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continuous metric space valued function on compact metric space is uniformly continuous (changes) in nLab
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continuous metric space valued function on compact metric space is uniformly continuous (changes) </h1> <div class="navigation"> <span class="skipNav"><a href='#navEnd'>Skip the Navigation Links</a> | </span> <span style="display:inline-block; width: 0.3em;"></span> <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/7860/#Item_5" 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> <span id='navEnd'></span> </div> <div id="revision"> <p class="show_diff"> Showing changes from revision #1 to #2: <ins class="diffins">Added</ins> | <del class="diffdel">Removed</del> | <del class="diffmod">Chan</del><ins class="diffmod">ged</ins> </p> <div class='num_prop'> <del class='diffmod'><h6 id='proposition'>Proposition</h6></del><ins class='diffmod'><div class='toc clickDown' tabindex='0'> <h3 id='context'>Context</h3> <h4 id='topology'>Topology</h4> <div class='hide'> <p><strong><a class='existingWikiWord' href='/nlab/show/diff/topology'>topology</a></strong> (<a class='existingWikiWord' href='/nlab/show/diff/general+topology'>point-set topology</a>, <a class='existingWikiWord' href='/nlab/show/diff/point-free+topology'>point-free topology</a>)</p> <p>see also <em><a class='existingWikiWord' href='/nlab/show/diff/differential+topology'>differential topology</a></em>, <em><a class='existingWikiWord' href='/nlab/show/diff/algebraic+topology'>algebraic topology</a></em>, <em><a class='existingWikiWord' href='/nlab/show/diff/functional+analysis'>functional analysis</a></em> and <em><a class='existingWikiWord' href='/nlab/show/diff/topological+homotopy+theory'>topological</a> <a class='existingWikiWord' href='/nlab/show/diff/homotopy+theory'>homotopy theory</a></em></p> <p><a class='existingWikiWord' href='/nlab/show/diff/Introduction+to+Topology'>Introduction</a></p> <p><strong>Basic concepts</strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/open+subspace'>open subset</a>, <a class='existingWikiWord' href='/nlab/show/diff/closed+subspace'>closed subset</a>, <a class='existingWikiWord' href='/nlab/show/diff/neighborhood'>neighbourhood</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/topological+space'>topological space</a>, <a class='existingWikiWord' href='/nlab/show/diff/locale'>locale</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/topological+base'>base for the topology</a>, <a class='existingWikiWord' href='/nlab/show/diff/neighborhood+base'>neighbourhood base</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/finer+topology'>finer/coarser topology</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/closed+subspace'>closure</a>, <a class='existingWikiWord' href='/nlab/show/diff/interior'>interior</a>, <a class='existingWikiWord' href='/nlab/show/diff/boundary'>boundary</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/separation+axioms'>separation</a>, <a class='existingWikiWord' href='/nlab/show/diff/sober+topological+space'>sobriety</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/continuous+map'>continuous function</a>, <a class='existingWikiWord' href='/nlab/show/diff/homeomorphism'>homeomorphism</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/uniformly+continuous+map'>uniformly continuous function</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/embedding+of+topological+spaces'>embedding</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/open+map'>open map</a>, <a class='existingWikiWord' href='/nlab/show/diff/closed+map'>closed map</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/sequence'>sequence</a>, <a class='existingWikiWord' href='/nlab/show/diff/net'>net</a>, <a class='existingWikiWord' href='/nlab/show/diff/subnet'>sub-net</a>, <a class='existingWikiWord' href='/nlab/show/diff/filter'>filter</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/convergence'>convergence</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/category'>category</a> <a class='existingWikiWord' href='/nlab/show/diff/Top'>Top</a></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/convenient+category+of+topological+spaces'>convenient category of topological spaces</a></li> </ul> </li> </ul> <p><strong><a href='Top#UniversalConstructions'>Universal constructions</a></strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/weak+topology'>initial topology</a>, <a class='existingWikiWord' href='/nlab/show/diff/weak+topology'>final topology</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/subspace'>subspace</a>, <a class='existingWikiWord' href='/nlab/show/diff/quotient+space'>quotient space</a>,</p> </li> <li> <p>fiber space, <a class='existingWikiWord' href='/nlab/show/diff/space+attachment'>space attachment</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/product+topological+space'>product space</a>, <a class='existingWikiWord' href='/nlab/show/diff/disjoint+union+topological+space'>disjoint union space</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/mapping+cylinder'>mapping cylinder</a>, <a class='existingWikiWord' href='/nlab/show/diff/cocylinder'>mapping cocylinder</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/mapping+cone'>mapping cone</a>, <a class='existingWikiWord' href='/nlab/show/diff/mapping+cocone'>mapping cocone</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/mapping+telescope'>mapping telescope</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/colimits+of+normal+spaces'>colimits of normal spaces</a></p> </li> </ul> <p><strong><a class='existingWikiWord' href='/nlab/show/diff/stuff%2C+structure%2C+property'>Extra stuff, structure, properties</a></strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/nice+topological+space'>nice topological space</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/metric+space'>metric space</a>, <a class='existingWikiWord' href='/nlab/show/diff/metric+topology'>metric topology</a>, <a class='existingWikiWord' href='/nlab/show/diff/metrisable+topological+space'>metrisable space</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Kolmogorov+topological+space'>Kolmogorov space</a>, <a class='existingWikiWord' href='/nlab/show/diff/Hausdorff+space'>Hausdorff space</a>, <a class='existingWikiWord' href='/nlab/show/diff/regular+space'>regular space</a>, <a class='existingWikiWord' href='/nlab/show/diff/normal+space'>normal space</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/sober+topological+space'>sober space</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/compact+space'>compact space</a>, <a class='existingWikiWord' href='/nlab/show/diff/proper+map'>proper map</a></p> <p><a class='existingWikiWord' href='/nlab/show/diff/sequentially+compact+topological+space'>sequentially compact</a>, <a class='existingWikiWord' href='/nlab/show/diff/countably+compact+topological+space'>countably compact</a>, <a class='existingWikiWord' href='/nlab/show/diff/locally+compact+topological+space'>locally compact</a>, <a class='existingWikiWord' href='/nlab/show/diff/sigma-compact+topological+space'>sigma-compact</a>, <a class='existingWikiWord' href='/nlab/show/diff/paracompact+topological+space'>paracompact</a>, <a class='existingWikiWord' href='/nlab/show/diff/countably+paracompact+topological+space'>countably paracompact</a>, <a class='existingWikiWord' href='/nlab/show/diff/strongly+compact+topological+space'>strongly compact</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/compactly+generated+topological+space'>compactly generated space</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/second-countable+space'>second-countable space</a>, <a class='existingWikiWord' href='/nlab/show/diff/first-countable+space'>first-countable space</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/contractible+space'>contractible space</a>, <a class='existingWikiWord' href='/nlab/show/diff/locally+contractible+space'>locally contractible space</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/connected+space'>connected space</a>, <a class='existingWikiWord' href='/nlab/show/diff/locally+connected+topological+space'>locally connected space</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/simply+connected+space'>simply-connected space</a>, <a class='existingWikiWord' href='/nlab/show/diff/semi-locally+simply-connected+topological+space'>locally simply-connected space</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/cell+complex'>cell complex</a>, <a class='existingWikiWord' href='/nlab/show/diff/CW+complex'>CW-complex</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/pointed+topological+space'>pointed space</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/topological+vector+space'>topological vector space</a>, <a class='existingWikiWord' href='/nlab/show/diff/Banach+space'>Banach space</a>, <a class='existingWikiWord' href='/nlab/show/diff/Hilbert+space'>Hilbert space</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/topological+group'>topological group</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/topological+vector+bundle'>topological vector bundle</a>, <a class='existingWikiWord' href='/nlab/show/diff/topological+K-theory'>topological K-theory</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/topological+manifold'>topological manifold</a></p> </li> </ul> <p><strong>Examples</strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/empty+space'>empty space</a>, <a class='existingWikiWord' href='/nlab/show/diff/point+space'>point space</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/discrete+object'>discrete space</a>, <a class='existingWikiWord' href='/nlab/show/diff/codiscrete+space'>codiscrete space</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Sierpinski+space'>Sierpinski space</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/order+topology'>order topology</a>, <a class='existingWikiWord' href='/nlab/show/diff/specialization+topology'>specialization topology</a>, <a class='existingWikiWord' href='/nlab/show/diff/Scott+topology'>Scott topology</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Euclidean+space'>Euclidean space</a></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/real+number'>real line</a>, <a class='existingWikiWord' href='/nlab/show/diff/plane'>plane</a></li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/cylinder+object'>cylinder</a>, <a class='existingWikiWord' href='/nlab/show/diff/cone'>cone</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/sphere'>sphere</a>, <a class='existingWikiWord' href='/nlab/show/diff/ball'>ball</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/circle'>circle</a>, <a class='existingWikiWord' href='/nlab/show/diff/torus'>torus</a>, <a class='existingWikiWord' href='/nlab/show/diff/annulus'>annulus</a>, <a class='existingWikiWord' href='/nlab/show/diff/M%C3%B6bius+strip'>Moebius strip</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/polytope'>polytope</a>, <a class='existingWikiWord' href='/nlab/show/diff/polyhedron'>polyhedron</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/projective+space'>projective space</a> (<a class='existingWikiWord' href='/nlab/show/diff/real+projective+space'>real</a>, <a class='existingWikiWord' href='/nlab/show/diff/complex+projective+space'>complex</a>)</p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/classifying+space'>classifying space</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/configuration+space+of+points'>configuration space</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/path'>path</a>, <a class='existingWikiWord' href='/nlab/show/diff/loop'>loop</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/compact-open+topology'>mapping spaces</a>: <a class='existingWikiWord' href='/nlab/show/diff/compact-open+topology'>compact-open topology</a>, <a class='existingWikiWord' href='/nlab/show/diff/topology+of+uniform+convergence'>topology of uniform convergence</a></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/loop+space'>loop space</a>, <a class='existingWikiWord' href='/nlab/show/diff/path+space'>path space</a></li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Zariski+topology'>Zariski topology</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Cantor+space'>Cantor space</a>, <a class='existingWikiWord' href='/nlab/show/diff/Mandelbrot+set'>Mandelbrot space</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Peano+curve'>Peano curve</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/line+with+two+origins'>line with two origins</a>, <a class='existingWikiWord' href='/nlab/show/diff/long+line'>long line</a>, <a class='existingWikiWord' href='/nlab/show/diff/Sorgenfrey+line'>Sorgenfrey line</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/K-topology'>K-topology</a>, <a class='existingWikiWord' href='/nlab/show/diff/Dowker+space'>Dowker space</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Warsaw+circle'>Warsaw circle</a>, <a class='existingWikiWord' href='/nlab/show/diff/Hawaiian+earring+space'>Hawaiian earring space</a></p> </li> </ul> <p><strong>Basic statements</strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Hausdorff+implies+sober'>Hausdorff spaces are sober</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/schemes+are+sober'>schemes are sober</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/continuous+images+of+compact+spaces+are+compact'>continuous images of compact spaces are compact</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/closed+subspaces+of+compact+Hausdorff+spaces+are+equivalently+compact+subspaces'>closed subspaces of compact Hausdorff spaces are equivalently compact subspaces</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/open+subspaces+of+compact+Hausdorff+spaces+are+locally+compact'>open subspaces of compact Hausdorff spaces are locally compact</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/quotient+projections+out+of+compact+Hausdorff+spaces+are+closed+precisely+if+the+codomain+is+Hausdorff'>quotient projections out of compact Hausdorff spaces are closed precisely if the codomain is Hausdorff</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/compact+spaces+equivalently+have+converging+subnets'>compact spaces equivalently have converging subnet of every net</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Lebesgue+number+lemma'>Lebesgue number lemma</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/sequentially+compact+metric+spaces+are+equivalently+compact+metric+spaces'>sequentially compact metric spaces are equivalently compact metric spaces</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/compact+spaces+equivalently+have+converging+subnets'>compact spaces equivalently have converging subnet of every net</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/sequentially+compact+metric+spaces+are+totally+bounded'>sequentially compact metric spaces are totally bounded</a></p> </li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/continuous+metric+space+valued+function+on+compact+metric+space+is+uniformly+continuous'>continuous metric space valued function on compact metric space is uniformly continuous</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/paracompact+Hausdorff+spaces+are+normal'>paracompact Hausdorff spaces are normal</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/paracompact+Hausdorff+spaces+equivalently+admit+subordinate+partitions+of+unity'>paracompact Hausdorff spaces equivalently admit subordinate partitions of unity</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/closed+injections+are+embeddings'>closed injections are embeddings</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/proper+maps+to+locally+compact+spaces+are+closed'>proper maps to locally compact spaces are closed</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/injective+proper+maps+to+locally+compact+spaces+are+equivalently+the+closed+embeddings'>injective proper maps to locally compact spaces are equivalently the closed embeddings</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/locally+compact+and+sigma-compact+spaces+are+paracompact'>locally compact and sigma-compact spaces are paracompact</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/locally+compact+and+second-countable+spaces+are+sigma-compact'>locally compact and second-countable spaces are sigma-compact</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/second-countable+regular+spaces+are+paracompact'>second-countable regular spaces are paracompact</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/CW-complexes+are+paracompact+Hausdorff+spaces'>CW-complexes are paracompact Hausdorff spaces</a></p> </li> </ul> <p><strong>Theorems</strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Urysohn%27s+lemma'>Urysohn's lemma</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Tietze+extension+theorem'>Tietze extension theorem</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Tychonoff+theorem'>Tychonoff theorem</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/tube+lemma'>tube lemma</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Michael%27s+theorem'>Michael's theorem</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Brouwer%27s+fixed+point+theorem'>Brouwer's fixed point theorem</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/topological+invariance+of+dimension'>topological invariance of dimension</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Jordan+curve+theorem'>Jordan curve theorem</a></p> </li> </ul> <p><strong>Analysis Theorems</strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Heine-Borel+theorem'>Heine-Borel theorem</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/intermediate+value+theorem'>intermediate value theorem</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/extreme+value+theorem'>extreme value theorem</a></p> </li> </ul> <p><strong><a class='existingWikiWord' href='/nlab/show/diff/topological+homotopy+theory'>topological homotopy theory</a></strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/homotopy'>left homotopy</a>, <a class='existingWikiWord' href='/nlab/show/diff/homotopy'>right homotopy</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/homotopy+equivalence'>homotopy equivalence</a>, <a class='existingWikiWord' href='/nlab/show/diff/deformation+retract'>deformation retract</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/fundamental+group'>fundamental group</a>, <a class='existingWikiWord' href='/nlab/show/diff/covering+space'>covering space</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/fundamental+theorem+of+covering+spaces'>fundamental theorem of covering spaces</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/homotopy+group'>homotopy group</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/weak+homotopy+equivalence'>weak homotopy equivalence</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Whitehead+theorem'>Whitehead's theorem</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Freudenthal+suspension+theorem'>Freudenthal suspension theorem</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/nerve+theorem'>nerve theorem</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/homotopy+extension+property'>homotopy extension property</a>, <a class='existingWikiWord' href='/nlab/show/diff/Hurewicz+cofibration'>Hurewicz cofibration</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/topological+cofiber+sequence'>cofiber sequence</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Str%C3%B8m+model+structure'>Strøm model category</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/classical+model+structure+on+topological+spaces'>classical model structure on topological spaces</a></p> </li> </ul> </div> <h4 id='analysis'>Analysis</h4> <div class='hide'> <p><strong><a class='existingWikiWord' href='/nlab/show/diff/analysis'>analysis</a></strong> (<a class='existingWikiWord' href='/nlab/show/diff/differential+calculus'>differential</a>/<a class='existingWikiWord' href='/nlab/show/diff/integral+calculus'>integral</a> <a class='existingWikiWord' href='/nlab/show/diff/calculus'>calculus</a>, <a class='existingWikiWord' href='/nlab/show/diff/functional+analysis'>functional analysis</a>, <a class='existingWikiWord' href='/nlab/show/diff/topology'>topology</a>)</p> <p><a class='existingWikiWord' href='/nlab/show/diff/epsilontic+analysis'>epsilontic analysis</a></p> <p><a class='existingWikiWord' href='/nlab/show/diff/infinitesimal+analysis'>infinitesimal analysis</a></p> <p><a class='existingWikiWord' href='/nlab/show/diff/computable+analysis'>computable analysis</a></p> <p><em><a class='existingWikiWord' href='/nlab/show/diff/Introduction+to+Topology+--+1'>Introduction</a></em></p> <h2 id='basic_concepts'>Basic concepts</h2> <p><a class='existingWikiWord' href='/nlab/show/diff/triangle+inequality'>triangle inequality</a></p> <p><a class='existingWikiWord' href='/nlab/show/diff/metric+space'>metric space</a>, <a class='existingWikiWord' href='/nlab/show/diff/norm'>normed vector space</a></p> <p><a class='existingWikiWord' href='/nlab/show/diff/ball'>open ball</a>, <a class='existingWikiWord' href='/nlab/show/diff/open+subspace'>open subset</a>, <a class='existingWikiWord' href='/nlab/show/diff/neighborhood'>neighbourhood</a></p> <p><a class='existingWikiWord' href='/nlab/show/diff/metric+topology'>metric topology</a></p> <p><a class='existingWikiWord' href='/nlab/show/diff/sequence'>sequence</a>, <a class='existingWikiWord' href='/nlab/show/diff/net'>net</a></p> <p><a class='existingWikiWord' href='/nlab/show/diff/convergence'>convergence</a>, <a class='existingWikiWord' href='/nlab/show/diff/convergence'>limit of a sequence</a></p> <p><a class='existingWikiWord' href='/nlab/show/diff/compact+space'>compactness</a>, <a class='existingWikiWord' href='/nlab/show/diff/sequentially+compact+topological+space'>sequential compactness</a></p> <p><a class='existingWikiWord' href='/nlab/show/diff/differentiation'>differentiation</a>, <a class='existingWikiWord' href='/nlab/show/diff/integral'>integration</a></p> <p><a class='existingWikiWord' href='/nlab/show/diff/topological+vector+space'>topological vector space</a></p> <h2 id='basic_facts'>Basic facts</h2> <p><a class='existingWikiWord' href='/nlab/show/diff/continuous+metric+space+valued+function+on+compact+metric+space+is+uniformly+continuous'>continuous metric space valued function on compact metric space is uniformly continuous</a></p> <p>…</p> <h2 id='theorems'>Theorems</h2> <p><a class='existingWikiWord' href='/nlab/show/diff/intermediate+value+theorem'>intermediate value theorem</a></p> <p><a class='existingWikiWord' href='/nlab/show/diff/extreme+value+theorem'>extreme value theorem</a></p> <p><a class='existingWikiWord' href='/nlab/show/diff/Heine-Borel+theorem'>Heine-Borel theorem</a></p> <p>…</p> </div> </div></ins><del class='diffdel'> </del><del class='diffdel'><p><strong>(continuous function on compact space is uniformly continuous)</strong></p></del><del class='diffdel'> </del><del class='diffdel'><p>Let <math class='maruku-mathml' display='inline' id='mathml_6ea6b99403f10ab12658d34f49162b1eceb63ecf_1' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>K</mi><mo>⊂</mo><msup><mi>ℝ</mi> <mi>n</mi></msup></mrow><annotation encoding='application/x-tex'>K \subset \mathbb{R}^n</annotation></semantics></math> be a compact <a class='existingWikiWord' href='/nlab/show/diff/subspace'>subspace</a> of <a class='existingWikiWord' href='/nlab/show/diff/Euclidean+space'>Euclidean space</a>. Then every <a class='existingWikiWord' href='/nlab/show/diff/continuous+map'>continuous function</a> of the form <math class='maruku-mathml' display='inline' id='mathml_6ea6b99403f10ab12658d34f49162b1eceb63ecf_2' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>K</mi><mo>→</mo><mi>ℝ</mi></mrow><annotation encoding='application/x-tex'>K \to \mathbb{R}</annotation></semantics></math> is <a class='existingWikiWord' href='/nlab/show/diff/uniformly+continuous+map'>uniformly continuous</a>.</p></del> </div><ins class='diffins'> </ins><ins class='diffins'><h1 id='contents'>Contents</h1></ins><ins class='diffins'> </ins><ins class='diffins'><div class='maruku_toc'><ul><li><a href='#statement'>Statement</a></li><li><a href='#related_statements'>Related statements</a></li></ul></div></ins><ins class='diffins'> </ins><ins class='diffins'><h2 id='statement'>Statement</h2></ins><ins class='diffins'> </ins><ins class='diffins'><div class='num_prop'> <h6 id='proposition'>Proposition</h6> <p><strong>(continuous metric space valued function on compact metric space is uniformly continuous)</strong></p> <p>Let <math class='maruku-mathml' display='inline' id='mathml_6a5661976a03a008d8b7bd59297a986437fdf83e_1' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>K</mi></mrow><annotation encoding='application/x-tex'>K</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_6a5661976a03a008d8b7bd59297a986437fdf83e_2' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Y</mi></mrow><annotation encoding='application/x-tex'>Y</annotation></semantics></math> be two <a class='existingWikiWord' href='/nlab/show/diff/metric+space'>metric spaces</a> regarded as <a class='existingWikiWord' href='/nlab/show/diff/topological+space'>topological spaces</a> via their <a class='existingWikiWord' href='/nlab/show/diff/metric+topology'>metric topology</a>, such that <math class='maruku-mathml' display='inline' id='mathml_6a5661976a03a008d8b7bd59297a986437fdf83e_3' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>K</mi></mrow><annotation encoding='application/x-tex'>K</annotation></semantics></math> <a class='existingWikiWord' href='/nlab/show/diff/compact+space'>compact</a>.</p> <p>Then every <a class='existingWikiWord' href='/nlab/show/diff/continuous+map'>continuous function</a> of the form</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_6a5661976a03a008d8b7bd59297a986437fdf83e_4' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>K</mi><mover><mo>⟶</mo><mi>f</mi></mover><mi>Y</mi></mrow><annotation encoding='application/x-tex'> K \overset{f}{\longrightarrow} Y </annotation></semantics></math></div> <p>is <a class='existingWikiWord' href='/nlab/show/diff/uniformly+continuous+map'>uniformly continuous</a>.</p> <p>In particular since <a class='existingWikiWord' href='/nlab/show/diff/subspace'>subspaces</a> of metric spaces canonically are metric spaces, it follows that if <math class='maruku-mathml' display='inline' id='mathml_6a5661976a03a008d8b7bd59297a986437fdf83e_5' 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 metric space</p> </div></ins><ins class='diffins'> </ins><ins class='diffins'><div class='proof'> <h6 id='proof'>Proof</h6> <p>Assume on the contrary that it were not. This would mean that for all <math class='maruku-mathml' display='inline' id='mathml_6a5661976a03a008d8b7bd59297a986437fdf83e_6' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>ϵ</mi><mo>∈</mo><mo stretchy='false'>(</mo><mn>0</mn><mo>,</mo><mn>∞</mn><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>\epsilon \in (0,\infty)</annotation></semantics></math> there would for all <math class='maruku-mathml' display='inline' id='mathml_6a5661976a03a008d8b7bd59297a986437fdf83e_7' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>n</mi><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding='application/x-tex'>n \in \mathbb{N}</annotation></semantics></math> be points <math class='maruku-mathml' display='inline' id='mathml_6a5661976a03a008d8b7bd59297a986437fdf83e_8' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>x</mi> <mi>n</mi></msub><mo>,</mo><msub><mi>y</mi> <mi>n</mi></msub><mo>∈</mo><mi>K</mi></mrow><annotation encoding='application/x-tex'>x_n,y_n \in K</annotation></semantics></math> with <math class='maruku-mathml' display='inline' id='mathml_6a5661976a03a008d8b7bd59297a986437fdf83e_9' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>d</mi> <mi>K</mi></msub><mo stretchy='false'>(</mo><msub><mi>x</mi> <mi>n</mi></msub><mo>,</mo><msub><mi>y</mi> <mi>n</mi></msub><mo stretchy='false'>)</mo><mo><</mo><mn>1</mn><mo stretchy='false'>/</mo><mo stretchy='false'>(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>d_K(x_n,y_n) \lt 1/(n+1)</annotation></semantics></math> but <math class='maruku-mathml' display='inline' id='mathml_6a5661976a03a008d8b7bd59297a986437fdf83e_10' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>d</mi> <mi>Y</mi></msub><mo stretchy='false'>(</mo><mi>f</mi><mo stretchy='false'>(</mo><msub><mi>x</mi> <mi>n</mi></msub><mo stretchy='false'>)</mo><mo>,</mo><mi>f</mi><mo stretchy='false'>(</mo><msub><mi>y</mi> <mi>n</mi></msub><mo stretchy='false'>)</mo><mo stretchy='false'>)</mo><mo>></mo><mi>ϵ</mi></mrow><annotation encoding='application/x-tex'>d_Y(f(x_n),f(y_n)) \gt \epsilon</annotation></semantics></math>.</p> <p>Since in compact metric spaces every <a class='existingWikiWord' href='/nlab/show/diff/sequence'>sequence</a> has a <a class='existingWikiWord' href='/nlab/show/diff/convergence'>converging</a> subsequence (<a class='existingWikiWord' href='/nlab/show/diff/sequentially+compact+metric+spaces+are+equivalently+compact+metric+spaces'>here</a>), it follows that there is a converging subsequences <math class='maruku-mathml' display='inline' id='mathml_6a5661976a03a008d8b7bd59297a986437fdf83e_11' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><msub><mi>x</mi> <mrow><msub><mi>n</mi> <mi>k</mi></msub></mrow></msub><msub><mo stretchy='false'>)</mo> <mrow><mi>k</mi><mo>∈</mo><mi>ℕ</mi></mrow></msub></mrow><annotation encoding='application/x-tex'>(x_{n_k})_{k \in \mathbb{N}}</annotation></semantics></math>. Since <math class='maruku-mathml' display='inline' id='mathml_6a5661976a03a008d8b7bd59297a986437fdf83e_12' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>d</mi> <mi>K</mi></msub><mo stretchy='false'>(</mo><msub><mi>x</mi> <mrow><msub><mi>n</mi> <mi>k</mi></msub></mrow></msub><msub><mi>y</mi> <mrow><msub><mi>n</mi> <mi>k</mi></msub></mrow></msub><mo stretchy='false'>)</mo><mo><</mo><mn>1</mn><mo stretchy='false'>/</mo><mo stretchy='false'>(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>d_K(x_{n_k} y_{n_k}) \lt 1/(n+1)</annotation></semantics></math> it follows that also <math class='maruku-mathml' display='inline' id='mathml_6a5661976a03a008d8b7bd59297a986437fdf83e_13' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><msub><mi>y</mi> <mrow><msub><mi>n</mi> <mi>k</mi></msub></mrow></msub><msub><mo stretchy='false'>)</mo> <mrow><mi>k</mi><mo>∈</mo><mi>ℕ</mi></mrow></msub></mrow><annotation encoding='application/x-tex'>(y_{n_k})_{k \in \mathbb{N}}</annotation></semantics></math> is a converging subsequence which converges to the same point:</p> <p>But since continuous functions preserve <a class='existingWikiWord' href='/nlab/show/diff/convergence'>limits of sequences</a>, it would follow that</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_6a5661976a03a008d8b7bd59297a986437fdf83e_14' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><munder><mi>lim</mi><mrow><mi>k</mi><mo>→</mo><mn>∞</mn></mrow></munder><mi>f</mi><mo stretchy='false'>(</mo><msub><mi>x</mi> <mrow><msub><mi>n</mi> <mi>k</mi></msub></mrow></msub><mo stretchy='false'>)</mo><mspace width='thickmathspace' /><mo>=</mo><mspace width='thickmathspace' /><munder><mi>lim</mi><mrow><mi>k</mi><mo>→</mo><mn>∞</mn></mrow></munder><mi>f</mi><mo stretchy='false'>(</mo><msub><mi>y</mi> <mrow><msub><mi>n</mi> <mi>k</mi></msub></mrow></msub><mo stretchy='false'>)</mo><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> \underset{k \to \infty}{\lim} f(x_{n_k}) \;=\; \underset{k \to \infty}{\lim} f(y_{n_k}) \,. </annotation></semantics></math></div> <p>This however would contradict the assumption that <math class='maruku-mathml' display='inline' id='mathml_6a5661976a03a008d8b7bd59297a986437fdf83e_15' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>d</mi> <mi>Y</mi></msub><mo stretchy='false'>(</mo><mi>f</mi><mo stretchy='false'>(</mo><msub><mi>x</mi> <mi>k</mi></msub><mo stretchy='false'>)</mo><mo>,</mo><mi>f</mi><mo stretchy='false'>(</mo><msub><mi>y</mi> <mi>k</mi></msub><mo stretchy='false'>)</mo><mo stretchy='false'>)</mo><mo>></mo><mi>ϵ</mi></mrow><annotation encoding='application/x-tex'>d_Y(f(x_k), f(y_k)) \gt \epsilon</annotation></semantics></math>. Hence we have a <a class='existingWikiWord' href='/nlab/show/diff/proof+by+contradiction'>proof by contradiction</a>.</p> </div></ins><ins class='diffins'> </ins><ins class='diffins'><h2 id='related_statements'>Related statements</h2></ins><ins class='diffins'> </ins><ins class='diffins'><ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/maps+from+compact+spaces+to+Hausdorff+spaces+are+closed+and+proper'>maps from compact spaces to Hausdorff spaces are closed and proper</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/sequentially+compact+metric+spaces+are+equivalently+compact+metric+spaces'>sequentially compact metric spaces are equivalently compact metric spaces</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/compact+spaces+equivalently+have+converging+subnets'>compact spaces equivalently have converging subnet of every net</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/countably+compact+metric+spaces+are+equivalently+compact+metric+spaces'>countably compact metric spaces are equivalently compact metric spaces</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/sequentially+compact+metric+spaces+are+totally+bounded'>sequentially compact metric spaces are totally bounded</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Lebesgue+number+lemma'>Lebesgue number lemma</a></p> </li> </ul></ins><ins class='diffins'> </ins><ins class='diffins'><p> </p></ins> </div> <div class="revisedby"> <p> Last revised on June 21, 2017 at 08:48:49. See the <a href="/nlab/history/continuous+metric+space+valued+function+on+compact+metric+space+is+uniformly+continuous" 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/continuous+metric+space+valued+function+on+compact+metric+space+is+uniformly+continuous" accesskey="E" class="navlink" id="edit" rel="nofollow">Edit</a><a href="https://nforum.ncatlab.org/discussion/7860/#Item_5">Discuss</a><span class="backintime"><a href="/nlab/revision/diff/continuous+metric+space+valued+function+on+compact+metric+space+is+uniformly+continuous/1" accesskey="B" class="navlinkbackintime" id="to_previous_revision" rel="nofollow">Previous revision</a></span><a href="/nlab/show/continuous+metric+space+valued+function+on+compact+metric+space+is+uniformly+continuous" accesskey="C" class="navlink" id="see_changes" rel="nofollow">Hide changes</a><a href="/nlab/history/continuous+metric+space+valued+function+on+compact+metric+space+is+uniformly+continuous" accesskey="S" class="navlink" id="history" rel="nofollow">History (1 revision)</a> <a href="/nlab/show/continuous+metric+space+valued+function+on+compact+metric+space+is+uniformly+continuous/cite" style="color: black">Cite</a> <a href="/nlab/print/continuous+metric+space+valued+function+on+compact+metric+space+is+uniformly+continuous" accesskey="p" id="view_print" rel="nofollow">Print</a> <a href="/nlab/source/continuous+metric+space+valued+function+on+compact+metric+space+is+uniformly+continuous" id="view_source" rel="nofollow">Source</a> </div> </div> <!-- Content --> </div> <!-- Container --> </body> </html>