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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="continuous_maps">Continuous maps</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <ul> <li><a href='#in_traditional_topology'>In traditional topology</a></li> <li><a href='#in_cohesive_homotopy_type_theory'>In cohesive homotopy type theory</a></li> </ul> <li><a href='#definitions'>Definitions</a></li> <ul> <li><a href='#for_the_set_of_real_numbers'>For the set of real numbers</a></li> <li><a href='#EpsilonticDefinition'>The epsilontic definition for metric spaces</a></li> <li><a href='#for_topological_spaces'>For topological spaces</a></li> <li><a href='#further_variants'>Further variants</a></li> </ul> <li><a href='#properties'>Properties</a></li> <ul> <li><a href='#properties_preserved'>Properties preserved</a></li> <li><a href='#special_maps'>Special maps</a></li> <li><a href='#special_cases_in_specific_contexts'>Special cases in specific contexts</a></li> </ul> <li><a href='#in_constructive_mathematics'>In constructive mathematics</a></li> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#literature'>Literature</a></li> </ul> </div> <h2 id="idea">Idea</h2> <p>There are multiple different concepts of continuous maps, even in classical mathematics, depending upon which foundation one is using.</p> <h3 id="in_traditional_topology">In traditional topology</h3> <p>A <a class="existingWikiWord" href="/nlab/show/function">function</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> is called <em>continuous</em> if its values <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(x)</annotation></semantics></math> do not “jump” with variation of its argument <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>, unless <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 “jumps”. Roughly speaking, if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>x</mi> <mn>1</mn></msub><mo>≈</mo><msub><mi>x</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">x_1 \approx x_2</annotation></semantics></math>, then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><msub><mi>x</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mo>≈</mo><mi>f</mi><mo stretchy="false">(</mo><msub><mi>x</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(x_1) \approx f(x_2)</annotation></semantics></math>. (This can be made into a precise definition in <a class="existingWikiWord" href="/nlab/show/nonstandard+analysis">nonstandard analysis</a> if care is taken about the domains of these variables.)</p> <p>In order to make this precise (in standard analysis) one needs some concept of <a class="existingWikiWord" href="/nlab/show/neighbourhoods">neighbourhoods</a> of elements 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> and <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>.</p> <p>For instance 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> and <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> carry structure of <a class="existingWikiWord" href="/nlab/show/metric+spaces">metric spaces</a>, then one may say that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> is continuous if for every point <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>∈</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">x \in X</annotation></semantics></math> and for every small <a class="existingWikiWord" href="/nlab/show/open+ball">open ball</a> around its image <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(x)</annotation></semantics></math> in <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>, there exists a sufficiently small open ball around <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>∈</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">x \in X</annotation></semantics></math> which is still mapped by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> into that target open ball. This definition turns out to have more elegant formulation that needs to mention neither the points 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> nor the radii of open balls around points: the <a class="existingWikiWord" href="/nlab/show/metric">metric</a> induces a concept of <a class="existingWikiWord" href="/nlab/show/open+subsets">open subsets</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> is continuous precisely if <a class="existingWikiWord" href="/nlab/show/preimages">preimages</a> under <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/open+subsets">open subsets</a> in <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> are still open subsets 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>.</p> <p>This then is the general definition of continuity of a function <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> between <a class="existingWikiWord" href="/nlab/show/topological+spaces">topological spaces</a>:</p> <p>A function between <a class="existingWikiWord" href="/nlab/show/topological+spaces">topological spaces</a> is <em>continuous</em> precisely if its <a class="existingWikiWord" href="/nlab/show/preimages">preimages</a> of <a class="existingWikiWord" href="/nlab/show/open+subsets">open subsets</a> are again open subsets.</p> <p>Continuous maps are the <a class="existingWikiWord" href="/nlab/show/homomorphisms">homomorphisms</a> between <a class="existingWikiWord" href="/nlab/show/topological+spaces">topological spaces</a>. In other words, the collection of <a class="existingWikiWord" href="/nlab/show/topological+spaces">topological spaces</a> forms a <a class="existingWikiWord" href="/nlab/show/category">category</a>, often denoted <em><a class="existingWikiWord" href="/nlab/show/Top">Top</a></em>, whose <a class="existingWikiWord" href="/nlab/show/morphisms">morphisms</a> are the continuous functions.</p> <p>Further generalization of the concept of continuity exists, for instance to <em><a class="existingWikiWord" href="/nlab/show/locales">locales</a></em> (and then to <a class="existingWikiWord" href="/nlab/show/toposes">toposes</a>) or to <em><a class="existingWikiWord" href="/nlab/show/convergence+spaces">convergence spaces</a></em>. (See also at <em><a class="existingWikiWord" href="/nlab/show/continuous+space">continuous space</a></em>.)</p> <h3 id="in_cohesive_homotopy_type_theory">In cohesive homotopy type theory</h3> <p>There is also a concept of continuous function in <a class="existingWikiWord" href="/nlab/show/cohesive+homotopy+type+theory">cohesive homotopy type theory</a>, a two-tiered type theory with two types (usual types) and (cohesive types), where the continuous functions are precisely the functions between cohesive types, and the not-necessarily continuous functions are the functions between usual types.</p> <h2 id="definitions">Definitions</h2> <h3 id="for_the_set_of_real_numbers">For the set of real numbers</h3> <p>Given a bounded or unbounded <a class="existingWikiWord" href="/nlab/show/open+interval">open</a> or <a class="existingWikiWord" href="/nlab/show/closed+interval">closed interval</a> in the <a class="existingWikiWord" href="/nlab/show/Dedekind+real+numbers">Dedekind real numbers</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>I</mi></mrow><annotation encoding="application/x-tex">I</annotation></semantics></math> with <a class="existingWikiWord" href="/nlab/show/injection">injection</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi><mo>:</mo><mi>I</mi><mo>→</mo><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">i:I \to \mathbb{R}</annotation></semantics></math> A function <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><mi>I</mi><mo>→</mo><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">f:I \to \mathbb{R}</annotation></semantics></math> is continuous if the <a class="existingWikiWord" href="/nlab/show/graph+of+a+function">graph</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> defined by the <a class="existingWikiWord" href="/nlab/show/function">function</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>r</mi> <mi>f</mi></msub><mo>:</mo><mi>I</mi><mo>→</mo><msup><mi>ℝ</mi> <mn>2</mn></msup></mrow><annotation encoding="application/x-tex">r_f:I \to \mathbb{R}^2</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>r</mi> <mi>f</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>≔</mo><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>,</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">r_f(x) \coloneqq (i(x), f(x))</annotation></semantics></math>, has an <a class="existingWikiWord" href="/nlab/show/image">image</a> whose <a class="existingWikiWord" href="/nlab/show/shape">shape</a> is <a class="existingWikiWord" href="/nlab/show/contractible">contractible</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo lspace="0em" rspace="thinmathspace">ʃ</mo><mi mathvariant="normal">im</mi><mo stretchy="false">(</mo><msub><mi>r</mi> <mi>f</mi></msub><mo stretchy="false">)</mo><mo>≃</mo><mi>𝟙</mi></mrow><annotation encoding="application/x-tex">\esh \mathrm{im}(r_f) \simeq \mathbb{1}</annotation></semantics></math></div> <p>This definition is the formal definition that most closely adheres to the intuitive idea of a continuous function as introduced in an introductory college algebra class or textbook: as a function which could be drawn on a sheet of paper without picking the writing utensil up.</p> <h3 id="EpsilonticDefinition">The epsilontic definition for metric spaces</h3> <p>We state the definition of continuity in terms of <a class="existingWikiWord" href="/nlab/show/epsilontic+analysis">epsilontic analysis</a>, definition <a class="maruku-ref" href="#EpsilonDeltaDefinitionOfContinuity"></a> below. First recall the relevant concepts:</p> <div class="num_defn" id="MetricSpace"> <h6 id="definition">Definition</h6> <p>A <em><a class="existingWikiWord" href="/nlab/show/metric+space">metric space</a></em> is</p> <ol> <li> <p>a <a class="existingWikiWord" href="/nlab/show/set">set</a> <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> (the “underlying set”);</p> </li> <li> <p>a <a class="existingWikiWord" href="/nlab/show/function">function</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>X</mi><mo>×</mo><mi>X</mi><mo>→</mo><mo stretchy="false">[</mo><mn>0</mn><mo>,</mo><mn>∞</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">d \;\colon\; X \times X \to [0,\infty)</annotation></semantics></math> (the “distance function”) from the <a class="existingWikiWord" href="/nlab/show/Cartesian+product">Cartesian product</a> of the set with itself to the <a class="existingWikiWord" href="/nlab/show/nonnegative+number">non-negative</a> <a class="existingWikiWord" href="/nlab/show/real+numbers">real numbers</a></p> </li> </ol> <p>such that for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>z</mi><mo>∈</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">x,y,z \in X</annotation></semantics></math>:</p> <ol> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn><mspace width="thickmathspace"></mspace><mo>⇔</mo><mspace width="thickmathspace"></mspace><mi>x</mi><mo>=</mo><mi>y</mi></mrow><annotation encoding="application/x-tex">d(x,y) = 0 \;\Leftrightarrow\; x = y</annotation></semantics></math></p> </li> <li> <p>(symmetry) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo><mo>=</mo><mi>d</mi><mo stretchy="false">(</mo><mi>y</mi><mo>,</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">d(x,y) = d(y,x)</annotation></semantics></math></p> </li> <li> <p>(<a class="existingWikiWord" href="/nlab/show/triangle+inequality">triangle inequality</a>) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo><mo>+</mo><mi>d</mi><mo stretchy="false">(</mo><mi>y</mi><mo>,</mo><mi>z</mi><mo stretchy="false">)</mo><mo>≥</mo><mi>d</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>z</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">d(x,y)+ d(y,z) \geq d(x,z)</annotation></semantics></math>.</p> </li> </ol> </div> <div class="num_example"> <h6 id="example">Example</h6> <p>Every <a class="existingWikiWord" href="/nlab/show/normed+vector+space">normed vector space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>V</mi><mo>,</mo><mrow><mo stretchy="false">‖</mo><mo>−</mo><mo stretchy="false">‖</mo></mrow><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(V, {\Vert -\Vert})</annotation></semantics></math> becomes a <a class="existingWikiWord" href="/nlab/show/metric+space">metric space</a> according to def. <a class="maruku-ref" href="#MetricSpace"></a> by setting</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>d</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo><mo>≔</mo><mrow><mo stretchy="false">‖</mo><mi>x</mi><mo>−</mo><mi>y</mi><mo stretchy="false">‖</mo></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> d(x,y) \coloneqq {\Vert x-y\Vert} \,. </annotation></semantics></math></div></div> <div class="num_defn" id="OpenBalls"> <h6 id="definition_2">Definition</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>d</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(X,d)</annotation></semantics></math>, be a <a class="existingWikiWord" href="/nlab/show/metric+space">metric space</a>. Then for every element <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>∈</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">x \in X</annotation></semantics></math> and every <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϵ</mi><mo>∈</mo><msub><mi>ℝ</mi> <mo>+</mo></msub></mrow><annotation encoding="application/x-tex">\epsilon \in \mathbb{R}_+</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/positive+number">positive</a> <a class="existingWikiWord" href="/nlab/show/real+number">real number</a>, write</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" 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><mspace width="thickmathspace"></mspace><mo>≔</mo><mspace width="thickmathspace"></mspace><mrow><mo>{</mo><mi>y</mi><mo>∈</mo><mi>X</mi><mspace width="thickmathspace"></mspace><mo stretchy="false">|</mo><mspace width="thickmathspace"></mspace><mi>d</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo><mo><</mo><mi>ϵ</mi><mo>}</mo></mrow></mrow><annotation encoding="application/x-tex"> B^\circ_x(\epsilon) \;\coloneqq\; \left\{ y \in X \;\vert\; d(x,y) \lt \epsilon \right\} </annotation></semantics></math></div> <p>for the <a class="existingWikiWord" href="/nlab/show/open+ball">open ball</a> of <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></mrow><annotation encoding="application/x-tex">\epsilon</annotation></semantics></math> around <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> </div> <div class="num_defn" id="EpsilonDeltaDefinitionOfContinuity"> <h6 id="definition_3">Definition</h6> <p><strong>(epsilontic definition of continuity)</strong></p> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><msub><mi>d</mi> <mi>X</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(X,d_X)</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>Y</mi><mo>,</mo><msub><mi>d</mi> <mi>Y</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(Y,d_Y)</annotation></semantics></math> two <a class="existingWikiWord" href="/nlab/show/metric+spaces">metric spaces</a> (def. <a class="maruku-ref" href="#MetricSpace"></a>), then a <a class="existingWikiWord" href="/nlab/show/function">function</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>f</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>X</mi><mo>⟶</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex"> f \;\colon\; X \longrightarrow Y </annotation></semantics></math></div> <p>is said to be <em>continuous at a point <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>∈</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">x \in X</annotation></semantics></math></em> if for every <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϵ</mi><mo>></mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\epsilon \gt 0</annotation></semantics></math> there exists <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>δ</mi><mo>></mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\delta\gt 0</annotation></semantics></math> such that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>d</mi> <mi>X</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo><mo><</mo><mi>δ</mi><mspace width="thickmathspace"></mspace><mo>⇒</mo><mspace width="thickmathspace"></mspace><msub><mi>d</mi> <mi>Y</mi></msub><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>,</mo><mi>f</mi><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo><</mo><mi>ϵ</mi></mrow><annotation encoding="application/x-tex"> d_X(x,y) \lt \delta \;\Rightarrow\; d_Y(f(x), f(y)) \lt \epsilon </annotation></semantics></math></div> <p>or equivalently such that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><msubsup><mi>B</mi> <mi>x</mi> <mo>∘</mo></msubsup><mo stretchy="false">(</mo><mi>δ</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>⊂</mo><msubsup><mi>B</mi> <mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow> <mo>∘</mo></msubsup><mo stretchy="false">(</mo><mi>ϵ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> f(B_x^\circ(\delta)) \subset B^\circ_{f(x)}(\epsilon) </annotation></semantics></math></div> <p>where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>B</mi> <mo>∘</mo></msup></mrow><annotation encoding="application/x-tex">B^\circ</annotation></semantics></math> denotes the <a class="existingWikiWord" href="/nlab/show/open+ball">open ball</a> (definition <a class="maruku-ref" href="#OpenBalls"></a>).</p> <p>The function <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> is called just <em>continuous</em> if it is continuous at every point <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>∈</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">x \in X</annotation></semantics></math>.</p> </div> <p>This definition is equivalent to a more abstract one, which does not explicitly refer to points or radii anymore:</p> <div class="num_defn" id="OpenSubsetsOfAMetricSpace"> <h6 id="definition_4">Definition</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>d</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(X,d)</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/metric+space">metric space</a> (def. <a class="maruku-ref" href="#MetricSpace"></a>). Say that</p> <ol> <li> <p>A <em><a class="existingWikiWord" href="/nlab/show/neighbourhood">neighbourhood</a></em> of a point <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>∈</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">x \in X</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/subset">subset</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>∈</mo><mi>U</mi><mo>⊂</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">x \in U \subset X</annotation></semantics></math> which contains some <a class="existingWikiWord" href="/nlab/show/open+ball">open ball</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_x^\circ(\epsilon)</annotation></semantics></math> around <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> (def. <a class="maruku-ref" href="#OpenBalls"></a>).</p> </li> <li> <p>An <em><a class="existingWikiWord" href="/nlab/show/open+subset">open subset</a></em> 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 a <a class="existingWikiWord" href="/nlab/show/subset">subset</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo>⊂</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">U \subset X</annotation></semantics></math> such that for every for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>∈</mo><mi>U</mi></mrow><annotation encoding="application/x-tex">x \in U</annotation></semantics></math> it also contains a <a class="existingWikiWord" href="/nlab/show/neighbourhood">neighbourhood</a> 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> </ol> </div> <div class="num_remark"> <h6 id="remark">Remark</h6> <p>The collection of open subsets in def. <a class="maruku-ref" href="#OpenSubsetsOfAMetricSpace"></a> constitutes a <em><a class="existingWikiWord" href="/nlab/show/topological+space">topology</a></em> on the set <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>, making it a <em><a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a></em>. This is called the <em><a class="existingWikiWord" href="/nlab/show/metric+topology">metric topology</a></em>. Stated more concisely: the <a class="existingWikiWord" href="/nlab/show/open+balls">open balls</a> in a metric space constitute the <a class="existingWikiWord" href="/nlab/show/basis+of+a+topology">basis of a topology</a> for the <a class="existingWikiWord" href="/nlab/show/metric+topology">metric topology</a>.</p> </div> <div class="num_prop"> <h6 id="proposition">Proposition</h6> <p>A <a class="existingWikiWord" href="/nlab/show/function">function</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> between <a class="existingWikiWord" href="/nlab/show/metric+spaces">metric spaces</a> (def. <a class="maruku-ref" href="#MetricSpace"></a>) is continuous in the <a class="existingWikiWord" href="/nlab/show/epsilontic+analysis">epsilontic</a> sense of def. <a class="maruku-ref" href="#EpsilonDeltaDefinitionOfContinuity"></a> precisely if it has the property that its <a class="existingWikiWord" href="/nlab/show/pre-images">pre-images</a> of <a class="existingWikiWord" href="/nlab/show/open+subsets">open subsets</a> of <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> (in the sense of def. <a class="maruku-ref" href="#OpenSubsetsOfAMetricSpace"></a>) are open subsets 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> </div> <div class="proof"> <h6 id="proof">Proof</h6> <p>First assume that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> is continuous in the epsilontic sense. Then for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>O</mi> <mi>Y</mi></msub><mo>⊂</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">O_Y \subset Y</annotation></semantics></math> any <a class="existingWikiWord" href="/nlab/show/open+subset">open subset</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>∈</mo><msup><mi>f</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn><mo stretchy="false">(</mo><msub><mi>O</mi> <mi>Y</mi></msub><mo stretchy="false">)</mo></mrow></msup></mrow><annotation encoding="application/x-tex">x \in f^{-1(O_Y)}</annotation></semantics></math> any point in the pre-image, we need to show that there exists a <a class="existingWikiWord" href="/nlab/show/neighbourhood">neighbourhood</a> 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 <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi></mrow><annotation encoding="application/x-tex">U</annotation></semantics></math>. But by assumption there exists an <a class="existingWikiWord" href="/nlab/show/open+ball">open ball</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_x^\circ(\epsilon)</annotation></semantics></math> with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><msubsup><mi>B</mi> <mi>X</mi> <mo>∘</mo></msubsup><mo stretchy="false">(</mo><mi>ϵ</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>⊂</mo><msub><mi>O</mi> <mi>Y</mi></msub></mrow><annotation encoding="application/x-tex">f(B_X^\circ(\epsilon)) \subset O_Y</annotation></semantics></math>. Since this is true for all <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>, by definition this means that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>f</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><msub><mi>O</mi> <mi>Y</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f^{-1}(O_Y)</annotation></semantics></math> is open 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>.</p> <p>Conversely, assume that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>f</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup></mrow><annotation encoding="application/x-tex">f^{-1}</annotation></semantics></math> takes open subsets to open subsets. Then for every <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>∈</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">x \in X</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>B</mi> <mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow> <mo>∘</mo></msubsup><mo stretchy="false">(</mo><mi>ϵ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">B_{f(x)}^\circ(\epsilon)</annotation></semantics></math> an <a class="existingWikiWord" href="/nlab/show/open+ball">open ball</a> around its image, we need to produce an open ball <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>δ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">B_x^\circ(\delta)</annotation></semantics></math> in its pre-image. But by assumption <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>f</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><msubsup><mi>B</mi> <mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow> <mo>∘</mo></msubsup><mo stretchy="false">(</mo><mi>ϵ</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f^{-1}(B_{f(x)}^\circ(\epsilon))</annotation></semantics></math> contains a <a class="existingWikiWord" href="/nlab/show/neighbourhood">neighbourhood</a> 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> which by definition means that it contains such an open ball around <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> </div> <h3 id="for_topological_spaces">For topological spaces</h3> <div class="num_defn" id="topological"> <h6 id="definition_5">Definition</h6> <p>A <a class="existingWikiWord" href="/nlab/show/function">function</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>X</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">f \;\colon\; X\to Y</annotation></semantics></math> between <a class="existingWikiWord" href="/nlab/show/topological+spaces">topological spaces</a> is a <strong>continuous map</strong> (or is said to be <em>continuous</em>) if for every <a class="existingWikiWord" href="/nlab/show/open+subset">open subset</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo>⊂</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">U \subset Y</annotation></semantics></math>, the <a class="existingWikiWord" href="/nlab/show/preimage">preimage</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>f</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>U</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f^{-1}(U)</annotation></semantics></math> is an open subset <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> </div> <p>In <a class="existingWikiWord" href="/nlab/show/nonstandard+analysis">nonstandard analysis</a>, this is equivalent to</p> <div class="num_defn" id="nonstandard"> <h6 id="definition_6">Definition</h6> <p>A <a class="existingWikiWord" href="/nlab/show/function">function</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>X</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">f \;\colon\; X\to Y</annotation></semantics></math> between <a class="existingWikiWord" href="/nlab/show/topological+spaces">topological spaces</a> is a <strong>continuous map</strong> (or is said to be <em>continuous</em>) if for every <span class="newWikiWord">standard point<a href="/nlab/new/standard+point">?</a></span> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>x</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">x_1</annotation></semantics></math> and every <span class="newWikiWord">hyperpoint<a href="/nlab/new/hyperpoint">?</a></span> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>x</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">x_2</annotation></semantics></math>, if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>x</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">x_1</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>x</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">x_2</annotation></semantics></math> are <span class="newWikiWord">adequal<a href="/nlab/new/adequality">?</a></span> (infinitely close, or in other words if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>x</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">x_2</annotation></semantics></math> is in the <a class="existingWikiWord" href="/nlab/show/halo">halo</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>x</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">x_1</annotation></semantics></math>), then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><msub><mi>x</mi> <mn>1</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(x_1)</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mmultiscripts><mi>f</mi><mprescripts></mprescripts><none></none> <mo>*</mo></mmultiscripts><mo stretchy="false">(</mo><msub><mi>x</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\multiscripts{^*}f{}(x_2)</annotation></semantics></math> are adequal (where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mmultiscripts><mi>f</mi><mprescripts></mprescripts><none></none> <mo>*</mo></mmultiscripts></mrow><annotation encoding="application/x-tex">\multiscripts{^*}f{}</annotation></semantics></math> is the <span class="newWikiWord">nonstandard extension<a href="/nlab/new/nonstandard+extension">?</a></span> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math>). Equivalently, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> is continuous iff <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mmultiscripts><mi>f</mi><mprescripts></mprescripts><none></none> <mo>*</mo></mmultiscripts></mrow><annotation encoding="application/x-tex">\multiscripts{^*}f{}</annotation></semantics></math> is <span class="newWikiWord">microcontinuous<a href="/nlab/new/microcontinuous+function">?</a></span>.</p> </div> <h3 id="further_variants">Further variants</h3> <p>A function <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> between <a class="existingWikiWord" href="/nlab/show/convergence+spaces">convergence spaces</a> is <strong>continuous</strong> if for any <a class="existingWikiWord" href="/nlab/show/filter">filter</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo>→</mo><mi>x</mi></mrow><annotation encoding="application/x-tex">F \to x</annotation></semantics></math>, it follows that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>F</mi><mo stretchy="false">)</mo><mo>→</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(F) \to f(x)</annotation></semantics></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>F</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(F)</annotation></semantics></math> is the filter generated by the filterbase <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><mi>F</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo stretchy="false">|</mo><mspace width="thickmathspace"></mspace><mi>A</mi><mo>∈</mo><mi>F</mi><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{F(A) \;|\; A \in F\}</annotation></semantics></math>.</p> <p>A <strong>continuous map</strong> between <a class="existingWikiWord" href="/nlab/show/locales">locales</a> is simply a <a class="existingWikiWord" href="/nlab/show/frame">frame</a> <a class="existingWikiWord" href="/nlab/show/homomorphism">homomorphism</a> in the opposite direction. Equivalently (via the <a class="existingWikiWord" href="/nlab/show/adjoint+functor+theorem">adjoint functor theorem</a>), it may be defined as a homomorphism of <a class="existingWikiWord" href="/nlab/show/inflattices">inflattices</a> whose <a class="existingWikiWord" href="/nlab/show/left+adjoint">left adjoint</a> preserves finitary <a class="existingWikiWord" href="/nlab/show/meets">meets</a> (and hence is a frame homomorphism).</p> <h2 id="properties">Properties</h2> <p>Since continuity is defined in terms of <em>preservation of property</em> (namely preserving “openness” under preimages), it is natural to ask what other properties they preserve.<br />Also, when a property is not always preserved it is useful to label those maps which do preserve it for closer study.</p> <h3 id="properties_preserved">Properties preserved</h3> <p> <div class='num_prop' id='PropertiesPreservedByAContinuousFunction'> <h6>Proposition</h6> <p></p> <p>Under a <a class="existingWikiWord" href="/nlab/show/continuous+function">continuous function</a>:</p> <ol> <li> <p>the <a class="existingWikiWord" href="/nlab/show/preimage">preimage</a> of an <a class="existingWikiWord" href="/nlab/show/open+subset">open subset</a> is open;</p> </li> <li> <p>the <a class="existingWikiWord" href="/nlab/show/preimage">preimage</a> of a <a class="existingWikiWord" href="/nlab/show/closed+subset">closed subset</a> is closed;</p> </li> <li> <p>the <a class="existingWikiWord" href="/nlab/show/image">image</a> of a <a class="existingWikiWord" href="/nlab/show/connected+space">connected subset</a> is again connected;</p> </li> <li> <p>theimage of a <a class="existingWikiWord" href="/nlab/show/compact+space">compact subset</a> is again compact (see at <em><a class="existingWikiWord" href="/nlab/show/continuous+images+of+compact+spaces+are+compact">continuous images of compact spaces are compact</a></em>).</p> </li> </ol> <p></p> </div> </p> <h3 id="special_maps">Special maps</h3> <ol> <li> <p>The preimage of a compact set need not be compact; a continuous map for which this is true is known as a <strong><a class="existingWikiWord" href="/nlab/show/proper+map">proper map</a></strong>.</p> </li> <li> <p>The image of an open set need not be open; a continuous map for which this is true is said to be an <strong><a class="existingWikiWord" href="/nlab/show/open+map">open map</a></strong>. (Technically, an open map is any <a class="existingWikiWord" href="/nlab/show/function">function</a> with just this property.)</p> </li> <li> <p>The image of an closed set need not be closed; a continuous map for which this is true is said to be an <strong><a class="existingWikiWord" href="/nlab/show/closed+map">closed map</a></strong>. (Technically, a closed map is any function with just this property.)</p> </li> <li> <p>A continuous map of topological spaces which is invertible as a function of sets is a <strong><a class="existingWikiWord" href="/nlab/show/homeomorphism">homeomorphism</a></strong> if the <a class="existingWikiWord" href="/nlab/show/inverse+function">inverse function</a> is a continuous map as well.</p> </li> </ol> <h3 id="special_cases_in_specific_contexts">Special cases in specific contexts</h3> <p>Although these don’t make sense for arbitrary topological spaces (convergence spaces, locales, etc), they are special kinds of continuous maps in contexts such as <a class="existingWikiWord" href="/nlab/show/metric+spaces">metric spaces</a>:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/uniformly+continuous+maps">uniformly continuous maps</a>,</li> <li><a class="existingWikiWord" href="/nlab/show/Lipschitz+maps">Lipschitz maps</a>,</li> <li><a class="existingWikiWord" href="/nlab/show/short+maps">short maps</a>,</li> <li><a class="existingWikiWord" href="/nlab/show/differentiable+maps">differentiable maps</a>,</li> <li><a class="existingWikiWord" href="/nlab/show/smooth+maps">smooth maps</a>.</li> </ul> <h2 id="in_constructive_mathematics">In constructive mathematics</h2> <p>Various notions of continuous function are used in <a class="existingWikiWord" href="/nlab/show/constructive+mathematics">constructive mathematics</a>. A function <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> (say <a class="existingWikiWord" href="/nlab/show/real+number">real</a>-valued and defined on a real <a class="existingWikiWord" href="/nlab/show/interval">interval</a>) is:</p> <ul> <li><em>pointwise-continuous</em> if it <a class="existingWikiWord" href="/nlab/show/pointwise+continuous+function">pointwise continuous</a> in the usual <a class="existingWikiWord" href="/nlab/show/epsilon-delta">epsilon-delta</a> (or equivalently <span class="newWikiWord">open-subset<a href="/nlab/new/open-subset">?</a></span>) sense;</li> <li><em>uniformly continuous</em> if it <a class="existingWikiWord" href="/nlab/show/uniformly+continuous+map">uniformly continuous</a> in the usual epsilon-delta (or equivalently <a class="existingWikiWord" href="/nlab/show/entourage">entourage</a>-theoretic) sense;</li> <li><em>Bishop-continuous</em> or <em>locally uniformly continuous</em> if it is pointwise continuous and furthermore, the restriction to any closed and bounded interval is uniformly continuous;</li> <li><em>Bridges-continuous</em> if … (this one's kind of complicated).</li> </ul> <p>In <a class="existingWikiWord" href="/nlab/show/classical+mathematics">classical mathematics</a>, these are all equivalent when the domain is itself a closed and bounded interval, and all of them except for uniform continuity are equivalent in general. The same equivalences hold in <a class="existingWikiWord" href="/nlab/show/intuitionistic+mathematics">intuitionistic mathematics</a>, thanks to the <a class="existingWikiWord" href="/nlab/show/fan+theorem">fan theorem</a>. But no two of these are equivalent in <a class="existingWikiWord" href="/nlab/show/Russian+constructivism">Russian constructivism</a>.</p> <p>In fact, assuming that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">\mathbb{R}</annotation></semantics></math> is defined as the set of located <a class="existingWikiWord" href="/nlab/show/Dedekind+cuts">Dedekind cuts</a>, there is the following negative result by <a class="existingWikiWord" href="/nlab/show/Frank+Waaldijk">Frank Waaldijk</a> (<a href="#Waaldijk2003">Waaldijk2003</a>): Without the <a class="existingWikiWord" href="/nlab/show/fan+theorem">fan theorem</a>, there is no notion of continuity for set-theoretic functions in <a class="existingWikiWord" href="/nlab/show/constructive+mathematics">constructive mathematics</a>, spelled “kontinuity” in the following, such that all of the following desiderata are met:</p> <ul> <li>A function <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy="false">]</mo><mo>→</mo><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">[0,1] \to \mathbb{R}</annotation></semantics></math> is kontinuous if and only if it is <a class="existingWikiWord" href="/nlab/show/uniformly+continuous">uniformly continuous</a> in the usual sense.</li> <li>The composition of kontinuous functions is kontinuous.</li> <li>The function <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mo>+</mo></msup><mo>→</mo><mi>ℝ</mi><mo>,</mo><mi>x</mi><mo>↦</mo><mn>1</mn><mo stretchy="false">/</mo><mi>x</mi></mrow><annotation encoding="application/x-tex">\mathbb{R}^+ \to \mathbb{R}, x \mapsto 1/x</annotation></semantics></math> is kontinuous.</li> </ul> <p>The key problem is that a uniformly continuous, <a class="existingWikiWord" href="/nlab/show/positive+number">positive</a>-valued function defined on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[0,1]</annotation></semantics></math> might fail to be bounded below by a positive number, since the interval <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[0,1]</annotation></semantics></math> might fail to be <a class="existingWikiWord" href="/nlab/show/compact+space">compact</a>, yet its reciprocal (if also uniformly continuous) must be bounded above.</p> <p>Waaldijk’s negative result can be circumvented by dropping the insistence on points and instead working with maps between <a class="existingWikiWord" href="/nlab/show/locales">locales</a>, <a class="existingWikiWord" href="/nlab/show/toposes">toposes</a>, or formal spaces as studied in <a class="existingWikiWord" href="/nlab/show/formal+topology">formal topology</a>.</p> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/pointwise+continuous+function">pointwise continuous function</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/degree+of+a+continuous+function">degree of a continuous function</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/equicontinuous+family+of+functions">equicontinuous family of functions</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/analytic+function">analytic function</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/differentiable+function">differentiable function</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/smooth+function">smooth function</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/convex+function">convex function</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/integrable+function">integrable function</a>, <a class="existingWikiWord" href="/nlab/show/square-integrable+function">square-integrable function</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/bounded+function">bounded function</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/compactly+supported+function">compactly supported function</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/measurable+function">measurable function</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/rapidly+decreasing+function">rapidly decreasing function</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/function+with+rapidly+decreasing+partial+derivatives">function with rapidly decreasing partial derivatives</a></p> </li> </ul> <h2 id="literature">Literature</h2> <p>Discussion in <a class="existingWikiWord" href="/nlab/show/constructive+mathematics">constructive mathematics</a>:</p> <ul> <li id="Waaldijk2003"><a class="existingWikiWord" href="/nlab/show/Frank+Waaldijk">Frank Waaldijk</a>, <em>On the foundations of constructive mathematics – especially in relation to the theory of continuous functions</em>, 2003 (<a href="http://www.fwaaldijk.nl/foundations%20of%20constructive%20mathematics.pdf">pdf</a>)</li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on May 26, 2023 at 17:27:42. 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