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href="/nlab/show/differential+topology">differential topology</a></em>, <em><a class="existingWikiWord" href="/nlab/show/algebraic+topology">algebraic topology</a></em>, <em><a class="existingWikiWord" href="/nlab/show/functional+analysis">functional analysis</a></em> and <em><a class="existingWikiWord" href="/nlab/show/topological+homotopy+theory">topological</a> <a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopy theory</a></em></p> <p><a class="existingWikiWord" href="/nlab/show/Introduction+to+Topology">Introduction</a></p> <p><strong>Basic concepts</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/open+subset">open subset</a>, <a class="existingWikiWord" href="/nlab/show/closed+subset">closed subset</a>, <a class="existingWikiWord" href="/nlab/show/neighbourhood">neighbourhood</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a>, <a class="existingWikiWord" href="/nlab/show/locale">locale</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/base+for+the+topology">base for the topology</a>, <a class="existingWikiWord" href="/nlab/show/neighbourhood+base">neighbourhood base</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/finer+topology">finer/coarser topology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+closure">closure</a>, <a class="existingWikiWord" href="/nlab/show/topological+interior">interior</a>, <a class="existingWikiWord" href="/nlab/show/topological+boundary">boundary</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/separation+axiom">separation</a>, <a class="existingWikiWord" href="/nlab/show/sober+topological+space">sobriety</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/continuous+function">continuous function</a>, <a class="existingWikiWord" href="/nlab/show/homeomorphism">homeomorphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/uniformly+continuous+function">uniformly continuous function</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+embedding">embedding</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/open+map">open map</a>, <a class="existingWikiWord" href="/nlab/show/closed+map">closed map</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/sequence">sequence</a>, <a class="existingWikiWord" href="/nlab/show/net">net</a>, <a class="existingWikiWord" href="/nlab/show/sub-net">sub-net</a>, <a class="existingWikiWord" href="/nlab/show/filter">filter</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/convergence">convergence</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/category">category</a><a class="existingWikiWord" href="/nlab/show/Top">Top</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/convenient+category+of+topological+spaces">convenient category of topological spaces</a></li> </ul> </li> </ul> <p><strong><a href="Top#UniversalConstructions">Universal constructions</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/initial+topology">initial topology</a>, <a class="existingWikiWord" href="/nlab/show/final+topology">final topology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/subspace">subspace</a>, <a class="existingWikiWord" href="/nlab/show/quotient+space">quotient space</a>,</p> </li> <li> <p>fiber space, <a class="existingWikiWord" href="/nlab/show/space+attachment">space attachment</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/product+space">product space</a>, <a class="existingWikiWord" href="/nlab/show/disjoint+union+space">disjoint union space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/mapping+cylinder">mapping cylinder</a>, <a class="existingWikiWord" href="/nlab/show/mapping+cocylinder">mapping cocylinder</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/mapping+cone">mapping cone</a>, <a class="existingWikiWord" href="/nlab/show/mapping+cocone">mapping cocone</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/mapping+telescope">mapping telescope</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/colimits+of+normal+spaces">colimits of normal spaces</a></p> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/stuff%2C+structure%2C+property">Extra stuff, structure, properties</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/nice+topological+space">nice topological space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/metric+space">metric space</a>, <a class="existingWikiWord" href="/nlab/show/metric+topology">metric topology</a>, <a class="existingWikiWord" href="/nlab/show/metrisable+space">metrisable space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Kolmogorov+space">Kolmogorov space</a>, <a class="existingWikiWord" href="/nlab/show/Hausdorff+space">Hausdorff space</a>, <a class="existingWikiWord" href="/nlab/show/regular+space">regular space</a>, <a class="existingWikiWord" href="/nlab/show/normal+space">normal space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/sober+space">sober space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/compact+space">compact space</a>, <a class="existingWikiWord" href="/nlab/show/proper+map">proper map</a></p> <p><a class="existingWikiWord" href="/nlab/show/sequentially+compact+topological+space">sequentially compact</a>, <a class="existingWikiWord" href="/nlab/show/countably+compact+topological+space">countably compact</a>, <a class="existingWikiWord" href="/nlab/show/locally+compact+topological+space">locally compact</a>, <a class="existingWikiWord" href="/nlab/show/sigma-compact+topological+space">sigma-compact</a>, <a class="existingWikiWord" href="/nlab/show/paracompact+space">paracompact</a>, <a class="existingWikiWord" href="/nlab/show/countably+paracompact+topological+space">countably paracompact</a>, <a class="existingWikiWord" href="/nlab/show/strongly+compact+topological+space">strongly compact</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/compactly+generated+space">compactly generated space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/second-countable+space">second-countable space</a>, <a class="existingWikiWord" href="/nlab/show/first-countable+space">first-countable space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/contractible+space">contractible space</a>, <a class="existingWikiWord" href="/nlab/show/locally+contractible+space">locally contractible space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/connected+space">connected space</a>, <a class="existingWikiWord" href="/nlab/show/locally+connected+space">locally connected space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/simply-connected+space">simply-connected space</a>, <a class="existingWikiWord" href="/nlab/show/locally+simply-connected+space">locally simply-connected space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cell+complex">cell complex</a>, <a class="existingWikiWord" href="/nlab/show/CW-complex">CW-complex</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/pointed+topological+space">pointed space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+vector+space">topological vector space</a>, <a class="existingWikiWord" href="/nlab/show/Banach+space">Banach space</a>, <a class="existingWikiWord" href="/nlab/show/Hilbert+space">Hilbert space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+group">topological group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+vector+bundle">topological vector bundle</a>, <a class="existingWikiWord" href="/nlab/show/topological+K-theory">topological K-theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+manifold">topological manifold</a></p> </li> </ul> <p><strong>Examples</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/empty+space">empty space</a>, <a class="existingWikiWord" href="/nlab/show/point+space">point space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/discrete+space">discrete space</a>, <a class="existingWikiWord" href="/nlab/show/codiscrete+space">codiscrete space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Sierpinski+space">Sierpinski space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/order+topology">order topology</a>, <a class="existingWikiWord" href="/nlab/show/specialization+topology">specialization topology</a>, <a class="existingWikiWord" href="/nlab/show/Scott+topology">Scott topology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Euclidean+space">Euclidean space</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/real+line">real line</a>, <a class="existingWikiWord" href="/nlab/show/plane">plane</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cylinder">cylinder</a>, <a class="existingWikiWord" href="/nlab/show/cone">cone</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/sphere">sphere</a>, <a class="existingWikiWord" href="/nlab/show/ball">ball</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/circle">circle</a>, <a class="existingWikiWord" href="/nlab/show/torus">torus</a>, <a class="existingWikiWord" href="/nlab/show/annulus">annulus</a>, <a class="existingWikiWord" href="/nlab/show/Moebius+strip">Moebius strip</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/polytope">polytope</a>, <a class="existingWikiWord" href="/nlab/show/polyhedron">polyhedron</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/projective+space">projective space</a> (<a class="existingWikiWord" href="/nlab/show/real+projective+space">real</a>, <a class="existingWikiWord" href="/nlab/show/complex+projective+space">complex</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/classifying+space">classifying space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/configuration+space+%28mathematics%29">configuration space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/path">path</a>, <a class="existingWikiWord" href="/nlab/show/loop">loop</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/mapping+spaces">mapping spaces</a>: <a class="existingWikiWord" href="/nlab/show/compact-open+topology">compact-open topology</a>, <a class="existingWikiWord" href="/nlab/show/topology+of+uniform+convergence">topology of uniform convergence</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/loop+space">loop space</a>, <a class="existingWikiWord" href="/nlab/show/path+space">path space</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Zariski+topology">Zariski topology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Cantor+space">Cantor space</a>, <a class="existingWikiWord" href="/nlab/show/Mandelbrot+space">Mandelbrot space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Peano+curve">Peano curve</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/line+with+two+origins">line with two origins</a>, <a class="existingWikiWord" href="/nlab/show/long+line">long line</a>, <a class="existingWikiWord" href="/nlab/show/Sorgenfrey+line">Sorgenfrey line</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/K-topology">K-topology</a>, <a class="existingWikiWord" href="/nlab/show/Dowker+space">Dowker space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Warsaw+circle">Warsaw circle</a>, <a class="existingWikiWord" href="/nlab/show/Hawaiian+earring+space">Hawaiian earring space</a></p> </li> </ul> <p><strong>Basic statements</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Hausdorff+spaces+are+sober">Hausdorff spaces are sober</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/schemes+are+sober">schemes are sober</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/continuous+images+of+compact+spaces+are+compact">continuous images of compact spaces are compact</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/closed+subspaces+of+compact+Hausdorff+spaces+are+equivalently+compact+subspaces">closed subspaces of compact Hausdorff spaces are equivalently compact subspaces</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/open+subspaces+of+compact+Hausdorff+spaces+are+locally+compact">open subspaces of compact Hausdorff spaces are locally compact</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quotient+projections+out+of+compact+Hausdorff+spaces+are+closed+precisely+if+the+codomain+is+Hausdorff">quotient projections out of compact Hausdorff spaces are closed precisely if the codomain is Hausdorff</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/compact+spaces+equivalently+have+converging+subnet+of+every+net">compact spaces equivalently have converging subnet of every net</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Lebesgue+number+lemma">Lebesgue number lemma</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/sequentially+compact+metric+spaces+are+equivalently+compact+metric+spaces">sequentially compact metric spaces are equivalently compact metric spaces</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/compact+spaces+equivalently+have+converging+subnet+of+every+net">compact spaces equivalently have converging subnet of every net</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/sequentially+compact+metric+spaces+are+totally+bounded">sequentially compact metric spaces are totally bounded</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/continuous+metric+space+valued+function+on+compact+metric+space+is+uniformly+continuous">continuous metric space valued function on compact metric space is uniformly continuous</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/paracompact+Hausdorff+spaces+are+normal">paracompact Hausdorff spaces are normal</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/paracompact+Hausdorff+spaces+equivalently+admit+subordinate+partitions+of+unity">paracompact Hausdorff spaces equivalently admit subordinate partitions of unity</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/closed+injections+are+embeddings">closed injections are embeddings</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/proper+maps+to+locally+compact+spaces+are+closed">proper maps to locally compact spaces are closed</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/injective+proper+maps+to+locally+compact+spaces+are+equivalently+the+closed+embeddings">injective proper maps to locally compact spaces are equivalently the closed embeddings</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/locally+compact+and+sigma-compact+spaces+are+paracompact">locally compact and sigma-compact spaces are paracompact</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/locally+compact+and+second-countable+spaces+are+sigma-compact">locally compact and second-countable spaces are sigma-compact</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/second-countable+regular+spaces+are+paracompact">second-countable regular spaces are paracompact</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/CW-complexes+are+paracompact+Hausdorff+spaces">CW-complexes are paracompact Hausdorff spaces</a></p> </li> </ul> <p><strong>Theorems</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Urysohn%27s+lemma">Urysohn's lemma</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Tietze+extension+theorem">Tietze extension theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Tychonoff+theorem">Tychonoff theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/tube+lemma">tube lemma</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Michael%27s+theorem">Michael's theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Brouwer%27s+fixed+point+theorem">Brouwer's fixed point theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+invariance+of+dimension">topological invariance of dimension</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Jordan+curve+theorem">Jordan curve theorem</a></p> </li> </ul> <p><strong>Analysis Theorems</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Heine-Borel+theorem">Heine-Borel theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/intermediate+value+theorem">intermediate value theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/extreme+value+theorem">extreme value theorem</a></p> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/topological+homotopy+theory">topological homotopy theory</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/left+homotopy">left homotopy</a>, <a class="existingWikiWord" href="/nlab/show/right+homotopy">right homotopy</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+equivalence">homotopy equivalence</a>, <a class="existingWikiWord" href="/nlab/show/deformation+retract">deformation retract</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+group">fundamental group</a>, <a class="existingWikiWord" href="/nlab/show/covering+space">covering space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+theorem+of+covering+spaces">fundamental theorem of covering spaces</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+group">homotopy group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/weak+homotopy+equivalence">weak homotopy equivalence</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Whitehead%27s+theorem">Whitehead's theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Freudenthal+suspension+theorem">Freudenthal suspension theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/nerve+theorem">nerve theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+extension+property">homotopy extension property</a>, <a class="existingWikiWord" href="/nlab/show/Hurewicz+cofibration">Hurewicz cofibration</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+cofiber+sequence">cofiber sequence</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Str%C3%B8m+model+category">Strøm model category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/classical+model+structure+on+topological+spaces">classical model structure on topological spaces</a></p> </li> </ul> </div></div> </div> </div> <h1 id="secondcountable_spaces">Second-countable spaces</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#definitions'>Definitions</a></li> <li><a href='#generalisations'>Generalisations</a></li> <li><a href='#examples'>Examples</a></li> <li><a href='#properties'>Properties</a></li> <li><a href='#related_countability_properties'>Related countability properties</a></li> <ul> <li><a href='#properties'>Properties</a></li> <li><a href='#implications'>Implications</a></li> </ul> </ul> </div> <h2 id="idea">Idea</h2> <p>A <a class="existingWikiWord" href="/nlab/show/space">space</a> (such as a <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a>) is <em>second-countable</em> if, in a certain sense, there is only a <a class="existingWikiWord" href="/nlab/show/countable+set">countable</a> amount of information globally in its topology. (Change ‘globally’ to ‘locally’ to get a <a class="existingWikiWord" href="/nlab/show/first-countable+space">first-countable space</a>.)</p> <h2 id="definitions">Definitions</h2> <div class="num_defn" id="SecondContablTopologicalSpace"> <h6 id="definition">Definition</h6> <p><strong>(second-countable topological space)</strong></p> <p>A <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a> is <strong>second-countable</strong> if it has a <a class="existingWikiWord" href="/nlab/show/base+for+a+topology">base for its topology</a> consisting of a <a class="existingWikiWord" href="/nlab/show/countable+set">countable set</a> of <a class="existingWikiWord" href="/nlab/show/subsets">subsets</a>.</p> </div> <div class="num_defn"> <h6 id="definition_2">Definition</h6> <p>A <a class="existingWikiWord" href="/nlab/show/locale">locale</a> is <strong>second-countable</strong> if there is a <a class="existingWikiWord" href="/nlab/show/countable+set">countable set</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/open+subspaces">open subspaces</a> (elements of the <a class="existingWikiWord" href="/nlab/show/frame+of+opens">frame of opens</a>) such that every open <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/join">join</a> of some <a class="existingWikiWord" href="/nlab/show/subset">subset</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math>. That is, we have</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>G</mi><mo>=</mo><mo lspace="thinmathspace" rspace="thinmathspace">⋁</mo><mo stretchy="false">{</mo><mi>U</mi><mo lspace="verythinmathspace">:</mo><mi>B</mi><mspace width="thickmathspace"></mspace><mo stretchy="false">|</mo><mspace width="thickmathspace"></mspace><mi>U</mi><mo>⊆</mo><mi>G</mi><mo stretchy="false">}</mo><mo>.</mo></mrow><annotation encoding="application/x-tex"> G = \bigvee \{ U\colon B \;|\; U \subseteq G \} .</annotation></semantics></math></div></div> <h2 id="generalisations">Generalisations</h2> <p>The <strong>weight</strong> of a space is the <a class="existingWikiWord" href="/nlab/show/minimum">minimum</a> of the <a class="existingWikiWord" href="/nlab/show/cardinalities">cardinalities</a> of the possible bases <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math>. We are implicitly using the <a class="existingWikiWord" href="/nlab/show/axiom+of+choice">axiom of choice</a> here, to suppose that this set of cardinalities (which really is a <a class="existingWikiWord" href="/nlab/show/small+set">small set</a> because bounded above by the number of open subspaces, and <a class="existingWikiWord" href="/nlab/show/inhabited+set">inhabited</a> by this number as well) has a minimum. But without Choice, we can still consider this collection of cardinalities.</p> <p>Then a second-countable space is simply one with a countable weight.</p> <h2 id="examples">Examples</h2> <div class="num_example" id="SecondCountableEuclideanSpace"> <h6 id="example">Example</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/Euclidean+space">Euclidean space</a> is second-countable)</strong></p> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">n \in \mathbb{N}</annotation></semantics></math>. Consider the <a class="existingWikiWord" href="/nlab/show/Euclidean+space">Euclidean space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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 <a class="existingWikiWord" href="/nlab/show/Euclidean+norm">Euclidean</a> <a class="existingWikiWord" href="/nlab/show/metric+topology">metric topology</a>. Then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^n</annotation></semantics></math> is second countable.</p> <p>A <a class="existingWikiWord" href="/nlab/show/countable+set">countable set</a> of <a class="existingWikiWord" href="/nlab/show/topological+base">base open subsets</a> is given by the <a class="existingWikiWord" href="/nlab/show/open+balls">open balls</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>B</mi> <mi>x</mi> <mo>∘</mo></msubsup><mo stretchy="false">(</mo><mi>ϵ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">B^\circ_x(\epsilon)</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/rational+number">rational</a> <a class="existingWikiWord" href="/nlab/show/radius">radius</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϵ</mi><mo>∈</mo><msub><mi>ℚ</mi> <mrow><mo>≥</mo><mn>0</mn></mrow></msub><mo>⊂</mo><msub><mi>ℝ</mi> <mrow><mo>≥</mo><mn>0</mn></mrow></msub></mrow><annotation encoding="application/x-tex">\epsilon \in \mathbb{Q}_{\geq 0} \subset \mathbb{R}_{\geq 0}</annotation></semantics></math> and centered at points with <a class="existingWikiWord" href="/nlab/show/rational+number">rational</a> <a class="existingWikiWord" href="/nlab/show/coordinates">coordinates</a>: <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>∈</mo><msup><mi>ℚ</mi> <mi>n</mi></msup><mo>⊂</mo><msup><mi>ℝ</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">x \in \mathbb{Q}^n \subset \mathbb{R}^n</annotation></semantics></math>.</p> </div> <div class="num_example"> <h6 id="example_2">Example</h6> <p>A <a class="existingWikiWord" href="/nlab/show/compact+space">compact</a> <a class="existingWikiWord" href="/nlab/show/metric+space">metric space</a> is second-countable.</p> </div> <div class="num_example"> <h6 id="example_3">Example</h6> <p>A <a class="existingWikiWord" href="/nlab/show/separable+space">separable</a> <a class="existingWikiWord" href="/nlab/show/metric+space">metric space</a>, e.g., a <a class="existingWikiWord" href="/nlab/show/Polish+space">Polish space</a>, is second-countable.</p> </div> <div class="num_remark"> <h6 id="remark">Remark</h6> <p>It is <em>not</em> true that separable <a class="existingWikiWord" href="/nlab/show/first-countable+spaces">first-countable spaces</a> are second-countable; a counterexample is the <a class="existingWikiWord" href="/nlab/show/real+line">real line</a> equipped with the half-open or lower limit topology that has as basis the collection of half-open intervals <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">[a, b)</annotation></semantics></math>.</p> </div> <div class="num_example"> <h6 id="example_4">Example</h6> <p>A <a class="existingWikiWord" href="/nlab/show/Hausdorff+space">Hausdorff</a> <a class="existingWikiWord" href="/nlab/show/locally+Euclidean+space">locally Euclidean space</a> is second-countable precisely it is <a class="existingWikiWord" href="/nlab/show/paracompact+space">paracompact</a> and has a <a class="existingWikiWord" href="/nlab/show/countable+set">countable set</a> of <a class="existingWikiWord" href="/nlab/show/connected+components">connected components</a>. In this case it is called a <em><a class="existingWikiWord" href="/nlab/show/topological+manifold">topological manifold</a></em>.</p> <p>See at <em><a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a></em> <a href="#RegularityConditionsForTopologicalManifoldsComparison">this prop.</a>.</p> </div> <div class="num_example"> <h6 id="example_5">Example</h6> <p>A <a class="existingWikiWord" href="/nlab/show/countable+set">countable</a> <a class="existingWikiWord" href="/nlab/show/coproduct">coproduct</a> (<a class="existingWikiWord" href="/nlab/show/disjoint+union+space">disjoint union space</a>) of second-countable spaces is second-countable.</p> <p>Countable <a class="existingWikiWord" href="/nlab/show/products">products</a> (<a class="existingWikiWord" href="/nlab/show/product+topological+spaces">product topological spaces</a>) of second-countable spaces are second-countable.</p> <p><a class="existingWikiWord" href="/nlab/show/subspace">Subspaces</a> of second-countable spaces are second-countable.</p> </div> <div class="num_exampl"> <h6 id="example_6">Example</h6> <p>If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> is second-countable and there is an <a class="existingWikiWord" href="/nlab/show/open+map">open</a> <a class="existingWikiWord" href="/nlab/show/surjection">surjection</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo lspace="verythinmathspace">:</mo><mi>X</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">f \colon X \to Y</annotation></semantics></math>, then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math> is second-countable.</p> </div> <div class="num_example"> <h6 id="example_7">Example</h6> <p>For second-countable <a class="existingWikiWord" href="/nlab/show/separation+axiom">T_3</a> spaces <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>,</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">X, Y</annotation></semantics></math>, if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/locally+compact+topological+space">locally compact</a>, then the <a class="existingWikiWord" href="/nlab/show/mapping+space">mapping space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Y</mi> <mi>X</mi></msup></mrow><annotation encoding="application/x-tex">Y^X</annotation></semantics></math> with the <a class="existingWikiWord" href="/nlab/show/compact-open+topology">compact-open topology</a> is second-countable.</p> <p>Cf. <a class="existingWikiWord" href="/nlab/show/Urysohn+metrization+theorem">Urysohn metrization theorem</a> and <a class="existingWikiWord" href="/nlab/show/Polish+space">Polish space</a>. I (<a class="existingWikiWord" href="/nlab/show/Todd+Trimble">Todd Trimble</a>) am uncertain to what extent the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>T</mi> <mn>3</mn></msub></mrow><annotation encoding="application/x-tex">T_3</annotation></semantics></math> assumption can be removed.</p> </div> <h2 id="properties">Properties</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/second-countable+regular+spaces+are+paracompact">second-countable regular spaces are paracompact</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/locally+compact+and+second-countable+spaces+are+sigma-compact">locally compact and second-countable spaces are sigma-compact</a></p> </li> </ul> <h2 id="related_countability_properties">Related countability properties</h2> <div> <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" width="678.574pt" height="203.568pt" viewBox="0 0 678.574 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1.206173 1.02194 0.0488286 1.989779 0.0012661 C 1.02239 -0.0474574 -0.665601 -1.205445 -1.292715 -2.559727 " transform="matrix(0.93478,-0.36075,-0.36075,-0.93478,270.68342,114.24634)"></path> <path style=" stroke:none;fill-rule:nonzero;fill:rgb(100%,100%,100%);fill-opacity:1;" d="M 217.125 136.425781 L 232.765625 136.425781 L 232.765625 127.371094 L 217.125 127.371094 Z M 217.125 136.425781 "></path> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#454609461-glyph0-27" x="219.27" y="134.281"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#454609461-glyph0-25" x="224.934818" y="134.281"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#454609461-glyph0-5" x="131.727" y="175.967"></use> <use xlink:href="#454609461-glyph0-24" x="133.982327" y="175.967"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#454609461-glyph1-2" x="139.137" y="175.967"></use> </g> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#454609461-glyph0-26" x="148.897" y="175.967"></use> <use xlink:href="#454609461-glyph0-7" x="153.297468" y="175.967"></use> <use xlink:href="#454609461-glyph0-4" x="157.269047" y="175.967"></use> <use xlink:href="#454609461-glyph0-7" x="160.382151" y="175.967"></use> <use xlink:href="#454609461-glyph0-12" x="164.35373" y="175.967"></use> <use xlink:href="#454609461-glyph0-11" x="167.895723" y="175.967"></use> <use xlink:href="#454609461-glyph0-1" x="171.867302" y="175.967"></use> <use xlink:href="#454609461-glyph0-26" x="178.412214" y="175.967"></use> <use xlink:href="#454609461-glyph0-7" x="182.812682" y="175.967"></use> <use xlink:href="#454609461-glyph0-12" x="186.784261" y="175.967"></use> <use xlink:href="#454609461-glyph0-3" x="190.326254" y="175.967"></use> </g> </g> </svg> <p>Axioms: <a class="existingWikiWord" href="/nlab/show/axiom+of+choice">axiom of choice</a> (AC), <a class="existingWikiWord" href="/nlab/show/countable+choice">countable choice</a> (CC).</p> <h3 id="properties">Properties</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/second-countable+space">second-countable</a>: there is a <a class="existingWikiWord" href="/nlab/show/countable+set">countable</a> <a class="existingWikiWord" href="/nlab/show/topological+base">base</a> of the topology.</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/metrisable+topological+space">metrisable</a>: the topology is induced by a metric.</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>σ</mi></mrow><annotation encoding="application/x-tex">\sigma</annotation></semantics></math>-locally discrete base: the topology of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> is generated by a <a class="existingWikiWord" href="/nlab/show/countably+locally+discrete+set+of+subsets"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"> <semantics> <mrow> <mi>σ</mi> </mrow> <annotation encoding="application/x-tex">\sigma</annotation> </semantics> </math>-locally discrete</a> <a class="existingWikiWord" href="/nlab/show/topological+base">base</a>.</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>σ</mi></mrow><annotation encoding="application/x-tex">\sigma</annotation></semantics></math>-locally finite base: the topology of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> is generated by a <a class="existingWikiWord" href="/nlab/show/countably+locally+finite+set+of+subsets">countably locally finite</a> <a class="existingWikiWord" href="/nlab/show/topological+base">base</a>.</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/separable+space">separable</a>: there is a countable <a class="existingWikiWord" href="/nlab/show/dense+subspace">dense</a> subset.</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Lindel%C3%B6f+topological+space">Lindelöf</a>: every <a class="existingWikiWord" href="/nlab/show/open+cover">open cover</a> has a <a class="existingWikiWord" href="/nlab/show/countable+cover">countable</a> sub-cover.</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/weakly+Lindel%C3%B6f+topological+space">weakly Lindelöf</a>: every <a class="existingWikiWord" href="/nlab/show/open+cover">open cover</a> has a <a class="existingWikiWord" href="/nlab/show/countable+set">countable</a> subcollection the union of which is dense.</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/metacompact+space">metacompact</a>: every open cover has a point-finite open refinement.</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/countable+chain+condition">countable chain condition</a>: A family of pairwise disjoint open subsets is at most countable.</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/first-countable+space">first-countable</a>: every point has a countable <a class="existingWikiWord" href="/nlab/show/neighborhood+base">neighborhood base</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Frechet-Uryson+space">Frechet-Uryson space</a>: the closure of a set <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> consists precisely of all limit points of sequences in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/sequential+topological+space">sequential topological space</a>: a set <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> is closed if it contains all limit points of sequences in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/countably+tight+space">countably tight</a>: for each subset <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> and each point <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>∈</mo><mover><mi>A</mi><mo>¯</mo></mover></mrow><annotation encoding="application/x-tex">x\in \overline A</annotation></semantics></math> there is a countable subset <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi><mo>⊆</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">D\subseteq A</annotation></semantics></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>∈</mo><mover><mi>D</mi><mo>¯</mo></mover></mrow><annotation encoding="application/x-tex">x\in \overline D</annotation></semantics></math>.</p> </li> </ul> <h3 id="implications">Implications</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/open+covers+of+metric+spaces+have+open+countably+locally+discrete+refinements">a metric space has a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"> <semantics> <mrow> <mi>σ</mi> </mrow> <annotation encoding="application/x-tex">\sigma</annotation> </semantics> </math>-locally discrete base</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Nagata-Smirnov+metrization+theorem">Nagata-Smirnov metrization theorem</a></p> </li> <li> <p>a second-countable space has a <a class="existingWikiWord" href="/nlab/show/countably+locally+finite+set+of+subsets"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"> <semantics> <mrow> <mi>σ</mi> </mrow> <annotation encoding="application/x-tex">\sigma</annotation> </semantics> </math>-locally finite</a> <a class="existingWikiWord" href="/nlab/show/topological+base">base</a>: take the the collection of singeltons of all elements of a countable cover of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>.</p> </li> <li> <p>second-countable spaces are separable: use the axiom of <a class="existingWikiWord" href="/nlab/show/countable+choice">countable choice</a> to choose a point in each set of a countable cover.</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/second-countable+spaces+are+Lindel%C3%B6f">second-countable spaces are Lindelöf</a>.</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/weakly+Lindel%C3%B6f+spaces+with+countably+locally+finite+base+are+second+countable">weakly Lindelöf spaces with countably locally finite base are second countable</a>.</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/separable+metacompact+spaces+are+Lindel%C3%B6f">separable metacompact spaces are Lindelöf</a>.</p> </li> <li> <p>separable spaces satisfy the countable chain condition: given a dense set <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math> and a family <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><msub><mi>U</mi> <mi>α</mi></msub><mo>:</mo><mi>α</mi><mo>∈</mo><mi>A</mi><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{U_\alpha : \alpha \in A\}</annotation></semantics></math>, the map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi><mo>∩</mo><msub><mo lspace="thinmathspace" rspace="thinmathspace">⋃</mo> <mrow><mi>α</mi><mo>∈</mo><mi>A</mi></mrow></msub><msub><mi>U</mi> <mi>α</mi></msub><mo>→</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">D \cap \bigcup_{\alpha \in A} U_\alpha \to A</annotation></semantics></math> assigning <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi></mrow><annotation encoding="application/x-tex">d</annotation></semantics></math> to the unique <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>α</mi><mo>∈</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">\alpha \in A</annotation></semantics></math> with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi><mo>∈</mo><msub><mi>U</mi> <mi>α</mi></msub></mrow><annotation encoding="application/x-tex">d \in U_\alpha</annotation></semantics></math> is surjective.</p> </li> <li> <p>separable spaces are weakly Lindelöf: given a countable dense subset and an open cover <a class="existingWikiWord" href="/nlab/show/axiom+of+choice">choose</a> for each point of the subset an open from the cover.</p> </li> <li> <p>Lindelöf spaces are trivially also weakly Lindelöf.</p> </li> <li> <p>a space with a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>σ</mi></mrow><annotation encoding="application/x-tex">\sigma</annotation></semantics></math>-locally finite base is first countable: obviously, every point is contained in at most countably many sets of a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>σ</mi></mrow><annotation encoding="application/x-tex">\sigma</annotation></semantics></math>-locally finite base.</p> </li> <li> <p>a first-countable space is obviously Fréchet-Urysohn.</p> </li> <li> <p>a Fréchet-Uryson space is obviously sequential.</p> </li> <li> <p>a sequential space is obviously countably tight.</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/paracompact+spaces+satisfying+the+countable+chain+condition+are+Lindel%C3%B6f">paracompact spaces satisfying the countable chain condition are Lindelöf</a>.</p> </li> </ul> </div></body></html> </div> <div class="revisedby"> <p> Last revised on June 1, 2024 at 01:02:54. See the <a href="/nlab/history/second-countable+space" style="color: #005c19">history</a> of this page for a list of all contributions to it. </p> </div> <div class="navigation navfoot"> <a href="/nlab/edit/second-countable+space" accesskey="E" class="navlink" id="edit" rel="nofollow">Edit</a><a href="https://nforum.ncatlab.org/discussion/18067/#Item_1">Discuss</a><span class="backintime"><a href="/nlab/revision/second-countable+space/17" accesskey="B" class="navlinkbackintime" id="to_previous_revision" rel="nofollow">Previous revision</a></span><a href="/nlab/show/diff/second-countable+space" accesskey="C" class="navlink" id="see_changes" rel="nofollow">Changes from previous revision</a><a href="/nlab/history/second-countable+space" accesskey="S" class="navlink" id="history" rel="nofollow">History (17 revisions)</a> <a href="/nlab/show/second-countable+space/cite" style="color: black">Cite</a> <a href="/nlab/print/second-countable+space" accesskey="p" id="view_print" rel="nofollow">Print</a> <a href="/nlab/source/second-countable+space" id="view_source" rel="nofollow">Source</a> </div> </div>  </div>  </body> </html>