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It does not yet have a dedicated thread; feel free to create one, giving it the same name as the title of this page" style="color:black">Discuss this page</a> | <form accept-charset="utf-8" action="/nlab/search" id="navigationSearchForm" method="get"> <fieldset class="search"><input type="text" id="searchField" name="query" value="Search" style="display:inline-block; float: left;" onfocus="this.value == 'Search' ? this.value = '' : true" onblur="this.value == '' ? this.value = 'Search' : true" /></fieldset> </form> </div> <div id="revision"> Showing changes from revision #5 to #6: <ins class="diffins">Added</ins> | <del class="diffdel">Removed</del> | <del class="diffmod">Chan</del><ins class="diffmod">ged</ins> <div class='rightHandSide'> <div class='toc clickDown' tabindex='0'> <h3 id='context'>Context</h3> <h4 id='topology'>Topology</h4> <div class='hide'> <a class='existingWikiWord' href='/nlab/show/diff/topology'>topology</a> (<a class='existingWikiWord' href='/nlab/show/diff/general+topology'>point-set topology</a>, <a class='existingWikiWord' href='/nlab/show/diff/point-free+topology'>point-free topology</a>) see also <a class='existingWikiWord' href='/nlab/show/diff/differential+topology'>differential topology</a>, <a class='existingWikiWord' href='/nlab/show/diff/algebraic+topology'>algebraic topology</a>, <a class='existingWikiWord' href='/nlab/show/diff/functional+analysis'>functional analysis</a> and <a class='existingWikiWord' href='/nlab/show/diff/topological+homotopy+theory'>topological</a> <a class='existingWikiWord' href='/nlab/show/diff/homotopy+theory'>homotopy theory</a> <a class='existingWikiWord' href='/nlab/show/diff/Introduction+to+Topology'>Introduction</a> Basic concepts <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/open+subspace'>open subset</a>, <a class='existingWikiWord' href='/nlab/show/diff/closed+subspace'>closed subset</a>, <a class='existingWikiWord' href='/nlab/show/diff/neighborhood'>neighbourhood</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/topological+space'>topological space</a>, <a class='existingWikiWord' href='/nlab/show/diff/locale'>locale</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/topological+base'>base for the topology</a>, <a class='existingWikiWord' href='/nlab/show/diff/neighborhood+base'>neighbourhood base</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/finer+topology'>finer/coarser topology</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/closed+subspace'>closure</a>, <a class='existingWikiWord' href='/nlab/show/diff/interior'>interior</a>, <a class='existingWikiWord' href='/nlab/show/diff/boundary'>boundary</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/separation+axioms'>separation</a>, <a class='existingWikiWord' href='/nlab/show/diff/sober+topological+space'>sobriety</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/continuous+map'>continuous function</a>, <a class='existingWikiWord' href='/nlab/show/diff/homeomorphism'>homeomorphism</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/uniformly+continuous+map'>uniformly continuous function</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/embedding+of+topological+spaces'>embedding</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/open+map'>open map</a>, <a class='existingWikiWord' href='/nlab/show/diff/closed+map'>closed map</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/sequence'>sequence</a>, <a class='existingWikiWord' href='/nlab/show/diff/net'>net</a>, <a class='existingWikiWord' href='/nlab/show/diff/subnet'>sub-net</a>, <a class='existingWikiWord' href='/nlab/show/diff/filter'>filter</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/convergence'>convergence</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/category'>category</a> <a class='existingWikiWord' href='/nlab/show/diff/Top'>Top</a> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/convenient+category+of+topological+spaces'>convenient category of topological spaces</a></li> </ul> </li> </ul> <a href='Top#UniversalConstructions'>Universal constructions</a> <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/weak+topology'>initial topology</a>, <a class='existingWikiWord' href='/nlab/show/diff/weak+topology'>final topology</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/subspace'>subspace</a>, <a class='existingWikiWord' href='/nlab/show/diff/quotient+space'>quotient space</a>, </li> <li> fiber space, <a class='existingWikiWord' href='/nlab/show/diff/space+attachment'>space attachment</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/product+topological+space'>product space</a>, <a class='existingWikiWord' href='/nlab/show/diff/disjoint+union+topological+space'>disjoint union space</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/mapping+cylinder'>mapping cylinder</a>, <a class='existingWikiWord' href='/nlab/show/diff/cocylinder'>mapping cocylinder</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/mapping+cone'>mapping cone</a>, <a class='existingWikiWord' href='/nlab/show/diff/mapping+cocone'>mapping cocone</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/mapping+telescope'>mapping telescope</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/colimits+of+normal+spaces'>colimits of normal spaces</a> </li> </ul> <a class='existingWikiWord' href='/nlab/show/diff/stuff%2C+structure%2C+property'>Extra stuff, structure, properties</a> <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/nice+topological+space'>nice topological space</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/metric+space'>metric space</a>, <a class='existingWikiWord' href='/nlab/show/diff/metric+topology'>metric topology</a>, <a class='existingWikiWord' href='/nlab/show/diff/metrisable+topological+space'>metrisable space</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/Kolmogorov+topological+space'>Kolmogorov space</a>, <a class='existingWikiWord' href='/nlab/show/diff/Hausdorff+space'>Hausdorff space</a>, <a class='existingWikiWord' href='/nlab/show/diff/regular+space'>regular space</a>, <a class='existingWikiWord' href='/nlab/show/diff/normal+space'>normal space</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/sober+topological+space'>sober space</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/compact+space'>compact space</a>, <a class='existingWikiWord' href='/nlab/show/diff/proper+map'>proper map</a> <a class='existingWikiWord' href='/nlab/show/diff/sequentially+compact+topological+space'>sequentially compact</a>, <a class='existingWikiWord' href='/nlab/show/diff/countably+compact+topological+space'>countably compact</a>, <a class='existingWikiWord' href='/nlab/show/diff/locally+compact+topological+space'>locally compact</a>, <a class='existingWikiWord' href='/nlab/show/diff/sigma-compact+topological+space'>sigma-compact</a>, <a class='existingWikiWord' href='/nlab/show/diff/paracompact+topological+space'>paracompact</a>, <a class='existingWikiWord' href='/nlab/show/diff/countably+paracompact+topological+space'>countably paracompact</a>, <a class='existingWikiWord' href='/nlab/show/diff/strongly+compact+topological+space'>strongly compact</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/compactly+generated+topological+space'>compactly generated space</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/second-countable+space'>second-countable space</a>, <a class='existingWikiWord' href='/nlab/show/diff/first-countable+space'>first-countable space</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/contractible+space'>contractible space</a>, <a class='existingWikiWord' href='/nlab/show/diff/locally+contractible+space'>locally contractible space</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/connected+space'>connected space</a>, <a class='existingWikiWord' href='/nlab/show/diff/locally+connected+topological+space'>locally connected space</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/simply+connected+space'>simply-connected space</a>, <a class='existingWikiWord' href='/nlab/show/diff/semi-locally+simply-connected+topological+space'>locally simply-connected space</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/cell+complex'>cell complex</a>, <a class='existingWikiWord' href='/nlab/show/diff/CW+complex'>CW-complex</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/pointed+topological+space'>pointed space</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/topological+vector+space'>topological vector space</a>, <a class='existingWikiWord' href='/nlab/show/diff/Banach+space'>Banach space</a>, <a class='existingWikiWord' href='/nlab/show/diff/Hilbert+space'>Hilbert space</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/topological+group'>topological group</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/topological+vector+bundle'>topological vector bundle</a>, <a class='existingWikiWord' href='/nlab/show/diff/topological+K-theory'>topological K-theory</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/topological+manifold'>topological manifold</a> </li> </ul> Examples <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/empty+space'>empty space</a>, <a class='existingWikiWord' href='/nlab/show/diff/point+space'>point space</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/discrete+object'>discrete space</a>, <a class='existingWikiWord' href='/nlab/show/diff/codiscrete+space'>codiscrete space</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/Sierpinski+space'>Sierpinski space</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/order+topology'>order topology</a>, <a class='existingWikiWord' href='/nlab/show/diff/specialization+topology'>specialization topology</a>, <a class='existingWikiWord' href='/nlab/show/diff/Scott+topology'>Scott topology</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/Euclidean+space'>Euclidean space</a> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/real+number'>real line</a>, <a class='existingWikiWord' href='/nlab/show/diff/plane'>plane</a></li> </ul> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/cylinder+object'>cylinder</a>, <a class='existingWikiWord' href='/nlab/show/diff/cone'>cone</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/sphere'>sphere</a>, <a class='existingWikiWord' href='/nlab/show/diff/ball'>ball</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/circle'>circle</a>, <a class='existingWikiWord' href='/nlab/show/diff/torus'>torus</a>, <a class='existingWikiWord' href='/nlab/show/diff/annulus'>annulus</a>, <a class='existingWikiWord' href='/nlab/show/diff/M%C3%B6bius+strip'>Moebius strip</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/polytope'>polytope</a>, <a class='existingWikiWord' href='/nlab/show/diff/polyhedron'>polyhedron</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/projective+space'>projective space</a> (<a class='existingWikiWord' href='/nlab/show/diff/real+projective+space'>real</a>, <a class='existingWikiWord' href='/nlab/show/diff/complex+projective+space'>complex</a>) </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/classifying+space'>classifying space</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/configuration+space+of+points'>configuration space</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/path'>path</a>, <a class='existingWikiWord' href='/nlab/show/diff/loop'>loop</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/compact-open+topology'>mapping spaces</a>: <a class='existingWikiWord' href='/nlab/show/diff/compact-open+topology'>compact-open topology</a>, <a class='existingWikiWord' href='/nlab/show/diff/topology+of+uniform+convergence'>topology of uniform convergence</a> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/loop+space'>loop space</a>, <a class='existingWikiWord' href='/nlab/show/diff/path+space'>path space</a></li> </ul> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/Zariski+topology'>Zariski topology</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/Cantor+space'>Cantor space</a>, <a class='existingWikiWord' href='/nlab/show/diff/Mandelbrot+set'>Mandelbrot space</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/Peano+curve'>Peano curve</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/line+with+two+origins'>line with two origins</a>, <a class='existingWikiWord' href='/nlab/show/diff/long+line'>long line</a>, <a class='existingWikiWord' href='/nlab/show/diff/Sorgenfrey+line'>Sorgenfrey line</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/K-topology'>K-topology</a>, <a class='existingWikiWord' href='/nlab/show/diff/Dowker+space'>Dowker space</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/Warsaw+circle'>Warsaw circle</a>, <a class='existingWikiWord' href='/nlab/show/diff/Hawaiian+earring+space'>Hawaiian earring space</a> </li> </ul> Basic statements <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/Hausdorff+implies+sober'>Hausdorff spaces are sober</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/schemes+are+sober'>schemes are sober</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/continuous+images+of+compact+spaces+are+compact'>continuous images of compact spaces are compact</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/closed+subspaces+of+compact+Hausdorff+spaces+are+equivalently+compact+subspaces'>closed subspaces of compact Hausdorff spaces are equivalently compact subspaces</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/open+subspaces+of+compact+Hausdorff+spaces+are+locally+compact'>open subspaces of compact Hausdorff spaces are locally compact</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/quotient+projections+out+of+compact+Hausdorff+spaces+are+closed+precisely+if+the+codomain+is+Hausdorff'>quotient projections out of compact Hausdorff spaces are closed precisely if the codomain is Hausdorff</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/compact+spaces+equivalently+have+converging+subnets'>compact spaces equivalently have converging subnet of every net</a> <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/Lebesgue+number+lemma'>Lebesgue number lemma</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/sequentially+compact+metric+spaces+are+equivalently+compact+metric+spaces'>sequentially compact metric spaces are equivalently compact metric spaces</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/compact+spaces+equivalently+have+converging+subnets'>compact spaces equivalently have converging subnet of every net</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/sequentially+compact+metric+spaces+are+totally+bounded'>sequentially compact metric spaces are totally bounded</a> </li> </ul> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/continuous+metric+space+valued+function+on+compact+metric+space+is+uniformly+continuous'>continuous metric space valued function on compact metric space is uniformly continuous</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/paracompact+Hausdorff+spaces+are+normal'>paracompact Hausdorff spaces are normal</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/paracompact+Hausdorff+spaces+equivalently+admit+subordinate+partitions+of+unity'>paracompact Hausdorff spaces equivalently admit subordinate partitions of unity</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/closed+injections+are+embeddings'>closed injections are embeddings</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/proper+maps+to+locally+compact+spaces+are+closed'>proper maps to locally compact spaces are closed</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/injective+proper+maps+to+locally+compact+spaces+are+equivalently+the+closed+embeddings'>injective proper maps to locally compact spaces are equivalently the closed embeddings</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/locally+compact+and+sigma-compact+spaces+are+paracompact'>locally compact and sigma-compact spaces are paracompact</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/locally+compact+and+second-countable+spaces+are+sigma-compact'>locally compact and second-countable spaces are sigma-compact</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/second-countable+regular+spaces+are+paracompact'>second-countable regular spaces are paracompact</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/CW-complexes+are+paracompact+Hausdorff+spaces'>CW-complexes are paracompact Hausdorff spaces</a> </li> </ul> Theorems <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/Urysohn%27s+lemma'>Urysohn's lemma</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/Tietze+extension+theorem'>Tietze extension theorem</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/Tychonoff+theorem'>Tychonoff theorem</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/tube+lemma'>tube lemma</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/Michael%27s+theorem'>Michael's theorem</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/Brouwer%27s+fixed+point+theorem'>Brouwer's fixed point theorem</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/topological+invariance+of+dimension'>topological invariance of dimension</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/Jordan+curve+theorem'>Jordan curve theorem</a> </li> </ul> Analysis Theorems <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/Heine-Borel+theorem'>Heine-Borel theorem</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/intermediate+value+theorem'>intermediate value theorem</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/extreme+value+theorem'>extreme value theorem</a> </li> </ul> <a class='existingWikiWord' href='/nlab/show/diff/topological+homotopy+theory'>topological homotopy theory</a> <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/homotopy'>left homotopy</a>, <a class='existingWikiWord' href='/nlab/show/diff/homotopy'>right homotopy</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/homotopy+equivalence'>homotopy equivalence</a>, <a class='existingWikiWord' href='/nlab/show/diff/deformation+retract'>deformation retract</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/fundamental+group'>fundamental group</a>, <a class='existingWikiWord' href='/nlab/show/diff/covering+space'>covering space</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/fundamental+theorem+of+covering+spaces'>fundamental theorem of covering spaces</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/homotopy+group'>homotopy group</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/weak+homotopy+equivalence'>weak homotopy equivalence</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/Whitehead+theorem'>Whitehead's theorem</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/Freudenthal+suspension+theorem'>Freudenthal suspension theorem</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/nerve+theorem'>nerve theorem</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/homotopy+extension+property'>homotopy extension property</a>, <a class='existingWikiWord' href='/nlab/show/diff/Hurewicz+cofibration'>Hurewicz cofibration</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/topological+cofiber+sequence'>cofiber sequence</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/Str%C3%B8m+model+structure'>Strøm model category</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/classical+model+structure+on+topological+spaces'>classical model structure on topological spaces</a> </li> </ul> </div> <h4 id='topos_theory'>Topos Theory</h4> <div class='hide'> <a class='existingWikiWord' href='/nlab/show/diff/sheaf+and+topos+theory'>topos theory</a> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/Topos'>Toposes</a></li> </ul> <h2 id='background'>Background</h2> <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/category+theory'>category theory</a> <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/category'>category</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/functor'>functor</a> </li> </ul> </li> </ul> <h2 id='toposes'>Toposes</h2> <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/%280%2C1%29-topos'>(0,1)-topos</a>, <a class='existingWikiWord' href='/nlab/show/diff/Heyting+algebra'>Heyting algebra</a>, <a class='existingWikiWord' href='/nlab/show/diff/locale'>locale</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/pretopos'>pretopos</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/topos'>topos</a> <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/Grothendieck+topos'>Grothendieck topos</a> <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/category+of+presheaves'>category of presheaves</a> <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/presheaf'>presheaf</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/representable+functor'>representable presheaf</a> </li> </ul> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/category+of+sheaves'>category of sheaves</a> <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/site'>site</a> <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/sieve'>sieve</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/coverage'>coverage</a>, <a class='existingWikiWord' href='/nlab/show/diff/Grothendieck+pretopology'>pretopology</a>, <a class='existingWikiWord' href='/nlab/show/diff/Grothendieck+topology'>topology</a> </li> </ul> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/sheaf'>sheaf</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/sheafification'>sheafification</a> </li> </ul> </li> </ul> </li> </ul> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/quasitopos'>quasitopos</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/base+topos'>base topos</a>, <a class='existingWikiWord' href='/nlab/show/diff/indexed+topos'>indexed topos</a> </li> </ul> <h2 id='internal_logic'>Internal Logic</h2> <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/categorical+semantics'>categorical semantics</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/internal+logic'>internal logic</a> <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/subobject+classifier'>subobject classifier</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/natural+numbers+object'>natural numbers object</a> </li> </ul> </li> </ul> <h2 id='topos_morphisms'>Topos morphisms</h2> <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/logical+functor'>logical morphism</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/geometric+morphism'>geometric morphism</a> <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/direct+image'>direct image</a>/<a class='existingWikiWord' href='/nlab/show/diff/inverse+image'>inverse image</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/global+section'>global sections</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/geometric+embedding'>geometric embedding</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/surjective+geometric+morphism'>surjective geometric morphism</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/essential+geometric+morphism'>essential geometric morphism</a> <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/locally+connected+geometric+morphism'>locally connected geometric morphism</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/connected+topos'>connected geometric morphism</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/totally+connected+geometric+morphism'>totally connected geometric morphism</a> </li> </ul> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/%C3%A9tale+geometric+morphism'>étale geometric morphism</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/open+geometric+morphism'>open geometric morphism</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/proper+geometric+morphism'>proper geometric morphism</a>, <a class='existingWikiWord' href='/nlab/show/diff/proper+geometric+morphism'>compact topos</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/separated+geometric+morphism'>separated geometric morphism</a>, <a class='existingWikiWord' href='/nlab/show/diff/separated+geometric+morphism'>Hausdorff topos</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/local+geometric+morphism'>local geometric morphism</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/bounded+geometric+morphism'>bounded geometric morphism</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/base+change'>base change</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/localic+geometric+morphism'>localic geometric morphism</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/hyperconnected+geometric+morphism'>hyperconnected geometric morphism</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/atomic+geometric+morphism'>atomic geometric morphism</a> </li> </ul> </li> </ul> <h2 id='extra_stuff_structure_properties'>Extra stuff, structure, properties</h2> <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/spatial+locale'>topological locale</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/localic+topos'>localic topos</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/big+and+little+toposes'>petit topos/gros topos</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/locally+connected+topos'>locally connected topos</a>, <a class='existingWikiWord' href='/nlab/show/diff/connected+topos'>connected topos</a>, <a class='existingWikiWord' href='/nlab/show/diff/totally+connected+geometric+morphism'>totally connected topos</a>, <a class='existingWikiWord' href='/nlab/show/diff/stably+locally+connected+topos'>strongly connected topos</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/local+geometric+morphism'>local topos</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/cohesive+topos'>cohesive topos</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/classifying+topos'>classifying topos</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/smooth+topos'>smooth topos</a> </li> </ul> <h2 id='cohomology_and_homotopy'>Cohomology and homotopy</h2> <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/cohomology'>cohomology</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/homotopy+groups+in+an+%28infinity%2C1%29-topos'>homotopy</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/abelian+sheaf+cohomology'>abelian sheaf cohomology</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/model+structure+on+simplicial+presheaves'>model structure on simplicial presheaves</a> </li> </ul> <h2 id='in_higher_category_theory'>In higher category theory</h2> <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/higher+topos+theory'>higher topos theory</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/%280%2C1%29-topos'>(0,1)-topos</a> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/posite'>(0,1)-site</a></li> </ul> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/2-topos'>2-topos</a> <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/2-site'>2-site</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/2-sheaf'>2-sheaf</a>, <a class='existingWikiWord' href='/nlab/show/diff/stack'>stack</a> </li> </ul> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-topos'>(∞,1)-topos</a> <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-site'>(∞,1)-site</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-sheaf'>(∞,1)-sheaf</a>, <a class='existingWikiWord' href='/nlab/show/diff/infinity-stack'>∞-stack</a>, <a class='existingWikiWord' href='/nlab/show/diff/derived+stack'>derived stack</a> </li> </ul> </li> </ul> <h2 id='theorems'>Theorems</h2> <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/Diaconescu%27s+theorem'>Diaconescu's theorem</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/Barr%27s+theorem'>Barr's theorem</a> </li> </ul> <div> <a href='/nlab/edit/topos+theory+-+contents'>Edit this sidebar</a> </div></div> </div> </div> Given a <a class='existingWikiWord' href='/nlab/show/diff/topological+space'>topological space</a> <math class='maruku-mathml' display='inline' id='mathml_b6adf993a7711dfec3419c83c5b86f2ece9c7b4a_1' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math>, the <a class='existingWikiWord' href='/nlab/show/diff/open+subspace'>open subspaces</a> of <math class='maruku-mathml' display='inline' id='mathml_b6adf993a7711dfec3419c83c5b86f2ece9c7b4a_2' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> form a <a class='existingWikiWord' href='/nlab/show/diff/partial+order'>poset</a> which is in fact a <a class='existingWikiWord' href='/nlab/show/diff/frame'>frame</a>. This is the frame of open subspaces of <math class='maruku-mathml' display='inline' id='mathml_b6adf993a7711dfec3419c83c5b86f2ece9c7b4a_3' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math>. When thought of as a <a class='existingWikiWord' href='/nlab/show/diff/locale'>locale</a>, this is the <a class='existingWikiWord' href='/nlab/show/diff/spatial+locale'>topological locale</a> <math class='maruku-mathml' display='inline' id='mathml_b6adf993a7711dfec3419c83c5b86f2ece9c7b4a_4' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Ω</mi><mo stretchy='false'>(</mo><mi>X</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>\Omega(X)</annotation></semantics></math>. When thought of as a <a class='existingWikiWord' href='/nlab/show/diff/category'>category</a>, this is the <a class='existingWikiWord' href='/nlab/show/diff/category+of+open+subsets'>category of open subsets</a> of <math class='maruku-mathml' display='inline' id='mathml_b6adf993a7711dfec3419c83c5b86f2ece9c7b4a_5' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math>. Similarly, given a <a class='existingWikiWord' href='/nlab/show/diff/locale'>locale</a> <math class='maruku-mathml' display='inline' id='mathml_b6adf993a7711dfec3419c83c5b86f2ece9c7b4a_6' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math>, the open subspaces of <math class='maruku-mathml' display='inline' id='mathml_b6adf993a7711dfec3419c83c5b86f2ece9c7b4a_7' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> form a poset which is in fact a frame. This is the frame of open subspaces of <math class='maruku-mathml' display='inline' id='mathml_b6adf993a7711dfec3419c83c5b86f2ece9c7b4a_8' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math>. When thought of as a locale, this is simply <math class='maruku-mathml' display='inline' id='mathml_b6adf993a7711dfec3419c83c5b86f2ece9c7b4a_9' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> all over again. When thought of as a category, this is a <a class='existingWikiWord' href='/nlab/show/diff/site'>site</a> whose <a class='existingWikiWord' href='/nlab/show/diff/category+of+sheaves'>topos of sheaves</a> is a <a class='existingWikiWord' href='/nlab/show/diff/localic+topos'>localic topos</a>.<ins class='diffins'> </ins><ins class='diffins'>The frame of open subsets of the <a class='existingWikiWord' href='/nlab/show/diff/point'>point</a> is given by the power set of a singleton, or more generally by the <a class='existingWikiWord' href='/nlab/show/diff/subobject+classifier'>object of truth values</a> of the ambient topos.</ins> </div> <div class="revisedby"> Last revised on December 30, 2013 at 11:41:24. See the <a href="/nlab/history/frame+of+opens" style="color: #005c19">history</a> of this page for a list of all contributions to it. </div> <div class="navigation navfoot"> <a href="/nlab/edit/frame+of+opens" accesskey="E" class="navlink" id="edit" rel="nofollow">Edit</a><a href="https://nforum.ncatlab.org/discussions/?CategoryID=0">Discuss</a><a href="/nlab/revision/diff/frame+of+opens/5" accesskey="B" class="navlinkbackintime" id="to_previous_revision" rel="nofollow">Previous revision</a><a href="/nlab/show/frame+of+opens" accesskey="C" class="navlink" id="see_changes" rel="nofollow">Hide changes</a><a href="/nlab/history/frame+of+opens" accesskey="S" class="navlink" id="history" rel="nofollow">History (5 revisions)</a> <a href="/nlab/show/frame+of+opens/cite" style="color: black">Cite</a> <a href="/nlab/print/frame+of+opens" accesskey="p" id="view_print" rel="nofollow">Print</a> <a href="/nlab/source/frame+of+opens" id="view_source" rel="nofollow">Source</a> </div> </div>  </div>  </body> </html>