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1.1.4-5.2-10-8.2-13.5-10-11.1-5.2-30-15.3-35-27.3-2.5-6 2.8-13.8 9.4-13.6 6.9.2 13.4 7 17.5 12C70.9 34 75 43.8 86.3 47.4z"/> </svg> nLab topological space (changes) </h1> <div class="navigation"> <a href='#navEnd'>Skip the Navigation Links</a> | <a href="/nlab/show/diff/HomePage" accesskey="H" title="Home page">Home Page</a> | <a href="/nlab/all_pages" accesskey="A" title="List of all pages">All Pages</a> | <a href="/nlab/latest_revisions" accesskey="U" title="Latest edits and page creations">Latest Revisions</a> | <a href="https://nforum.ncatlab.org/discussion/4118/#Item_25" title="Discuss this page in its dedicated thread on the nForum" style="color: black">Discuss this page</a> | <form accept-charset="utf-8" action="/nlab/search" id="navigationSearchForm" method="get"> <fieldset class="search"><input type="text" id="searchField" name="query" value="Search" style="display:inline-block; float: left;" onfocus="this.value == 'Search' ? this.value = '' : true" onblur="this.value == '' ? this.value = 'Search' : true" /></fieldset> </form> </div> <div id="revision"> Showing changes from revision #72 to #73: <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> </div> </div> <h1 id='topological_spaces'>Topological spaces</h1> <div class='maruku_toc'><ul><li><a href='#Idea'>Idea</a></li><li><a href='#Definitions'>Definitions</a><ul><li><a href='#StandardDefinition'>Standard definition</a></li><li><a href='#AlternateDefinitions'>Alternate equivalent definitions</a></li><li><a href='#Variants'>Variations</a></li><li><a href='#DependentTypeTheory'> In dependent type theory</a></li></ul></li><li><a href='#examples'>Examples</a><ul><li><a href='#special_cases'>Special cases</a></li><li><a href='#specific_examples'>Specific examples</a></li></ul></li><li><a href='#related_concepts'>Related concepts</a></li><li><a href='#References'>References</a><ul><li><a href='#ReferencesHistoricalOrigins'>Historical origins</a></li><li><a href='#further'>Further</a></li></ul></li></ul></div> <h2 id='Idea'>Idea</h2> The notion of topological space aims to axiomatize the idea of a <a class='existingWikiWord' href='/nlab/show/diff/space'>space</a> as a collection of <a class='existingWikiWord' href='/nlab/show/diff/point'>points</a> that hang together (“<a class='existingWikiWord' href='/nlab/show/diff/cohesive+topos'>cohere</a>”) in a <a class='existingWikiWord' href='/nlab/show/diff/continuous+map'>continuous</a> way. Roughly speaking, a topology on a <a class='existingWikiWord' href='/nlab/show/diff/set'>set</a> “of points” prescribes which <a class='existingWikiWord' href='/nlab/show/diff/subset'>subsets</a> are to be considered “<a class='existingWikiWord' href='/nlab/show/diff/neighborhood'>neighborhoods</a>” of the points they contain. Various conditions or <a class='existingWikiWord' href='/nlab/show/diff/axiom'>axioms</a> must be satisfied in order for such neighborhood systems to form a topology, but one of the most important is that for any two neighborhoods of a point, their <a class='existingWikiWord' href='/nlab/show/diff/intersection'>intersection</a> must also be a neighborhood of that point. Many notions of <a class='existingWikiWord' href='/nlab/show/diff/space'>spaces</a> used in <a class='existingWikiWord' href='/nlab/show/diff/mathematics'>mathematics</a> have <a class='existingWikiWord' href='/nlab/show/diff/underlying+topological+space'>underlying topological spaces</a>, such as: <a class='existingWikiWord' href='/nlab/show/diff/manifold'>manifolds</a>, <a class='existingWikiWord' href='/nlab/show/diff/scheme'>schemes</a>, <a class='existingWikiWord' href='/nlab/show/diff/probability+space'>probability spaces</a>, etc. The concept of a topology, gradually refined over the latter half of the 19th century and the first two decades of the 20th, was developed to capture what it means abstractly for a mapping between sets of points to be “<a class='existingWikiWord' href='/nlab/show/diff/continuous+map'>continuous</a>”. Intuitively, the idea of bending, twisting or crumpling a continuous body applies to continuous mappings, because they preserve neighborhood relations (in a suitable sense), but tearing, for instance, does not. For example, the surface of a <a class='existingWikiWord' href='/nlab/show/diff/torus'>torus</a> or doughnut is topologically equivalent to the surface of a mug: the surface of the mug can be deformed continuously into the surface of a torus. Abstractly speaking: the continuous <a class='existingWikiWord' href='/nlab/show/diff/cohesive'>cohesion</a> among the collections of points of the two surfaces is the same. Similarly, a <a class='existingWikiWord' href='/nlab/show/diff/circle'>circle</a> and a <a class='existingWikiWord' href='/nlab/show/diff/square'>square</a> are considered equivalent from the standpoint of their topologies. Some one-dimensional shapes with different topologies: the Mercedes-Benz symbol, a line, a circle, a complete graph with 5 nodes, the skeleton of a cube, and an asterisk (or, if you’ll permit the one-dimensional approximation, a starfish). On the other hand, a circle has the same topology as a line segment with a wormhole at its finish which teleports you to its start; or more prosaically: The <a class='existingWikiWord' href='/nlab/show/diff/circle'>circle</a> is <a class='existingWikiWord' href='/nlab/show/diff/homeomorphism'>homeomorphic</a> to the <a class='existingWikiWord' href='/nlab/show/diff/interval'>closed interval</a> with endpoints identified. There is a generalization of the notion of topological spaces to that of <a class='existingWikiWord' href='/nlab/show/diff/locale'>locales</a>, which consists of dropping the assumption that all <a class='existingWikiWord' href='/nlab/show/diff/neighborhood'>neighbourhoods</a> are explicitly or even necessarily supported by points. For this reason, the theory of locales is sometimes called “pointless topology”. In this form, the definition turns out to be quite fundamental and can be naturally motivated from just pure <a class='existingWikiWord' href='/nlab/show/diff/logic'>logic</a> – as the formal dual of <a class='existingWikiWord' href='/nlab/show/diff/frame'>frames</a> – as well as, and <a class='existingWikiWord' href='/nlab/show/diff/duality'>dually</a>, from <a class='existingWikiWord' href='/nlab/show/diff/category+theory'>category theory</a> in its variant as <a class='existingWikiWord' href='/nlab/show/diff/sheaf+and+topos+theory'>topos theory</a> – by the notion of <a class='existingWikiWord' href='/nlab/show/diff/%280%2C1%29-topos'>(0,1)-toposes</a>. Topological spaces are the objects studied in <a class='existingWikiWord' href='/nlab/show/diff/topology'>topology</a>. But types of topological spaces exist in such great and wild profusion that in practice it is often more convenient to replace strict topological equivalence by a notion of <a class='existingWikiWord' href='/nlab/show/diff/weak+equivalence'>weak equivalence</a>, namely of <a class='existingWikiWord' href='/nlab/show/diff/weak+homotopy+equivalence'>weak homotopy equivalence</a>. From this point of view, topological spaces support also <a class='existingWikiWord' href='/nlab/show/diff/homotopy+theory'>homotopy theory</a>. Topological spaces equipped with extra <a class='existingWikiWord' href='/nlab/show/diff/stuff%2C+structure%2C+property'>properties</a> and <a class='existingWikiWord' href='/nlab/show/diff/structure'>structure</a> form the fundament of much of <a class='existingWikiWord' href='/nlab/show/diff/geometry'>geometry</a>. For instance a topological space locally isomorphic to a <a class='existingWikiWord' href='/nlab/show/diff/cartesian+space'>Cartesian space</a> is a <a class='existingWikiWord' href='/nlab/show/diff/manifold'>manifold</a>. A topological space equipped with a notion of <a class='existingWikiWord' href='/nlab/show/diff/smooth+map'>smooth functions</a> into it is a <a class='existingWikiWord' href='/nlab/show/diff/diffeological+space'>diffeological space</a>. The intersection of these two notions is that of a <a class='existingWikiWord' href='/nlab/show/diff/smooth+manifold'>smooth manifold</a> on which <a class='existingWikiWord' href='/nlab/show/diff/differential+geometry'>differential geometry</a> is based. And so on. <h2 id='Definitions'>Definitions</h2> We present first the <ul> <li><a href='#StandardDefinition'>standard definition</a></li> </ul> and then a list of different <ul> <li><a href='#AlternateDefinitions'>equivalent definitions</a>.</li> </ul> Finally we mention genuine <ul> <li><a href='#Variants'>variants of the notion</a>.</li> </ul> <h3 id='StandardDefinition'>Standard definition</h3> \begin{definition}\label{TheStandardDefinition} A topological space is a <a class='existingWikiWord' href='/nlab/show/diff/set'>set</a> <math class='maruku-mathml' display='inline' id='mathml_7bf8dc58f8c6751947ec2d84d79ad90256896f51_1' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> equipped with a set of <a class='existingWikiWord' href='/nlab/show/diff/subset'>subsets</a> <math class='maruku-mathml' display='inline' id='mathml_7bf8dc58f8c6751947ec2d84d79ad90256896f51_2' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>U</mi><mo>⊂</mo><mi>X</mi></mrow><annotation encoding='application/x-tex'>U \subset X</annotation></semantics></math>, called the <a class='existingWikiWord' href='/nlab/show/diff/open+subspace'>open sets</a>, which are closed under <ol> <li> finite <a class='existingWikiWord' href='/nlab/show/diff/intersection'>intersections</a>, </li> <li> arbitrary <a class='existingWikiWord' href='/nlab/show/diff/union'>unions</a>. </li> </ol> \end{definition} <div class='num_remark'> <h6 id='remark'>Remark</h6> The word ‘topology’ sometimes means the <a class='existingWikiWord' href='/nlab/show/diff/topology'>study of topological spaces</a> but here it means the collection of open sets in a topological space. In particular, if someone says ‘Let <math class='maruku-mathml' display='inline' id='mathml_7bf8dc58f8c6751947ec2d84d79ad90256896f51_3' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>T</mi></mrow><annotation encoding='application/x-tex'>T</annotation></semantics></math> be a topology on <math class='maruku-mathml' display='inline' id='mathml_7bf8dc58f8c6751947ec2d84d79ad90256896f51_4' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math>’, then they mean ‘Let <math class='maruku-mathml' display='inline' id='mathml_7bf8dc58f8c6751947ec2d84d79ad90256896f51_5' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> be equipped with the structure of a topological space, and let <math class='maruku-mathml' display='inline' id='mathml_7bf8dc58f8c6751947ec2d84d79ad90256896f51_6' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>T</mi></mrow><annotation encoding='application/x-tex'>T</annotation></semantics></math> be the collection of open sets in this space’. </div> <div class='num_remark'> <h6 id='remark_2'>Remark</h6> Since <math class='maruku-mathml' display='inline' id='mathml_7bf8dc58f8c6751947ec2d84d79ad90256896f51_7' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> itself is the intersection of zero subsets, it is open, and since the <a class='existingWikiWord' href='/nlab/show/diff/empty+set'>empty set</a> <math class='maruku-mathml' display='inline' id='mathml_7bf8dc58f8c6751947ec2d84d79ad90256896f51_8' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>∅</mi></mrow><annotation encoding='application/x-tex'>\emptyset</annotation></semantics></math> is the union of zero subsets, it is also open. Moreover, every open subset <math class='maruku-mathml' display='inline' id='mathml_7bf8dc58f8c6751947ec2d84d79ad90256896f51_9' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>U</mi></mrow><annotation encoding='application/x-tex'>U</annotation></semantics></math> of <math class='maruku-mathml' display='inline' id='mathml_7bf8dc58f8c6751947ec2d84d79ad90256896f51_10' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> contains the empty set and is contained in <math class='maruku-mathml' display='inline' id='mathml_7bf8dc58f8c6751947ec2d84d79ad90256896f51_11' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_7bf8dc58f8c6751947ec2d84d79ad90256896f51_12' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>∅</mi><mo>⊂</mo><mi>U</mi><mo>⊂</mo><mi>X</mi><mspace width='thinmathspace' /><mo>,</mo></mrow><annotation encoding='application/x-tex'> \emptyset \subset U \subset X \,, </annotation></semantics></math></div> so that the topology of <math class='maruku-mathml' display='inline' id='mathml_7bf8dc58f8c6751947ec2d84d79ad90256896f51_13' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> is determined by a <a class='existingWikiWord' href='/nlab/show/diff/category+of+open+subsets'>poset of open subsets</a> <math class='maruku-mathml' display='inline' id='mathml_7bf8dc58f8c6751947ec2d84d79ad90256896f51_14' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Op</mi><mo stretchy='false'>(</mo><mi>X</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>Op(X)</annotation></semantics></math> with <a class='existingWikiWord' href='/nlab/show/diff/bottom'>bottom</a> element <math class='maruku-mathml' display='inline' id='mathml_7bf8dc58f8c6751947ec2d84d79ad90256896f51_15' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo>⊥</mo><mo>=</mo><mi>∅</mi></mrow><annotation encoding='application/x-tex'>\bot = \emptyset</annotation></semantics></math> and <a class='existingWikiWord' href='/nlab/show/diff/top'>top</a> element <math class='maruku-mathml' display='inline' id='mathml_7bf8dc58f8c6751947ec2d84d79ad90256896f51_16' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo>⊤</mo><mo>=</mo><mi>X</mi></mrow><annotation encoding='application/x-tex'>\top = X</annotation></semantics></math>. Since by definition the elements in this <a class='existingWikiWord' href='/nlab/show/diff/partial+order'>poset</a> are closed under finite <a class='existingWikiWord' href='/nlab/show/diff/meet'>meets</a> (intersection) and arbitrary <a class='existingWikiWord' href='/nlab/show/diff/join'>joins</a> (unions), this poset of open subsets defining a topology is a <a class='existingWikiWord' href='/nlab/show/diff/frame'>frame</a>, the <a class='existingWikiWord' href='/nlab/show/diff/frame+of+opens'>frame of opens</a> of <math class='maruku-mathml' display='inline' id='mathml_7bf8dc58f8c6751947ec2d84d79ad90256896f51_17' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math>. </div> <div class='num_defn'> <h6 id='definition'>Definition</h6> A <a class='existingWikiWord' href='/nlab/show/diff/homomorphism'>homomorphism</a> between topological spaces <math class='maruku-mathml' display='inline' id='mathml_7bf8dc58f8c6751947ec2d84d79ad90256896f51_18' 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>: a <a class='existingWikiWord' href='/nlab/show/diff/function'>function</a> <math class='maruku-mathml' display='inline' id='mathml_7bf8dc58f8c6751947ec2d84d79ad90256896f51_19' 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> of the underlying <a class='existingWikiWord' href='/nlab/show/diff/set'>sets</a> such that the <a class='existingWikiWord' href='/nlab/show/diff/preimage'>preimage</a> of every open set of <math class='maruku-mathml' display='inline' id='mathml_7bf8dc58f8c6751947ec2d84d79ad90256896f51_20' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Y</mi></mrow><annotation encoding='application/x-tex'>Y</annotation></semantics></math> is an open set of <math class='maruku-mathml' display='inline' id='mathml_7bf8dc58f8c6751947ec2d84d79ad90256896f51_21' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math>. </div> Topological spaces with continuous maps between them form a <a class='existingWikiWord' href='/nlab/show/diff/category'>category</a>, usually denoted <a class='existingWikiWord' href='/nlab/show/diff/Top'>Top</a>. <div class='num_remark'> <h6 id='remark_3'>Remark</h6> The definition of <a class='existingWikiWord' href='/nlab/show/diff/continuous+map'>continuous function</a> <math class='maruku-mathml' display='inline' id='mathml_7bf8dc58f8c6751947ec2d84d79ad90256896f51_22' 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 such that it induces a homomorphism of the corresponding <a class='existingWikiWord' href='/nlab/show/diff/frame+of+opens'>frames of opens</a> the other way around <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_7bf8dc58f8c6751947ec2d84d79ad90256896f51_23' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Op</mi><mo stretchy='false'>(</mo><mi>X</mi><mo stretchy='false'>)</mo><mo>←</mo><mi>Op</mi><mo stretchy='false'>(</mo><mi>Y</mi><mo stretchy='false'>)</mo><mo>:</mo><msup><mi>f</mi> <mrow><mo lspace='verythinmathspace' rspace='0em'>−</mo><mn>1</mn></mrow></msup><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> Op(X) \leftarrow Op(Y) : f^{-1} \,. </annotation></semantics></math></div> And this is not just a morphism of <a class='existingWikiWord' href='/nlab/show/diff/partial+order'>posets</a> but even of <a class='existingWikiWord' href='/nlab/show/diff/frame'>frames</a>. For more on this see at <a class='existingWikiWord' href='/nlab/show/diff/locale'>locale</a>. </div> <h3 id='AlternateDefinitions'>Alternate equivalent definitions</h3> There are many equivalent ways to define a topological space. A non-exhaustive list follows: <ul> <li> A set <math class='maruku-mathml' display='inline' id='mathml_7bf8dc58f8c6751947ec2d84d79ad90256896f51_24' 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 <a class='existingWikiWord' href='/nlab/show/diff/frame'>frame</a> of open sets (the standard definition, given above), called a topology on <math class='maruku-mathml' display='inline' id='mathml_7bf8dc58f8c6751947ec2d84d79ad90256896f51_25' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math>. </li> <li> A set <math class='maruku-mathml' display='inline' id='mathml_7bf8dc58f8c6751947ec2d84d79ad90256896f51_26' 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 co-frame of <a class='existingWikiWord' href='/nlab/show/diff/closed+subspace'>closed sets</a> (the complements of the open sets), satisfying dual axioms: closure under finite unions and arbitrary intersections. This is sometimes called a co-topology on <math class='maruku-mathml' display='inline' id='mathml_7bf8dc58f8c6751947ec2d84d79ad90256896f51_27' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math>. </li> <li> A pair <math class='maruku-mathml' display='inline' id='mathml_7bf8dc58f8c6751947ec2d84d79ad90256896f51_28' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mi>X</mi><mo>,</mo><mi>int</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(X, int)</annotation></semantics></math>, where <math class='maruku-mathml' display='inline' id='mathml_7bf8dc58f8c6751947ec2d84d79ad90256896f51_29' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>int</mi><mo lspace='verythinmathspace'>:</mo><mi>P</mi><mo stretchy='false'>(</mo><mi>X</mi><mo stretchy='false'>)</mo><mo>→</mo><mi>P</mi><mo stretchy='false'>(</mo><mi>X</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>int\colon P(X) \to P(X)</annotation></semantics></math> is a <a class='existingWikiWord' href='/nlab/show/diff/exact+functor'>left exact</a> <a class='existingWikiWord' href='/nlab/show/diff/comonad'>comonad</a> on the <a class='existingWikiWord' href='/nlab/show/diff/power+set'>power set</a> of <math class='maruku-mathml' display='inline' id='mathml_7bf8dc58f8c6751947ec2d84d79ad90256896f51_30' 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/interior'>interior</a> operator”). In more nuts-and-bolts terms, this means for all subsets <math class='maruku-mathml' display='inline' id='mathml_7bf8dc58f8c6751947ec2d84d79ad90256896f51_31' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>A</mi><mo>,</mo><mi>B</mi></mrow><annotation encoding='application/x-tex'>A, B</annotation></semantics></math> of <math class='maruku-mathml' display='inline' id='mathml_7bf8dc58f8c6751947ec2d84d79ad90256896f51_32' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> we have <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_7bf8dc58f8c6751947ec2d84d79ad90256896f51_33' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>A</mi><mo>⊆</mo><mi>B</mi><mo>⇒</mo><mi>int</mi><mo stretchy='false'>(</mo><mi>A</mi><mo stretchy='false'>)</mo><mo>⊆</mo><mi>int</mi><mo stretchy='false'>(</mo><mi>B</mi><mo stretchy='false'>)</mo><mo>,</mo><mspace width='thickmathspace' /><mspace width='thickmathspace' /><mspace width='thickmathspace' /><mi>int</mi><mo stretchy='false'>(</mo><mi>A</mi><mo stretchy='false'>)</mo><mo>⊆</mo><mi>A</mi><mo>,</mo><mspace width='thickmathspace' /><mspace width='thickmathspace' /><mspace width='thickmathspace' /><mi>int</mi><mo stretchy='false'>(</mo><mi>A</mi><mo stretchy='false'>)</mo><mo>⊆</mo><mi>int</mi><mo stretchy='false'>(</mo><mi>int</mi><mo stretchy='false'>(</mo><mi>A</mi><mo stretchy='false'>)</mo><mo stretchy='false'>)</mo><mo>,</mo><mspace width='thickmathspace' /><mspace width='thickmathspace' /><mspace width='thickmathspace' /><mi>int</mi><mo stretchy='false'>(</mo><mi>A</mi><mo>∩</mo><mi>B</mi><mo stretchy='false'>)</mo><mo>=</mo><mi>int</mi><mo stretchy='false'>(</mo><mi>A</mi><mo stretchy='false'>)</mo><mo>∩</mo><mi>int</mi><mo stretchy='false'>(</mo><mi>B</mi><mo stretchy='false'>)</mo><mo>,</mo><mspace width='thickmathspace' /><mspace width='thickmathspace' /><mspace width='thickmathspace' /><mi>int</mi><mo stretchy='false'>(</mo><mi>X</mi><mo stretchy='false'>)</mo><mo>=</mo><mi>X</mi><mo>.</mo></mrow><annotation encoding='application/x-tex'>A \subseteq B \Rightarrow int(A) \subseteq int(B), \;\;\; int(A) \subseteq A, \;\;\; int(A) \subseteq int(int(A)), \;\;\; int(A \cap B) = int(A) \cap int(B), \;\;\; int(X) = X.</annotation></semantics></math></div> The open sets are exactly the fixed points of <math class='maruku-mathml' display='inline' id='mathml_7bf8dc58f8c6751947ec2d84d79ad90256896f51_34' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>int</mi></mrow><annotation encoding='application/x-tex'>int</annotation></semantics></math>. The first three of these conditions say <math class='maruku-mathml' display='inline' id='mathml_7bf8dc58f8c6751947ec2d84d79ad90256896f51_35' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>int</mi></mrow><annotation encoding='application/x-tex'>int</annotation></semantics></math> is a <a class='existingWikiWord' href='/nlab/show/diff/closure+operator'>coclosure operator</a>. </li> <li> A pair <math class='maruku-mathml' display='inline' id='mathml_7bf8dc58f8c6751947ec2d84d79ad90256896f51_36' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mi>X</mi><mo>,</mo><mi>cl</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(X, cl)</annotation></semantics></math> where <math class='maruku-mathml' display='inline' id='mathml_7bf8dc58f8c6751947ec2d84d79ad90256896f51_37' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>cl</mi></mrow><annotation encoding='application/x-tex'>cl</annotation></semantics></math> is a <a class='existingWikiWord' href='/nlab/show/diff/exact+functor'>right exact</a> <a class='existingWikiWord' href='/nlab/show/diff/Moore+closure'>Moore closure</a> operator satisfying axioms dual to those of <math class='maruku-mathml' display='inline' id='mathml_7bf8dc58f8c6751947ec2d84d79ad90256896f51_38' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>int</mi></mrow><annotation encoding='application/x-tex'>int</annotation></semantics></math>. The closed sets are the fixed points of <math class='maruku-mathml' display='inline' id='mathml_7bf8dc58f8c6751947ec2d84d79ad90256896f51_39' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>cl</mi></mrow><annotation encoding='application/x-tex'>cl</annotation></semantics></math>. Such an operator is sometimes called a Kuratowski closure operator (compare Kuratowski’s closure-complement problem at <a class='existingWikiWord' href='/nlab/show/diff/closed+subspace'>closed subspace</a>). </li> <li> A set <math class='maruku-mathml' display='inline' id='mathml_7bf8dc58f8c6751947ec2d84d79ad90256896f51_40' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> together with, for each <math class='maruku-mathml' display='inline' id='mathml_7bf8dc58f8c6751947ec2d84d79ad90256896f51_41' 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>, a <a class='existingWikiWord' href='/nlab/show/diff/filter'>filter</a> <math class='maruku-mathml' display='inline' id='mathml_7bf8dc58f8c6751947ec2d84d79ad90256896f51_42' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>N</mi> <mi>x</mi></msub></mrow><annotation encoding='application/x-tex'>N_x</annotation></semantics></math> on <math class='maruku-mathml' display='inline' id='mathml_7bf8dc58f8c6751947ec2d84d79ad90256896f51_43' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math>, i.e., a collection of inhabited subsets of <math class='maruku-mathml' display='inline' id='mathml_7bf8dc58f8c6751947ec2d84d79ad90256896f51_44' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> closed under finite intersections and also upward-closed (<math class='maruku-mathml' display='inline' id='mathml_7bf8dc58f8c6751947ec2d84d79ad90256896f51_45' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>U</mi><mo>∈</mo><msub><mi>N</mi> <mi>x</mi></msub></mrow><annotation encoding='application/x-tex'>U \in N_x</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_7bf8dc58f8c6751947ec2d84d79ad90256896f51_46' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>U</mi><mo>⊆</mo><mi>V</mi></mrow><annotation encoding='application/x-tex'>U \subseteq V</annotation></semantics></math> together imply <math class='maruku-mathml' display='inline' id='mathml_7bf8dc58f8c6751947ec2d84d79ad90256896f51_47' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>V</mi><mo>⊆</mo><msub><mi>N</mi> <mi>x</mi></msub></mrow><annotation encoding='application/x-tex'>V \subseteq N_x</annotation></semantics></math>). If <math class='maruku-mathml' display='inline' id='mathml_7bf8dc58f8c6751947ec2d84d79ad90256896f51_48' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>U</mi><mo>∈</mo><msub><mi>N</mi> <mi>x</mi></msub></mrow><annotation encoding='application/x-tex'>U \in N_x</annotation></semantics></math>, we call <math class='maruku-mathml' display='inline' id='mathml_7bf8dc58f8c6751947ec2d84d79ad90256896f51_49' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>U</mi></mrow><annotation encoding='application/x-tex'>U</annotation></semantics></math> a neighborhood of <math class='maruku-mathml' display='inline' id='mathml_7bf8dc58f8c6751947ec2d84d79ad90256896f51_50' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>x</mi></mrow><annotation encoding='application/x-tex'>x</annotation></semantics></math>. The remaining conditions on these neighborhood systems are that <math class='maruku-mathml' display='inline' id='mathml_7bf8dc58f8c6751947ec2d84d79ad90256896f51_51' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>x</mi><mo>∈</mo><mi>U</mi></mrow><annotation encoding='application/x-tex'>x \in U</annotation></semantics></math> for every <math class='maruku-mathml' display='inline' id='mathml_7bf8dc58f8c6751947ec2d84d79ad90256896f51_52' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>U</mi><mo>∈</mo><msub><mi>N</mi> <mi>x</mi></msub></mrow><annotation encoding='application/x-tex'>U \in N_x</annotation></semantics></math>, and that for every <math class='maruku-mathml' display='inline' id='mathml_7bf8dc58f8c6751947ec2d84d79ad90256896f51_53' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>U</mi><mo>∈</mo><msub><mi>N</mi> <mi>x</mi></msub></mrow><annotation encoding='application/x-tex'>U \in N_x</annotation></semantics></math>, there exists <math class='maruku-mathml' display='inline' id='mathml_7bf8dc58f8c6751947ec2d84d79ad90256896f51_54' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>V</mi><mo>∈</mo><msub><mi>N</mi> <mi>x</mi></msub></mrow><annotation encoding='application/x-tex'>V \in N_x</annotation></semantics></math> such that <math class='maruku-mathml' display='inline' id='mathml_7bf8dc58f8c6751947ec2d84d79ad90256896f51_55' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>V</mi><mo>⊆</mo><mi>U</mi></mrow><annotation encoding='application/x-tex'>V \subseteq U</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_7bf8dc58f8c6751947ec2d84d79ad90256896f51_56' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>V</mi></mrow><annotation encoding='application/x-tex'>V</annotation></semantics></math> is a neighborhood of each point it contains. In this formulation, a subset <math class='maruku-mathml' display='inline' id='mathml_7bf8dc58f8c6751947ec2d84d79ad90256896f51_57' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>U</mi><mo>⊆</mo><mi>X</mi></mrow><annotation encoding='application/x-tex'>U \subseteq X</annotation></semantics></math> is open if it is a neighborhood of every point it contains. </li> </ul> The next two definitions of topological space are at a higher level of abstraction, but the underlying idea that connects them with the neighborhood system formulation is that we say a filter <math class='maruku-mathml' display='inline' id='mathml_7bf8dc58f8c6751947ec2d84d79ad90256896f51_58' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>F</mi></mrow><annotation encoding='application/x-tex'>F</annotation></semantics></math> on <math class='maruku-mathml' display='inline' id='mathml_7bf8dc58f8c6751947ec2d84d79ad90256896f51_59' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> converges to a point <math class='maruku-mathml' display='inline' id='mathml_7bf8dc58f8c6751947ec2d84d79ad90256896f51_60' 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> if <math class='maruku-mathml' display='inline' id='mathml_7bf8dc58f8c6751947ec2d84d79ad90256896f51_61' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>N</mi> <mi>x</mi></msub><mo>⊆</mo><mi>F</mi></mrow><annotation encoding='application/x-tex'>N_x \subseteq F</annotation></semantics></math>. The point then is to characterize properties of convergence abstractly. <ul> <li> A <a class='existingWikiWord' href='/nlab/show/diff/relational+beta-module'>relational beta-module</a>; that is, a <a class='existingWikiWord' href='/nlab/show/diff/lax+algebra+for+a+2-monad'>lax algebra</a> of the <a class='existingWikiWord' href='/nlab/show/diff/monad'>monad</a> <math class='maruku-mathml' display='inline' id='mathml_7bf8dc58f8c6751947ec2d84d79ad90256896f51_62' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>β</mi></mrow><annotation encoding='application/x-tex'>\beta</annotation></semantics></math> of <a class='existingWikiWord' href='/nlab/show/diff/ultrafilter'>ultrafilters</a> on the <a class='existingWikiWord' href='/nlab/show/diff/2-poset'>(1,2)-category</a> <a class='existingWikiWord' href='/nlab/show/diff/Rel'>Rel</a> of sets and <a class='existingWikiWord' href='/nlab/show/diff/relation'>binary relations</a>. More explicitly, this means a set <math class='maruku-mathml' display='inline' id='mathml_7bf8dc58f8c6751947ec2d84d79ad90256896f51_63' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> together with a relation called “<a class='existingWikiWord' href='/nlab/show/diff/convergence'>convergence</a>” between ultrafilters and points satisfying certain axioms. This exhibits it as a special sort of <a class='existingWikiWord' href='/nlab/show/diff/generalized+multicategory'>generalized multicategory</a>, and also as a special sort of <a class='existingWikiWord' href='/nlab/show/diff/pseudotopological+space'>pseudotopological space</a>. However, the equivalence of this definition to the traditional definition of a topological space requires the <a class='existingWikiWord' href='/nlab/show/diff/ultrafilter+theorem'>ultrafilter principle</a> to be true. </li> <li> A set with a <a class='existingWikiWord' href='/nlab/show/diff/convergence+space'>convergence relation</a> between <a class='existingWikiWord' href='/nlab/show/diff/net'>nets</a> or <a class='existingWikiWord' href='/nlab/show/diff/filter'>filters</a> (not just ultrafilters) and points, or even between transfinite <a class='existingWikiWord' href='/nlab/show/diff/sequence'>sequences</a> and points, satisfying appropriate axioms. </li> </ul> The following are not definitions, but they provide alternative ways to present a topological space. <ul> <li>A <a class='existingWikiWord' href='/nlab/show/diff/topological+base'>(topological) base</a> on a set <math class='maruku-mathml' display='inline' id='mathml_7bf8dc58f8c6751947ec2d84d79ad90256896f51_64' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> is a collection <math class='maruku-mathml' display='inline' id='mathml_7bf8dc58f8c6751947ec2d84d79ad90256896f51_65' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>ℬ</mi></mrow><annotation encoding='application/x-tex'>\mathcal{B}</annotation></semantics></math> of subsets of <math class='maruku-mathml' display='inline' id='mathml_7bf8dc58f8c6751947ec2d84d79ad90256896f51_66' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> whose union is all of <math class='maruku-mathml' display='inline' id='mathml_7bf8dc58f8c6751947ec2d84d79ad90256896f51_67' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math>, and such that whenever <math class='maruku-mathml' display='inline' id='mathml_7bf8dc58f8c6751947ec2d84d79ad90256896f51_68' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>B</mi><mo>,</mo><mi>C</mi><mo>∈</mo><mi>ℬ</mi></mrow><annotation encoding='application/x-tex'>B, C \in \mathcal{B}</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_7bf8dc58f8c6751947ec2d84d79ad90256896f51_69' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>x</mi><mo>∈</mo><mi>B</mi><mo>∩</mo><mi>C</mi></mrow><annotation encoding='application/x-tex'>x \in B \cap C</annotation></semantics></math>, there exists <math class='maruku-mathml' display='inline' id='mathml_7bf8dc58f8c6751947ec2d84d79ad90256896f51_70' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>D</mi><mo>∈</mo><mi>ℬ</mi></mrow><annotation encoding='application/x-tex'>D \in \mathcal{B}</annotation></semantics></math> such that <math class='maruku-mathml' display='inline' id='mathml_7bf8dc58f8c6751947ec2d84d79ad90256896f51_71' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>D</mi><mo>⊆</mo><mi>B</mi><mo>∩</mo><mi>C</mi></mrow><annotation encoding='application/x-tex'>D \subseteq B \cap C</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_7bf8dc58f8c6751947ec2d84d79ad90256896f51_72' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>x</mi><mo>∈</mo><mi>D</mi></mrow><annotation encoding='application/x-tex'>x \in D</annotation></semantics></math>.</li> </ul> If <math class='maruku-mathml' display='inline' id='mathml_7bf8dc58f8c6751947ec2d84d79ad90256896f51_73' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>ℬ</mi></mrow><annotation encoding='application/x-tex'>\mathcal{B}</annotation></semantics></math> is a base on <math class='maruku-mathml' display='inline' id='mathml_7bf8dc58f8c6751947ec2d84d79ad90256896f51_74' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math>, then it is easily shown that the collection of all unions of subcollections of <math class='maruku-mathml' display='inline' id='mathml_7bf8dc58f8c6751947ec2d84d79ad90256896f51_75' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>ℬ</mi></mrow><annotation encoding='application/x-tex'>\mathcal{B}</annotation></semantics></math> is a topology on <math class='maruku-mathml' display='inline' id='mathml_7bf8dc58f8c6751947ec2d84d79ad90256896f51_76' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math>. <ul> <li>A set <math class='maruku-mathml' display='inline' id='mathml_7bf8dc58f8c6751947ec2d84d79ad90256896f51_77' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> with any collection of subsets whatsoever, to be thought of as a <a class='existingWikiWord' href='/nlab/show/diff/base'>subbase</a> for a topology.</li> </ul> From the fact that the intersection of any collection of topologies is also a topology, there is a smallest topology that contains a given subbase <math class='maruku-mathml' display='inline' id='mathml_7bf8dc58f8c6751947ec2d84d79ad90256896f51_78' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>𝒮</mi></mrow><annotation encoding='application/x-tex'>\mathcal{S}</annotation></semantics></math>. It consists of all possible unions of all possible finite intersections of members of <math class='maruku-mathml' display='inline' id='mathml_7bf8dc58f8c6751947ec2d84d79ad90256896f51_79' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>𝒮</mi></mrow><annotation encoding='application/x-tex'>\mathcal{S}</annotation></semantics></math>. This is called the topology generated by the subbase. <h3 id='Variants'>Variations</h3> Historically, the notion of topological space (see the historical references given <a href='topology#ReferencesHistoricalOrigins'>there</a>) involving <a class='existingWikiWord' href='/nlab/show/diff/neighborhood'>neighbourhoods</a> was first developed by <a class='existingWikiWord' href='/nlab/show/diff/Felix+Hausdorff'>Felix Hausdorff</a> in 1914 in his seminal text on <a class='existingWikiWord' href='/nlab/show/diff/set+theory'>set theory</a> and <a class='existingWikiWord' href='/nlab/show/diff/topology'>topology</a>, Fundamentals of Set Theory (<a class='existingWikiWord' href='/nlab/show/diff/Grundz%C3%BCge+der+Mengenlehre'>Grundzüge der Mengenlehre</a>). Hausdorff’s definition originally contained the <math class='maruku-mathml' display='inline' id='mathml_7bf8dc58f8c6751947ec2d84d79ad90256896f51_80' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>T</mi> <mn>2</mn></msub></mrow><annotation encoding='application/x-tex'>T_2</annotation></semantics></math>-<a class='existingWikiWord' href='/nlab/show/diff/separation+axioms'>separation axiom</a> (now known as the definition of <a class='existingWikiWord' href='/nlab/show/diff/Hausdorff+space'>Hausdorff spaces</a>). This axiom was in effect removed by <a class='existingWikiWord' href='/nlab/show/diff/Kazimierz+Kuratowski'>Kazimierz Kuratowski</a> in 1922, who defined general topological spaces in terms of <a class='existingWikiWord' href='/nlab/show/diff/closure+operator'>closure operators</a> that preserve finite <a class='existingWikiWord' href='/nlab/show/diff/union'>unions</a>. The usual <a class='existingWikiWord' href='/nlab/show/diff/open+subspace'>open set</a> formulation was widely popularized by Bourbaki in their 1940 treatise (without identifying a single author behind this notion). However, in more modern treatments that emphasize <a class='existingWikiWord' href='/nlab/show/diff/category+theory'>category theoretic</a> methods, particularly to address needs of <a class='existingWikiWord' href='/nlab/show/diff/homotopy+theory'>homotopy theory</a>, it becomes important to consider not just the category <a class='existingWikiWord' href='/nlab/show/diff/Top'>Top</a> of all topological spaces, but <a class='existingWikiWord' href='/nlab/show/diff/convenient+category+of+topological+spaces'>convenient categories of topological spaces</a> that are better behaved, especially with regard to function spaces and <a class='existingWikiWord' href='/nlab/show/diff/cartesian+closed+category'>cartesian closure</a>. Thus many texts work with <a class='existingWikiWord' href='/nlab/show/diff/nice+topological+space'>nice topological spaces</a> (such as <a class='existingWikiWord' href='/nlab/show/diff/sequential+topological+space'>sequential topological spaces</a>) and/or a <a class='existingWikiWord' href='/nlab/show/diff/nice+category+of+spaces'>nice-</a> or <a class='existingWikiWord' href='/nlab/show/diff/convenient+category+of+topological+spaces'>convenient category of topological spaces</a> (such as <a class='existingWikiWord' href='/nlab/show/diff/compactly+generated+topological+space'>compactly generated spaces</a>), or indeed to directly use a model of <math class='maruku-mathml' display='inline' id='mathml_7bf8dc58f8c6751947ec2d84d79ad90256896f51_81' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>∞</mn></mrow><annotation encoding='application/x-tex'>\infty</annotation></semantics></math>-<a class='existingWikiWord' href='/nlab/show/diff/infinity-groupoid'>groupoids</a> (such as <a class='existingWikiWord' href='/nlab/show/diff/simplicial+set'>simplicial sets</a>). On the other hand, when doing <a class='existingWikiWord' href='/nlab/show/diff/sheaf+and+topos+theory'>topos theory</a> or working in <a class='existingWikiWord' href='/nlab/show/diff/constructive+mathematics'>constructive mathematics</a>, it is often more appropriate to use <a class='existingWikiWord' href='/nlab/show/diff/locale'>locales</a> than topological spaces. Some applications to <a class='existingWikiWord' href='/nlab/show/diff/analysis'>analysis</a> require more general <a class='existingWikiWord' href='/nlab/show/diff/convergence+space'>convergence spaces</a> or other generalisations. There are also <a class='existingWikiWord' href='/nlab/show/diff/unified+topological+space'>unified topological spaces</a> which contains notions of apartness and nearness in addition to the usual notion of neighbourhood in a topological space. In <a class='existingWikiWord' href='/nlab/show/diff/dependent+type+theory'>dependent type theory</a>, one could also have a <a class='existingWikiWord' href='/nlab/show/diff/topological+space'>topological space</a> be a general <a class='existingWikiWord' href='/nlab/show/diff/type'>type</a> instead of an <a class='existingWikiWord' href='/nlab/show/diff/h-set'>h-set</a>. For most kinds of topological spaces in dependent type theory, the <a class='existingWikiWord' href='/nlab/show/diff/Kolmogorov+topological+space'>$T_0$</a>-<a class='existingWikiWord' href='/nlab/show/diff/separation+axioms'>separation axiom</a> forces the type to be an h-set. <h3 id='DependentTypeTheory'> In dependent type theory</h3> In <a class='existingWikiWord' href='/nlab/show/diff/dependent+type+theory'>dependent type theory</a>, given a type <math class='maruku-mathml' display='inline' id='mathml_7bf8dc58f8c6751947ec2d84d79ad90256896f51_82' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math>, the type of all <a class='existingWikiWord' href='/nlab/show/diff/subtype'>subtypes</a> of <math class='maruku-mathml' display='inline' id='mathml_7bf8dc58f8c6751947ec2d84d79ad90256896f51_83' 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/power+set'>powerset</a> of <math class='maruku-mathml' display='inline' id='mathml_7bf8dc58f8c6751947ec2d84d79ad90256896f51_84' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math>, is defined as the <a class='existingWikiWord' href='/nlab/show/diff/function+type'>function type</a> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_7bf8dc58f8c6751947ec2d84d79ad90256896f51_85' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>𝒫</mi><mo stretchy='false'>(</mo><mi>X</mi><mo stretchy='false'>)</mo><mo>≔</mo><mi>X</mi><mo>→</mo><mi>Ω</mi></mrow><annotation encoding='application/x-tex'>\mathcal{P}(X) \coloneqq X \to \Omega</annotation></semantics></math></div> where <math class='maruku-mathml' display='inline' id='mathml_7bf8dc58f8c6751947ec2d84d79ad90256896f51_86' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Ω</mi></mrow><annotation encoding='application/x-tex'>\Omega</annotation></semantics></math> is the <a class='existingWikiWord' href='/nlab/show/diff/type+of+propositions'>type of all propositions</a> with the type reflector type family <math class='maruku-mathml' display='inline' id='mathml_7bf8dc58f8c6751947ec2d84d79ad90256896f51_87' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>P</mi><mo>:</mo><mi>Ω</mi><mo>⊢</mo><msub><mi mathvariant='normal'>El</mi> <mi>Ω</mi></msub><mo stretchy='false'>(</mo><mi>P</mi><mo stretchy='false'>)</mo><mspace width='thickmathspace' /><mi mathvariant='normal'>type</mi></mrow><annotation encoding='application/x-tex'>P:\Omega \vdash \mathrm{El}_\Omega(P) \; \mathrm{type}</annotation></semantics></math>. In the <a class='existingWikiWord' href='/nlab/show/diff/deductive+system'>inference rules</a> for the <a class='existingWikiWord' href='/nlab/show/diff/type+of+propositions'>type of all propositions</a>, one has an operation <math class='maruku-mathml' display='inline' id='mathml_7bf8dc58f8c6751947ec2d84d79ad90256896f51_88' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><msub><mo stretchy='false'>)</mo> <mi>Ω</mi></msub></mrow><annotation encoding='application/x-tex'>(-)_\Omega</annotation></semantics></math> which takes a proposition <math class='maruku-mathml' display='inline' id='mathml_7bf8dc58f8c6751947ec2d84d79ad90256896f51_89' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>P</mi></mrow><annotation encoding='application/x-tex'>P</annotation></semantics></math> and turns it into an element of the <a class='existingWikiWord' href='/nlab/show/diff/type+of+propositions'>type of all propositions</a> <math class='maruku-mathml' display='inline' id='mathml_7bf8dc58f8c6751947ec2d84d79ad90256896f51_90' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>P</mi> <mi>Ω</mi></msub><mo>:</mo><mi>Ω</mi></mrow><annotation encoding='application/x-tex'>P_\Omega:\Omega</annotation></semantics></math>. The local membership relation <math class='maruku-mathml' display='inline' id='mathml_7bf8dc58f8c6751947ec2d84d79ad90256896f51_91' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>x</mi><msub><mo>∈</mo> <mi>A</mi></msub><mi>B</mi></mrow><annotation encoding='application/x-tex'>x \in_A B</annotation></semantics></math> between elements <math class='maruku-mathml' display='inline' id='mathml_7bf8dc58f8c6751947ec2d84d79ad90256896f51_92' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>x</mi><mo>:</mo><mi>A</mi></mrow><annotation encoding='application/x-tex'>x:A</annotation></semantics></math> and material subtypes <math class='maruku-mathml' display='inline' id='mathml_7bf8dc58f8c6751947ec2d84d79ad90256896f51_93' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>B</mi><mo>:</mo><mi>𝒫</mi><mo stretchy='false'>(</mo><mi>A</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>B:\mathcal{P}(A)</annotation></semantics></math> is defined as <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_7bf8dc58f8c6751947ec2d84d79ad90256896f51_94' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>x</mi><msub><mo>∈</mo> <mi>A</mi></msub><mi>B</mi><mo>≔</mo><msub><mi mathvariant='normal'>El</mi> <mi>Ω</mi></msub><mo stretchy='false'>(</mo><mi>B</mi><mo stretchy='false'>(</mo><mi>x</mi><mo stretchy='false'>)</mo><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>x \in_A B \coloneqq \mathrm{El}_\Omega(B(x))</annotation></semantics></math></div> Arbitrary unions and intersections of subtypes could be defined in <a class='existingWikiWord' href='/nlab/show/diff/dependent+type+theory'>dependent type theory</a>: <ul> <li> Given a type <math class='maruku-mathml' display='inline' id='mathml_7bf8dc58f8c6751947ec2d84d79ad90256896f51_95' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>A</mi></mrow><annotation encoding='application/x-tex'>A</annotation></semantics></math> and a type <math class='maruku-mathml' display='inline' id='mathml_7bf8dc58f8c6751947ec2d84d79ad90256896f51_96' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>I</mi></mrow><annotation encoding='application/x-tex'>I</annotation></semantics></math>, there is an function <math class='maruku-mathml' display='inline' id='mathml_7bf8dc58f8c6751947ec2d84d79ad90256896f51_97' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo lspace='thinmathspace' rspace='thinmathspace'>⋂</mo><mo>:</mo><mo stretchy='false'>(</mo><mi>I</mi><mo>→</mo><mi>𝒫</mi><mo stretchy='false'>(</mo><mi>A</mi><mo stretchy='false'>)</mo><mo stretchy='false'>)</mo><mo>→</mo><mi>𝒫</mi><mo stretchy='false'>(</mo><mi>A</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>\bigcap:(I \to \mathcal{P}(A)) \to \mathcal{P}(A)</annotation></semantics></math> called the <math class='maruku-mathml' display='inline' id='mathml_7bf8dc58f8c6751947ec2d84d79ad90256896f51_98' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>I</mi></mrow><annotation encoding='application/x-tex'>I</annotation></semantics></math>-indexed <a class='existingWikiWord' href='/nlab/show/diff/intersection'>intersection</a>, such that for all families of subtypes <math class='maruku-mathml' display='inline' id='mathml_7bf8dc58f8c6751947ec2d84d79ad90256896f51_99' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>B</mi><mo>:</mo><mi>I</mi><mo>→</mo><mi>𝒫</mi><mo stretchy='false'>(</mo><mi>A</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>B:I \to \mathcal{P}(A)</annotation></semantics></math>, <math class='maruku-mathml' display='inline' id='mathml_7bf8dc58f8c6751947ec2d84d79ad90256896f51_100' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mo lspace='thinmathspace' rspace='thinmathspace'>⋂</mo> <mrow><mi>i</mi><mo>:</mo><mi>I</mi></mrow></msub><mi>B</mi><mo stretchy='false'>(</mo><mi>i</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>\bigcap_{i:I} B(i)</annotation></semantics></math> is defined as <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_7bf8dc58f8c6751947ec2d84d79ad90256896f51_101' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mrow><mo>(</mo><munder><mo lspace='thinmathspace' rspace='thinmathspace'>⋂</mo> <mrow><mi>i</mi><mo>:</mo><mi>I</mi></mrow></munder><mi>B</mi><mo stretchy='false'>(</mo><mi>i</mi><mo stretchy='false'>)</mo><mo>)</mo></mrow><mo stretchy='false'>(</mo><mi>x</mi><mo stretchy='false'>)</mo><mo>≔</mo><mo stretchy='false'>(</mo><mo>∀</mo><mi>i</mi><mo>:</mo><mi>I</mi><mo>.</mo><mi>x</mi><msub><mo>∈</mo> <mi>A</mi></msub><mi>B</mi><mo stretchy='false'>(</mo><mi>i</mi><mo stretchy='false'>)</mo><msub><mo stretchy='false'>)</mo> <mi>Ω</mi></msub></mrow><annotation encoding='application/x-tex'>\left(\bigcap_{i:I} B(i)\right)(x) \coloneqq (\forall i:I.x \in_A B(i))_\Omega</annotation></semantics></math></div> for all <math class='maruku-mathml' display='inline' id='mathml_7bf8dc58f8c6751947ec2d84d79ad90256896f51_102' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>x</mi><mo>:</mo><mi>A</mi></mrow><annotation encoding='application/x-tex'>x:A</annotation></semantics></math>, where <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_7bf8dc58f8c6751947ec2d84d79ad90256896f51_103' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo>∀</mo><mi>x</mi><mo>:</mo><mi>A</mi><mo>.</mo><mi>B</mi><mo stretchy='false'>(</mo><mi>x</mi><mo stretchy='false'>)</mo><mo>≔</mo><mrow><mo>[</mo><munder><mo lspace='thinmathspace' rspace='thinmathspace'>∏</mo> <mrow><mi>x</mi><mo>:</mo><mi>A</mi></mrow></munder><mi>B</mi><mo stretchy='false'>(</mo><mi>x</mi><mo stretchy='false'>)</mo><mo>]</mo></mrow></mrow><annotation encoding='application/x-tex'>\forall x:A.B(x) \coloneqq \left[\prod_{x:A} B(x)\right]</annotation></semantics></math></div> is the <a class='existingWikiWord' href='/nlab/show/diff/universal+quantifier'>universal quantification</a> of a <a class='existingWikiWord' href='/nlab/show/diff/dependent+type'>type family</a> and <math class='maruku-mathml' display='inline' id='mathml_7bf8dc58f8c6751947ec2d84d79ad90256896f51_104' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>[</mo><mi>T</mi><mo stretchy='false'>]</mo></mrow><annotation encoding='application/x-tex'>[T]</annotation></semantics></math> is the propositional truncation of <math class='maruku-mathml' display='inline' id='mathml_7bf8dc58f8c6751947ec2d84d79ad90256896f51_105' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>T</mi></mrow><annotation encoding='application/x-tex'>T</annotation></semantics></math>. </li> <li> Given a type <math class='maruku-mathml' display='inline' id='mathml_7bf8dc58f8c6751947ec2d84d79ad90256896f51_106' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>A</mi></mrow><annotation encoding='application/x-tex'>A</annotation></semantics></math> and a type <math class='maruku-mathml' display='inline' id='mathml_7bf8dc58f8c6751947ec2d84d79ad90256896f51_107' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>I</mi></mrow><annotation encoding='application/x-tex'>I</annotation></semantics></math>, there is an function <math class='maruku-mathml' display='inline' id='mathml_7bf8dc58f8c6751947ec2d84d79ad90256896f51_108' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo lspace='thinmathspace' rspace='thinmathspace'>⋃</mo><mo>:</mo><mo stretchy='false'>(</mo><mi>I</mi><mo>→</mo><mi>𝒫</mi><mo stretchy='false'>(</mo><mi>A</mi><mo stretchy='false'>)</mo><mo stretchy='false'>)</mo><mo>→</mo><mi>𝒫</mi><mo stretchy='false'>(</mo><mi>A</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>\bigcup:(I \to \mathcal{P}(A)) \to \mathcal{P}(A)</annotation></semantics></math> called the <math class='maruku-mathml' display='inline' id='mathml_7bf8dc58f8c6751947ec2d84d79ad90256896f51_109' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>I</mi></mrow><annotation encoding='application/x-tex'>I</annotation></semantics></math>-indexed <a class='existingWikiWord' href='/nlab/show/diff/union'>union</a>, such that for all families of subtypes <math class='maruku-mathml' display='inline' id='mathml_7bf8dc58f8c6751947ec2d84d79ad90256896f51_110' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>B</mi><mo>:</mo><mi>I</mi><mo>→</mo><mi>𝒫</mi><mo stretchy='false'>(</mo><mi>A</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>B:I \to \mathcal{P}(A)</annotation></semantics></math>, <math class='maruku-mathml' display='inline' id='mathml_7bf8dc58f8c6751947ec2d84d79ad90256896f51_111' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mo lspace='thinmathspace' rspace='thinmathspace'>⋃</mo> <mrow><mi>i</mi><mo>:</mo><mi>I</mi></mrow></msub><mi>B</mi><mo stretchy='false'>(</mo><mi>i</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>\bigcup_{i:I} B(i)</annotation></semantics></math> is defined as <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_7bf8dc58f8c6751947ec2d84d79ad90256896f51_112' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mrow><mo>(</mo><munder><mo lspace='thinmathspace' rspace='thinmathspace'>⋃</mo> <mrow><mi>i</mi><mo>:</mo><mi>I</mi></mrow></munder><mi>B</mi><mo stretchy='false'>(</mo><mi>i</mi><mo stretchy='false'>)</mo><mo>)</mo></mrow><mo stretchy='false'>(</mo><mi>x</mi><mo stretchy='false'>)</mo><mo>≔</mo><mo stretchy='false'>(</mo><mo>∃</mo><mi>i</mi><mo>:</mo><mi>I</mi><mo>.</mo><mi>x</mi><msub><mo>∈</mo> <mi>A</mi></msub><mi>B</mi><mo stretchy='false'>(</mo><mi>i</mi><mo stretchy='false'>)</mo><msub><mo stretchy='false'>)</mo> <mi>Ω</mi></msub></mrow><annotation encoding='application/x-tex'>\left(\bigcup_{i:I} B(i)\right)(x) \coloneqq (\exists i:I.x \in_A B(i))_\Omega</annotation></semantics></math></div> for all <math class='maruku-mathml' display='inline' id='mathml_7bf8dc58f8c6751947ec2d84d79ad90256896f51_113' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>x</mi><mo>:</mo><mi>A</mi></mrow><annotation encoding='application/x-tex'>x:A</annotation></semantics></math>, where <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_7bf8dc58f8c6751947ec2d84d79ad90256896f51_114' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo>∃</mo><mi>x</mi><mo>:</mo><mi>A</mi><mo>.</mo><mi>B</mi><mo stretchy='false'>(</mo><mi>x</mi><mo stretchy='false'>)</mo><mo>≔</mo><mrow><mo>[</mo><munder><mo lspace='thinmathspace' rspace='thinmathspace'>∑</mo> <mrow><mi>x</mi><mo>:</mo><mi>A</mi></mrow></munder><mi>B</mi><mo stretchy='false'>(</mo><mi>x</mi><mo stretchy='false'>)</mo><mo>]</mo></mrow></mrow><annotation encoding='application/x-tex'>\exists x:A.B(x) \coloneqq \left[\sum_{x:A} B(x)\right]</annotation></semantics></math></div> is the <a class='existingWikiWord' href='/nlab/show/diff/existential+quantifier'>existential quantification</a> of a <a class='existingWikiWord' href='/nlab/show/diff/dependent+type'>type family</a> and <math class='maruku-mathml' display='inline' id='mathml_7bf8dc58f8c6751947ec2d84d79ad90256896f51_115' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>[</mo><mi>T</mi><mo stretchy='false'>]</mo></mrow><annotation encoding='application/x-tex'>[T]</annotation></semantics></math> is the propositional truncation of <math class='maruku-mathml' display='inline' id='mathml_7bf8dc58f8c6751947ec2d84d79ad90256896f51_116' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>T</mi></mrow><annotation encoding='application/x-tex'>T</annotation></semantics></math>. </li> </ul> In <a class='existingWikiWord' href='/nlab/show/diff/dependent+type+theory'>dependent type theory</a>, however, one cannot quantify over arbitrary types, since one could only quantify over elements of a type. Instead, one has to use a <a class='existingWikiWord' href='/nlab/show/diff/Tarski+universe'>Tarski universe</a> <math class='maruku-mathml' display='inline' id='mathml_7bf8dc58f8c6751947ec2d84d79ad90256896f51_117' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mi>U</mi><mo>,</mo><msub><mi mathvariant='normal'>El</mi> <mi>U</mi></msub><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(U, \mathrm{El}_U)</annotation></semantics></math>, where the elements of <math class='maruku-mathml' display='inline' id='mathml_7bf8dc58f8c6751947ec2d84d79ad90256896f51_118' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>U</mi></mrow><annotation encoding='application/x-tex'>U</annotation></semantics></math> represent <math class='maruku-mathml' display='inline' id='mathml_7bf8dc58f8c6751947ec2d84d79ad90256896f51_119' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>U</mi></mrow><annotation encoding='application/x-tex'>U</annotation></semantics></math>-<a class='existingWikiWord' href='/nlab/show/diff/small+set'>small types</a>, and then quantify over <math class='maruku-mathml' display='inline' id='mathml_7bf8dc58f8c6751947ec2d84d79ad90256896f51_120' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>U</mi></mrow><annotation encoding='application/x-tex'>U</annotation></semantics></math>. In the case of topological spaces, instead of the <a class='existingWikiWord' href='/nlab/show/diff/open+subspace'>open sets</a> being closed under arbitrary unions, the open sets are only closed under all <math class='maruku-mathml' display='inline' id='mathml_7bf8dc58f8c6751947ec2d84d79ad90256896f51_121' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>U</mi></mrow><annotation encoding='application/x-tex'>U</annotation></semantics></math>-small unions <math class='maruku-mathml' display='inline' id='mathml_7bf8dc58f8c6751947ec2d84d79ad90256896f51_122' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mo lspace='thinmathspace' rspace='thinmathspace'>⋃</mo> <mrow><mi>i</mi><mo>:</mo><msub><mi mathvariant='normal'>El</mi> <mi>U</mi></msub><mo stretchy='false'>(</mo><mi>I</mi><mo stretchy='false'>)</mo></mrow></msub><mi>B</mi><mo stretchy='false'>(</mo><mi>i</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>\bigcup_{i:\mathrm{El}_U(I)} B(i)</annotation></semantics></math> for <math class='maruku-mathml' display='inline' id='mathml_7bf8dc58f8c6751947ec2d84d79ad90256896f51_123' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>I</mi><mo>:</mo><mi>U</mi></mrow><annotation encoding='application/x-tex'>I:U</annotation></semantics></math>. \begin{definition} Given a <a class='existingWikiWord' href='/nlab/show/diff/Tarski+universe'>Tarski universe</a> <math class='maruku-mathml' display='inline' id='mathml_7bf8dc58f8c6751947ec2d84d79ad90256896f51_124' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mi>U</mi><mo>,</mo><msub><mi mathvariant='normal'>El</mi> <mi>U</mi></msub><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(U, \mathrm{El}_U)</annotation></semantics></math>, a <a class='existingWikiWord' href='/nlab/show/diff/topological+space'>topological space</a> is a type <math class='maruku-mathml' display='inline' id='mathml_7bf8dc58f8c6751947ec2d84d79ad90256896f51_125' 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 <math class='maruku-mathml' display='inline' id='mathml_7bf8dc58f8c6751947ec2d84d79ad90256896f51_126' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>U</mi></mrow><annotation encoding='application/x-tex'>U</annotation></semantics></math>-small topology, a type of subtypes <math class='maruku-mathml' display='inline' id='mathml_7bf8dc58f8c6751947ec2d84d79ad90256896f51_127' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>O</mi><mo stretchy='false'>(</mo><mi>X</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>O(X)</annotation></semantics></math> with canonical <a class='existingWikiWord' href='/nlab/show/diff/embedding'>embedding</a> <math class='maruku-mathml' display='inline' id='mathml_7bf8dc58f8c6751947ec2d84d79ad90256896f51_128' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>i</mi> <mi>O</mi></msub><mo>:</mo><mi>O</mi><mo stretchy='false'>(</mo><mi>X</mi><mo stretchy='false'>)</mo><mo>↪</mo><mi>𝒫</mi><mo stretchy='false'>(</mo><mi>X</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>i_O:O(X) \hookrightarrow \mathcal{P}(X)</annotation></semantics></math>, called the open sets of <math class='maruku-mathml' display='inline' id='mathml_7bf8dc58f8c6751947ec2d84d79ad90256896f51_129' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math>, which are closed under <a class='existingWikiWord' href='/nlab/show/diff/finite+type'>finite</a> <a class='existingWikiWord' href='/nlab/show/diff/intersection'>intersections</a> and <math class='maruku-mathml' display='inline' id='mathml_7bf8dc58f8c6751947ec2d84d79ad90256896f51_130' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>U</mi></mrow><annotation encoding='application/x-tex'>U</annotation></semantics></math>-<a class='existingWikiWord' href='/nlab/show/diff/small+set'>small</a> unions. \end{definition} Given a topological space <math class='maruku-mathml' display='inline' id='mathml_7bf8dc58f8c6751947ec2d84d79ad90256896f51_131' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mi>X</mi><mo>,</mo><mi>O</mi><mo stretchy='false'>(</mo><mi>X</mi><mo stretchy='false'>)</mo><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(X, O(X))</annotation></semantics></math>, we define the membership <a class='existingWikiWord' href='/nlab/show/diff/relation'>relation</a> between elements <math class='maruku-mathml' display='inline' id='mathml_7bf8dc58f8c6751947ec2d84d79ad90256896f51_132' 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:X</annotation></semantics></math> and open sets <math class='maruku-mathml' display='inline' id='mathml_7bf8dc58f8c6751947ec2d84d79ad90256896f51_133' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>V</mi><mo>:</mo><mi>O</mi><mo stretchy='false'>(</mo><mi>X</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>V:O(X)</annotation></semantics></math>: <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_7bf8dc58f8c6751947ec2d84d79ad90256896f51_134' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>x</mi><mo>:</mo><mi>X</mi><mo>,</mo><mi>V</mi><mo>:</mo><mi>O</mi><mo stretchy='false'>(</mo><mi>X</mi><mo stretchy='false'>)</mo><mo>⊢</mo><mi>x</mi><mo>∈</mo><mi>V</mi><mspace width='thickmathspace' /><mi mathvariant='normal'>type</mi></mrow><annotation encoding='application/x-tex'>x:X, V:O(X) \vdash x \in V \; \mathrm{type}</annotation></semantics></math></div> by <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_7bf8dc58f8c6751947ec2d84d79ad90256896f51_135' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>x</mi><mo>∈</mo><mi>V</mi><mo>≔</mo><msub><mi mathvariant='normal'>El</mi> <mi>Ω</mi></msub><mo stretchy='false'>(</mo><mo stretchy='false'>(</mo><msub><mi>i</mi> <mi>O</mi></msub><mo stretchy='false'>(</mo><mi>V</mi><mo stretchy='false'>)</mo><mo stretchy='false'>)</mo><mo stretchy='false'>(</mo><mi>x</mi><mo stretchy='false'>)</mo><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>x \in V \coloneqq \mathrm{El}_\Omega((i_O(V))(x))</annotation></semantics></math></div> By definition of the type of all propositions and its type reflector, <math class='maruku-mathml' display='inline' id='mathml_7bf8dc58f8c6751947ec2d84d79ad90256896f51_136' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>x</mi><mo>∈</mo><mi>V</mi></mrow><annotation encoding='application/x-tex'>x \in V</annotation></semantics></math> is always a <a class='existingWikiWord' href='/nlab/show/diff/mere+proposition'>h-proposition</a> for all <math class='maruku-mathml' display='inline' id='mathml_7bf8dc58f8c6751947ec2d84d79ad90256896f51_137' 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:X</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_7bf8dc58f8c6751947ec2d84d79ad90256896f51_138' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>V</mi><mo>:</mo><mi>O</mi><mo stretchy='false'>(</mo><mi>X</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>V:O(X)</annotation></semantics></math>. <h2 id='examples'>Examples</h2> <h3 id='special_cases'>Special cases</h3> <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/finite+topological+space'>finite topological space</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/discrete+and+indiscrete+topology'>discrete topological space</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/discrete+and+indiscrete+topology'>codiscrete topological space</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/cofinite+topology'>cofinite topology</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/compact+space'>compact topological space</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/paracompact+topological+space'>paracompact topological space</a>, <a class='existingWikiWord' href='/nlab/show/diff/sigma-compact+topological+space'>sigma-compact topological space</a>, <a class='existingWikiWord' href='/nlab/show/diff/Lindel%C3%B6f+topological+space'>Lindelöf topological space</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/coherent+space'>spectral topological space</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/CW+complex'>CW-complex</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/manifold'>manifold</a> </li> </ul> … <h3 id='specific_examples'>Specific examples</h3> <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/empty+space'>empty space</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/point+space'>point 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/simplex'>simplex</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/circle'>circle</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/Cantor+space'>Cantor space</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/long+line'>long line</a>, <a class='existingWikiWord' href='/nlab/show/diff/line+with+two+origins'>line with two origins</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/K-topology'>K-topology</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 projective space</a>, <a class='existingWikiWord' href='/nlab/show/diff/complex+projective+space'>complex projective space</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/classifying+space'>classifying space</a> </li> </ul> … <div class='num_example'> <h6 id='example'>Example</h6> The <a class='existingWikiWord' href='/nlab/show/diff/cartesian+space'>Cartesian space</a> <math class='maruku-mathml' display='inline' id='mathml_7bf8dc58f8c6751947ec2d84d79ad90256896f51_139' 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> with its standard notion of open subsets <a class='existingWikiWord' href='/nlab/show/diff/topological+base'>generated from</a>: <a class='existingWikiWord' href='/nlab/show/diff/union'>unions</a> of <a class='existingWikiWord' href='/nlab/show/diff/ball'>open balls</a> <math class='maruku-mathml' display='inline' id='mathml_7bf8dc58f8c6751947ec2d84d79ad90256896f51_140' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>D</mi> <mi>n</mi></msup><mo>⊂</mo><msup><mi>ℝ</mi> <mi>n</mi></msup></mrow><annotation encoding='application/x-tex'>D^n \subset \mathbb{R}^n</annotation></semantics></math>. </div> <h2 id='related_concepts'>Related concepts</h2> <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/separation+axioms'>separation axioms</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/finer+topology'>finer topology</a>, <a class='existingWikiWord' href='/nlab/show/diff/finer+topology'>coarser topology</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/first-countable+space'>first countable topological space</a>, <a class='existingWikiWord' href='/nlab/show/diff/second-countable+space'>second countable topological space</a>, <a class='existingWikiWord' href='/nlab/show/diff/separable+space'>separable topological space</a>, <a class='existingWikiWord' href='/nlab/show/diff/Hausdorff+space'>Hausdorff topological space</a>, <a class='existingWikiWord' href='/nlab/show/diff/topological+manifold'>topological manifold</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/sigma-topological+space'>$\sigma$-topological space</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/unified+topological+space'>unified topological space</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/topological+property'>topological property</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/locale'>locale</a>, <a class='existingWikiWord' href='/nlab/show/diff/topos'>topos</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/ionad'>ionad</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/proximity+space'>proximity space</a>, <a class='existingWikiWord' href='/nlab/show/diff/uniform+space'>uniform space</a>, <a class='existingWikiWord' href='/nlab/show/diff/syntopogenous+space'>syntopogenous space</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/quotient+space'>quotient topology</a>, <a class='existingWikiWord' href='/nlab/show/diff/quotient+space'>quotient topological space</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/subspace+topology'>subspace topology</a>, <a class='existingWikiWord' href='/nlab/show/diff/subspace+topology'>topological subspace</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/product+topological+space'>product topological space</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/effective+topological+space'>effective topological space</a>, <a class='existingWikiWord' href='/nlab/show/diff/equilogical+space'>equilogical space</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/topological+G-space'>topological G-space</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/filter+space'>filter space</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/connected+space'>connected topological spaces</a>, <a class='existingWikiWord' href='/nlab/show/diff/simply+connected+space'>simply connected topological space</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/nilpotent+topological+space'>nilpotent topological space</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/fractal'>fractal</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/sheaf+on+a+topological+space'>sheaf on a topological space</a> </li><ins class='diffins'> </ins><ins class='diffins'><li> <a class='existingWikiWord' href='/nlab/show/diff/pretopological+space'>pretopological space</a> </li></ins> </ul> <h2 id='References'>References</h2> <h3 id='ReferencesHistoricalOrigins'>Historical origins</h3> The general idea of <a class='existingWikiWord' href='/nlab/show/diff/topology'>topology</a> goes back to: <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/Henri+Poincar%C3%A9'>Henri Poincaré</a>, <a class='existingWikiWord' href='/nlab/show/diff/Analysis+Situs'>Analysis Situs</a>, Journal de l’École Polytechnique 2 1 (1895) 1–123 [<a href='https://gallica.bnf.fr/ark:/12148/bpt6k4337198/f7'>gallica:12148/bpt6k4337198/f7</a>, Engl: <a href='https://www.maths.ed.ac.uk/~v1ranick/papers/poincare2009.pdf'>pdf</a>, <a class='existingWikiWord' href='/nlab/files/Stillwell_AnalysisSitus.pdf' title='pdf'>pdf</a>]</li> </ul> The notion of <a class='existingWikiWord' href='/nlab/show/diff/topological+space'>topological space</a> involving <a class='existingWikiWord' href='/nlab/show/diff/neighborhood'>neighbourhoods</a> was first developed, for the special case now known as <a class='existingWikiWord' href='/nlab/show/diff/Hausdorff+space'>Hausdorff spaces</a>, in: <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/Felix+Hausdorff'>Felix Hausdorff</a>, <a class='existingWikiWord' href='/nlab/show/diff/Grundz%C3%BCge+der+Mengenlehre'>Grundzüge der Mengenlehre</a>, Leipzig: Veit (1914), Reprinted by Chelsea Publishing Company (1944, 1949, 1965) [ISBN:978-0-8284-0061-9, <a href='https://archive.org/details/grundzgedermen00hausuoft/page/n5/mode/2up'>ark:/13960/t2891gn8g</a>]</li> </ul> The more general definition – dropping Hausdorff’s <math class='maruku-mathml' display='inline' id='mathml_7bf8dc58f8c6751947ec2d84d79ad90256896f51_141' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>T</mi> <mn>2</mn></msub></mrow><annotation encoding='application/x-tex'>T_2</annotation></semantics></math>-<a class='existingWikiWord' href='/nlab/show/diff/separation+axioms'>separation axiom</a> and formulated in terms of <a class='existingWikiWord' href='/nlab/show/diff/closure+operator'>closure operators</a> that preserve finite <a class='existingWikiWord' href='/nlab/show/diff/union'>unions</a> – is due to: <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/Kazimierz+Kuratowski'>Kazimierz Kuratowski</a>, Sur l’opération Ā de l’Analysis Situs, Fundamenta Mathematicae 3 (1922) 182–199 [<a href='https://www.impan.pl/pl/wydawnictwa/czasopisma-i-serie-wydawnicze/fundamenta-mathematicae/all/3/0/92454/sur-l-operation-a-de-l-analysis-situs'>doi:10.4064/fm-3-1-182-199</a>]</li> </ul> The modern formulation via <a class='existingWikiWord' href='/nlab/show/diff/open+subspace'>open set</a> was widely popularized by: <ul> <li id='Bourbak71'> <a class='existingWikiWord' href='/nlab/show/diff/Bourbaki'>Nicolas Bourbaki</a>], Eléments de mathématique II. Première partie. Les structures fondamentales de l’analyse. Livre III. Topologie générale. Chapitre I. Structures topologiques. Actualités scientifiques et industrielles, vol. 858. Hermann, Paris (1940) General topology, Elements of Mathematics III, Springer (1971, 1990, 1995) [<a href='https://doi.org/10.1007/978-3-642-61701-0'>doi:10.1007/978-3-642-61701-0</a>] </li> </ul> <h3 id='further'>Further</h3> Further textbook accounts: <ul> <li id='Kelley55'> <a class='existingWikiWord' href='/nlab/show/diff/John+L.+Kelley'>John Kelley</a>, General topology, D. van Nostrand, New York (1955), reprinted as: Graduate Texts in Mathematics, Springer (1975) [<a href='https://www.springer.com/gp/book/9780387901251'>ISBN:978-0-387-90125-1</a>] </li> <li id='Dugundji66'> <a class='existingWikiWord' href='/nlab/show/diff/James+Dugundji'>James Dugundji</a>, Topology, Allyn and Bacon 1966 (<a href='https://www.southalabama.edu/mathstat/personal_pages/carter/Dugundji.pdf'>pdf</a>) </li> <li id='Munkres75'> <a class='existingWikiWord' href='/nlab/show/diff/James+Munkres'>James Munkres</a>, Topology, Prentice Hall (1975, 2000) [ISBN:0-13-181629-2, <a href='http://mathcenter.spb.ru/nikaan/2019/topology/4.pdf'>pdf</a>] </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/Richard+E.+Hodel'>Richard E. Hodel</a> (ed.), Set-Theoretic Topology, Academic Press (1977) [<a href='https://doi.org/10.1016/C2013-0-11355-4'>doi:10.1016/C2013-0-11355-4</a>] </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/Klaus+J%C3%A4nich'>Klaus Jänich</a>, Topology, Undergraduate Texts in Mathematics, Springer (1984, 1999) [<a href='https://link.springer.com/book/9780387908922'>ISBN:9780387908922</a>, <a href='https://doi.org/10.1007/978-3-662-10574-0'>doi:10.1007/978-3-662-10574-0</a>, Chapters 1-2: <a href='http://topologicalmedialab.net/xinwei/classes/readings/Janich/Janich_Topology_ch1-2.pdf'>pdf</a>] </li> <li id='Engelking89'> <a class='existingWikiWord' href='/nlab/show/diff/Ryszard+Engelking'>Ryszard Engelking</a>, General Topology, Sigma series in pure mathematics 6, Heldermann 1989 (<a href='https://www.heldermann.de/SSPM/SSPM06/sspm06.htm'>ISBN 388538-006-4</a>) </li> <li id='Vickers89'> <a class='existingWikiWord' href='/nlab/show/diff/Steve+Vickers'>Steven Vickers</a>, Topology via Logic, Cambridge University Press (1989) (<a href='http://www.gbv.de/dms/ilmenau/toc/21309293X.PDF'>toc pdf</a>) </li> <li id='Bredon93'> <a class='existingWikiWord' href='/nlab/show/diff/Glen+Bredon'>Glen Bredon</a>, Topology and Geometry, Graduate Texts in Mathematics 139, Springer (1993) [<a href='https://link.springer.com/book/10.1007/978-1-4757-6848-0'>doi:10.1007/978-1-4757-6848-0</a>, <a href='http://virtualmath1.stanford.edu/~ralph/math215b/Bredon.pdf'>pdf</a>] </li> <li id='Lawson03'> Terry Lawson, Topology: A Geometric Approach, Oxford University Press (2003) (<a href='http://users.metu.edu.tr/serge/courses/422-2014/supplementary/TGeometric.pdf'>pdf</a>) </li> <li> Anatole Katok, Alexey Sossinsky, Introduction to Modern Topology and Geometry (2010) [<a href='http://akatok.s3-website-us-east-1.amazonaws.com/TOPOLOGY/Contents.pdf'>toc pdf</a>, <a class='existingWikiWord' href='/nlab/files/KatokSossinsky-Topology-Ch1.pdf' title='pdf'>pdf</a>] </li> </ul> and leading over to <a class='existingWikiWord' href='/nlab/show/diff/homotopy+theory'>homotopy theory</a>: <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/Ioan+James'>Ioan Mackenzie James</a>, General Topology and Homotopy Theory, Springer 1984 (<a href='https://doi.org/10.1007/978-1-4613-8283-6'>doi:10.1007/978-1-4613-8283-6</a>)</li> </ul> On <a class='existingWikiWord' href='/nlab/show/diff/counterexample'>counterexamples</a> in topology: <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/Lynn+Arthur+Steen'>Lynn Arthur Steen</a>, <a class='existingWikiWord' href='/nlab/show/diff/J.+Arthur+Seebach'>J. Arthur Seebach</a>, <a class='existingWikiWord' href='/nlab/show/diff/Counterexamples+in+Topology'>Counterexamples in Topology</a>, Springer 1971 (<a href='https://link.springer.com/book/10.1007/978-1-4612-6290-9'>doi:10.1007/978-1-4612-6290-9</a>)</li> </ul> With emphasis on <a class='existingWikiWord' href='/nlab/show/diff/category+theory'>category theoretic</a> aspects of <a class='existingWikiWord' href='/nlab/show/diff/general+topology'>general topology</a>, notably on <a href='separation+axioms#Reflection'><math class='maruku-mathml' display='inline' id='mathml_7bf8dc58f8c6751947ec2d84d79ad90256896f51_142' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>T</mi> <mi>n</mi></msub></mrow><annotation encoding='application/x-tex'>T_n</annotation></semantics></math>-reflections</a>: <ul> <li id='HerrlichStrecker71'> <a class='existingWikiWord' href='/nlab/show/diff/Horst+Herrlich'>Horst Herrlich</a>, <a class='existingWikiWord' href='/nlab/show/diff/George+Strecker'>George Strecker</a>, Categorical topology – Its origins as exemplified by the unfolding of the theory of topological reflections and coreflections before 1971 (<a href='https://link.springer.com/content/pdf/10.1007%2F978-94-017-0468-7_15.pdf'>pdf</a>), pages 255-341 in: C. E. Aull, R Lowen (eds.), Handbook of the History of General Topology. Vol. 1, Kluwer 1997 (<a href='https://link.springer.com/book/10.1007/978-94-017-0468-7'>doi:10.1007/978-94-017-0468-7</a>) </li> <li id='BradleyBrysonTerilla20'> <a class='existingWikiWord' href='/nlab/show/diff/Tai-Danae+Bradley'>Tai-Danae Bradley</a>, <a class='existingWikiWord' href='/nlab/show/diff/Tyler+Bryson'>Tyler Bryson</a>, <a class='existingWikiWord' href='/nlab/show/diff/John+Terilla'>John Terilla</a>, Topology – A categorical approach, MIT Press 2020 (<a href='https://mitpress.mit.edu/books/topology'>ISBN:9780262539357</a>, <a href='https://topology.pubpub.org/'>web version</a>) </li> </ul> See also: <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/Allen+Hatcher'>Alan Hatcher</a>, <a href='https://www.math.cornell.edu/~hatcher/AT/ATpage.html'>Algebraic Topology</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/Neil+Strickland'>Neil Strickland</a>, A Bestiary of Topological Objects [<a href='https://strickland1.org/courses/bestiary/bestiary.pdf'>pdf</a>, <a class='existingWikiWord' href='/nlab/files/Strickland-BestiaryTopological.pdf' title='pdf'>pdf</a>] </li> </ul> and see further references at <a class='existingWikiWord' href='/nlab/show/diff/algebraic+topology'>algebraic topology</a>. Lecture notes: <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/Friedhelm+Waldhausen'>Friedhelm Waldhausen</a>, Topologie (<a href='https://www.math.uni-bielefeld.de/~fw/ein.pdf'>pdf</a>) </li> <li> Alex Kuronya, Introduction to topology, 2010 (<a href='https://www.uni- frankfurt.de/64271720/TopNotes_Spring10.pdf'>pdf</a>) </li> <li> Anatole Katok, Alexey Sossinsky, Introduction to modern topology and geometry (<a href='http://www.personal.psu.edu/axk29/MASS-07/Background-forMASS.pdf'>pdf</a>) </li> <li id='Schreiber17'> <a class='existingWikiWord' href='/nlab/show/diff/Urs+Schreiber'>Urs Schreiber</a>, <a class='existingWikiWord' href='/nlab/show/diff/Introduction+to+Topology'>Introduction to Topology</a>, Bonn 2017 </li> <li id='Mueger18'> <a class='existingWikiWord' href='/nlab/show/diff/Michael+M%C3%BCger'>Michael Müger</a>, Topology for the working mathematician, Nijmegen 2018 (<a href='https://www.math.ru.nl/~mueger/topology.pdf'>pdf</a>) </li> </ul> Basic topology set up in <a class='existingWikiWord' href='/nlab/show/diff/intuitionistic+mathematics'>intuitionistic mathematics</a> is discussed in <ul> <li id='Waaldijk96'><a class='existingWikiWord' href='/nlab/show/diff/Franka+Waaldijk'>Franka Waaldijk</a>, modern intuitionistic topology, 1996 (<a href='http://www.fwaaldijk.nl/modern%20intuitionistic%20topology.pdf'>pdf</a>)</li> </ul> See also: <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/Topospaces'>Topospaces</a>, a Wiki with basic material on topology.</li> </ul> </div> <div class="revisedby"> Last revised on November 10, 2024 at 18:16:18. 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