo><mo stretchy='false'>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy='false'>]</mo><mo>,</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>I = ([0, 1], 0, 1)</annotation></semantics></math> is the terminal <math class='maruku-mathml' display='inline' id='mathml_959a5257e1bdaf21e8ec816c6597738862b6180a_53' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>F</mi></mrow><annotation encoding='application/x-tex'>F</annotation></semantics></math>-<a class='existingWikiWord' href='/nlab/show/diff/terminal+coalgebra+for+an+endofunctor'>coalgebra</a>.</p></del><del class='diffdel'> </del></div><span /><p><span><del class='diffmod'> For</del><ins class='diffmod'> where</ins><del class='diffdel'> more</del><del class='diffdel'> information,</del><del class='diffdel'> see</del></span><del class='diffmod'><a class='existingWikiWord' href='/nlab/show/diff/coalgebra+of+the+real+interval'>coalgebra of the real interval</a></del><ins class='diffmod'><math class='maruku-mathml' display='inline' id='mathml_5d694da69bac01d48a59559292f479be75bae55e_37' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi><mo>∨</mo><mi>X</mi></mrow><annotation encoding='application/x-tex'>X \vee X</annotation></semantics></math></ins><span><del class='diffmod'> ,</del><ins class='diffmod'> </ins><del class='diffmod'> which</del><ins class='diffmod'> denotes</ins><del class='diffdel'> shows</del><del class='diffdel'> in</del><del class='diffdel'> particular</del><del class='diffdel'> how</del> the<del class='diffdel'> interval</del><del class='diffdel'> structure</del><del class='diffdel'> of</del></span><del class='diffmod'><math class='maruku-mathml' display='inline' id='mathml_959a5257e1bdaf21e8ec816c6597738862b6180a_54' xmlns='http://www.w3.org/1998/Math/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></del><ins class='diffmod'><a class='existingWikiWord' href='/nlab/show/diff/quotient+space'>quotient space</a></ins><span> <del class='diffmod'> may</del><ins class='diffmod'> of</ins><del class='diffmod'> be</del><ins class='diffmod'> the</ins><del class='diffdel'> defined</del><del class='diffdel'> by</del><del class='diffdel'> coinduction.</del></span><ins class='diffins'><a class='existingWikiWord' href='/nlab/show/diff/disjoint+union+topological+space'>disjoint union space</a></ins><ins class='diffins'> </ins><ins class='diffins'><math class='maruku-mathml' display='inline' id='mathml_5d694da69bac01d48a59559292f479be75bae55e_38' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi><mo>⊔</mo><mi>X</mi></mrow><annotation encoding='application/x-tex'>X \sqcup X</annotation></semantics></math></ins><ins class='diffins'> formed by identifying the element </ins><ins class='diffins'><math class='maruku-mathml' display='inline' id='mathml_5d694da69bac01d48a59559292f479be75bae55e_39' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>x</mi> <mn>1</mn></msub></mrow><annotation encoding='application/x-tex'>x_1</annotation></semantics></math></ins><ins class='diffins'> of the first copy of </ins><ins class='diffins'><math class='maruku-mathml' display='inline' id='mathml_5d694da69bac01d48a59559292f479be75bae55e_40' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math></ins><ins class='diffins'> with </ins><ins class='diffins'><math class='maruku-mathml' display='inline' id='mathml_5d694da69bac01d48a59559292f479be75bae55e_41' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>x</mi> <mn>0</mn></msub></mrow><annotation encoding='application/x-tex'>x_0</annotation></semantics></math></ins><ins class='diffins'> of the second copy of </ins><ins class='diffins'><math class='maruku-mathml' display='inline' id='mathml_5d694da69bac01d48a59559292f479be75bae55e_42' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math></ins><ins class='diffins'>, where </ins><ins class='diffins'><math class='maruku-mathml' display='inline' id='mathml_5d694da69bac01d48a59559292f479be75bae55e_43' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>y</mi> <mn>0</mn></msub></mrow><annotation encoding='application/x-tex'>y_0</annotation></semantics></math></ins><ins class='diffins'> is identified with </ins><ins class='diffins'><math class='maruku-mathml' display='inline' id='mathml_5d694da69bac01d48a59559292f479be75bae55e_44' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>x</mi> <mn>0</mn></msub></mrow><annotation encoding='application/x-tex'>x_0</annotation></semantics></math></ins><ins class='diffins'> of the first copy, and </ins><ins class='diffins'><math class='maruku-mathml' display='inline' id='mathml_5d694da69bac01d48a59559292f479be75bae55e_45' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>y</mi> <mn>1</mn></msub></mrow><annotation encoding='application/x-tex'>y_1</annotation></semantics></math></ins><ins class='diffins'> is identified with </ins><ins class='diffins'><math class='maruku-mathml' display='inline' id='mathml_5d694da69bac01d48a59559292f479be75bae55e_46' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>x</mi> <mn>1</mn></msub></mrow><annotation encoding='application/x-tex'>x_1</annotation></semantics></math></ins><ins class='diffins'> of the second copy.</ins></p><ins class='diffins'> </ins><ins class='diffins'><p>For <math class='maruku-mathml' display='inline' id='mathml_5d694da69bac01d48a59559292f479be75bae55e_47' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>I</mi><mo>=</mo><mo stretchy='false'>(</mo><mo stretchy='false'>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy='false'>]</mo><mo>;</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>I = ([0, 1]; 0, 1)</annotation></semantics></math> there is an evident identification <math class='maruku-mathml' display='inline' id='mathml_5d694da69bac01d48a59559292f479be75bae55e_48' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>F</mi><mo stretchy='false'>(</mo><mi>I</mi><mo stretchy='false'>)</mo><mo>=</mo><mo stretchy='false'>(</mo><mo stretchy='false'>[</mo><mn>0</mn><mo>,</mo><mn>2</mn><mo stretchy='false'>]</mo><mo>;</mo><mn>0</mn><mo>,</mo><mn>2</mn><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>F(I) = ([0, 2]; 0, 2)</annotation></semantics></math>, and moreover there is an <math class='maruku-mathml' display='inline' id='mathml_5d694da69bac01d48a59559292f479be75bae55e_49' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>F</mi></mrow><annotation encoding='application/x-tex'>F</annotation></semantics></math>-<a class='existingWikiWord' href='/nlab/show/diff/coalgebra+for+an+endofunctor'>coalgebra</a> structure <math class='maruku-mathml' display='inline' id='mathml_5d694da69bac01d48a59559292f479be75bae55e_50' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>I</mi><mo>→</mo><mi>F</mi><mo stretchy='false'>(</mo><mi>I</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>I \to F(I)</annotation></semantics></math> given by “multiplication by <math class='maruku-mathml' display='inline' id='mathml_5d694da69bac01d48a59559292f479be75bae55e_51' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>2</mn></mrow><annotation encoding='application/x-tex'>2</annotation></semantics></math>”.</p></ins><ins class='diffins'> </ins><ins class='diffins'><div class='num_theorem'> <h6 id='theorem'>Theorem</h6> <p><strong>(Freyd)</strong></p> <p>The topological interval <math class='maruku-mathml' display='inline' id='mathml_5d694da69bac01d48a59559292f479be75bae55e_52' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>I</mi><mo>=</mo><mo stretchy='false'>(</mo><mo stretchy='false'>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy='false'>]</mo><mo>,</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>I = ([0, 1], 0, 1)</annotation></semantics></math> is the <a class='existingWikiWord' href='/nlab/show/diff/terminal+coalgebra+for+an+endofunctor'>terminal F-coalgebra</a>.</p> </div></ins><ins class='diffins'> </ins><ins class='diffins'><p>For more information, see <a class='existingWikiWord' href='/nlab/show/diff/coalgebra+of+the+real+interval'>coalgebra of the real interval</a>, which shows in particular how the interval structure of <math class='maruku-mathml' display='inline' id='mathml_5d694da69bac01d48a59559292f479be75bae55e_53' xmlns='http://www.w3.org/1998/Math/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> may be defined by coinduction.</p></ins> <p> </p> </div> <div class="revisedby"> <p> Last revised on July 2, 2017 at 11:54:08. See the <a href="/nlab/history/topological+interval" 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/topological+interval" accesskey="E" class="navlink" id="edit" rel="nofollow">Edit</a><a href="https://nforum.ncatlab.org/discussion/7877/#Item_9">Discuss</a><span class="backintime"><a href="/nlab/revision/diff/topological+interval/6" accesskey="B" class="navlinkbackintime" id="to_previous_revision" rel="nofollow">Previous revision</a></span><a href="/nlab/show/topological+interval" accesskey="C" class="navlink" id="see_changes" rel="nofollow">Hide changes</a><a href="/nlab/history/topological+interval" accesskey="S" class="navlink" id="history" rel="nofollow">History (6 revisions)</a> <a href="/nlab/show/topological+interval/cite" style="color: black">Cite</a> <a href="/nlab/print/topological+interval" accesskey="p" id="view_print" rel="nofollow">Print</a> <a href="/nlab/source/topological+interval" id="view_source" rel="nofollow">Source</a> </div> </div>  </div>  </body> </html>