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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="complete_spaces">Complete spaces</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#definitions'>Definitions</a></li> <li><a href='#completion'>Completion</a></li> <li><a href='#properties'>Properties</a></li> <ul> <li><a href='#relation_to_compact_spaces'>Relation to compact spaces</a></li> <li><a href='#relation_to_baire_spaces'>Relation to Baire spaces</a></li> </ul> <li><a href='#generalization'>Generalization</a></li> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> </ul> </div> <h2 id="idea">Idea</h2> <p>A <a class="existingWikiWord" href="/nlab/show/space">space</a> (with “space” taken in a sense relevant to the field of <a class="existingWikiWord" href="/nlab/show/topology">topology</a>) is <em>complete</em> (or <em>Cauchy-complete</em>) if every <a class="existingWikiWord" href="/nlab/show/sequence">sequence</a>, <a class="existingWikiWord" href="/nlab/show/net">net</a>, or <a class="existingWikiWord" href="/nlab/show/filter">filter</a> that should converge really does <a class="existingWikiWord" href="/nlab/show/convergence">converge</a>. We identify the sequences, nets, or filters that “should” converge as the <em><a class="existingWikiWord" href="/nlab/show/Cauchy+sequence">Cauchy</a></em> ones.</p> <p>A space that is not complete has “gaps” that may be filled to form its <em>completion</em>; it is rather natural to make the space (or equivalently its underlying topological space) <a class="existingWikiWord" href="/nlab/show/Hausdorff+space">Hausdorff</a> at the same time. Forming the completion of a Hausdorff space is an important example of <a class="existingWikiWord" href="/nlab/show/completion">completion</a> in the general abstract sense.</p> <h2 id="definitions">Definitions</h2> <p>A space (which may be a <a class="existingWikiWord" href="/nlab/show/metric+space">metric space</a>, a <a class="existingWikiWord" href="/nlab/show/Cauchy+space">Cauchy space</a>, or anything in between) is <strong>Cauchy-complete</strong>, or simply <strong>complete</strong>, if every <a class="existingWikiWord" href="/nlab/show/Cauchy+filter">Cauchy filter</a> converges, equivalently if every <a class="existingWikiWord" href="/nlab/show/Cauchy+net">Cauchy net</a> converges. A space is <strong><a class="existingWikiWord" href="/nlab/show/sequentially+Cauchy-complete">sequentially Cauchy-complete</a></strong> if every <a class="existingWikiWord" href="/nlab/show/Cauchy+sequence">Cauchy sequence</a> converges. Note that a sequentially complete metric space must be complete (in <a class="existingWikiWord" href="/nlab/show/classical+mathematics">classical mathematics</a>), but this does not hold for more general spaces (nor even for metric spaces in <a class="existingWikiWord" href="/nlab/show/constructive+mathematics">constructive mathematics</a>).</p> <p>A space is <strong>topologically complete</strong> if its <a class="existingWikiWord" href="/nlab/show/underlying+topological+space">underlying topological space</a> is completely <a class="existingWikiWord" href="/nlab/show/metrizable+space">metrizable</a>. There are various other notions related to this. See <a class="existingWikiWord" href="/nlab/show/topologically+complete+space">topologically complete space</a>.</p> <h2 id="completion">Completion</h2> <p>The set <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi><mi>X</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}X</annotation></semantics></math> of Cauchy filters on a space <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> may generally be given the same sort of structure as <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> itself has, and this space will be complete. Exactly how to do this depends on what structure <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 supposed to have, of course, and one can make the general statement false by requiring something artificial as the structure in question, most extremely the structure of being a specific non-complete space. But it works for most natural categories of spaces.</p> <p>The general idea is this: every point in <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> generates a principal <a class="existingWikiWord" href="/nlab/show/ultrafilter">ultrafilter</a> (consisting of those sets to which the point belongs), so there is a natural map from <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> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi><mi>X</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}X</annotation></semantics></math>. Furthermore, this map is a morphism of the appropriate structure, which in particular makes it <a class="existingWikiWord" href="/nlab/show/Cauchy-continuous+map">Cauchy-continuous</a> (preserving Cauchy filters) and continuous (preserving limits). So all of the limits in <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> still exist in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi><mi>X</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}X</annotation></semantics></math>, but now each Cauchy filter in <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> (having become both a Cauchy filter in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math> and a point in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math>) has a limit as well. The additional Cauchy filters based on the additional points in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi><mi>X</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}X</annotation></semantics></math> will also have a limit in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi><mi>X</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}X</annotation></semantics></math>, essentially because <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/monad">monad</a> (so a Cauchy filter of Cauchy filters folds into a single Cauchy filter).</p> <p>There is a problem that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi><mi>X</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}X</annotation></semantics></math> is rather larger than necessary; for example, all of the filters that converge to a given point in <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> (not just the free ultrafilter at that point) exist in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi><mi>X</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}X</annotation></semantics></math> and converge to one another. But you can take a <a class="existingWikiWord" href="/nlab/show/quotient+space">quotient</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi><mi>X</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}X</annotation></semantics></math> to make it Hausdorff, obtaining the <strong>Hausdorff completion</strong> 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>. In case <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> was not Hausdorff to begin with, one can sometimes also force the quotient to leave in just as much redundancy as <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> has but no more, obtain a straight <strong>completion</strong> 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>. But really, it's most natural to make the space Hausdorff at the same time.</p> <div class="standout"> <p>Details to come, if I get around to it.</p> </div> <p>We have a picture like this, where <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 the original space, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℋ</mi></mrow><annotation encoding="application/x-tex">\mathcal{H}</annotation></semantics></math> gives a Hausdorff quotient, and an overline indicates completion:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd></mtd> <mtd></mtd> <mtd><mover><mi>X</mi><mo>¯</mo></mover></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>↪</mo></mtd> <mtd></mtd> <mtd><mi>↠</mi></mtd></mtr> <mtr><mtd><mi>X</mi></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><mi>ℋ</mi><mover><mi>X</mi><mo>¯</mo></mover><mo>≅</mo><mover><mrow><mi>ℋ</mi><mi>X</mi></mrow><mo>¯</mo></mover></mtd></mtr> <mtr><mtd></mtd> <mtd><mi>↠</mi></mtd> <mtd></mtd> <mtd><mo>↪</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mi>ℋ</mi><mi>X</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array { & & \overline{X} \\ & \hookrightarrow & & &#8608; \\ X & & & & \mathcal{H}\overline{X} \cong \overline{\mathcal{H}X} \\ & &#8608; & & \hookrightarrow \\ & & \mathcal{H}X } </annotation></semantics></math></div> <p>(Here the arrows are drawn horizontally to put styles on them; they should all be diagonal in the only possible way.)</p> <p>At least 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 <a class="existingWikiWord" href="/nlab/show/metric+space">metric space</a>, then we can also construct its completion as a <a class="existingWikiWord" href="/nlab/show/locale">locale</a>, the <a class="existingWikiWord" href="/nlab/show/localic+completion">localic completion</a>, whose spatial part is the above space, but which in <a class="existingWikiWord" href="/nlab/show/constructive+mathematics">constructive mathematics</a> may not be spatial. This is useful to have even 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 already complete.</p> <h2 id="properties">Properties</h2> <h3 id="relation_to_compact_spaces">Relation to compact spaces</h3> <p>A <a class="existingWikiWord" href="/nlab/show/compact+space">compact space</a> is necessarily complete. A space is called <strong>precompact</strong> if its completion is compact. For metric spaces (or even <a class="existingWikiWord" href="/nlab/show/uniform+spaces">uniform spaces</a>), there is a natural notion of a <a class="existingWikiWord" href="/nlab/show/totally+bounded+space">totally bounded space</a>; in <a class="existingWikiWord" href="/nlab/show/classical+mathematics">classical mathematics</a>, we have the theorem that a space is totally bounded if and only if it is precompact. Similarly, a space is compact if and only if it is both complete and totally bounded (or in <a class="existingWikiWord" href="/nlab/show/constructive+mathematics">constructive mathematics</a>, both complete and precompact). Thus the purely topological property of compactness is the conjunction of the nontopological properties of completeness and total boundedness.</p> <p>In some <a class="existingWikiWord" href="/nlab/show/constructive+analysis">constructive approaches to analysis</a> (including most of Brouwer's school and some of Bishop's school), ‘complete and totally bounded’ is taken as the <em>definition</em> of ‘compact’, because it holds of examples such as the <a class="existingWikiWord" href="/nlab/show/unit+interval">unit interval</a> that fail to be compact (in the usual sense) without the <a class="existingWikiWord" href="/nlab/show/fan+theorem">fan theorem</a>. However, in this case, compactness is no longer a <a class="existingWikiWord" href="/nlab/show/topological+property">topological property</a>; see <a class="existingWikiWord" href="/nlab/show/Bishop-compact+space">Bishop-compact space</a> for more information. This is reconciled somewhat with the theory of <a class="existingWikiWord" href="/nlab/show/localic+completion">localic completion</a>, in which a uniform space is totally bounded if and only if its localic completion is compact (in the usual sense).</p> <h3 id="relation_to_baire_spaces">Relation to Baire spaces</h3> <p>Every complete metric space is a <a class="existingWikiWord" href="/nlab/show/Baire+space">Baire space</a>. Since being a Baire space is a <a class="existingWikiWord" href="/nlab/show/topological+property">topological property</a>, it follows that every <a class="existingWikiWord" href="/nlab/show/topologically+complete+space">topologically complete space</a> is a Baire space.</p> <p>There is also a topological property of <span class="newWikiWord">Čech-completeness<a href="/nlab/new/Cech-complete+space">?</a></span> that is related to this; in particular, a metric space is Čech-complete if and only if it is complete, and every Čech-complete space is a Baire space. In general, we have these proper implications: topologically complete → Čech-complete → Baire.</p> <p>It is not the case that a complete uniform space (meaning complete with respect to its uniformity) is a Baire space. See MathOverflow <a href="https://mathoverflow.net/questions/212308/baire-category-theorem-for-complete-uniform-spaces">here</a> for some examples.</p> <h2 id="generalization">Generalization</h2> <p>When <a class="existingWikiWord" href="/nlab/show/Bill+Lawvere">Bill Lawvere</a> interpreted (in <a href="#Lawvere1973">Lawvere 1973</a>) <a class="existingWikiWord" href="/nlab/show/metric+spaces">metric spaces</a> as certain <a class="existingWikiWord" href="/nlab/show/enriched+categories">enriched categories</a>, he found that a metric space was complete if and only if every <a class="existingWikiWord" href="/nlab/show/adjunction">adjunction</a> of <a class="existingWikiWord" href="/nlab/show/bimodules">bimodules</a> over the enriched category is induced by an <a class="existingWikiWord" href="/nlab/show/enriched+functor">enriched functor</a>. Accordingly, this becomes the notion of <a class="existingWikiWord" href="/nlab/show/Cauchy-complete+category">Cauchy-complete category</a>. (Note that one <em>must</em> say ‘Cauchy’ here, since this is <em>weaker</em> than being a <a class="existingWikiWord" href="/nlab/show/complete+category">complete category</a>, which is based on an incompatible analogy.)</p> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p>a complete <a class="existingWikiWord" href="/nlab/show/normed+vector+space">normed vector space</a> is a <em><a class="existingWikiWord" href="/nlab/show/Banach+space">Banach space</a></em></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/completion+of+a+space">completion of a space</a></p> </li> </ul> <h2 id="references">References</h2> <ul> <li id="Ark1977"> <p>A.V. Arkhangel′skii (1977). <em>Complete space</em>. Matematicheskaya entsiklopediya. <a href="https://www.encyclopediaofmath.org/index.php/Complete_space">Updated English version</a>.</p> </li> <li id="Lawvere1973"> <p><a class="existingWikiWord" href="/nlab/show/Bill+Lawvere">Bill Lawvere</a> (1973). <em>Metric spaces, generalized logic and closed categories</em>. Reprinted in <a class="existingWikiWord" href="/nlab/show/TAC">TAC</a>, 1986. <a href="http://www.tac.mta.ca/tac/reprints/articles/1/tr1abs.html">Web</a>.</p> </li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on August 13, 2023 at 18:07:28. See the <a href="/nlab/history/complete+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/complete+space" accesskey="E" class="navlink" id="edit" rel="nofollow">Edit</a><a href="https://nforum.ncatlab.org/discussion/15759/#Item_2">Discuss</a><span class="backintime"><a href="/nlab/revision/complete+space/24" accesskey="B" class="navlinkbackintime" id="to_previous_revision" rel="nofollow">Previous revision</a></span><a href="/nlab/show/diff/complete+space" accesskey="C" class="navlink" id="see_changes" rel="nofollow">Changes from previous revision</a><a href="/nlab/history/complete+space" accesskey="S" class="navlink" id="history" rel="nofollow">History (24 revisions)</a> <a href="/nlab/show/complete+space/cite" style="color: black">Cite</a> <a href="/nlab/print/complete+space" accesskey="p" id="view_print" rel="nofollow">Print</a> <a href="/nlab/source/complete+space" id="view_source" rel="nofollow">Source</a> </div> </div>  </div>  </body> </html>