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style="display:inline-block; float: left;" onfocus="this.value == 'Search' ? this.value = '' : true" onblur="this.value == '' ? this.value = 'Search' : true" /></fieldset> </form> <span id='navEnd'></span> </div> <div id="revision"> <p class="show_diff"> Showing changes from revision #45 to #46: <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> <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='#UniversalConstructions'>Universal constructions</a></li><li><a href='#relation_with_'>Relation with <math class='maruku-mathml' display='inline' id='mathml_fb0302666679a9db8a44d8a779257e05038cbef3_1' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Set</mi></mrow><annotation encoding='application/x-tex'>Set</annotation></semantics></math></a></li><li><a href='#MonoEpiMorphisms'>Mono-/Epimorphisms</a></li><li><a href='#intersections_and_quotients'>Intersections and quotients</a></li><ins class='diffins'><li><a href='#closed_monoidal_structure'>Closed monoidal structure</a></li></ins></ul></li><li><a href='#related_concepts'>Related concepts</a></li><li><a href='#references'>References</a></li></ul></div> <h2 id='definition'>Definition</h2> <p><strong>Top</strong> denotes the <a class='existingWikiWord' href='/nlab/show/diff/category'>category</a> whose <a class='existingWikiWord' href='/nlab/show/diff/object'>objects</a> are <a class='existingWikiWord' href='/nlab/show/diff/topological+space'>topological spaces</a> and whose <a class='existingWikiWord' href='/nlab/show/diff/morphism'>morphisms</a> are <a class='existingWikiWord' href='/nlab/show/diff/continuous+map'>continuous functions</a> between them. Its <a class='existingWikiWord' href='/nlab/show/diff/isomorphism'>isomorphisms</a> are the <a class='existingWikiWord' href='/nlab/show/diff/homeomorphism'>homeomorphisms</a>.</p> <p>For exposition see <em><a class='existingWikiWord' href='/nlab/show/diff/Introduction+to+Topology+--+1'>Introduction to point-set topology</a></em>.</p> <p>Often one considers (sometimes by default) <a class='existingWikiWord' href='/nlab/show/diff/subcategory'>subcategories</a> of <a class='existingWikiWord' href='/nlab/show/diff/nice+topological+space'>nice topological spaces</a> such as <a class='existingWikiWord' href='/nlab/show/diff/compactly+generated+topological+space'>compactly generated topological spaces</a>, notably because these are <a class='existingWikiWord' href='/nlab/show/diff/cartesian+closed+category'>cartesian closed</a>. There other other <a class='existingWikiWord' href='/nlab/show/diff/convenient+category+of+topological+spaces'>convenient categories of topological spaces</a>. With any one such choice understood, it is often useful to regard it as “the” category of topological spaces.</p> <p>The <a class='existingWikiWord' href='/nlab/show/diff/homotopy+category'>homotopy category</a> of <math class='maruku-mathml' display='inline' id='mathml_fb0302666679a9db8a44d8a779257e05038cbef3_2' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Top</mi></mrow><annotation encoding='application/x-tex'>Top</annotation></semantics></math> given by its <a class='existingWikiWord' href='/nlab/show/diff/localization'>localization</a> at the <a class='existingWikiWord' href='/nlab/show/diff/weak+homotopy+equivalence'>weak homotopy equivalences</a> is the <a class='existingWikiWord' href='/nlab/show/diff/Ho%28Top%29'>classical homotopy category</a> <a class='existingWikiWord' href='/nlab/show/diff/Ho%28Top%29'>Ho(Top)</a>. This is the central object of study in <a class='existingWikiWord' href='/nlab/show/diff/homotopy+theory'>homotopy theory</a>, see also at <em><a class='existingWikiWord' href='/nlab/show/diff/classical+model+structure+on+topological+spaces'>classical model structure on topological spaces</a></em>. The <a class='existingWikiWord' href='/nlab/show/diff/simplicial+localization'>simplicial localization</a> of <a class='existingWikiWord' href='/nlab/show/diff/Top'>Top</a> at the <a class='existingWikiWord' href='/nlab/show/diff/weak+homotopy+equivalence'>weak homotopy equivalences</a> is the archetypical <a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-category'>(∞,1)-category</a>, <a class='existingWikiWord' href='/nlab/show/diff/equivalence+of+%28infinity%2C1%29-categories'>equivalent</a> to <a class='existingWikiWord' href='/nlab/show/diff/Infinity-Grpd'>∞Grpd</a> (see at <em><a class='existingWikiWord' href='/nlab/show/diff/homotopy+hypothesis'>homotopy hypothesis</a></em>).</p> <h2 id='properties'>Properties</h2> <h3 id='UniversalConstructions'>Universal constructions</h3> <p>We discuss <a class='existingWikiWord' href='/nlab/show/diff/universal+construction'>universal constructions</a> in <a class='existingWikiWord' href='/nlab/show/diff/Top'>Top</a>, such as <a class='existingWikiWord' href='/nlab/show/diff/limit'>limits</a>/<a class='existingWikiWord' href='/nlab/show/diff/colimit'>colimits</a>, etc. The following definition suggests that universal constructions be seen in the context of <math class='maruku-mathml' display='inline' id='mathml_fb0302666679a9db8a44d8a779257e05038cbef3_3' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Top</mi></mrow><annotation encoding='application/x-tex'>Top</annotation></semantics></math> as a <a class='existingWikiWord' href='/nlab/show/diff/topological+concrete+category'>topological concrete category</a> (see Proposition <a class='maruku-ref' href='#topcat'>4</a> below).</p> <p><math class='maruku-mathml' display='inline' id='mathml_fb0302666679a9db8a44d8a779257e05038cbef3_4' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mspace width='thinmathspace' /></mrow><annotation encoding='application/x-tex'>\,</annotation></semantics></math></p> <p><strong>examples of <a href='Top#UniversalConstructions'>universal constructions of topological spaces</a>:</strong></p> <table><thead><tr><th><math class='maruku-mathml' display='inline' id='mathml_fb0302666679a9db8a44d8a779257e05038cbef3_5' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>AAAA</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{AAAA}</annotation></semantics></math><a class='existingWikiWord' href='/nlab/show/diff/limit'>limits</a></th><th><math class='maruku-mathml' display='inline' id='mathml_fb0302666679a9db8a44d8a779257e05038cbef3_6' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>AAAA</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{AAAA}</annotation></semantics></math><a class='existingWikiWord' href='/nlab/show/diff/colimit'>colimits</a></th></tr></thead><tbody><tr><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_fb0302666679a9db8a44d8a779257e05038cbef3_7' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mspace width='thinmathspace' /></mrow><annotation encoding='application/x-tex'>\,</annotation></semantics></math> <a class='existingWikiWord' href='/nlab/show/diff/point+space'>point space</a><math class='maruku-mathml' display='inline' id='mathml_fb0302666679a9db8a44d8a779257e05038cbef3_8' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mspace width='thinmathspace' /></mrow><annotation encoding='application/x-tex'>\,</annotation></semantics></math></td><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_fb0302666679a9db8a44d8a779257e05038cbef3_9' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mspace width='thinmathspace' /></mrow><annotation encoding='application/x-tex'>\,</annotation></semantics></math> <a class='existingWikiWord' href='/nlab/show/diff/empty+space'>empty space</a> <math class='maruku-mathml' display='inline' id='mathml_fb0302666679a9db8a44d8a779257e05038cbef3_10' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mspace width='thinmathspace' /></mrow><annotation encoding='application/x-tex'>\,</annotation></semantics></math></td></tr> <tr><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_fb0302666679a9db8a44d8a779257e05038cbef3_11' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mspace width='thinmathspace' /></mrow><annotation encoding='application/x-tex'>\,</annotation></semantics></math> <a class='existingWikiWord' href='/nlab/show/diff/product+topological+space'>product topological space</a> <math class='maruku-mathml' display='inline' id='mathml_fb0302666679a9db8a44d8a779257e05038cbef3_12' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mspace width='thinmathspace' /></mrow><annotation encoding='application/x-tex'>\,</annotation></semantics></math></td><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_fb0302666679a9db8a44d8a779257e05038cbef3_13' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mspace width='thinmathspace' /></mrow><annotation encoding='application/x-tex'>\,</annotation></semantics></math> <a class='existingWikiWord' href='/nlab/show/diff/disjoint+union+topological+space'>disjoint union topological space</a> <math class='maruku-mathml' display='inline' id='mathml_fb0302666679a9db8a44d8a779257e05038cbef3_14' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mspace width='thinmathspace' /></mrow><annotation encoding='application/x-tex'>\,</annotation></semantics></math></td></tr> <tr><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_fb0302666679a9db8a44d8a779257e05038cbef3_15' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mspace width='thinmathspace' /></mrow><annotation encoding='application/x-tex'>\,</annotation></semantics></math> <a class='existingWikiWord' href='/nlab/show/diff/subspace+topology'>topological subspace</a> <math class='maruku-mathml' display='inline' id='mathml_fb0302666679a9db8a44d8a779257e05038cbef3_16' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mspace width='thinmathspace' /></mrow><annotation encoding='application/x-tex'>\,</annotation></semantics></math></td><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_fb0302666679a9db8a44d8a779257e05038cbef3_17' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mspace width='thinmathspace' /></mrow><annotation encoding='application/x-tex'>\,</annotation></semantics></math> <a class='existingWikiWord' href='/nlab/show/diff/quotient+space'>quotient topological space</a> <math class='maruku-mathml' display='inline' id='mathml_fb0302666679a9db8a44d8a779257e05038cbef3_18' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mspace width='thinmathspace' /></mrow><annotation encoding='application/x-tex'>\,</annotation></semantics></math></td></tr> <tr><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_fb0302666679a9db8a44d8a779257e05038cbef3_19' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mspace width='thinmathspace' /></mrow><annotation encoding='application/x-tex'>\,</annotation></semantics></math> fiber space <math class='maruku-mathml' display='inline' id='mathml_fb0302666679a9db8a44d8a779257e05038cbef3_20' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mspace width='thinmathspace' /></mrow><annotation encoding='application/x-tex'>\,</annotation></semantics></math></td><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_fb0302666679a9db8a44d8a779257e05038cbef3_21' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mspace width='thinmathspace' /></mrow><annotation encoding='application/x-tex'>\,</annotation></semantics></math> <a class='existingWikiWord' href='/nlab/show/diff/space+attachment'>space attachment</a> <math class='maruku-mathml' display='inline' id='mathml_fb0302666679a9db8a44d8a779257e05038cbef3_22' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mspace width='thinmathspace' /></mrow><annotation encoding='application/x-tex'>\,</annotation></semantics></math></td></tr> <tr><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_fb0302666679a9db8a44d8a779257e05038cbef3_23' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mspace width='thinmathspace' /></mrow><annotation encoding='application/x-tex'>\,</annotation></semantics></math> <a class='existingWikiWord' href='/nlab/show/diff/cocylinder'>mapping cocylinder</a>, <a class='existingWikiWord' href='/nlab/show/diff/mapping+cocone'>mapping cocone</a> <math class='maruku-mathml' display='inline' id='mathml_fb0302666679a9db8a44d8a779257e05038cbef3_24' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mspace width='thinmathspace' /></mrow><annotation encoding='application/x-tex'>\,</annotation></semantics></math></td><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_fb0302666679a9db8a44d8a779257e05038cbef3_25' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mspace width='thinmathspace' /></mrow><annotation encoding='application/x-tex'>\,</annotation></semantics></math> <a class='existingWikiWord' href='/nlab/show/diff/mapping+cylinder'>mapping cylinder</a>, <a class='existingWikiWord' href='/nlab/show/diff/mapping+cone'>mapping cone</a>, <a class='existingWikiWord' href='/nlab/show/diff/mapping+telescope'>mapping telescope</a> <math class='maruku-mathml' display='inline' id='mathml_fb0302666679a9db8a44d8a779257e05038cbef3_26' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mspace width='thinmathspace' /></mrow><annotation encoding='application/x-tex'>\,</annotation></semantics></math></td></tr> <tr><td style='text-align: left;' /><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_fb0302666679a9db8a44d8a779257e05038cbef3_27' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mspace width='thinmathspace' /></mrow><annotation encoding='application/x-tex'>\,</annotation></semantics></math> <a class='existingWikiWord' href='/nlab/show/diff/cell+complex'>cell complex</a>, <a class='existingWikiWord' href='/nlab/show/diff/CW+complex'>CW-complex</a> <math class='maruku-mathml' display='inline' id='mathml_fb0302666679a9db8a44d8a779257e05038cbef3_28' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mspace width='thinmathspace' /></mrow><annotation encoding='application/x-tex'>\,</annotation></semantics></math></td></tr> </tbody></table> <p><math class='maruku-mathml' display='inline' id='mathml_fb0302666679a9db8a44d8a779257e05038cbef3_29' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mspace width='thinmathspace' /></mrow><annotation encoding='application/x-tex'>\,</annotation></semantics></math></p> <div class='num_defn' id='InitialAndFinalTopologies'> <h6 id='definition_2'>Definition</h6> <p><strong>(<a class='existingWikiWord' href='/nlab/show/diff/weak+topology'>weak topology</a> and <a class='existingWikiWord' href='/nlab/show/diff/weak+topology'>strong topology</a>)</strong></p> <p>Let <math class='maruku-mathml' display='inline' id='mathml_fb0302666679a9db8a44d8a779257e05038cbef3_30' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>{</mo><msub><mi>X</mi> <mi>i</mi></msub><mo>=</mo><mo stretchy='false'>(</mo><msub><mi>S</mi> <mi>i</mi></msub><mo>,</mo><msub><mi>τ</mi> <mi>i</mi></msub><mo stretchy='false'>)</mo><mo>∈</mo><mi>Top</mi><msub><mo stretchy='false'>}</mo> <mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></msub></mrow><annotation encoding='application/x-tex'>\{X_i = (S_i,\tau_i) \in Top\}_{i \in I}</annotation></semantics></math> be a <a class='existingWikiWord' href='/nlab/show/diff/class'>class</a> of <a class='existingWikiWord' href='/nlab/show/diff/topological+space'>topological spaces</a>, and let <math class='maruku-mathml' display='inline' id='mathml_fb0302666679a9db8a44d8a779257e05038cbef3_31' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>S</mi><mo>∈</mo><mi>Set</mi></mrow><annotation encoding='application/x-tex'>S \in Set</annotation></semantics></math> be a bare <a class='existingWikiWord' href='/nlab/show/diff/set'>set</a>. Then</p> <ul> <li> <p>For <math class='maruku-mathml' display='inline' id='mathml_fb0302666679a9db8a44d8a779257e05038cbef3_32' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>{</mo><mi>S</mi><mover><mo>→</mo><mrow><msub><mi>f</mi> <mi>i</mi></msub></mrow></mover><msub><mi>S</mi> <mi>i</mi></msub><msub><mo stretchy='false'>}</mo> <mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></msub></mrow><annotation encoding='application/x-tex'>\{S \stackrel{f_i}{\to} S_i \}_{i \in I}</annotation></semantics></math> a set of <a class='existingWikiWord' href='/nlab/show/diff/function'>functions</a> out of <math class='maruku-mathml' display='inline' id='mathml_fb0302666679a9db8a44d8a779257e05038cbef3_33' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>S</mi></mrow><annotation encoding='application/x-tex'>S</annotation></semantics></math>, the <em><a class='existingWikiWord' href='/nlab/show/diff/weak+topology'>initial topology</a></em> <math class='maruku-mathml' display='inline' id='mathml_fb0302666679a9db8a44d8a779257e05038cbef3_34' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>τ</mi> <mi>initial</mi></msub><mo stretchy='false'>(</mo><mo stretchy='false'>{</mo><msub><mi>f</mi> <mi>i</mi></msub><msub><mo stretchy='false'>}</mo> <mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></msub><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>\tau_{initial}(\{f_i\}_{i \in I})</annotation></semantics></math> is the topology on <math class='maruku-mathml' display='inline' id='mathml_fb0302666679a9db8a44d8a779257e05038cbef3_35' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>S</mi></mrow><annotation encoding='application/x-tex'>S</annotation></semantics></math> with the <a class='existingWikiWord' href='/nlab/show/diff/extremum'>minimum</a> collection of <a class='existingWikiWord' href='/nlab/show/diff/open+subspace'>open subsets</a> such that all <math class='maruku-mathml' display='inline' id='mathml_fb0302666679a9db8a44d8a779257e05038cbef3_36' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>f</mi> <mi>i</mi></msub><mo lspace='verythinmathspace'>:</mo><mo stretchy='false'>(</mo><mi>S</mi><mo>,</mo><msub><mi>τ</mi> <mi>initial</mi></msub><mo stretchy='false'>(</mo><mo stretchy='false'>{</mo><msub><mi>f</mi> <mi>i</mi></msub><msub><mo stretchy='false'>}</mo> <mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></msub><mo stretchy='false'>)</mo><mo stretchy='false'>)</mo><mo>→</mo><msub><mi>X</mi> <mi>i</mi></msub></mrow><annotation encoding='application/x-tex'>f_i \colon (S,\tau_{initial}(\{f_i\}_{i \in I}))\to X_i</annotation></semantics></math> are <a class='existingWikiWord' href='/nlab/show/diff/continuous+map'>continuous</a>.</p> </li> <li> <p>For <math class='maruku-mathml' display='inline' id='mathml_fb0302666679a9db8a44d8a779257e05038cbef3_37' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>{</mo><msub><mi>S</mi> <mi>i</mi></msub><mover><mo>→</mo><mrow><msub><mi>f</mi> <mi>i</mi></msub></mrow></mover><mi>S</mi><msub><mo stretchy='false'>}</mo> <mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></msub></mrow><annotation encoding='application/x-tex'>\{S_i \stackrel{f_i}{\to} S\}_{i \in I}</annotation></semantics></math> a set of <a class='existingWikiWord' href='/nlab/show/diff/function'>functions</a> into <math class='maruku-mathml' display='inline' id='mathml_fb0302666679a9db8a44d8a779257e05038cbef3_38' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>S</mi></mrow><annotation encoding='application/x-tex'>S</annotation></semantics></math>, the <em><a class='existingWikiWord' href='/nlab/show/diff/weak+topology'>final topology</a></em> <math class='maruku-mathml' display='inline' id='mathml_fb0302666679a9db8a44d8a779257e05038cbef3_39' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>τ</mi> <mi>final</mi></msub><mo stretchy='false'>(</mo><mo stretchy='false'>{</mo><msub><mi>f</mi> <mi>i</mi></msub><msub><mo stretchy='false'>}</mo> <mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></msub><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>\tau_{final}(\{f_i\}_{i \in I})</annotation></semantics></math> is the topology on <math class='maruku-mathml' display='inline' id='mathml_fb0302666679a9db8a44d8a779257e05038cbef3_40' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>S</mi></mrow><annotation encoding='application/x-tex'>S</annotation></semantics></math> with the <a class='existingWikiWord' href='/nlab/show/diff/extremum'>maximum</a> collection of <a class='existingWikiWord' href='/nlab/show/diff/open+subspace'>open subsets</a> such that all <math class='maruku-mathml' display='inline' id='mathml_fb0302666679a9db8a44d8a779257e05038cbef3_41' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>f</mi> <mi>i</mi></msub><mo lspace='verythinmathspace'>:</mo><msub><mi>X</mi> <mi>i</mi></msub><mo>→</mo><mo stretchy='false'>(</mo><mi>S</mi><mo>,</mo><msub><mi>τ</mi> <mi>final</mi></msub><mo stretchy='false'>(</mo><mo stretchy='false'>{</mo><msub><mi>f</mi> <mi>i</mi></msub><msub><mo stretchy='false'>}</mo> <mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></msub><mo stretchy='false'>)</mo><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>f_i \colon X_i \to (S,\tau_{final}(\{f_i\}_{i \in I}))</annotation></semantics></math> are <a class='existingWikiWord' href='/nlab/show/diff/continuous+map'>continuous</a>.</p> </li> </ul> </div> <div class='num_example' id='TopologicalSubspace'> <h6 id='example'>Example</h6> <p>For <math class='maruku-mathml' display='inline' id='mathml_fb0302666679a9db8a44d8a779257e05038cbef3_42' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> a single topological space, and <math class='maruku-mathml' display='inline' id='mathml_fb0302666679a9db8a44d8a779257e05038cbef3_43' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>ι</mi> <mi>S</mi></msub><mo lspace='verythinmathspace'>:</mo><mi>S</mi><mo>↪</mo><mi>U</mi><mo stretchy='false'>(</mo><mi>X</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>\iota_S \colon S \hookrightarrow U(X)</annotation></semantics></math> a subset of its underlying set, then the initial topology <math class='maruku-mathml' display='inline' id='mathml_fb0302666679a9db8a44d8a779257e05038cbef3_44' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>τ</mi> <mi>intial</mi></msub><mo stretchy='false'>(</mo><msub><mi>ι</mi> <mi>S</mi></msub><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>\tau_{intial}(\iota_S)</annotation></semantics></math>, def. <a class='maruku-ref' href='#InitialAndFinalTopologies'>1</a>, is the <a class='existingWikiWord' href='/nlab/show/diff/subspace+topology'>subspace topology</a>, making</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_fb0302666679a9db8a44d8a779257e05038cbef3_45' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>ι</mi> <mi>S</mi></msub><mspace width='thickmathspace' /><mo lspace='verythinmathspace'>:</mo><mspace width='thickmathspace' /><mo stretchy='false'>(</mo><mi>S</mi><mo>,</mo><msub><mi>τ</mi> <mi>initial</mi></msub><mo stretchy='false'>(</mo><msub><mi>ι</mi> <mi>S</mi></msub><mo stretchy='false'>)</mo><mo stretchy='false'>)</mo><mo>↪</mo><mi>X</mi></mrow><annotation encoding='application/x-tex'> \iota_S \;\colon\; (S, \tau_{initial}(\iota_S)) \hookrightarrow X </annotation></semantics></math></div> <p>a <a class='existingWikiWord' href='/nlab/show/diff/subspace+topology'>topological subspace</a> inclusion.</p> </div> <div class='num_example' id='QuotientTopology'> <h6 id='example_2'>Example</h6> <p>Conversely, for <math class='maruku-mathml' display='inline' id='mathml_fb0302666679a9db8a44d8a779257e05038cbef3_46' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>p</mi> <mi>S</mi></msub><mo lspace='verythinmathspace'>:</mo><mi>U</mi><mo stretchy='false'>(</mo><mi>X</mi><mo stretchy='false'>)</mo><mo>⟶</mo><mi>S</mi></mrow><annotation encoding='application/x-tex'>p_S \colon U(X) \longrightarrow S</annotation></semantics></math> an <a class='existingWikiWord' href='/nlab/show/diff/epimorphism'>epimorphism</a>, then the final topology <math class='maruku-mathml' display='inline' id='mathml_fb0302666679a9db8a44d8a779257e05038cbef3_47' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>τ</mi> <mi>final</mi></msub><mo stretchy='false'>(</mo><msub><mi>p</mi> <mi>S</mi></msub><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>\tau_{final}(p_S)</annotation></semantics></math> on <math class='maruku-mathml' display='inline' id='mathml_fb0302666679a9db8a44d8a779257e05038cbef3_48' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>S</mi></mrow><annotation encoding='application/x-tex'>S</annotation></semantics></math> is the <em><a class='existingWikiWord' href='/nlab/show/diff/quotient+space'>quotient topology</a></em>.</p> </div> <div class='num_prop' id='DescriptionOfLimitsAndColimitsInTop'> <h6 id='proposition'>Proposition</h6> <p>Let <math class='maruku-mathml' display='inline' id='mathml_fb0302666679a9db8a44d8a779257e05038cbef3_49' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>I</mi></mrow><annotation encoding='application/x-tex'>I</annotation></semantics></math> be a <a class='existingWikiWord' href='/nlab/show/diff/small+category'>small category</a> and let <math class='maruku-mathml' display='inline' id='mathml_fb0302666679a9db8a44d8a779257e05038cbef3_50' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>X</mi> <mo>•</mo></msub><mo lspace='verythinmathspace'>:</mo><mi>I</mi><mo>⟶</mo><mi>Top</mi></mrow><annotation encoding='application/x-tex'>X_\bullet \colon I \longrightarrow Top</annotation></semantics></math> be an <math class='maruku-mathml' display='inline' id='mathml_fb0302666679a9db8a44d8a779257e05038cbef3_51' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>I</mi></mrow><annotation encoding='application/x-tex'>I</annotation></semantics></math>-<a class='existingWikiWord' href='/nlab/show/diff/diagram'>diagram</a> in <a class='existingWikiWord' href='/nlab/show/diff/Top'>Top</a> (a <a class='existingWikiWord' href='/nlab/show/diff/functor'>functor</a> from <math class='maruku-mathml' display='inline' id='mathml_fb0302666679a9db8a44d8a779257e05038cbef3_52' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>I</mi></mrow><annotation encoding='application/x-tex'>I</annotation></semantics></math> to <math class='maruku-mathml' display='inline' id='mathml_fb0302666679a9db8a44d8a779257e05038cbef3_53' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Top</mi></mrow><annotation encoding='application/x-tex'>Top</annotation></semantics></math>), with components denoted <math class='maruku-mathml' display='inline' id='mathml_fb0302666679a9db8a44d8a779257e05038cbef3_54' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>X</mi> <mi>i</mi></msub><mo>=</mo><mo stretchy='false'>(</mo><msub><mi>S</mi> <mi>i</mi></msub><mo>,</mo><msub><mi>τ</mi> <mi>i</mi></msub><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>X_i = (S_i, \tau_i)</annotation></semantics></math>, where <math class='maruku-mathml' display='inline' id='mathml_fb0302666679a9db8a44d8a779257e05038cbef3_55' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>S</mi> <mi>i</mi></msub><mo>∈</mo><mi>Set</mi></mrow><annotation encoding='application/x-tex'>S_i \in Set</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_fb0302666679a9db8a44d8a779257e05038cbef3_56' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>τ</mi> <mi>i</mi></msub></mrow><annotation encoding='application/x-tex'>\tau_i</annotation></semantics></math> a topology on <math class='maruku-mathml' display='inline' id='mathml_fb0302666679a9db8a44d8a779257e05038cbef3_57' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>S</mi> <mi>i</mi></msub></mrow><annotation encoding='application/x-tex'>S_i</annotation></semantics></math>. Then:</p> <ol> <li> <p>The <a class='existingWikiWord' href='/nlab/show/diff/limit'>limit</a> of <math class='maruku-mathml' display='inline' id='mathml_fb0302666679a9db8a44d8a779257e05038cbef3_58' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>X</mi> <mo>•</mo></msub></mrow><annotation encoding='application/x-tex'>X_\bullet</annotation></semantics></math> exists and is given by <a class='existingWikiWord' href='/nlab/show/diff/generalized+the'>the</a> topological space whose underlying set is <a class='existingWikiWord' href='/nlab/show/diff/generalized+the'>the</a> limit in <a class='existingWikiWord' href='/nlab/show/diff/Set'>Set</a> of the underlying sets in the diagram, and whose topology is the <a class='existingWikiWord' href='/nlab/show/diff/weak+topology'>initial topology</a>, def. <a class='maruku-ref' href='#InitialAndFinalTopologies'>1</a>, for the functions <math class='maruku-mathml' display='inline' id='mathml_fb0302666679a9db8a44d8a779257e05038cbef3_59' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>p</mi> <mi>i</mi></msub></mrow><annotation encoding='application/x-tex'>p_i</annotation></semantics></math> which are the limiting <a class='existingWikiWord' href='/nlab/show/diff/cone'>cone</a> components:</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_fb0302666679a9db8a44d8a779257e05038cbef3_60' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mrow><mtable><mtr><mtd /> <mtd /> <mtd><msub><munder><mi>lim</mi><mo>⟵</mo></munder> <mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></msub><msub><mi>S</mi> <mi>i</mi></msub></mtd></mtr> <mtr><mtd /> <mtd><msup><mrow /> <mpadded lspace='-100%width' width='0'><mrow><msub><mi>p</mi> <mi>i</mi></msub></mrow></mpadded></msup><mo>↙</mo></mtd> <mtd /> <mtd><msup><mo>↘</mo> <mpadded width='0'><mrow><msub><mi>p</mi> <mi>j</mi></msub></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><msub><mi>S</mi> <mi>i</mi></msub></mtd> <mtd /> <mtd><munder><mo>⟶</mo><mrow /></munder></mtd> <mtd /> <mtd><msub><mi>S</mi> <mi>j</mi></msub></mtd></mtr></mtable></mrow><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> \array{ && \underset{\longleftarrow}{\lim}_{i \in I} S_i \\ & {}^{\mathllap{p_i}}\swarrow && \searrow^{\mathrlap{p_j}} \\ S_i && \underset{}{\longrightarrow} && S_j } \,. </annotation></semantics></math></div> <p>Hence</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_fb0302666679a9db8a44d8a779257e05038cbef3_61' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><munder><mi>lim</mi><mo>⟵</mo></munder> <mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></msub><msub><mi>X</mi> <mi>i</mi></msub><mo>≃</mo><mrow><mo>(</mo><msub><munder><mi>lim</mi><mo>⟵</mo></munder> <mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></msub><msub><mi>S</mi> <mi>i</mi></msub><mo>,</mo><mspace width='thickmathspace' /><msub><mi>τ</mi> <mi>initial</mi></msub><mo stretchy='false'>(</mo><mo stretchy='false'>{</mo><msub><mi>p</mi> <mi>i</mi></msub><msub><mo stretchy='false'>}</mo> <mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></msub><mo stretchy='false'>)</mo><mo>)</mo></mrow></mrow><annotation encoding='application/x-tex'> \underset{\longleftarrow}{\lim}_{i \in I} X_i \simeq \left(\underset{\longleftarrow}{\lim}_{i \in I} S_i,\; \tau_{initial}(\{p_i\}_{i \in I})\right) </annotation></semantics></math></div></li> <li> <p>The <a class='existingWikiWord' href='/nlab/show/diff/colimit'>colimit</a> of <math class='maruku-mathml' display='inline' id='mathml_fb0302666679a9db8a44d8a779257e05038cbef3_62' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>X</mi> <mo>•</mo></msub></mrow><annotation encoding='application/x-tex'>X_\bullet</annotation></semantics></math> exists and is the topological space whose underlying set is the colimit in <a class='existingWikiWord' href='/nlab/show/diff/Set'>Set</a> of the underlying diagram of sets, and whose topology is the <a class='existingWikiWord' href='/nlab/show/diff/weak+topology'>final topology</a>, def. <a class='maruku-ref' href='#InitialAndFinalTopologies'>1</a> for the component maps <math class='maruku-mathml' display='inline' id='mathml_fb0302666679a9db8a44d8a779257e05038cbef3_63' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>ι</mi> <mi>i</mi></msub></mrow><annotation encoding='application/x-tex'>\iota_i</annotation></semantics></math> of the colimiting <a class='existingWikiWord' href='/nlab/show/diff/cocone'>cocone</a></p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_fb0302666679a9db8a44d8a779257e05038cbef3_64' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mrow><mtable><mtr><mtd><msub><mi>S</mi> <mi>i</mi></msub></mtd> <mtd /> <mtd><mo>⟶</mo></mtd> <mtd /> <mtd><msub><mi>S</mi> <mi>j</mi></msub></mtd></mtr> <mtr><mtd /> <mtd><msub><mrow /> <mpadded lspace='-100%width' width='0'><mrow><msub><mi>ι</mi> <mi>i</mi></msub></mrow></mpadded></msub><mo>↘</mo></mtd> <mtd /> <mtd><msub><mo>↙</mo> <mpadded width='0'><mrow><msub><mi>ι</mi> <mi>j</mi></msub></mrow></mpadded></msub></mtd></mtr> <mtr><mtd /> <mtd /> <mtd><msub><munder><mi>lim</mi><mo>⟶</mo></munder> <mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></msub><msub><mi>S</mi> <mi>i</mi></msub></mtd></mtr></mtable></mrow><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> \array{ S_i && \longrightarrow && S_j \\ & {}_{\mathllap{\iota_i}}\searrow && \swarrow_{\mathrlap{\iota_j}} \\ && \underset{\longrightarrow}{\lim}_{i \in I} S_i } \,. </annotation></semantics></math></div> <p>Hence</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_fb0302666679a9db8a44d8a779257e05038cbef3_65' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><munder><mi>lim</mi><mo>⟶</mo></munder> <mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></msub><msub><mi>X</mi> <mi>i</mi></msub><mo>≃</mo><mrow><mo>(</mo><msub><munder><mi>lim</mi><mo>⟶</mo></munder> <mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></msub><msub><mi>S</mi> <mi>i</mi></msub><mo>,</mo><mspace width='thickmathspace' /><msub><mi>τ</mi> <mi>final</mi></msub><mo stretchy='false'>(</mo><mo stretchy='false'>{</mo><msub><mi>ι</mi> <mi>i</mi></msub><msub><mo stretchy='false'>}</mo> <mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></msub><mo stretchy='false'>)</mo><mo>)</mo></mrow></mrow><annotation encoding='application/x-tex'> \underset{\longrightarrow}{\lim}_{i \in I} X_i \simeq \left(\underset{\longrightarrow}{\lim}_{i \in I} S_i,\; \tau_{final}(\{\iota_i\}_{i \in I})\right) </annotation></semantics></math></div></li> </ol> </div> <p>(e.g. <a href='#Bourbaki71'>Bourbaki 71, section I.4</a>)</p> <div class='proof'> <h6 id='proof'>Proof</h6> <p>The required <a class='existingWikiWord' href='/nlab/show/diff/universal+construction'>universal property</a> of <math class='maruku-mathml' display='inline' id='mathml_fb0302666679a9db8a44d8a779257e05038cbef3_66' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mrow><mo>(</mo><msub><munder><mi>lim</mi><mo>⟵</mo></munder> <mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></msub><msub><mi>S</mi> <mi>i</mi></msub><mo>,</mo><mspace width='thickmathspace' /><msub><mi>τ</mi> <mi>initial</mi></msub><mo stretchy='false'>(</mo><mo stretchy='false'>{</mo><msub><mi>p</mi> <mi>i</mi></msub><msub><mo stretchy='false'>}</mo> <mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></msub><mo stretchy='false'>)</mo><mo>)</mo></mrow></mrow><annotation encoding='application/x-tex'>\left(\underset{\longleftarrow}{\lim}_{i \in I} S_i,\; \tau_{initial}(\{p_i\}_{i \in I})\right)</annotation></semantics></math> is immediate: for</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_fb0302666679a9db8a44d8a779257e05038cbef3_67' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mrow><mtable><mtr><mtd /> <mtd /> <mtd><mo stretchy='false'>(</mo><mi>S</mi><mo>,</mo><mi>τ</mi><mo stretchy='false'>)</mo></mtd></mtr> <mtr><mtd /> <mtd><msup><mrow /> <mpadded lspace='-100%width' width='0'><mrow><msub><mi>f</mi> <mi>i</mi></msub></mrow></mpadded></msup><mo>↙</mo></mtd> <mtd /> <mtd><msup><mo>↘</mo> <mpadded width='0'><mrow><msub><mi>f</mi> <mi>j</mi></msub></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><msub><mi>X</mi> <mi>i</mi></msub></mtd> <mtd /> <mtd><munder><mo>⟶</mo><mrow /></munder></mtd> <mtd /> <mtd><msub><mi>X</mi> <mi>j</mi></msub></mtd></mtr></mtable></mrow></mrow><annotation encoding='application/x-tex'> \array{ && (S,\tau) \\ & {}^{\mathllap{f_i}}\swarrow && \searrow^{\mathrlap{f_j}} \\ X_i && \underset{}{\longrightarrow} && X_j } </annotation></semantics></math></div> <p>any <a class='existingWikiWord' href='/nlab/show/diff/cone'>cone</a> over the diagram, then by construction there is a unique function of underlying sets <math class='maruku-mathml' display='inline' id='mathml_fb0302666679a9db8a44d8a779257e05038cbef3_68' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>S</mi><mo>⟶</mo><msub><munder><mi>lim</mi><mo>⟵</mo></munder> <mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></msub><msub><mi>S</mi> <mi>i</mi></msub></mrow><annotation encoding='application/x-tex'>S \longrightarrow \underset{\longleftarrow}{\lim}_{i \in I} S_i</annotation></semantics></math> making the required diagrams commute, and so all that is required is that this unique function is always <a class='existingWikiWord' href='/nlab/show/diff/continuous+map'>continuous</a>. But this is precisely what the initial topology ensures.</p> <p>The case of the colimit is <a class='existingWikiWord' href='/nlab/show/diff/duality'>formally dual</a>.</p> </div> <div class='num_example' id='PointTopologicalSpaceAsEmptyLimit'> <h6 id='example_3'>Example</h6> <p>The limit over the empty diagram in <math class='maruku-mathml' display='inline' id='mathml_fb0302666679a9db8a44d8a779257e05038cbef3_69' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Top</mi></mrow><annotation encoding='application/x-tex'>Top</annotation></semantics></math> is the <a class='existingWikiWord' href='/nlab/show/diff/point'>point</a> <math class='maruku-mathml' display='inline' id='mathml_fb0302666679a9db8a44d8a779257e05038cbef3_70' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo>*</mo></mrow><annotation encoding='application/x-tex'>\ast</annotation></semantics></math> with its unique topology.</p> </div> <div class='num_example' id='DisjointUnionOfTopologicalSpacesIsCoproduct'> <h6 id='example_4'>Example</h6> <p>For <math class='maruku-mathml' display='inline' id='mathml_fb0302666679a9db8a44d8a779257e05038cbef3_71' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>{</mo><msub><mi>X</mi> <mi>i</mi></msub><msub><mo stretchy='false'>}</mo> <mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></msub></mrow><annotation encoding='application/x-tex'>\{X_i\}_{i \in I}</annotation></semantics></math> a set of topological spaces, their <a class='existingWikiWord' href='/nlab/show/diff/coproduct'>coproduct</a> <math class='maruku-mathml' display='inline' id='mathml_fb0302666679a9db8a44d8a779257e05038cbef3_72' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><munder><mo>⊔</mo><mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></munder><msub><mi>X</mi> <mi>i</mi></msub><mo>∈</mo><mi>Top</mi></mrow><annotation encoding='application/x-tex'>\underset{i \in I}{\sqcup} X_i \in Top</annotation></semantics></math> is their <em><a class='existingWikiWord' href='/nlab/show/diff/disjoint+union'>disjoint union</a></em>.</p> </div> <p>In particular:</p> <div class='num_example' id='DiscreteTopologicalSpaceAsCoproduct'> <h6 id='example_5'>Example</h6> <p>For <math class='maruku-mathml' display='inline' id='mathml_fb0302666679a9db8a44d8a779257e05038cbef3_73' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>S</mi><mo>∈</mo><mi>Set</mi></mrow><annotation encoding='application/x-tex'>S \in Set</annotation></semantics></math>, the <math class='maruku-mathml' display='inline' id='mathml_fb0302666679a9db8a44d8a779257e05038cbef3_74' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>S</mi></mrow><annotation encoding='application/x-tex'>S</annotation></semantics></math>-indexed <a class='existingWikiWord' href='/nlab/show/diff/coproduct'>coproduct</a> of the point, <math class='maruku-mathml' display='inline' id='mathml_fb0302666679a9db8a44d8a779257e05038cbef3_75' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><munder><mo lspace='thinmathspace' rspace='thinmathspace'>∐</mo><mrow><mi>s</mi><mo>∈</mo><mi>S</mi></mrow></munder><mo>*</mo></mrow><annotation encoding='application/x-tex'>\underset{s \in S}{\coprod}\ast </annotation></semantics></math>, is the set <math class='maruku-mathml' display='inline' id='mathml_fb0302666679a9db8a44d8a779257e05038cbef3_76' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>S</mi></mrow><annotation encoding='application/x-tex'>S</annotation></semantics></math> itself equipped with the <a class='existingWikiWord' href='/nlab/show/diff/weak+topology'>final topology</a>, hence is the <a class='existingWikiWord' href='/nlab/show/diff/discrete+and+indiscrete+topology'>discrete topological space</a> on <math class='maruku-mathml' display='inline' id='mathml_fb0302666679a9db8a44d8a779257e05038cbef3_77' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>S</mi></mrow><annotation encoding='application/x-tex'>S</annotation></semantics></math>.</p> </div> <div class='num_example' id='ProductTopologicalSpace'> <h6 id='example_6'>Example</h6> <p>For <math class='maruku-mathml' display='inline' id='mathml_fb0302666679a9db8a44d8a779257e05038cbef3_78' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>{</mo><msub><mi>X</mi> <mi>i</mi></msub><msub><mo stretchy='false'>}</mo> <mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></msub></mrow><annotation encoding='application/x-tex'>\{X_i\}_{i \in I}</annotation></semantics></math> a set of topological spaces, their <a class='existingWikiWord' href='/nlab/show/diff/cartesian+product'>product</a> <math class='maruku-mathml' display='inline' id='mathml_fb0302666679a9db8a44d8a779257e05038cbef3_79' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><munder><mo lspace='thinmathspace' rspace='thinmathspace'>∏</mo><mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></munder><msub><mi>X</mi> <mi>i</mi></msub><mo>∈</mo><mi>Top</mi></mrow><annotation encoding='application/x-tex'>\underset{i \in I}{\prod} X_i \in Top</annotation></semantics></math> is the <a class='existingWikiWord' href='/nlab/show/diff/cartesian+product'>Cartesian product</a> of the underlying sets equipped with the <em><a class='existingWikiWord' href='/nlab/show/diff/product+topological+space'>product topology</a></em>, also called the <em><a class='existingWikiWord' href='/nlab/show/diff/product+topological+space'>Tychonoff product</a></em>.</p> <p>In the case that <math class='maruku-mathml' display='inline' id='mathml_fb0302666679a9db8a44d8a779257e05038cbef3_80' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>S</mi></mrow><annotation encoding='application/x-tex'>S</annotation></semantics></math> is a <a class='existingWikiWord' href='/nlab/show/diff/finite+set'>finite set</a>, such as for binary product spaces <math class='maruku-mathml' display='inline' id='mathml_fb0302666679a9db8a44d8a779257e05038cbef3_81' 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 \times Y</annotation></semantics></math>, then a <a class='existingWikiWord' href='/nlab/show/diff/topological+base'>sub-basis</a> for the product topology is given by the <a class='existingWikiWord' href='/nlab/show/diff/cartesian+product'>Cartesian products</a> of the open subsets of (a basis for) each factor space.</p> </div> <div class='num_example' id='EqualizerInTop'> <h6 id='example_7'>Example</h6> <p>The <a class='existingWikiWord' href='/nlab/show/diff/equalizer'>equalizer</a> of two <a class='existingWikiWord' href='/nlab/show/diff/continuous+map'>continuous functions</a> <math class='maruku-mathml' display='inline' id='mathml_fb0302666679a9db8a44d8a779257e05038cbef3_82' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>f</mi><mo>,</mo><mi>g</mi><mo lspace='verythinmathspace'>:</mo><mi>X</mi><mover><mo>⟶</mo><mo>⟶</mo></mover><mi>Y</mi></mrow><annotation encoding='application/x-tex'>f, g \colon X \stackrel{\longrightarrow}{\longrightarrow} Y</annotation></semantics></math> in <math class='maruku-mathml' display='inline' id='mathml_fb0302666679a9db8a44d8a779257e05038cbef3_83' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Top</mi></mrow><annotation encoding='application/x-tex'>Top</annotation></semantics></math> is the equalizer of the underlying functions of sets</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_fb0302666679a9db8a44d8a779257e05038cbef3_84' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>eq</mi><mo stretchy='false'>(</mo><mi>f</mi><mo>,</mo><mi>g</mi><mo stretchy='false'>)</mo><mo>↪</mo><msub><mi>S</mi> <mi>X</mi></msub><mover><munder><mo>⟶</mo><mi>g</mi></munder><mover><mo>⟶</mo><mi>f</mi></mover></mover><msub><mi>S</mi> <mi>Y</mi></msub></mrow><annotation encoding='application/x-tex'> eq(f,g) \hookrightarrow S_X \stackrel{\overset{f}{\longrightarrow}}{\underset{g}{\longrightarrow}} S_Y </annotation></semantics></math></div> <p>(hence the largets subset of <math class='maruku-mathml' display='inline' id='mathml_fb0302666679a9db8a44d8a779257e05038cbef3_85' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>S</mi> <mi>X</mi></msub></mrow><annotation encoding='application/x-tex'>S_X</annotation></semantics></math> on which both functions coincide) and equipped with the <a class='existingWikiWord' href='/nlab/show/diff/subspace+topology'>subspace topology</a>, example <a class='maruku-ref' href='#TopologicalSubspace'>1</a>.</p> </div> <div class='num_example' id='CoequalizerInTop'> <h6 id='example_8'>Example</h6> <p>The <a class='existingWikiWord' href='/nlab/show/diff/coequalizer'>coequalizer</a> of two <a class='existingWikiWord' href='/nlab/show/diff/continuous+map'>continuous functions</a> <math class='maruku-mathml' display='inline' id='mathml_fb0302666679a9db8a44d8a779257e05038cbef3_86' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>f</mi><mo>,</mo><mi>g</mi><mo lspace='verythinmathspace'>:</mo><mi>X</mi><mover><mo>⟶</mo><mo>⟶</mo></mover><mi>Y</mi></mrow><annotation encoding='application/x-tex'>f, g \colon X \stackrel{\longrightarrow}{\longrightarrow} Y</annotation></semantics></math> in <math class='maruku-mathml' display='inline' id='mathml_fb0302666679a9db8a44d8a779257e05038cbef3_87' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Top</mi></mrow><annotation encoding='application/x-tex'>Top</annotation></semantics></math> is the coequalizer of the underlying functions of sets</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_fb0302666679a9db8a44d8a779257e05038cbef3_88' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>S</mi> <mi>X</mi></msub><mover><munder><mo>⟶</mo><mi>g</mi></munder><mover><mo>⟶</mo><mi>f</mi></mover></mover><msub><mi>S</mi> <mi>Y</mi></msub><mo>⟶</mo><mi>coeq</mi><mo stretchy='false'>(</mo><mi>f</mi><mo>,</mo><mi>g</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'> S_X \stackrel{\overset{f}{\longrightarrow}}{\underset{g}{\longrightarrow}} S_Y \longrightarrow coeq(f,g) </annotation></semantics></math></div> <p>(hence the <a class='existingWikiWord' href='/nlab/show/diff/quotient+set'>quotient set</a> by the <a class='existingWikiWord' href='/nlab/show/diff/equivalence+relation'>equivalence relation</a> generated by <math class='maruku-mathml' display='inline' id='mathml_fb0302666679a9db8a44d8a779257e05038cbef3_89' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>f</mi><mo stretchy='false'>(</mo><mi>x</mi><mo stretchy='false'>)</mo><mo>∼</mo><mi>g</mi><mo stretchy='false'>(</mo><mi>x</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>f(x) \sim g(x)</annotation></semantics></math> for all <math class='maruku-mathml' display='inline' id='mathml_fb0302666679a9db8a44d8a779257e05038cbef3_90' 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 \in X</annotation></semantics></math>) and equipped with the <a class='existingWikiWord' href='/nlab/show/diff/quotient+space'>quotient topology</a>, example <a class='maruku-ref' href='#QuotientTopology'>2</a>.</p> </div> <div class='num_example' id='PushoutInTop'> <h6 id='example_9'>Example</h6> <p>For</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_fb0302666679a9db8a44d8a779257e05038cbef3_91' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mrow><mtable><mtr><mtd><mi>A</mi></mtd> <mtd><mover><mo>⟶</mo><mi>g</mi></mover></mtd> <mtd><mi>Y</mi></mtd></mtr> <mtr><mtd><msup><mrow /> <mpadded lspace='-100%width' width='0'><mi>f</mi></mpadded></msup><mo stretchy='false'>↓</mo></mtd></mtr> <mtr><mtd><mi>X</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding='application/x-tex'> \array{ A &\overset{g}{\longrightarrow}& Y \\ {}^{\mathllap{f}}\downarrow \\ X } </annotation></semantics></math></div> <p>two <a class='existingWikiWord' href='/nlab/show/diff/continuous+map'>continuous functions</a> out of the same <a class='existingWikiWord' href='/nlab/show/diff/domain'>domain</a>, then the <a class='existingWikiWord' href='/nlab/show/diff/colimit'>colimit</a> under this diagram is also called the <em><a class='existingWikiWord' href='/nlab/show/diff/pushout'>pushout</a></em>, denoted</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_fb0302666679a9db8a44d8a779257e05038cbef3_92' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mrow><mtable><mtr><mtd><mi>A</mi></mtd> <mtd><mover><mo>⟶</mo><mi>g</mi></mover></mtd> <mtd><mi>Y</mi></mtd></mtr> <mtr><mtd><msup><mrow /> <mpadded lspace='-100%width' width='0'><mi>f</mi></mpadded></msup><mo stretchy='false'>↓</mo></mtd> <mtd /> <mtd><msup><mo stretchy='false'>↓</mo> <mpadded width='0'><mrow><msub><mi>g</mi> <mo>*</mo></msub><mi>f</mi></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><mi>X</mi></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>X</mi><msub><mo>⊔</mo> <mi>A</mi></msub><mi>Y</mi><mspace width='thinmathspace' /><mo>.</mo></mtd></mtr></mtable></mrow><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> \array{ A &\overset{g}{\longrightarrow}& Y \\ {}^{\mathllap{f}}\downarrow && \downarrow^{\mathrlap{g_\ast f}} \\ X &\longrightarrow& X \sqcup_A Y \,. } \,. </annotation></semantics></math></div> <p>(Here <math class='maruku-mathml' display='inline' id='mathml_fb0302666679a9db8a44d8a779257e05038cbef3_93' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>g</mi> <mo>*</mo></msub><mi>f</mi></mrow><annotation encoding='application/x-tex'>g_\ast f</annotation></semantics></math> is also called the pushout of <math class='maruku-mathml' display='inline' id='mathml_fb0302666679a9db8a44d8a779257e05038cbef3_94' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>f</mi></mrow><annotation encoding='application/x-tex'>f</annotation></semantics></math>, or the <em><a class='existingWikiWord' href='/nlab/show/diff/base+change'>cobase change</a></em> of <math class='maruku-mathml' display='inline' id='mathml_fb0302666679a9db8a44d8a779257e05038cbef3_95' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>f</mi></mrow><annotation encoding='application/x-tex'>f</annotation></semantics></math> along <math class='maruku-mathml' display='inline' id='mathml_fb0302666679a9db8a44d8a779257e05038cbef3_96' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>g</mi></mrow><annotation encoding='application/x-tex'>g</annotation></semantics></math>.) If <math class='maruku-mathml' display='inline' id='mathml_fb0302666679a9db8a44d8a779257e05038cbef3_97' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>g</mi></mrow><annotation encoding='application/x-tex'>g</annotation></semantics></math> is an inclusion, one also write <math class='maruku-mathml' display='inline' id='mathml_fb0302666679a9db8a44d8a779257e05038cbef3_98' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi><msub><mo>∪</mo> <mi>f</mi></msub><mi>Y</mi></mrow><annotation encoding='application/x-tex'>X \cup_f Y</annotation></semantics></math> and calls this the <em><a class='existingWikiWord' href='/nlab/show/diff/space+attachment'>attaching space</a></em>.</p> <div style='float: left; margin: 0 10px 10px 0;'><img src='http://ncatlab.org/nlab/files/AttachingSpace.jpg' width='450' /></div> <p>By example <a class='maruku-ref' href='#CoequalizerInTop'>8</a> the <a class='existingWikiWord' href='/nlab/show/diff/pushout'>pushout</a>/<a class='existingWikiWord' href='/nlab/show/diff/space+attachment'>attaching space</a> is the <a class='existingWikiWord' href='/nlab/show/diff/quotient+space'>quotient topological space</a></p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_fb0302666679a9db8a44d8a779257e05038cbef3_99' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi><msub><mo>⊔</mo> <mi>A</mi></msub><mi>Y</mi><mo>≃</mo><mo stretchy='false'>(</mo><mi>X</mi><mo>⊔</mo><mi>Y</mi><mo stretchy='false'>)</mo><mo stretchy='false'>/</mo><mo>∼</mo></mrow><annotation encoding='application/x-tex'> X \sqcup_A Y \simeq (X\sqcup Y)/\sim </annotation></semantics></math></div> <p>of the <a class='existingWikiWord' href='/nlab/show/diff/disjoint+union'>disjoint union</a> of <math class='maruku-mathml' display='inline' id='mathml_fb0302666679a9db8a44d8a779257e05038cbef3_100' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_fb0302666679a9db8a44d8a779257e05038cbef3_101' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Y</mi></mrow><annotation encoding='application/x-tex'>Y</annotation></semantics></math> subject to the <a class='existingWikiWord' href='/nlab/show/diff/equivalence+relation'>equivalence relation</a> which identifies a point in <math class='maruku-mathml' display='inline' id='mathml_fb0302666679a9db8a44d8a779257e05038cbef3_102' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> with a point in <math class='maruku-mathml' display='inline' id='mathml_fb0302666679a9db8a44d8a779257e05038cbef3_103' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Y</mi></mrow><annotation encoding='application/x-tex'>Y</annotation></semantics></math> if they have the same pre-image in <math class='maruku-mathml' display='inline' id='mathml_fb0302666679a9db8a44d8a779257e05038cbef3_104' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>A</mi></mrow><annotation encoding='application/x-tex'>A</annotation></semantics></math>.</p> <p>(graphics from <a href='#AguilarGitlerPrieto02'>Aguilar-Gitler-Prieto 02</a>)</p> </div> <div class='num_example' id='TopologicalnSphereIsPushoutOfBoundaryOfnBallInclusionAlongItself'> <h6 id='example_10'>Example</h6> <p>As an important special case of example <a class='maruku-ref' href='#PushoutInTop'>9</a>, let</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_fb0302666679a9db8a44d8a779257e05038cbef3_105' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>i</mi> <mi>n</mi></msub><mo lspace='verythinmathspace'>:</mo><msup><mi>S</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup><mo>⟶</mo><msup><mi>D</mi> <mi>n</mi></msup></mrow><annotation encoding='application/x-tex'> i_n \colon S^{n-1}\longrightarrow D^n </annotation></semantics></math></div> <p>be the canonical inclusion of the standard <a class='existingWikiWord' href='/nlab/show/diff/sphere'>(n-1)-sphere</a> as the <a class='existingWikiWord' href='/nlab/show/diff/boundary'>boundary</a> of the standard <a class='existingWikiWord' href='/nlab/show/diff/ball'>n-disk</a> (both regarded as <a class='existingWikiWord' href='/nlab/show/diff/topological+space'>topological spaces</a> with their <a class='existingWikiWord' href='/nlab/show/diff/subspace+topology'>subspace topology</a> as subspaces of the <a class='existingWikiWord' href='/nlab/show/diff/cartesian+space'>Cartesian space</a> <math class='maruku-mathml' display='inline' id='mathml_fb0302666679a9db8a44d8a779257e05038cbef3_106' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>ℝ</mi> <mi>n</mi></msup></mrow><annotation encoding='application/x-tex'>\mathbb{R}^n</annotation></semantics></math>).</p> <div style='float: left; margin: 0 10px 10px 0;'> <img src='http://ncatlab.org/nlab/files/GluingHemispheres.jpg' width='400' /></div> <p>Then the colimit in <a class='existingWikiWord' href='/nlab/show/diff/Top'>Top</a> under the diagram, i.e. the <a class='existingWikiWord' href='/nlab/show/diff/pushout'>pushout</a> of <math class='maruku-mathml' display='inline' id='mathml_fb0302666679a9db8a44d8a779257e05038cbef3_107' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>i</mi> <mi>n</mi></msub></mrow><annotation encoding='application/x-tex'>i_n</annotation></semantics></math> along itself,</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_fb0302666679a9db8a44d8a779257e05038cbef3_108' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mrow><mo>{</mo><msup><mi>D</mi> <mi>n</mi></msup><mover><mo>⟵</mo><mrow><msub><mi>i</mi> <mi>n</mi></msub></mrow></mover><msup><mi>S</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup><mover><mo>⟶</mo><mrow><msub><mi>i</mi> <mi>n</mi></msub></mrow></mover><msup><mi>D</mi> <mi>n</mi></msup><mo>}</mo></mrow><mspace width='thinmathspace' /><mo>,</mo></mrow><annotation encoding='application/x-tex'> \left\{ D^n \overset{i_n}{\longleftarrow} S^{n-1} \overset{i_n}{\longrightarrow} D^n \right\} \,, </annotation></semantics></math></div> <p>is the <a class='existingWikiWord' href='/nlab/show/diff/sphere'>n-sphere</a> <math class='maruku-mathml' display='inline' id='mathml_fb0302666679a9db8a44d8a779257e05038cbef3_109' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>S</mi> <mi>n</mi></msup></mrow><annotation encoding='application/x-tex'>S^n</annotation></semantics></math>:</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_fb0302666679a9db8a44d8a779257e05038cbef3_110' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mrow><mtable><mtr><mtd><msup><mi>S</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup></mtd> <mtd><mover><mo>⟶</mo><mrow><msub><mi>i</mi> <mi>n</mi></msub></mrow></mover></mtd> <mtd><msup><mi>D</mi> <mi>n</mi></msup></mtd></mtr> <mtr><mtd><msup><mrow /> <mpadded lspace='-100%width' width='0'><mrow><msub><mi>i</mi> <mi>n</mi></msub></mrow></mpadded></msup><mo stretchy='false'>↓</mo></mtd> <mtd><mo stretchy='false'>(</mo><mi>po</mi><mo stretchy='false'>)</mo></mtd> <mtd><mo stretchy='false'>↓</mo></mtd></mtr> <mtr><mtd><msup><mi>D</mi> <mi>n</mi></msup></mtd> <mtd><mo>⟶</mo></mtd> <mtd><msup><mi>S</mi> <mi>n</mi></msup></mtd></mtr></mtable></mrow><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> \array{ S^{n-1} &\overset{i_n}{\longrightarrow}& D^n \\ {}^{\mathllap{i_n}}\downarrow &(po)& \downarrow \\ D^n &\longrightarrow& S^n } \,. </annotation></semantics></math></div> <p>(graphics from Ueno-Shiga-Morita 95)</p> </div> <div class='num_example' id='ClosedSubspacesGluing'> <h6 id='example_11'>Example</h6> <p><strong>(<a class='existingWikiWord' href='/nlab/show/diff/union'>union</a> of two <a class='existingWikiWord' href='/nlab/show/diff/open+subspace'>open</a> or two <a class='existingWikiWord' href='/nlab/show/diff/closed+subspace'>closed</a> <a class='existingWikiWord' href='/nlab/show/diff/subspace'>subspaces</a> is <a class='existingWikiWord' href='/nlab/show/diff/pushout'>pushout</a>)</strong></p> <p>Let <math class='maruku-mathml' display='inline' id='mathml_fb0302666679a9db8a44d8a779257e05038cbef3_111' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> be a <a class='existingWikiWord' href='/nlab/show/diff/topological+space'>topological space</a> and let <math class='maruku-mathml' display='inline' id='mathml_fb0302666679a9db8a44d8a779257e05038cbef3_112' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>A</mi><mo>,</mo><mi>B</mi><mo>⊂</mo><mi>X</mi></mrow><annotation encoding='application/x-tex'>A,B \subset X</annotation></semantics></math> be <a class='existingWikiWord' href='/nlab/show/diff/subspace'>subspaces</a> such that</p> <ol> <li> <p><math class='maruku-mathml' display='inline' id='mathml_fb0302666679a9db8a44d8a779257e05038cbef3_113' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>A</mi><mo>,</mo><mi>B</mi><mo>⊂</mo><mi>X</mi></mrow><annotation encoding='application/x-tex'>A,B \subset X</annotation></semantics></math> are both <a class='existingWikiWord' href='/nlab/show/diff/open+subspace'>open subsets</a> or are both <a class='existingWikiWord' href='/nlab/show/diff/closed+subspace'>closed subsets</a>;</p> </li> <li> <p>they constitute a <a class='existingWikiWord' href='/nlab/show/diff/cover'>cover</a>: <math class='maruku-mathml' display='inline' id='mathml_fb0302666679a9db8a44d8a779257e05038cbef3_114' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi><mo>=</mo><mi>A</mi><mo>∪</mo><mi>B</mi></mrow><annotation encoding='application/x-tex'>X = A \cup B</annotation></semantics></math></p> </li> </ol> <p>Write <math class='maruku-mathml' display='inline' id='mathml_fb0302666679a9db8a44d8a779257e05038cbef3_115' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>i</mi> <mi>A</mi></msub><mo lspace='verythinmathspace'>:</mo><mi>A</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding='application/x-tex'>i_A \colon A \to X</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_fb0302666679a9db8a44d8a779257e05038cbef3_116' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>i</mi> <mi>B</mi></msub><mo lspace='verythinmathspace'>:</mo><mi>B</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding='application/x-tex'>i_B \colon B \to X</annotation></semantics></math> for the corresponding inclusion <a class='existingWikiWord' href='/nlab/show/diff/continuous+map'>continuous functions</a>.</p> <p>Then the <a class='existingWikiWord' href='/nlab/show/diff/commutative+square'>commuting square</a></p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_fb0302666679a9db8a44d8a779257e05038cbef3_117' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mrow><mtable><mtr><mtd><mi>A</mi><mo>∩</mo><mi>B</mi></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>A</mi></mtd></mtr> <mtr><mtd><mo stretchy='false'>↓</mo></mtd> <mtd /> <mtd><msup><mo stretchy='false'>↓</mo> <mpadded width='0'><mrow><msub><mi>i</mi> <mi>A</mi></msub></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><mi>B</mi></mtd> <mtd><munder><mo>⟶</mo><mrow><msub><mi>i</mi> <mi>B</mi></msub></mrow></munder></mtd> <mtd><mi>X</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding='application/x-tex'> \array{ A \cap B &\longrightarrow& A \\ \downarrow && \downarrow^{\mathrlap{i_A}} \\ B &\underset{i_B}{\longrightarrow}& X } </annotation></semantics></math></div> <p>is a <a class='existingWikiWord' href='/nlab/show/diff/pushout'>pushout</a> square in <math class='maruku-mathml' display='inline' id='mathml_fb0302666679a9db8a44d8a779257e05038cbef3_118' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Top</mi></mrow><annotation encoding='application/x-tex'>Top</annotation></semantics></math> (example <a class='maruku-ref' href='#PushoutInTop'>9</a>).</p> <p>By the <a class='existingWikiWord' href='/nlab/show/diff/universal+construction'>universal property</a> of the <a class='existingWikiWord' href='/nlab/show/diff/pushout'>pushout</a> this means in particular that for <math class='maruku-mathml' display='inline' id='mathml_fb0302666679a9db8a44d8a779257e05038cbef3_119' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Y</mi></mrow><annotation encoding='application/x-tex'>Y</annotation></semantics></math> any <a class='existingWikiWord' href='/nlab/show/diff/topological+space'>topological space</a> then a function of underlying sets</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_fb0302666679a9db8a44d8a779257e05038cbef3_120' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>f</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 \;\colon\; X \longrightarrow Y </annotation></semantics></math></div> <p>is a <a class='existingWikiWord' href='/nlab/show/diff/continuous+map'>continuous function</a> as soon as its two restrictions</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_fb0302666679a9db8a44d8a779257e05038cbef3_121' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>f</mi><msub><mo stretchy='false'>|</mo> <mi>A</mi></msub><mspace width='thickmathspace' /><mo lspace='verythinmathspace'>:</mo><mspace width='thickmathspace' /><mi>A</mi><mo>⟶</mo><mi>Y</mi><mphantom><mi>AAAA</mi></mphantom><mi>f</mi><msub><mo stretchy='false'>|</mo> <mi>A</mi></msub><mspace width='thickmathspace' /><mo lspace='verythinmathspace'>:</mo><mspace width='thickmathspace' /><mi>B</mi><mo>⟶</mo><mi>Y</mi></mrow><annotation encoding='application/x-tex'> f\vert_A \;\colon\; A \longrightarrow Y \phantom{AAAA} f\vert_A \;\colon\; B \longrightarrow Y </annotation></semantics></math></div> <p>are continuous.</p> </div> <div class='proof'> <h6 id='proof_2'>Proof</h6> <p>Clearly the underlying diagram of underlying <a class='existingWikiWord' href='/nlab/show/diff/set'>sets</a> is a pushout in <a class='existingWikiWord' href='/nlab/show/diff/Set'>Set</a>. Therefore by prop. <a class='maruku-ref' href='#DescriptionOfLimitsAndColimitsInTop'>1</a> we need to show that the <a class='existingWikiWord' href='/nlab/show/diff/topological+space'>topology</a> on <math class='maruku-mathml' display='inline' id='mathml_fb0302666679a9db8a44d8a779257e05038cbef3_122' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> is the <a class='existingWikiWord' href='/nlab/show/diff/weak+topology'>final topology</a> induced by the set of functions <math class='maruku-mathml' display='inline' id='mathml_fb0302666679a9db8a44d8a779257e05038cbef3_123' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>{</mo><msub><mi>i</mi> <mi>A</mi></msub><mo>,</mo><msub><mi>i</mi> <mi>B</mi></msub><mo stretchy='false'>}</mo></mrow><annotation encoding='application/x-tex'>\{i_A, i_B\}</annotation></semantics></math>, hence that a <a class='existingWikiWord' href='/nlab/show/diff/subset'>subset</a> <math class='maruku-mathml' display='inline' id='mathml_fb0302666679a9db8a44d8a779257e05038cbef3_124' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>S</mi><mo>⊂</mo><mi>X</mi></mrow><annotation encoding='application/x-tex'>S \subset X</annotation></semantics></math> is an <a class='existingWikiWord' href='/nlab/show/diff/open+subspace'>open subset</a> precisely if the <a class='existingWikiWord' href='/nlab/show/diff/preimage'>pre-images</a> (restrictions)</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_fb0302666679a9db8a44d8a779257e05038cbef3_125' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msubsup><mi>i</mi> <mi>A</mi> <mrow><mo lspace='verythinmathspace' rspace='0em'>−</mo><mn>1</mn></mrow></msubsup><mo stretchy='false'>(</mo><mi>S</mi><mo stretchy='false'>)</mo><mo>=</mo><mi>S</mi><mo>∩</mo><mi>A</mi><mphantom><mi>AAA</mi></mphantom><mtext>and</mtext><mphantom><mi>AAA</mi></mphantom><msubsup><mi>i</mi> <mi>B</mi> <mrow><mo lspace='verythinmathspace' rspace='0em'>−</mo><mn>1</mn></mrow></msubsup><mo stretchy='false'>(</mo><mi>S</mi><mo stretchy='false'>)</mo><mo>=</mo><mi>S</mi><mo>∩</mo><mi>B</mi></mrow><annotation encoding='application/x-tex'> i_A^{-1}(S) = S \cap A \phantom{AAA} \text{and} \phantom{AAA} i_B^{-1}(S) = S \cap B </annotation></semantics></math></div> <p>are open subsets of <math class='maruku-mathml' display='inline' id='mathml_fb0302666679a9db8a44d8a779257e05038cbef3_126' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>A</mi></mrow><annotation encoding='application/x-tex'>A</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_fb0302666679a9db8a44d8a779257e05038cbef3_127' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>B</mi></mrow><annotation encoding='application/x-tex'>B</annotation></semantics></math>, respectively.</p> <p>Now by definition of the <a class='existingWikiWord' href='/nlab/show/diff/subspace+topology'>subspace topology</a>, if <math class='maruku-mathml' display='inline' id='mathml_fb0302666679a9db8a44d8a779257e05038cbef3_128' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>S</mi><mo>⊂</mo><mi>X</mi></mrow><annotation encoding='application/x-tex'>S \subset X</annotation></semantics></math> is open, then the intersections <math class='maruku-mathml' display='inline' id='mathml_fb0302666679a9db8a44d8a779257e05038cbef3_129' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>A</mi><mo>∩</mo><mi>S</mi><mo>⊂</mo><mi>A</mi></mrow><annotation encoding='application/x-tex'>A \cap S \subset A</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_fb0302666679a9db8a44d8a779257e05038cbef3_130' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>B</mi><mo>∩</mo><mi>S</mi><mo>⊂</mo><mi>B</mi></mrow><annotation encoding='application/x-tex'>B \cap S \subset B</annotation></semantics></math> are open in these subspaces.</p> <p>Conversely, assume that <math class='maruku-mathml' display='inline' id='mathml_fb0302666679a9db8a44d8a779257e05038cbef3_131' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>A</mi><mo>∩</mo><mi>S</mi><mo>⊂</mo><mi>A</mi></mrow><annotation encoding='application/x-tex'>A \cap S \subset A</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_fb0302666679a9db8a44d8a779257e05038cbef3_132' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>B</mi><mo>∩</mo><mi>S</mi><mo>⊂</mo><mi>B</mi></mrow><annotation encoding='application/x-tex'>B \cap S \subset B</annotation></semantics></math> are open. We need to show that then <math class='maruku-mathml' display='inline' id='mathml_fb0302666679a9db8a44d8a779257e05038cbef3_133' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>S</mi><mo>⊂</mo><mi>X</mi></mrow><annotation encoding='application/x-tex'>S \subset X</annotation></semantics></math> is open.</p> <p>Consider now first the case that <math class='maruku-mathml' display='inline' id='mathml_fb0302666679a9db8a44d8a779257e05038cbef3_134' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>A</mi><mo>;</mo><mi>B</mi><mo>⊂</mo><mi>X</mi></mrow><annotation encoding='application/x-tex'>A;B \subset X</annotation></semantics></math> are both open open. Then by the nature of the <a class='existingWikiWord' href='/nlab/show/diff/subspace+topology'>subspace topology</a>, that <math class='maruku-mathml' display='inline' id='mathml_fb0302666679a9db8a44d8a779257e05038cbef3_135' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>A</mi><mo>∩</mo><mi>S</mi></mrow><annotation encoding='application/x-tex'>A \cap S</annotation></semantics></math> is open in <math class='maruku-mathml' display='inline' id='mathml_fb0302666679a9db8a44d8a779257e05038cbef3_136' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>A</mi></mrow><annotation encoding='application/x-tex'>A</annotation></semantics></math> means that there is an open subset <math class='maruku-mathml' display='inline' id='mathml_fb0302666679a9db8a44d8a779257e05038cbef3_137' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>S</mi> <mi>A</mi></msub><mo>⊂</mo><mi>X</mi></mrow><annotation encoding='application/x-tex'>S_A \subset X</annotation></semantics></math> such that <math class='maruku-mathml' display='inline' id='mathml_fb0302666679a9db8a44d8a779257e05038cbef3_138' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>A</mi><mo>∩</mo><mi>S</mi><mo>=</mo><mi>A</mi><mo>∩</mo><msub><mi>S</mi> <mi>A</mi></msub></mrow><annotation encoding='application/x-tex'>A \cap S = A \cap S_A</annotation></semantics></math>. Since the intersection of two open subsets is open, this implies that <math class='maruku-mathml' display='inline' id='mathml_fb0302666679a9db8a44d8a779257e05038cbef3_139' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>A</mi><mo>∩</mo><msub><mi>S</mi> <mi>A</mi></msub></mrow><annotation encoding='application/x-tex'>A \cap S_A</annotation></semantics></math> and hence <math class='maruku-mathml' display='inline' id='mathml_fb0302666679a9db8a44d8a779257e05038cbef3_140' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>A</mi><mo>∩</mo><mi>S</mi></mrow><annotation encoding='application/x-tex'>A \cap S</annotation></semantics></math> is open. Similarly <math class='maruku-mathml' display='inline' id='mathml_fb0302666679a9db8a44d8a779257e05038cbef3_141' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>B</mi><mo>∩</mo><mi>S</mi></mrow><annotation encoding='application/x-tex'>B \cap S</annotation></semantics></math>. Therefore</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_fb0302666679a9db8a44d8a779257e05038cbef3_142' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mrow><mtable columnalign='right left right left right left right left right left' columnspacing='0em' displaystyle='true'><mtr><mtd><mi>S</mi></mtd> <mtd><mo>=</mo><mi>S</mi><mo>∩</mo><mi>X</mi></mtd></mtr> <mtr><mtd /> <mtd><mo>=</mo><mi>S</mi><mo>∩</mo><mo stretchy='false'>(</mo><mi>A</mi><mo>∪</mo><mi>B</mi><mo stretchy='false'>)</mo></mtd></mtr> <mtr><mtd /> <mtd><mo>=</mo><mo stretchy='false'>(</mo><mi>S</mi><mo>∩</mo><mi>A</mi><mo stretchy='false'>)</mo><mo>∪</mo><mo stretchy='false'>(</mo><mi>S</mi><mo>∩</mo><mi>B</mi><mo stretchy='false'>)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding='application/x-tex'> \begin{aligned} S & = S \cap X \\ & = S \cap (A \cup B) \\ & = (S \cap A) \cup (S \cap B) \end{aligned} </annotation></semantics></math></div> <p>is the union of two open subsets and therefore open.</p> <p>Now consider the case that <math class='maruku-mathml' display='inline' id='mathml_fb0302666679a9db8a44d8a779257e05038cbef3_143' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>A</mi><mo>,</mo><mi>B</mi><mo>⊂</mo><mi>X</mi></mrow><annotation encoding='application/x-tex'>A,B \subset X</annotation></semantics></math> are both closed subsets.</p> <p>Again by the nature of the subspace topology, that <math class='maruku-mathml' display='inline' id='mathml_fb0302666679a9db8a44d8a779257e05038cbef3_144' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>A</mi><mo>∩</mo><mi>S</mi><mo>⊂</mo><mi>A</mi></mrow><annotation encoding='application/x-tex'>A \cap S \subset A</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_fb0302666679a9db8a44d8a779257e05038cbef3_145' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>B</mi><mo>∩</mo><mi>S</mi><mo>⊂</mo><mi>B</mi></mrow><annotation encoding='application/x-tex'>B \cap S \subset B</annotation></semantics></math> are open means that there exist open subsets <math class='maruku-mathml' display='inline' id='mathml_fb0302666679a9db8a44d8a779257e05038cbef3_146' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>S</mi> <mi>A</mi></msub><mo>,</mo><msub><mi>S</mi> <mi>B</mi></msub><mo>⊂</mo><mi>X</mi></mrow><annotation encoding='application/x-tex'>S_A, S_B \subset X</annotation></semantics></math> such that <math class='maruku-mathml' display='inline' id='mathml_fb0302666679a9db8a44d8a779257e05038cbef3_147' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>A</mi><mo>∩</mo><mi>S</mi><mo>=</mo><mi>A</mi><mo>∩</mo><msub><mi>S</mi> <mi>A</mi></msub></mrow><annotation encoding='application/x-tex'>A \cap S = A \cap S_A</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_fb0302666679a9db8a44d8a779257e05038cbef3_148' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>B</mi><mo>∩</mo><mi>S</mi><mo>=</mo><mi>B</mi><mo>∩</mo><msub><mi>S</mi> <mi>B</mi></msub></mrow><annotation encoding='application/x-tex'>B \cap S = B \cap S_B</annotation></semantics></math>. Since <math class='maruku-mathml' display='inline' id='mathml_fb0302666679a9db8a44d8a779257e05038cbef3_149' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>A</mi><mo>,</mo><mi>B</mi><mo>⊂</mo><mi>X</mi></mrow><annotation encoding='application/x-tex'>A,B \subset X</annotation></semantics></math> are closed by assumption, this means that <math class='maruku-mathml' display='inline' id='mathml_fb0302666679a9db8a44d8a779257e05038cbef3_150' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>A</mi><mo>∖</mo><mi>S</mi><mo>,</mo><mi>B</mi><mo>∖</mo><mi>S</mi><mo>⊂</mo><mi>X</mi></mrow><annotation encoding='application/x-tex'>A \setminus S, B \setminus S \subset X</annotation></semantics></math> are still closed, hence that <math class='maruku-mathml' display='inline' id='mathml_fb0302666679a9db8a44d8a779257e05038cbef3_151' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi><mo>∖</mo><mo stretchy='false'>(</mo><mi>A</mi><mo>∖</mo><mi>S</mi><mo stretchy='false'>)</mo><mo>,</mo><mi>X</mi><mo>∖</mo><mo stretchy='false'>(</mo><mi>B</mi><mo>∖</mo><mi>S</mi><mo stretchy='false'>)</mo><mo>⊂</mo><mi>X</mi></mrow><annotation encoding='application/x-tex'>X \setminus (A \setminus S), X \setminus (B \setminus S) \subset X</annotation></semantics></math> are open.</p> <p>Now observe that (by <a class='existingWikiWord' href='/nlab/show/diff/De+Morgan+duality'>de Morgan duality</a>)</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_fb0302666679a9db8a44d8a779257e05038cbef3_152' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mrow><mtable columnalign='right left right left right left right left right left' columnspacing='0em' displaystyle='true'><mtr><mtd><mi>S</mi></mtd> <mtd><mo>=</mo><mi>X</mi><mo>∖</mo><mo stretchy='false'>(</mo><mi>X</mi><mo>∖</mo><mi>S</mi><mo stretchy='false'>)</mo></mtd></mtr> <mtr><mtd /> <mtd><mo>=</mo><mi>X</mi><mo>∖</mo><mo stretchy='false'>(</mo><mo stretchy='false'>(</mo><mi>A</mi><mo>∪</mo><mi>B</mi><mo stretchy='false'>)</mo><mo>∖</mo><mi>S</mi><mo stretchy='false'>)</mo></mtd></mtr> <mtr><mtd /> <mtd><mo>=</mo><mi>X</mi><mo>∖</mo><mo stretchy='false'>(</mo><mo stretchy='false'>(</mo><mi>A</mi><mo>∖</mo><mi>S</mi><mo stretchy='false'>)</mo><mo>∪</mo><mo stretchy='false'>(</mo><mi>B</mi><mo>∖</mo><mi>S</mi><mo stretchy='false'>)</mo><mo stretchy='false'>)</mo></mtd></mtr> <mtr><mtd /> <mtd><mo>=</mo><mo stretchy='false'>(</mo><mi>X</mi><mo>∖</mo><mo stretchy='false'>(</mo><mi>A</mi><mo>∖</mo><mi>S</mi><mo stretchy='false'>)</mo><mo stretchy='false'>)</mo><mo>∩</mo><mo stretchy='false'>(</mo><mi>X</mi><mo>∖</mo><mo stretchy='false'>(</mo><mi>B</mi><mo>∖</mo><mi>S</mi><mo stretchy='false'>)</mo><mo stretchy='false'>)</mo><mspace width='thinmathspace' /><mo>.</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding='application/x-tex'> \begin{aligned} S & = X \setminus (X \setminus S) \\ & = X \setminus ( (A \cup B) \setminus S ) \\ & = X \setminus ( (A \setminus S) \cup (B \setminus S) ) \\ & = (X \setminus (A \setminus S)) \cap (X \setminus (B \setminus S)) \,. \end{aligned} </annotation></semantics></math></div> <p>This exhibits <math class='maruku-mathml' display='inline' id='mathml_fb0302666679a9db8a44d8a779257e05038cbef3_153' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>S</mi></mrow><annotation encoding='application/x-tex'>S</annotation></semantics></math> as the intersection of two open subsets, hence as open.</p> </div> <div class='num_example' id='attach'> <h6 id='example_12'>Example</h6> <p>If <math class='maruku-mathml' display='inline' id='mathml_fb0302666679a9db8a44d8a779257e05038cbef3_154' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi><mo>,</mo><mi>Y</mi><mo>,</mo><mi>Z</mi></mrow><annotation encoding='application/x-tex'>X, Y, Z</annotation></semantics></math> are <a class='existingWikiWord' href='/nlab/show/diff/normal+space'>normal topological spaces</a> and <math class='maruku-mathml' display='inline' id='mathml_fb0302666679a9db8a44d8a779257e05038cbef3_155' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>h</mi><mo>:</mo><mi>X</mi><mo>→</mo><mi>Z</mi></mrow><annotation encoding='application/x-tex'>h: X \to Z</annotation></semantics></math> is a <a class='existingWikiWord' href='/nlab/show/diff/embedding+of+topological+spaces'>closed embedding of topological spaces</a> and <math class='maruku-mathml' display='inline' id='mathml_fb0302666679a9db8a44d8a779257e05038cbef3_156' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>f</mi><mo>:</mo><mi>X</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding='application/x-tex'>f: X \to Y</annotation></semantics></math> is a <a class='existingWikiWord' href='/nlab/show/diff/continuous+map'>continuous function</a>, then in the <a class='existingWikiWord' href='/nlab/show/diff/pushout'>pushout</a> diagram in <math class='maruku-mathml' display='inline' id='mathml_fb0302666679a9db8a44d8a779257e05038cbef3_157' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Top</mi></mrow><annotation encoding='application/x-tex'>Top</annotation></semantics></math> (example <a class='maruku-ref' href='#PushoutInTop'>9</a>)</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_fb0302666679a9db8a44d8a779257e05038cbef3_158' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mrow><mtable><mtr><mtd><mi>X</mi></mtd> <mtd><mover><mo>→</mo><mi>h</mi></mover></mtd> <mtd><mi>Z</mi></mtd></mtr> <mtr><mtd><mpadded lspace='-100%width' width='0'><mi>f</mi></mpadded><mo stretchy='false'>↓</mo></mtd> <mtd /> <mtd><mo stretchy='false'>↓</mo><mpadded width='0'><mi>g</mi></mpadded></mtd></mtr> <mtr><mtd><mi>Y</mi></mtd> <mtd><munder><mo>→</mo><mi>k</mi></munder></mtd> <mtd><mi>W</mi><mo>,</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding='application/x-tex'>\array{ X & \stackrel{h}{\to} & Z \\ \mathllap{f} \downarrow & & \downarrow \mathrlap{g} \\ Y & \underset{k}{\to} & W, } </annotation></semantics></math></div> <p>the space <math class='maruku-mathml' display='inline' id='mathml_fb0302666679a9db8a44d8a779257e05038cbef3_159' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>W</mi></mrow><annotation encoding='application/x-tex'>W</annotation></semantics></math> is normal and <math class='maruku-mathml' display='inline' id='mathml_fb0302666679a9db8a44d8a779257e05038cbef3_160' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>k</mi><mo>:</mo><mi>Y</mi><mo>→</mo><mi>W</mi></mrow><annotation encoding='application/x-tex'>k: Y \to W</annotation></semantics></math> is a closed embedding.</p> </div> <p>For <strong>proof</strong> of this and related statements see at <em><a class='existingWikiWord' href='/nlab/show/diff/colimits+of+normal+spaces'>colimits of normal spaces</a></em>.</p> <h3 id='relation_with_'>Relation with <math class='maruku-mathml' display='inline' id='mathml_fb0302666679a9db8a44d8a779257e05038cbef3_161' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Set</mi></mrow><annotation encoding='application/x-tex'>Set</annotation></semantics></math></h3> <p>Write <a class='existingWikiWord' href='/nlab/show/diff/Set'>Set</a> for the <a class='existingWikiWord' href='/nlab/show/diff/category'>category</a> of <a class='existingWikiWord' href='/nlab/show/diff/set'>sets</a>.</p> <div class='num_defn' id='ForgetfulFunctorFromTopToSet'> <h6 id='definition_3'>Definition</h6> <p>Write</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_fb0302666679a9db8a44d8a779257e05038cbef3_162' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>U</mi><mo lspace='verythinmathspace'>:</mo><mi>Top</mi><mo>⟶</mo><mi>Set</mi></mrow><annotation encoding='application/x-tex'> U \colon Top \longrightarrow Set </annotation></semantics></math></div> <p>for the <a class='existingWikiWord' href='/nlab/show/diff/forgetful+functor'>forgetful functor</a> that sends a topological space <math class='maruku-mathml' display='inline' id='mathml_fb0302666679a9db8a44d8a779257e05038cbef3_163' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi><mo>=</mo><mo stretchy='false'>(</mo><mi>S</mi><mo>,</mo><mi>τ</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>X = (S,\tau)</annotation></semantics></math> to its underlying set <math class='maruku-mathml' display='inline' id='mathml_fb0302666679a9db8a44d8a779257e05038cbef3_164' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>U</mi><mo stretchy='false'>(</mo><mi>X</mi><mo stretchy='false'>)</mo><mo>=</mo><mi>S</mi><mo>∈</mo><mi>Set</mi></mrow><annotation encoding='application/x-tex'>U(X) = S \in Set</annotation></semantics></math> and which regards a <a class='existingWikiWord' href='/nlab/show/diff/continuous+map'>continuous function</a> as a plain <a class='existingWikiWord' href='/nlab/show/diff/function'>function</a> on the underlying sets.</p> </div> <p>Prop. <a class='maruku-ref' href='#DescriptionOfLimitsAndColimitsInTop'>1</a> means in particular that:</p> <div class='num_prop'> <h6 id='proposition_2'>Proposition</h6> <p>The category <a class='existingWikiWord' href='/nlab/show/diff/Top'>Top</a> has all small <a class='existingWikiWord' href='/nlab/show/diff/limit'>limits</a> and <a class='existingWikiWord' href='/nlab/show/diff/colimit'>colimits</a>. The <a class='existingWikiWord' href='/nlab/show/diff/forgetful+functor'>forgetful functor</a> <math class='maruku-mathml' display='inline' id='mathml_fb0302666679a9db8a44d8a779257e05038cbef3_165' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>U</mi><mo lspace='verythinmathspace'>:</mo><mi>Top</mi><mo>→</mo><mi>Set</mi></mrow><annotation encoding='application/x-tex'>U \colon Top \to Set</annotation></semantics></math> from def. <a class='maruku-ref' href='#ForgetfulFunctorFromTopToSet'>2</a> <a class='existingWikiWord' href='/nlab/show/diff/preserved+limit'>preserves</a> and <a class='existingWikiWord' href='/nlab/show/diff/lifted+limit'>lifts</a> limits and colimits.</p> </div> <p>(But it does not <a class='existingWikiWord' href='/nlab/show/diff/created+limit'>create</a> or <a class='existingWikiWord' href='/nlab/show/diff/reflected+limit'>reflect</a> them.)</p> <div class='num_prop'> <h6 id='proposition_3'>Proposition</h6> <p>The <a class='existingWikiWord' href='/nlab/show/diff/forgetful+functor'>forgetful functor</a> <math class='maruku-mathml' display='inline' id='mathml_fb0302666679a9db8a44d8a779257e05038cbef3_166' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>U</mi></mrow><annotation encoding='application/x-tex'>U</annotation></semantics></math> from def. <a class='maruku-ref' href='#ForgetfulFunctorFromTopToSet'>2</a> has a <a class='existingWikiWord' href='/nlab/show/diff/left+adjoint'>left adjoint</a> <math class='maruku-mathml' display='inline' id='mathml_fb0302666679a9db8a44d8a779257e05038cbef3_167' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>disc</mi></mrow><annotation encoding='application/x-tex'>disc</annotation></semantics></math>, given by sending a <a class='existingWikiWord' href='/nlab/show/diff/set'>set</a> <math class='maruku-mathml' display='inline' id='mathml_fb0302666679a9db8a44d8a779257e05038cbef3_168' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>S</mi></mrow><annotation encoding='application/x-tex'>S</annotation></semantics></math> to the corresponding <a class='existingWikiWord' href='/nlab/show/diff/discrete+and+indiscrete+topology'>discrete topological space</a>, example <a class='maruku-ref' href='#DiscreteTopologicalSpaceAsCoproduct'>5</a></p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_fb0302666679a9db8a44d8a779257e05038cbef3_169' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Top</mi><mover><munder><mo>⟶</mo><mi>U</mi></munder><mover><mo>⟵</mo><mi>disc</mi></mover></mover><mi>Set</mi><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> Top \stackrel{\overset{disc}{\longleftarrow}}{\underset{U}{\longrightarrow}} Set \,. </annotation></semantics></math></div></div> <div class='num_prop' id='topcat'> <h6 id='proposition_4'>Proposition</h6> <p>The <a class='existingWikiWord' href='/nlab/show/diff/forgetful+functor'>forgetful functor</a> <math class='maruku-mathml' display='inline' id='mathml_fb0302666679a9db8a44d8a779257e05038cbef3_170' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>U</mi></mrow><annotation encoding='application/x-tex'>U</annotation></semantics></math> from def. <a class='maruku-ref' href='#ForgetfulFunctorFromTopToSet'>2</a> exhibits <math class='maruku-mathml' display='inline' id='mathml_fb0302666679a9db8a44d8a779257e05038cbef3_171' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Top</mi></mrow><annotation encoding='application/x-tex'>Top</annotation></semantics></math> as</p> <ul> <li> <p>a <a class='existingWikiWord' href='/nlab/show/diff/concrete+category'>concrete category</a></p> </li> <li> <p>a <a class='existingWikiWord' href='/nlab/show/diff/topological+concrete+category'>topological concrete category</a>.</p> </li> </ul> </div> <h3 id='MonoEpiMorphisms'>Mono-/Epimorphisms</h3> <div class='num_prop' id='SubspaceInclusionsAreRegularMonos'> <h6 id='proposition_5'>Proposition</h6> <p><strong>(<a class='existingWikiWord' href='/nlab/show/diff/regular+monomorphism'>regular monomorphisms</a> of <a class='existingWikiWord' href='/nlab/show/diff/topological+space'>topological spaces</a>)</strong></p> <p>In the <a class='existingWikiWord' href='/nlab/show/diff/category'>category</a> <a class='existingWikiWord' href='/nlab/show/diff/Top'>Top</a> of <a class='existingWikiWord' href='/nlab/show/diff/topological+space'>topological spaces</a>,</p> <ol> <li> <p>the <a class='existingWikiWord' href='/nlab/show/diff/monomorphism'>monomorphisms</a> are those <a class='existingWikiWord' href='/nlab/show/diff/continuous+map'>continuous functions</a> which are <a class='existingWikiWord' href='/nlab/show/diff/injection'>injective functions</a>;</p> </li> <li> <p>the <a class='existingWikiWord' href='/nlab/show/diff/regular+monomorphism'>regular monomorphisms</a> are the <a class='existingWikiWord' href='/nlab/show/diff/embedding+of+topological+spaces'>topological embeddings</a> (i.e. those continuous functions which are <a class='existingWikiWord' href='/nlab/show/diff/homeomorphism'>homeomorphisms</a> onto their <a class='existingWikiWord' href='/nlab/show/diff/image'>images</a> equipped with the <a class='existingWikiWord' href='/nlab/show/diff/subspace+topology'>subspace topology</a>).</p> </li> </ol> </div> <div class='proof'> <h6 id='proof_3'>Proof</h6> <p>Regarding the first statement: An injective continuous function <math class='maruku-mathml' display='inline' id='mathml_fb0302666679a9db8a44d8a779257e05038cbef3_172' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>f</mi><mo lspace='verythinmathspace'>:</mo><mi>X</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding='application/x-tex'>f \colon X \to Y</annotation></semantics></math> clearly has the cancellation property that defines monomorphisms: for parallel continuous functions <math class='maruku-mathml' display='inline' id='mathml_fb0302666679a9db8a44d8a779257e05038cbef3_173' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>g</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>g</mi> <mn>2</mn></msub><mo lspace='verythinmathspace'>:</mo><mi>Z</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding='application/x-tex'>g_1,g_2 \colon Z \to X</annotation></semantics></math>, if <math class='maruku-mathml' display='inline' id='mathml_fb0302666679a9db8a44d8a779257e05038cbef3_174' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>f</mi><mo>∘</mo><msub><mi>g</mi> <mn>1</mn></msub><mo>=</mo><mi>f</mi><mo>∘</mo><msub><mi>g</mi> <mn>2</mn></msub></mrow><annotation encoding='application/x-tex'>f \circ g_1 = f \circ g_2</annotation></semantics></math>, then <math class='maruku-mathml' display='inline' id='mathml_fb0302666679a9db8a44d8a779257e05038cbef3_175' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>g</mi> <mn>1</mn></msub><mo>=</mo><msub><mi>g</mi> <mn>2</mn></msub></mrow><annotation encoding='application/x-tex'>g_1 = g_2</annotation></semantics></math>, because continuous functions are equal precisely if their underlying functions of sets are equal. Conversely, if <math class='maruku-mathml' display='inline' id='mathml_fb0302666679a9db8a44d8a779257e05038cbef3_176' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>f</mi></mrow><annotation encoding='application/x-tex'>f</annotation></semantics></math> has the cancellation property, then testing on points <math class='maruku-mathml' display='inline' id='mathml_fb0302666679a9db8a44d8a779257e05038cbef3_177' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>g</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>g</mi> <mn>2</mn></msub><mo lspace='verythinmathspace'>:</mo><mo>*</mo><mo>→</mo><mi>X</mi></mrow><annotation encoding='application/x-tex'>g_1, g_2 \colon \ast \to X</annotation></semantics></math> gives that <math class='maruku-mathml' display='inline' id='mathml_fb0302666679a9db8a44d8a779257e05038cbef3_178' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>f</mi></mrow><annotation encoding='application/x-tex'>f</annotation></semantics></math> is injective.</p> <p>Regarding the second statement: from the construction of <a class='existingWikiWord' href='/nlab/show/diff/equalizer'>equalizers</a> in <a class='existingWikiWord' href='/nlab/show/diff/Top'>Top</a> (example <a class='maruku-ref' href='#EqualizerInTop'>7</a>) we have that these are topological subspace inclusions.</p> <p>Conversely, let <math class='maruku-mathml' display='inline' id='mathml_fb0302666679a9db8a44d8a779257e05038cbef3_179' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>i</mi><mo lspace='verythinmathspace'>:</mo><mi>X</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding='application/x-tex'>i \colon X \to Y</annotation></semantics></math> be a <a class='existingWikiWord' href='/nlab/show/diff/embedding+of+topological+spaces'>topological subspace embedding</a>. We need to show that this is the equalizer of some pair of parallel morphisms.</p> <p>To that end, form the <a class='existingWikiWord' href='/nlab/show/diff/cokernel+pair'>cokernel pair</a> <math class='maruku-mathml' display='inline' id='mathml_fb0302666679a9db8a44d8a779257e05038cbef3_180' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><msub><mi>i</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>i</mi> <mn>2</mn></msub><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(i_1, i_2)</annotation></semantics></math> by taking the <a class='existingWikiWord' href='/nlab/show/diff/pushout'>pushout</a> of <math class='maruku-mathml' display='inline' id='mathml_fb0302666679a9db8a44d8a779257e05038cbef3_181' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>i</mi></mrow><annotation encoding='application/x-tex'>i</annotation></semantics></math> against itself (in the category of sets, and using the <a class='existingWikiWord' href='/nlab/show/diff/quotient+space'>quotient topology</a> on a <a class='existingWikiWord' href='/nlab/show/diff/disjoint+union+topological+space'>disjoint union space</a>). By <a href='regular+monomorphism#RegEquEff'>this prop.</a>, the equalizer of that pair is the set-theoretic equalizer of that pair of functions endowed with the <a class='existingWikiWord' href='/nlab/show/diff/subspace+topology'>subspace topology</a>. Since monomorphisms in <a class='existingWikiWord' href='/nlab/show/diff/Set'>Set</a> are regular, we get the function <math class='maruku-mathml' display='inline' id='mathml_fb0302666679a9db8a44d8a779257e05038cbef3_182' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>i</mi></mrow><annotation encoding='application/x-tex'>i</annotation></semantics></math> back, and again by example <a class='maruku-ref' href='#EqualizerInTop'>7</a>, it gets equipped with the subspace topology. This completes the proof.</p> </div> <h3 id='intersections_and_quotients'>Intersections and quotients</h3> <div class='num_lemma' id='pushout'> <h6 id='lemma'>Lemma</h6> <p>The <a class='existingWikiWord' href='/nlab/show/diff/pushout'>pushout</a> in <a class='existingWikiWord' href='/nlab/show/diff/Top'>Top</a> of any (closed/open) <a class='existingWikiWord' href='/nlab/show/diff/subspace+topology'>topological subspace</a> inclusion <math class='maruku-mathml' display='inline' id='mathml_fb0302666679a9db8a44d8a779257e05038cbef3_183' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>i</mi><mo lspace='verythinmathspace'>:</mo><mi>A</mi><mo>↪</mo><mi>B</mi></mrow><annotation encoding='application/x-tex'>i \colon A \hookrightarrow B</annotation></semantics></math>, example <a class='maruku-ref' href='#TopologicalSubspace'>1</a>, along any <a class='existingWikiWord' href='/nlab/show/diff/continuous+map'>continuous function</a> <math class='maruku-mathml' display='inline' id='mathml_fb0302666679a9db8a44d8a779257e05038cbef3_184' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>f</mi><mo lspace='verythinmathspace'>:</mo><mi>A</mi><mo>→</mo><mi>C</mi></mrow><annotation encoding='application/x-tex'>f \colon A \to C</annotation></semantics></math> is itself an a (closed/open) subspace <math class='maruku-mathml' display='inline' id='mathml_fb0302666679a9db8a44d8a779257e05038cbef3_185' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>j</mi><mo lspace='verythinmathspace'>:</mo><mi>C</mi><mo>↪</mo><mi>D</mi></mrow><annotation encoding='application/x-tex'>j \colon C \hookrightarrow D</annotation></semantics></math>.</p> </div> <p>For proof see <a href='subspace+topology#pushout'>there</a>.</p> <ins class='diffins'><h3 id='closed_monoidal_structure'>Closed monoidal structure</h3></ins><ins class='diffins'> </ins><ins class='diffins'><p>It is well known that <a class='existingWikiWord' href='/nlab/show/diff/Top'>Top</a> is not <a class='existingWikiWord' href='/nlab/show/diff/cartesian+closed+category'>cartesian closed</a> (see for example at <a class='existingWikiWord' href='/nlab/show/diff/convenient+category+of+topological+spaces'>convenient category of topological spaces</a>).</p></ins><ins class='diffins'> </ins><ins class='diffins'><p>It is however <a class='existingWikiWord' href='/nlab/show/diff/closed+monoidal+category'>closed monoidal</a>.</p></ins><ins class='diffins'> </ins><ins class='diffins'><ul> <li> <p>The <a class='existingWikiWord' href='/nlab/show/diff/tensor+product'>tensor product</a> <math class='maruku-mathml' display='inline' id='mathml_e7a4bbdd15d6eaca608fc8e6338f9f7c0578dbd4_186' 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\otimes Y</annotation></semantics></math> is given by the cartesian product of the underlying spaces, equipped with the <em>topology of separate continuity</em>, formed by the sets <math class='maruku-mathml' display='inline' id='mathml_e7a4bbdd15d6eaca608fc8e6338f9f7c0578dbd4_187' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>U</mi><mo>⊆</mo><mi>X</mi><mo>×</mo><mi>Y</mi></mrow><annotation encoding='application/x-tex'>U\subseteq X\times Y</annotation></semantics></math> such that</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_e7a4bbdd15d6eaca608fc8e6338f9f7c0578dbd4_188' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>U</mi> <mi>x</mi></msub><mspace width='thickmathspace' /><mo>≔</mo><mspace width='thickmathspace' /><mo stretchy='false'>{</mo><mi>y</mi><mo>∈</mo><mi>Y</mi><mo>:</mo><mo stretchy='false'>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>∈</mo><mi>U</mi><mo stretchy='false'>)</mo><mo stretchy='false'>}</mo></mrow><annotation encoding='application/x-tex'> U_x \;\coloneqq\; \{y\in Y : (x,y\in U)\} </annotation></semantics></math></div> <p>is an open subset of <math class='maruku-mathml' display='inline' id='mathml_e7a4bbdd15d6eaca608fc8e6338f9f7c0578dbd4_189' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Y</mi></mrow><annotation encoding='application/x-tex'>Y</annotation></semantics></math> and</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_e7a4bbdd15d6eaca608fc8e6338f9f7c0578dbd4_190' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>U</mi> <mi>x</mi></msub><mspace width='thickmathspace' /><mo>≔</mo><mspace width='thickmathspace' /><mo stretchy='false'>{</mo><mi>x</mi><mo>∈</mo><mi>X</mi><mo>:</mo><mo stretchy='false'>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy='false'>)</mo><mo>∈</mo><mi>U</mi><mo stretchy='false'>}</mo></mrow><annotation encoding='application/x-tex'> U_x \;\coloneqq\; \{x\in X : (x,y)\in U\} </annotation></semantics></math></div> <p>is an open subset of <math class='maruku-mathml' display='inline' id='mathml_e7a4bbdd15d6eaca608fc8e6338f9f7c0578dbd4_191' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math>. Equivalently, it is the topology such that for all spaces <math class='maruku-mathml' display='inline' id='mathml_e7a4bbdd15d6eaca608fc8e6338f9f7c0578dbd4_192' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Z</mi></mrow><annotation encoding='application/x-tex'>Z</annotation></semantics></math>, a function <math class='maruku-mathml' display='inline' id='mathml_e7a4bbdd15d6eaca608fc8e6338f9f7c0578dbd4_193' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>f</mi><mo>:</mo><mi>X</mi><mo>⊗</mo><mi>Y</mi><mo>→</mo><mi>Z</mi></mrow><annotation encoding='application/x-tex'>f:X\otimes Y\to Z</annotation></semantics></math> is continuous if and only if: for every <math class='maruku-mathml' display='inline' id='mathml_e7a4bbdd15d6eaca608fc8e6338f9f7c0578dbd4_194' 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\in X</annotation></semantics></math> the function <math class='maruku-mathml' display='inline' id='mathml_e7a4bbdd15d6eaca608fc8e6338f9f7c0578dbd4_195' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>y</mi><mo>↦</mo><mi>f</mi><mo stretchy='false'>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>y\mapsto f(x,y)</annotation></semantics></math> is continuous, and for every <math class='maruku-mathml' display='inline' id='mathml_e7a4bbdd15d6eaca608fc8e6338f9f7c0578dbd4_196' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>y</mi><mo>∈</mo><mi>Y</mi></mrow><annotation encoding='application/x-tex'>y\in Y</annotation></semantics></math>, the functions <math class='maruku-mathml' display='inline' id='mathml_e7a4bbdd15d6eaca608fc8e6338f9f7c0578dbd4_197' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>x</mi><mo>↦</mo><mi>f</mi><mo stretchy='false'>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>x\mapsto f(x,y)</annotation></semantics></math> is continuous.</p> </li> <li> <p>The <a class='existingWikiWord' href='/nlab/show/diff/internal+hom'>internal hom</a> <math class='maruku-mathml' display='inline' id='mathml_e7a4bbdd15d6eaca608fc8e6338f9f7c0578dbd4_198' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>[</mo><mi>X</mi><mo>,</mo><mi>Y</mi><mo stretchy='false'>]</mo></mrow><annotation encoding='application/x-tex'>[X,Y]</annotation></semantics></math> is given by the set of <a class='existingWikiWord' href='/nlab/show/diff/continuous+map'>continuous functions</a> <math class='maruku-mathml' display='inline' id='mathml_e7a4bbdd15d6eaca608fc8e6338f9f7c0578dbd4_199' 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\to Y</annotation></semantics></math>, together with the topology of pointwise convergence, generated by the (sub-basic) sets</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_e7a4bbdd15d6eaca608fc8e6338f9f7c0578dbd4_200' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>S</mi><mo stretchy='false'>(</mo><mi>x</mi><mo>,</mo><mi>V</mi><mo stretchy='false'>)</mo><mspace width='thickmathspace' /><mo>≔</mo><mspace width='thickmathspace' /><mo stretchy='false'>{</mo><mi>f</mi><mo>:</mo><mi>X</mi><mo>→</mo><mi>Y</mi><mo>:</mo><mi>f</mi><mo stretchy='false'>(</mo><mi>x</mi><mo stretchy='false'>)</mo><mo>∈</mo><mi>V</mi><mo stretchy='false'>}</mo></mrow><annotation encoding='application/x-tex'> S(x,V) \;\coloneqq\; \{f:X\to Y : f(x)\in V\} </annotation></semantics></math></div> <p>for each <math class='maruku-mathml' display='inline' id='mathml_e7a4bbdd15d6eaca608fc8e6338f9f7c0578dbd4_201' 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\in X</annotation></semantics></math> and each open <math class='maruku-mathml' display='inline' id='mathml_e7a4bbdd15d6eaca608fc8e6338f9f7c0578dbd4_202' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>V</mi><mo>⊆</mo><mi>Y</mi></mrow><annotation encoding='application/x-tex'>V\subseteq Y</annotation></semantics></math>. Equivalently, a net <math class='maruku-mathml' display='inline' id='mathml_e7a4bbdd15d6eaca608fc8e6338f9f7c0578dbd4_203' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><msub><mi>f</mi> <mi>α</mi></msub><mo>:</mo><mi>X</mi><mo>→</mo><mi>Y</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(f_\alpha:X\to Y)</annotation></semantics></math> tends to <math class='maruku-mathml' display='inline' id='mathml_e7a4bbdd15d6eaca608fc8e6338f9f7c0578dbd4_204' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>f</mi><mo>:</mo><mi>X</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding='application/x-tex'>f:X\to Y</annotation></semantics></math> if and only if for all <math class='maruku-mathml' display='inline' id='mathml_e7a4bbdd15d6eaca608fc8e6338f9f7c0578dbd4_205' 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\in X</annotation></semantics></math>, <math class='maruku-mathml' display='inline' id='mathml_e7a4bbdd15d6eaca608fc8e6338f9f7c0578dbd4_206' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>f</mi> <mi>α</mi></msub><mo stretchy='false'>(</mo><mi>x</mi><mo stretchy='false'>)</mo><mo>→</mo><mi>f</mi><mo stretchy='false'>(</mo><mi>x</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>f_\alpha(x)\to f(x)</annotation></semantics></math> in <math class='maruku-mathml' display='inline' id='mathml_e7a4bbdd15d6eaca608fc8e6338f9f7c0578dbd4_207' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Y</mi></mrow><annotation encoding='application/x-tex'>Y</annotation></semantics></math>.</p> </li> </ul></ins><ins class='diffins'> </ins><h2 id='related_concepts'>Related concepts</h2> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/topological+concrete+category'>topological concrete category</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Ho%28Top%29'>Ho(Top)</a>, <a class='existingWikiWord' href='/nlab/show/diff/Infinity-Grpd'>∞Grpd</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/convenient+category+of+topological+spaces'>convenient category of topological spaces</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/TopGrp'>TopGrp</a></p> </li> </ul> <h2 id='references'>References</h2> <p>For general references see those listed at <em><a class='existingWikiWord' href='/nlab/show/diff/topology'>topology</a></em>, such as</p> <ul> <li id='Bourbaki71'><a class='existingWikiWord' href='/nlab/show/diff/Bourbaki'>Nicolas Bourbaki</a>, chapter 1 <em>Topological Structures</em> of <em>Elements of Mathematics III: General topology</em>, Springer 1971, 1990</li> </ul> <p>See also</p> <ul> <li id='AguilarGitlerPrieto02'>Marcelo Aguilar, <a class='existingWikiWord' href='/nlab/show/diff/Samuel+Gitler'>Samuel Gitler</a>, Carlos Prieto, section 12 of <em>Algebraic topology from a homotopical viewpoint</em>, Springer (2002) (<a href='http://tocs.ulb.tu-darmstadt.de/106999419.pdf'>toc pdf</a>)</li> </ul> <p>An axiomatic desciption of <math class='maruku-mathml' display='inline' id='mathml_fb0302666679a9db8a44d8a779257e05038cbef3_186' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Top</mi></mrow><annotation encoding='application/x-tex'>Top</annotation></semantics></math> along the lines of <a class='existingWikiWord' href='/nlab/show/diff/ETCS'>ETCS</a> for <a class='existingWikiWord' href='/nlab/show/diff/Set'>Set</a> is discussed in</p> <ul> <li>Dana Schlomiuk, <em>An elementary theory of the category of topological space</em>, Transactions of the AMS, volume 149 (1970)</li> </ul><ins class='diffins'> </ins><ins class='diffins'><p>For its <a href='#closed_monoidal_structure'>closed monoidal structure</a>, see:</p></ins><ins class='diffins'> </ins><ins class='diffins'><ul> <li>Maria Cristina Pedicchio and Fabio Rossi, <em>Monoidal closed structures for topological spaces: counter-example to a question of Booth and Tillotson</em>, Cahiers de topologie et géométrie différentielle catégoriques, 24(4), 1983.</li> <li id='dagger_martingales'>Appendix A of <a class='existingWikiWord' href='/nlab/show/diff/Paolo+Perrone'>Paolo Perrone</a> and Ruben Van Belle, <em>Convergence of martingales via enriched dagger categories</em>, 2024. (<a href='https://arxiv.org/abs/2404.15191'>arXiv</a>)</li> </ul></ins> <p><div class='property'> category: <a class='category_link' href='/nlab/list/category'>category</a></div></p> <p> </p> <p> </p> </div> <div class="revisedby"> <p> Last revised on July 26, 2024 at 09:48:17. 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