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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 #27 to #28: <ins class="diffins">Added</ins> | <del class="diffdel">Removed</del> | <del class="diffmod">Chan</del><ins class="diffmod">ged</ins> </p> <div class='rightHandSide'> <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> </div> </div> <h1 id='contents'>Contents</h1> <div class='maruku_toc'><ul><li><a href='#idea'>Idea</a></li><li><a href='#Definition'>Definition</a><ul><li><a href='#AsAbstractTopologicalSpace'>As an abstract space</a></li><li><a href='#what_kind_of_space'>What kind of space?</a></li><li><a href='#AsSubspaceOfTheRealLine'>As a subspace of the real line</a></li></ul></li><li><a href='#properties'>Properties</a></li><li><a href='#related_concepts'>Related concepts</a></li><li><a href='#references'>References</a></li></ul></div> <h2 id='idea'>Idea</h2> <p>What is called <em>Cantor space</em>, after <a class='existingWikiWord' href='/nlab/show/diff/Georg+Cantor'>Georg Cantor</a>, is the <a class='existingWikiWord' href='/nlab/show/diff/topological+space'>topological space</a> obtained from the <a class='existingWikiWord' href='/nlab/show/diff/interval'>closed interval</a> <math class='maruku-mathml' display='inline' id='mathml_2db5d3e9193c4a4cbce879f4b5614c00693af78f_1' xmlns='http://www.w3.org/1998/Math/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> by</p> <ol> <li> <p>removing the middle third, retaining only <math class='maruku-mathml' display='inline' id='mathml_2db5d3e9193c4a4cbce879f4b5614c00693af78f_2' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>[</mo><mn>0</mn><mo>,</mo><mstyle displaystyle='false'><mfrac><mn>1</mn><mn>3</mn></mfrac></mstyle><mo stretchy='false'>]</mo></mrow><annotation encoding='application/x-tex'>[0,\tfrac{1}{3}]</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_2db5d3e9193c4a4cbce879f4b5614c00693af78f_3' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>[</mo><mstyle displaystyle='false'><mfrac><mn>2</mn><mn>3</mn></mfrac></mstyle><mo>,</mo><mn>1</mn><mo stretchy='false'>]</mo></mrow><annotation encoding='application/x-tex'>[\tfrac{2}{3}, 1]</annotation></semantics></math>;</p> </li> <li> <p>removing from these two pieces <em>their</em> middle thirds, to retain the pieces <math class='maruku-mathml' display='inline' id='mathml_2db5d3e9193c4a4cbce879f4b5614c00693af78f_4' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>[</mo><mn>0</mn><mo>,</mo><mstyle displaystyle='false'><mfrac><mn>1</mn><mn>9</mn></mfrac></mstyle><mo stretchy='false'>]</mo></mrow><annotation encoding='application/x-tex'>[0,\tfrac{1}{9}]</annotation></semantics></math>, <math class='maruku-mathml' display='inline' id='mathml_2db5d3e9193c4a4cbce879f4b5614c00693af78f_5' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>[</mo><mstyle displaystyle='false'><mfrac><mn>2</mn><mn>9</mn></mfrac></mstyle><mo>,</mo><mstyle displaystyle='false'><mfrac><mn>1</mn><mn>3</mn></mfrac></mstyle><mo stretchy='false'>]</mo></mrow><annotation encoding='application/x-tex'>[\tfrac{2}{9},\tfrac{1}{3}]</annotation></semantics></math>, <math class='maruku-mathml' display='inline' id='mathml_2db5d3e9193c4a4cbce879f4b5614c00693af78f_6' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>[</mo><mstyle displaystyle='false'><mfrac><mn>2</mn><mn>3</mn></mfrac></mstyle><mo>,</mo><mstyle displaystyle='false'><mfrac><mn>7</mn><mn>9</mn></mfrac></mstyle><mo stretchy='false'>]</mo></mrow><annotation encoding='application/x-tex'>[\tfrac{2}{3}, \tfrac{7}{9}]</annotation></semantics></math>, <math class='maruku-mathml' display='inline' id='mathml_2db5d3e9193c4a4cbce879f4b5614c00693af78f_7' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>[</mo><mstyle displaystyle='false'><mfrac><mn>8</mn><mn>9</mn></mfrac></mstyle><mo>,</mo><mn>1</mn><mo stretchy='false'>]</mo></mrow><annotation encoding='application/x-tex'>[\tfrac{8}{9},1]</annotation></semantics></math>;</p> </li> <li> <p>and so ever on.</p> </li> </ol> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_2db5d3e9193c4a4cbce879f4b5614c00693af78f_8' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mrow><mtable><mtr><mtd><mpadded width='0'><mrow><mo stretchy='false'>[</mo><mn>0</mn></mrow></mpadded></mtd> <mtd /> <mtd /> <mtd><mo>,</mo></mtd> <mtd /> <mtd /> <mtd><mpadded lspace='-100%width' width='0'><mrow><mn>1</mn><mo stretchy='false'>]</mo></mrow></mpadded></mtd></mtr> <mtr><mtd><mpadded width='0'><mrow><mo stretchy='false'>[</mo><mn>0</mn></mrow></mpadded></mtd> <mtd><mo>,</mo></mtd> <mtd><mpadded lspace='-100%width' width='0'><mrow><mstyle displaystyle='false'><mfrac><mn>1</mn><mn>3</mn></mfrac></mstyle><mo stretchy='false'>]</mo></mrow></mpadded></mtd> <mtd><mphantom><mrow><mo stretchy='false'>(</mo><mstyle displaystyle='false'><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo>,</mo><mstyle displaystyle='false'><mfrac><mn>2</mn><mn>3</mn></mfrac></mstyle><mo stretchy='false'>)</mo></mrow></mphantom></mtd> <mtd><mpadded width='0'><mrow><mo stretchy='false'>[</mo><mstyle displaystyle='false'><mfrac><mn>2</mn><mn>3</mn></mfrac></mstyle></mrow></mpadded></mtd> <mtd><mo>,</mo></mtd> <mtd><mpadded lspace='-100%width' width='0'><mrow><mn>1</mn><mo stretchy='false'>]</mo></mrow></mpadded></mtd></mtr> <mtr><mtd><mpadded width='0'><mrow><mo stretchy='false'>[</mo><mn>0</mn><mo>,</mo><mstyle displaystyle='false'><mfrac><mn>1</mn><mn>9</mn></mfrac></mstyle><mo stretchy='false'>]</mo></mrow></mpadded></mtd> <mtd><mphantom><mi>AAAAAAAA</mi></mphantom></mtd> <mtd><mpadded lspace='-100%width' width='0'><mrow><mo stretchy='false'>[</mo><mstyle displaystyle='false'><mfrac><mn>2</mn><mn>9</mn></mfrac></mstyle><mo>,</mo><mstyle displaystyle='false'><mfrac><mn>1</mn><mn>3</mn></mfrac></mstyle><mo stretchy='false'>]</mo></mrow></mpadded></mtd> <mtd><mphantom><mrow><mo stretchy='false'>(</mo><mstyle displaystyle='false'><mfrac><mn>1</mn><mn>3</mn></mfrac></mstyle><mo>,</mo><mstyle displaystyle='false'><mfrac><mn>2</mn><mn>3</mn></mfrac></mstyle><mo stretchy='false'>)</mo></mrow></mphantom></mtd> <mtd><mpadded width='0'><mrow><mo stretchy='false'>[</mo><mstyle displaystyle='false'><mfrac><mn>2</mn><mn>3</mn></mfrac></mstyle><mo>,</mo><mstyle displaystyle='false'><mfrac><mn>7</mn><mn>9</mn></mfrac></mstyle><mo stretchy='false'>]</mo></mrow></mpadded></mtd> <mtd><mphantom><mi>AAAAAAAA</mi></mphantom></mtd> <mtd><mpadded lspace='-100%width' width='0'><mrow><mo stretchy='false'>[</mo><mstyle displaystyle='false'><mfrac><mn>8</mn><mn>9</mn></mfrac></mstyle><mo>,</mo><mn>1</mn><mo stretchy='false'>]</mo></mrow></mpadded></mtd></mtr> <mtr><mtd /> <mtd /> <mtd /> <mtd><mi>⋮</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding='application/x-tex'> \array{ \mathrlap{[0} && &,& && \mathllap{1]} \\ \mathrlap{[0} &,& \mathllap{\tfrac{1}{3}]} &\phantom{(\tfrac{1}{2}, \tfrac{2}{3})} & \mathrlap{[\tfrac{2}{3}} &,& \mathllap{1]} \\ \mathrlap{[0,\tfrac{1}{9}]} & \phantom{AAAAAAAA} & \mathllap{[\tfrac{2}{9}, \tfrac{1}{3} ]} & \phantom{ (\tfrac{1}{3}, \tfrac{2}{3}) } & \mathrlap{[\tfrac{2}{3}, \tfrac{7}{9}]} & \phantom{AAAAAAAA} & \mathllap{[\tfrac{8}{9}, 1]} \\ &&& \vdots } </annotation></semantics></math></div> <p>Cantor space is a fundamental object of <a class='existingWikiWord' href='/nlab/show/diff/descriptive+set+theory'>descriptive set theory</a>; some indications of its use may be found at <a class='existingWikiWord' href='/nlab/show/diff/Polish+space'>Polish space</a>. Among its applications is a simple construction of a “<a class='existingWikiWord' href='/nlab/show/diff/Peano+curve'>space-filling curve</a>” (q.v.).</p> <h2 id='Definition'>Definition</h2> <p>Traditionally the Cantor space was conceived of</p> <ul> <li><a href='#AsSubspaceOfTheRealLine'>as a subspace of the real line</a></li> </ul> <p>but of course it may also be described</p> <ul> <li><a href='#AsAbstractTopologicalSpace'>as an abstract space</a></li> </ul> <p>in itself.</p> <h3 id='AsAbstractTopologicalSpace'>As an abstract space</h3> <p>In brief, Cantor space may be abstractly described as the <a class='existingWikiWord' href='/nlab/show/diff/topological+space'>topological</a> <a class='existingWikiWord' href='/nlab/show/diff/cartesian+product'>product</a> of countable many copies of the <a class='existingWikiWord' href='/nlab/show/diff/discrete+object'>discrete space</a> <math class='maruku-mathml' display='inline' id='mathml_2db5d3e9193c4a4cbce879f4b5614c00693af78f_9' xmlns='http://www.w3.org/1998/Math/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>. In more concrete detail:</p> <p>Recall that a <strong><a class='existingWikiWord' href='/nlab/show/diff/binary+digit'>binary digit</a></strong> is either <math class='maruku-mathml' display='inline' id='mathml_2db5d3e9193c4a4cbce879f4b5614c00693af78f_10' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>0</mn></mrow><annotation encoding='application/x-tex'>0</annotation></semantics></math> or <math class='maruku-mathml' display='inline' id='mathml_2db5d3e9193c4a4cbce879f4b5614c00693af78f_11' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>1</mn></mrow><annotation encoding='application/x-tex'>1</annotation></semantics></math>; the <a class='existingWikiWord' href='/nlab/show/diff/set'>set</a> (or <a class='existingWikiWord' href='/nlab/show/diff/discrete+object'>discrete space</a>) of binary digits is the <a class='existingWikiWord' href='/nlab/show/diff/boolean+domain'>Boolean domain</a> <math class='maruku-mathml' display='inline' id='mathml_2db5d3e9193c4a4cbce879f4b5614c00693af78f_12' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>𝔹</mi></mrow><annotation encoding='application/x-tex'>\mathbb{B}</annotation></semantics></math>.</p> <p>A <strong>point</strong> in Cantor space is an <a class='existingWikiWord' href='/nlab/show/diff/sequence'>infinite sequence</a> of binary digits. Accordingly, Cantor space may be denoted <math class='maruku-mathml' display='inline' id='mathml_2db5d3e9193c4a4cbce879f4b5614c00693af78f_13' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>𝔹</mi> <mi>ℕ</mi></msup></mrow><annotation encoding='application/x-tex'>\mathbb{B}^{\mathbb{N}}</annotation></semantics></math>, since its set of points is a <a class='existingWikiWord' href='/nlab/show/diff/function+set'>function set</a>.</p> <p>An <strong>open</strong> in Cantor space is a collection <math class='maruku-mathml' display='inline' id='mathml_2db5d3e9193c4a4cbce879f4b5614c00693af78f_14' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>G</mi></mrow><annotation encoding='application/x-tex'>G</annotation></semantics></math> of <a class='existingWikiWord' href='/nlab/show/diff/list'>finite sequences</a> of binary digits (that is a <a class='existingWikiWord' href='/nlab/show/diff/subset'>subset</a> of the <a class='existingWikiWord' href='/nlab/show/diff/free+monoid'>free monoid</a> <math class='maruku-mathml' display='inline' id='mathml_2db5d3e9193c4a4cbce879f4b5614c00693af78f_15' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>𝔹</mi> <mo>*</mo></msup></mrow><annotation encoding='application/x-tex'>\mathbb{B}^*</annotation></semantics></math>) such that:</p> <ul> <li> <p>If <math class='maruku-mathml' display='inline' id='mathml_2db5d3e9193c4a4cbce879f4b5614c00693af78f_16' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>u</mi><mo>∈</mo><mi>G</mi></mrow><annotation encoding='application/x-tex'>u \in G</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_2db5d3e9193c4a4cbce879f4b5614c00693af78f_17' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>v</mi></mrow><annotation encoding='application/x-tex'>v</annotation></semantics></math> is an extension of <math class='maruku-mathml' display='inline' id='mathml_2db5d3e9193c4a4cbce879f4b5614c00693af78f_18' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>u</mi></mrow><annotation encoding='application/x-tex'>u</annotation></semantics></math> (that is <math class='maruku-mathml' display='inline' id='mathml_2db5d3e9193c4a4cbce879f4b5614c00693af78f_19' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>u</mi></mrow><annotation encoding='application/x-tex'>u</annotation></semantics></math> with possibly additional digits added to the end), then <math class='maruku-mathml' display='inline' id='mathml_2db5d3e9193c4a4cbce879f4b5614c00693af78f_20' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>v</mi><mo>∈</mo><mi>G</mi></mrow><annotation encoding='application/x-tex'>v \in G</annotation></semantics></math>;</p> </li> <li> <p>If <math class='maruku-mathml' display='inline' id='mathml_2db5d3e9193c4a4cbce879f4b5614c00693af78f_21' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>u</mi><mo>:</mo><mn>0</mn><mo>∈</mo><mi>G</mi></mrow><annotation encoding='application/x-tex'>u:0 \in G</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_2db5d3e9193c4a4cbce879f4b5614c00693af78f_22' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>u</mi><mo>:</mo><mn>1</mn><mo>∈</mo><mi>G</mi></mrow><annotation encoding='application/x-tex'>u:1 \in G</annotation></semantics></math> (where <math class='maruku-mathml' display='inline' id='mathml_2db5d3e9193c4a4cbce879f4b5614c00693af78f_23' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>u</mi><mo>:</mo><mi>i</mi></mrow><annotation encoding='application/x-tex'>u:i</annotation></semantics></math> is the immediate extension of <math class='maruku-mathml' display='inline' id='mathml_2db5d3e9193c4a4cbce879f4b5614c00693af78f_24' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>u</mi></mrow><annotation encoding='application/x-tex'>u</annotation></semantics></math> by the digit <math class='maruku-mathml' display='inline' id='mathml_2db5d3e9193c4a4cbce879f4b5614c00693af78f_25' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>i</mi></mrow><annotation encoding='application/x-tex'>i</annotation></semantics></math>), then <math class='maruku-mathml' display='inline' id='mathml_2db5d3e9193c4a4cbce879f4b5614c00693af78f_26' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>u</mi><mo>∈</mo><mi>G</mi></mrow><annotation encoding='application/x-tex'>u \in G</annotation></semantics></math>.</p> </li> </ul> <p>A point <math class='maruku-mathml' display='inline' id='mathml_2db5d3e9193c4a4cbce879f4b5614c00693af78f_27' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>α</mi></mrow><annotation encoding='application/x-tex'>\alpha</annotation></semantics></math> <strong>belongs</strong> to an open <math class='maruku-mathml' display='inline' id='mathml_2db5d3e9193c4a4cbce879f4b5614c00693af78f_28' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>G</mi></mrow><annotation encoding='application/x-tex'>G</annotation></semantics></math> if, for some <math class='maruku-mathml' display='inline' id='mathml_2db5d3e9193c4a4cbce879f4b5614c00693af78f_29' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>u</mi></mrow><annotation encoding='application/x-tex'>u</annotation></semantics></math> in <math class='maruku-mathml' display='inline' id='mathml_2db5d3e9193c4a4cbce879f4b5614c00693af78f_30' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>G</mi></mrow><annotation encoding='application/x-tex'>G</annotation></semantics></math>, <math class='maruku-mathml' display='inline' id='mathml_2db5d3e9193c4a4cbce879f4b5614c00693af78f_31' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>α</mi></mrow><annotation encoding='application/x-tex'>\alpha</annotation></semantics></math> is an extension of <math class='maruku-mathml' display='inline' id='mathml_2db5d3e9193c4a4cbce879f4b5614c00693af78f_32' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>u</mi></mrow><annotation encoding='application/x-tex'>u</annotation></semantics></math>.</p> <p>An alternative characterization of Cantor space is as the <a class='existingWikiWord' href='/nlab/show/diff/terminal+coalgebra+for+an+endofunctor'>terminal coalgebra</a> for the endofunctor on <a class='existingWikiWord' href='/nlab/show/diff/Top'>Top</a>, <math class='maruku-mathml' display='inline' id='mathml_2db5d3e9193c4a4cbce879f4b5614c00693af78f_33' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi><mo>↦</mo><mi>X</mi><mo>+</mo><mi>X</mi></mrow><annotation encoding='application/x-tex'>X \mapsto X + X</annotation></semantics></math>.</p> <h3 id='what_kind_of_space'>What kind of space?</h3> <p>Traditionally, Cantor space is understood as a <a class='existingWikiWord' href='/nlab/show/diff/topological+space'>topological space</a>. We start with the points, as defined above, then specify which sets of points are <a class='existingWikiWord' href='/nlab/show/diff/open+subspace'>open</a>. Although there are other ways to state which sets are open, we may define a set to be open if it is the set of points that belong to some open <math class='maruku-mathml' display='inline' id='mathml_2db5d3e9193c4a4cbce879f4b5614c00693af78f_34' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>G</mi></mrow><annotation encoding='application/x-tex'>G</annotation></semantics></math> as defined above.</p> <p>A newer approach is to understand Cantor space as a <a class='existingWikiWord' href='/nlab/show/diff/locale'>locale</a>. Then we start with the opens and define an <a class='existingWikiWord' href='/nlab/show/diff/order'>order</a> relation on them to define a <a class='existingWikiWord' href='/nlab/show/diff/frame'>frame</a>. In this case, the order relation is the obvious one, that <math class='maruku-mathml' display='inline' id='mathml_2db5d3e9193c4a4cbce879f4b5614c00693af78f_35' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>G</mi><mo>≤</mo><mi>H</mi></mrow><annotation encoding='application/x-tex'>G \leq H</annotation></semantics></math> if <math class='maruku-mathml' display='inline' id='mathml_2db5d3e9193c4a4cbce879f4b5614c00693af78f_36' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>G</mi><mo>⊆</mo><mi>H</mi></mrow><annotation encoding='application/x-tex'>G \subseteq H</annotation></semantics></math> as subsets of <math class='maruku-mathml' display='inline' id='mathml_2db5d3e9193c4a4cbce879f4b5614c00693af78f_37' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>𝔹</mi> <mo>*</mo></msup></mrow><annotation encoding='application/x-tex'>\mathbb{B}^*</annotation></semantics></math>. Then the <a class='existingWikiWord' href='/nlab/show/diff/point+of+a+locale'>points</a> come for free, and correspond precisely to the points as defined above.</p> <p>In <a class='existingWikiWord' href='/nlab/show/diff/classical+mathematics'>classical mathematics</a>, these two approaches are equivalent; a point is determined by its opens, and an open is determined by its points. The theorem that a point is determined by its opens (so that Cantor space, as a topological space, is <a class='existingWikiWord' href='/nlab/show/diff/sober+topological+space'>sober</a>) is valid <a class='existingWikiWord' href='/nlab/show/diff/internalization'>internal</a> to any <a class='existingWikiWord' href='/nlab/show/diff/pretopos'>pretopos</a> with an <a class='existingWikiWord' href='/nlab/show/diff/exponential+object'>exponentiable</a> <a class='existingWikiWord' href='/nlab/show/diff/natural+numbers+object'>natural numbers object</a>; as such, it applies even in <a class='existingWikiWord' href='/nlab/show/diff/predicative+mathematics'>predicative</a> and <a class='existingWikiWord' href='/nlab/show/diff/constructive+mathematics'>constructive</a> mathematics. However, the theorem that an open is determined by its points (so that Cantor space, as a locale, is <a class='existingWikiWord' href='/nlab/show/diff/spatial+locale'>topological</a>) is equivalent to the <a class='existingWikiWord' href='/nlab/show/diff/fan+theorem'>fan theorem</a>; it is true in some pretoposes and accepted by some schools of constructivism but false in other pretoposes and rejected, or even refuted, by other constructivists.</p> <p>When the fan theorem is not valid, the localic approach is probably better; it allows more of the useful properties of Cantor space to hold.</p> <h3 id='AsSubspaceOfTheRealLine'>As a subspace of the real line</h3> <p>Cantor space is usually conceived of as a <a class='existingWikiWord' href='/nlab/show/diff/subspace+topology'>topological subspace</a> of the <a class='existingWikiWord' href='/nlab/show/diff/real+number'>real line</a>:</p> <p>Write <math class='maruku-mathml' display='inline' id='mathml_2db5d3e9193c4a4cbce879f4b5614c00693af78f_38' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Disc</mi><mo stretchy='false'>(</mo><mo stretchy='false'>{</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy='false'>}</mo><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>Disc(\{0,1\})</annotation></semantics></math> for the the <a class='existingWikiWord' href='/nlab/show/diff/discrete+and+indiscrete+topology'>discrete topological space</a> with two points. Write <math class='maruku-mathml' display='inline' id='mathml_2db5d3e9193c4a4cbce879f4b5614c00693af78f_39' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><munder><mo lspace='thinmathspace' rspace='thinmathspace'>∏</mo><mrow><mi>n</mi><mo>∈</mo><mi>ℕ</mi></mrow></munder><mi>Disc</mi><mo stretchy='false'>(</mo><mo stretchy='false'>{</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy='false'>}</mo><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>\underset{n \in \mathbb{N}}{\prod} Disc(\{0,1\})</annotation></semantics></math> for the <a class='existingWikiWord' href='/nlab/show/diff/product+topological+space'>product topological space</a> of a <a class='existingWikiWord' href='/nlab/show/diff/countable+set'>countable set</a> of copies of this discrete space with itself (i.e. the corresponding <a class='existingWikiWord' href='/nlab/show/diff/cartesian+product'>Cartesian product</a> of sets <math class='maruku-mathml' display='inline' id='mathml_2db5d3e9193c4a4cbce879f4b5614c00693af78f_40' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><munder><mo lspace='thinmathspace' rspace='thinmathspace'>∏</mo><mrow><mi>n</mi><mo>∈</mo><mi>ℕ</mi></mrow></munder><mo stretchy='false'>{</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy='false'>}</mo></mrow><annotation encoding='application/x-tex'>\underset{n \in \mathbb{N}}{\prod} \{0,1\}</annotation></semantics></math> equipped with the <a class='existingWikiWord' href='/nlab/show/diff/product+topological+space'>Tychonoff topology</a> induced from the <a class='existingWikiWord' href='/nlab/show/diff/discrete+and+indiscrete+topology'>discrete topology</a> of <math class='maruku-mathml' display='inline' id='mathml_2db5d3e9193c4a4cbce879f4b5614c00693af78f_41' xmlns='http://www.w3.org/1998/Math/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>).</p> <p>Then consider the <a class='existingWikiWord' href='/nlab/show/diff/function'>function</a></p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_2db5d3e9193c4a4cbce879f4b5614c00693af78f_42' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mrow><mtable><mtr><mtd><munder><mo lspace='thinmathspace' rspace='thinmathspace'>∏</mo><mrow><mi>n</mi><mo>∈</mo><mi>ℕ</mi></mrow></munder></mtd> <mtd><mover><mo>⟶</mo><mi>κ</mi></mover></mtd> <mtd><mo stretchy='false'>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy='false'>]</mo></mtd></mtr> <mtr><mtd><mo stretchy='false'>(</mo><msub><mi>a</mi> <mi>i</mi></msub><msub><mo stretchy='false'>)</mo> <mrow><mi>i</mi><mo>∈</mo><mi>ℕ</mi></mrow></msub></mtd> <mtd><mover><mo>↦</mo><mphantom><mi>AAAA</mi></mphantom></mover></mtd> <mtd><munderover><mo lspace='thinmathspace' rspace='thinmathspace'>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mn>∞</mn></munderover><mfrac><mrow><mn>2</mn><msub><mi>a</mi> <mi>i</mi></msub></mrow><mrow><msup><mn>3</mn> <mi>i</mi></msup></mrow></mfrac></mtd></mtr></mtable></mrow></mrow><annotation encoding='application/x-tex'> \array{ \underset{n \in \mathbb{N}}{\prod} &\overset{\kappa}{\longrightarrow}& [0,1] \\ (a_i)_{i \in \mathbb{N}} &\overset{\phantom{AAAA}}{\mapsto}& \underoverset{i = 1}{\infty}{\sum} \frac{2 a_i}{3^i} } </annotation></semantics></math></div> <p>which sends an element in the product space, hence a <a class='existingWikiWord' href='/nlab/show/diff/sequence'>sequence</a> of binary digits, to the value of the <a class='existingWikiWord' href='/nlab/show/diff/power+series'>power series</a> as shown on the right.</p> <p>One checks that this is a <a class='existingWikiWord' href='/nlab/show/diff/continuous+map'>continuous function</a> (from the <a class='existingWikiWord' href='/nlab/show/diff/product+topological+space'>product topology</a> to the <a class='existingWikiWord' href='/nlab/show/diff/Euclidean+space'>Euclidean</a> <a class='existingWikiWord' href='/nlab/show/diff/metric+topology'>metric topology</a> on the <a class='existingWikiWord' href='/nlab/show/diff/interval'>closed interval</a>). Moreover with its <a class='existingWikiWord' href='/nlab/show/diff/image'>image</a> <math class='maruku-mathml' display='inline' id='mathml_2db5d3e9193c4a4cbce879f4b5614c00693af78f_43' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>κ</mi><mrow><mo>(</mo><munder><mo lspace='thinmathspace' rspace='thinmathspace'>∏</mo><mrow><mi>n</mi><mo>∈</mo><mi>ℕ</mi></mrow></munder><mo stretchy='false'>{</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy='false'>}</mo><mo>)</mo></mrow><mo>⊂</mo><mo stretchy='false'>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy='false'>]</mo></mrow><annotation encoding='application/x-tex'>\kappa\left( \underset{n \in \mathbb{N}}{\prod} \{0,1\}\right) \subset [0,1]</annotation></semantics></math> equipped with its <a class='existingWikiWord' href='/nlab/show/diff/subspace+topology'>subspace topology</a>, then this is a <a class='existingWikiWord' href='/nlab/show/diff/homeomorphism'>homeomorphism</a> onto its image:</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_2db5d3e9193c4a4cbce879f4b5614c00693af78f_44' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><munder><mo lspace='thinmathspace' rspace='thinmathspace'>∏</mo><mrow><mi>n</mi><mo>∈</mo><mi>ℕ</mi></mrow></munder><mi>Disc</mi><mo stretchy='false'>(</mo><mo stretchy='false'>{</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy='false'>}</mo><mo stretchy='false'>)</mo><mover><mo>⟶</mo><mrow><mphantom><mi>AA</mi></mphantom><mo>≃</mo><mphantom><mi>AA</mi></mphantom></mrow></mover><mi>κ</mi><mrow><mo>(</mo><munder><mo lspace='thinmathspace' rspace='thinmathspace'>∏</mo><mrow><mi>n</mi><mo>∈</mo><mi>ℕ</mi></mrow></munder><mi>Disc</mi><mo stretchy='false'>(</mo><mo stretchy='false'>{</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy='false'>}</mo><mo stretchy='false'>)</mo><mo>)</mo></mrow><mover><mo>↪</mo><mphantom><mi>AAAA</mi></mphantom></mover><mo stretchy='false'>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy='false'>]</mo><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> \underset{n \in \mathbb{N}}{\prod} Disc(\{0,1\}) \overset{\phantom{AA}\simeq\phantom{AA}}{\longrightarrow} \kappa\left( \underset{n \in \mathbb{N}}{\prod} Disc(\{0,1\}) \right) \overset{\phantom{AAAA}}{\hookrightarrow} [0,1] \,. </annotation></semantics></math></div> <p>This image is the <em>Cantor space</em> as a subspace of the closed interval.</p> <p>From the localic perspective, a <a class='existingWikiWord' href='/nlab/show/diff/continuous+map'>continuous map</a> is given by a <a class='existingWikiWord' href='/nlab/show/diff/homomorphism'>homomorphism</a> of frames in the opposite direction. Given an open <math class='maruku-mathml' display='inline' id='mathml_2db5d3e9193c4a4cbce879f4b5614c00693af78f_45' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo>∼</mo></mrow><annotation encoding='application/x-tex'>\sim</annotation></semantics></math> in <math class='maruku-mathml' display='inline' id='mathml_2db5d3e9193c4a4cbce879f4b5614c00693af78f_46' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>ℝ</mi></mrow><annotation encoding='application/x-tex'>\mathbb{R}</annotation></semantics></math> (as a <a class='existingWikiWord' href='/nlab/show/diff/relation'>binary relation</a> on <a class='existingWikiWord' href='/nlab/show/diff/rational+number'>rational numbers</a>, as described at <a class='existingWikiWord' href='/nlab/show/diff/locale+of+real+numbers'>locale of real numbers</a>), this is mapped to the open <math class='maruku-mathml' display='inline' id='mathml_2db5d3e9193c4a4cbce879f4b5614c00693af78f_47' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>G</mi></mrow><annotation encoding='application/x-tex'>G</annotation></semantics></math> in Cantor space such that <math class='maruku-mathml' display='inline' id='mathml_2db5d3e9193c4a4cbce879f4b5614c00693af78f_48' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>u</mi><mo>∈</mo><mi>G</mi></mrow><annotation encoding='application/x-tex'>u \in G</annotation></semantics></math> if and only if</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_2db5d3e9193c4a4cbce879f4b5614c00693af78f_49' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><munderover><mo lspace='thinmathspace' rspace='thinmathspace'>∑</mo> <mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow> <mrow><mi>len</mi><mo stretchy='false'>(</mo><mi>u</mi><mo stretchy='false'>)</mo></mrow></munderover><mfrac><mrow><mn>2</mn><msub><mi>u</mi> <mi>i</mi></msub></mrow><mrow><msup><mn>3</mn> <mi>i</mi></msup></mrow></mfrac><mo>∼</mo><munderover><mo lspace='thinmathspace' rspace='thinmathspace'>∑</mo> <mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow> <mrow><mi>len</mi><mo stretchy='false'>(</mo><mi>u</mi><mo stretchy='false'>)</mo></mrow></munderover><mfrac><mrow><mn>2</mn><msub><mi>u</mi> <mi>i</mi></msub></mrow><mrow><msup><mn>3</mn> <mi>i</mi></msup></mrow></mfrac><mo>+</mo><mfrac><mn>1</mn><mrow><msup><mn>3</mn> <mrow><mo lspace='verythinmathspace' rspace='0em'>−</mo><mi>len</mi><mo stretchy='false'>(</mo><mi>u</mi><mo stretchy='false'>)</mo></mrow></msup></mrow></mfrac><mo>.</mo></mrow><annotation encoding='application/x-tex'> \sum_{i=1}^{len(u)} \frac { 2 u_i } { 3^i } \sim \sum_{i=1}^{len(u)} \frac { 2 u_i } { 3^i } + \frac 1 { 3^{-len(u)} } .</annotation></semantics></math></div> <p>One then checks that this is an embedding.</p> <div class='query'> <p>I should check this some day; for the moment, I am taking it on faith. —Toby</p> </div> <p>In either case, the idea is:</p> <ul> <li> <p>A point of Cantor space corresponds to a number written in base <math class='maruku-mathml' display='inline' id='mathml_2db5d3e9193c4a4cbce879f4b5614c00693af78f_50' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>3</mn></mrow><annotation encoding='application/x-tex'>3</annotation></semantics></math> with infinitely many digits, using only the digits <math class='maruku-mathml' display='inline' id='mathml_2db5d3e9193c4a4cbce879f4b5614c00693af78f_51' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>0</mn></mrow><annotation encoding='application/x-tex'>0</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_2db5d3e9193c4a4cbce879f4b5614c00693af78f_52' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>2</mn></mrow><annotation encoding='application/x-tex'>2</annotation></semantics></math> (which are the options for <math class='maruku-mathml' display='inline' id='mathml_2db5d3e9193c4a4cbce879f4b5614c00693af78f_53' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>2</mn><msub><mi>a</mi> <mi>i</mi></msub></mrow><annotation encoding='application/x-tex'>2 a_i</annotation></semantics></math> when <math class='maruku-mathml' display='inline' id='mathml_2db5d3e9193c4a4cbce879f4b5614c00693af78f_54' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>a</mi> <mi>i</mi></msub><mo>∈</mo><mo stretchy='false'>{</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy='false'>}</mo></mrow><annotation encoding='application/x-tex'>a_i \in \{0,1\}</annotation></semantics></math>); while</p> </li> <li> <p>An open corresponds to a union of intervals, each of which is given by approximating a number in base <math class='maruku-mathml' display='inline' id='mathml_2db5d3e9193c4a4cbce879f4b5614c00693af78f_55' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>3</mn></mrow><annotation encoding='application/x-tex'>3</annotation></semantics></math> to a finite number of digits, using only the digits <math class='maruku-mathml' display='inline' id='mathml_2db5d3e9193c4a4cbce879f4b5614c00693af78f_56' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>0</mn></mrow><annotation encoding='application/x-tex'>0</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_2db5d3e9193c4a4cbce879f4b5614c00693af78f_57' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>2</mn></mrow><annotation encoding='application/x-tex'>2</annotation></semantics></math>.</p> </li> </ul> <p>One sometimes speaks of the <strong>Cantor set</strong> to stress that one is considering Cantor space as a subspace of the real line.</p> <p>As we can also consider Cantor space as a product space <math class='maruku-mathml' display='inline' id='mathml_2db5d3e9193c4a4cbce879f4b5614c00693af78f_58' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mi>ℤ</mi><mo stretchy='false'>/</mo><mn>2</mn><msup><mo stretchy='false'>)</mo> <mi>n</mi></msup></mrow><annotation encoding='application/x-tex'>(\mathbb{Z}/2)^n</annotation></semantics></math> of countably many copies of <math class='maruku-mathml' display='inline' id='mathml_2db5d3e9193c4a4cbce879f4b5614c00693af78f_59' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>ℤ</mi><mo stretchy='false'>/</mo><mo stretchy='false'>(</mo><mn>2</mn><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>\mathbb{Z}/(2)</annotation></semantics></math>, which carries a group structure, we can view Cantor space <math class='maruku-mathml' display='inline' id='mathml_2db5d3e9193c4a4cbce879f4b5614c00693af78f_60' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi></mrow><annotation encoding='application/x-tex'>C</annotation></semantics></math> as a <a class='existingWikiWord' href='/nlab/show/diff/topological+group'>topological group</a>. In particular, it is a <a class='existingWikiWord' href='/nlab/show/diff/homogeneous+space'>homogeneous space</a> (its group of self-homeomorphisms acts transitively on the space).</p> <h2 id='properties'>Properties</h2> <p>Cantor space, especially in its guise as a subspace of the real line, is quite famous; see <a href='http://en.wikipedia.org/wiki/Cantor_set'>Wikipedia</a>. Here are some headline properties:</p> <ul> <li> <p>Cantor space is a <a class='existingWikiWord' href='/nlab/show/diff/compactum'>compact Hausdorff space</a>. (For the topological space, this statement is again equivalent to the <a class='existingWikiWord' href='/nlab/show/diff/fan+theorem'>fan theorem</a>; for the locale, it holds regardless.)</p> </li> <li> <p>Cantor space is <a class='existingWikiWord' href='/nlab/show/diff/connected+space'>totally disconnected</a>.</p> </li> <li> <p>Thus Cantor space is a <a class='existingWikiWord' href='/nlab/show/diff/Stone+space'>Stone space</a>.</p> </li> <li> <p>Cantor space is <a class='existingWikiWord' href='/nlab/show/diff/metric+space'>metrizable</a>, and every compact metrizable space is a <a class='existingWikiWord' href='/nlab/show/diff/quotient+space'>quotient space</a> of Cantor space (see Theorem <a class='maruku-ref' href='#HA'>2</a> below).</p> </li> <li> <p>As a subspace of <math class='maruku-mathml' display='inline' id='mathml_2db5d3e9193c4a4cbce879f4b5614c00693af78f_61' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>ℝ</mi></mrow><annotation encoding='application/x-tex'>\mathbb{R}</annotation></semantics></math>, the Cantor set is <a class='existingWikiWord' href='/nlab/show/diff/perfect+space'>perfect</a> and <a class='existingWikiWord' href='/nlab/show/diff/countable+set'>uncountable</a> but of <a class='existingWikiWord' href='/nlab/show/diff/Lebesgue+measure'>Lebesgue</a> <a class='existingWikiWord' href='/nlab/show/diff/null+subset'>measure zero</a>.</p> </li> <li> <p>The Cantor set is a precisely self-similar <a class='existingWikiWord' href='/nlab/show/diff/fractal'>fractal</a> with <a class='existingWikiWord' href='/nlab/show/diff/Hausdorff+dimension'>Hausdorff dimension</a> <math class='maruku-mathml' display='inline' id='mathml_2db5d3e9193c4a4cbce879f4b5614c00693af78f_62' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>log</mi> <mn>3</mn></msub><mn>2</mn><mo>≈</mo><mn>0.631</mn></mrow><annotation encoding='application/x-tex'>\log_3 2 \approx 0.631</annotation></semantics></math>.</p> </li> </ul> <div class='num_theorem' id='Theorem'> <h6 id='theorem'>Theorem</h6> <p>A <a class='existingWikiWord' href='/nlab/show/diff/topological+space'>topological space</a> is <a class='existingWikiWord' href='/nlab/show/diff/homeomorphism'>homeomorphic</a> to Cantor space if and only if it is <a class='existingWikiWord' href='/nlab/show/diff/inhabited+set'>nonempty</a>, <a class='existingWikiWord' href='/nlab/show/diff/compact+space'>compact</a>, <a class='existingWikiWord' href='/nlab/show/diff/totally+disconnected+space'>totally disconnected</a>, <a class='existingWikiWord' href='/nlab/show/diff/metric+space'>metrizable</a>, and <a class='existingWikiWord' href='/nlab/show/diff/perfect+space'>perfect</a>.</p> </div> <p>This result is sometimes called <strong><a class='existingWikiWord' href='/nlab/show/diff/L.E.J.+Brouwer'>Brouwer</a>’s theorem</strong>. It can be seen from the perspective of <a class='existingWikiWord' href='/nlab/show/diff/Stone+duality'>Stone duality</a>, where the dual result is that any two <a class='existingWikiWord' href='/nlab/show/diff/countable+set'>countable</a> <a class='existingWikiWord' href='/nlab/show/diff/atom'>atomless</a> <a class='existingWikiWord' href='/nlab/show/diff/Boolean+algebra'>Boolean algebras</a> are <a class='existingWikiWord' href='/nlab/show/diff/isomorphism'>isomorphic</a>; this dual result can be proven by a <a class='existingWikiWord' href='/nlab/show/diff/Ehrenfeucht-Fra%C3%AFss%C3%A9+games'>back-and-forth argument</a>.</p> <div class='num_cor' id='CorA'> <h6 id='corollary'>Corollary</h6> <p>The <a class='existingWikiWord' href='/nlab/show/diff/one-point+compactification'>one-point compactification</a> <math class='maruku-mathml' display='inline' id='mathml_2db5d3e9193c4a4cbce879f4b5614c00693af78f_63' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mover><mi>X</mi><mo>¯</mo></mover></mrow><annotation encoding='application/x-tex'>\widebar{X}</annotation></semantics></math> of a space <math class='maruku-mathml' display='inline' id='mathml_2db5d3e9193c4a4cbce879f4b5614c00693af78f_64' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> that is <a class='existingWikiWord' href='/nlab/show/diff/second-countable+space'>second-countable</a> <a class='existingWikiWord' href='/nlab/show/diff/locally+compact+topological+space'>locally compact</a> <a class='existingWikiWord' href='/nlab/show/diff/Hausdorff+space'>Hausdorff</a>, <a class='existingWikiWord' href='/nlab/show/diff/totally+disconnected+space'>totally disconnected</a> and perfect, is homeomorphic to Cantor space (provided <math class='maruku-mathml' display='inline' id='mathml_2db5d3e9193c4a4cbce879f4b5614c00693af78f_65' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> is not itself compact).</p> </div> <div class='proof'> <h6 id='proof'>Proof</h6> <p><math class='maruku-mathml' display='inline' id='mathml_2db5d3e9193c4a4cbce879f4b5614c00693af78f_66' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mover><mi>X</mi><mo>¯</mo></mover></mrow><annotation encoding='application/x-tex'>\widebar{X}</annotation></semantics></math> is also second-countable, compact Hausdorff and therefore compact regular, and so by the <a class='existingWikiWord' href='/nlab/show/diff/Urysohn+metrization+theorem'>Urysohn metrization theorem</a> it is compact metrizable. The point <math class='maruku-mathml' display='inline' id='mathml_2db5d3e9193c4a4cbce879f4b5614c00693af78f_67' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>p</mi></mrow><annotation encoding='application/x-tex'>p</annotation></semantics></math> at infinity is not isolated since we assume <math class='maruku-mathml' display='inline' id='mathml_2db5d3e9193c4a4cbce879f4b5614c00693af78f_68' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> is not compact, so <math class='maruku-mathml' display='inline' id='mathml_2db5d3e9193c4a4cbce879f4b5614c00693af78f_69' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mover><mi>X</mi><mo>¯</mo></mover></mrow><annotation encoding='application/x-tex'>\widebar{X}</annotation></semantics></math> is perfect. If <math class='maruku-mathml' display='inline' id='mathml_2db5d3e9193c4a4cbce879f4b5614c00693af78f_70' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>V</mi></mrow><annotation encoding='application/x-tex'>V</annotation></semantics></math> is any open neighborhood of <math class='maruku-mathml' display='inline' id='mathml_2db5d3e9193c4a4cbce879f4b5614c00693af78f_71' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>p</mi></mrow><annotation encoding='application/x-tex'>p</annotation></semantics></math>, so <math class='maruku-mathml' display='inline' id='mathml_2db5d3e9193c4a4cbce879f4b5614c00693af78f_72' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>V</mi><mo>=</mo><mo>¬</mo><mi>K</mi></mrow><annotation encoding='application/x-tex'>V = \neg K</annotation></semantics></math> for some compact <math class='maruku-mathml' display='inline' id='mathml_2db5d3e9193c4a4cbce879f4b5614c00693af78f_73' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>K</mi><mo>⊂</mo><mi>X</mi></mrow><annotation encoding='application/x-tex'>K \subset X</annotation></semantics></math>, then we claim there exists a clopen that contains <math class='maruku-mathml' display='inline' id='mathml_2db5d3e9193c4a4cbce879f4b5614c00693af78f_74' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>K</mi></mrow><annotation encoding='application/x-tex'>K</annotation></semantics></math>; in that case <math class='maruku-mathml' display='inline' id='mathml_2db5d3e9193c4a4cbce879f4b5614c00693af78f_75' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>V</mi></mrow><annotation encoding='application/x-tex'>V</annotation></semantics></math> contains a clopen whence <math class='maruku-mathml' display='inline' id='mathml_2db5d3e9193c4a4cbce879f4b5614c00693af78f_76' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>{</mo><mi>p</mi><mo stretchy='false'>}</mo></mrow><annotation encoding='application/x-tex'>\{p\}</annotation></semantics></math> is the quasi-component of <math class='maruku-mathml' display='inline' id='mathml_2db5d3e9193c4a4cbce879f4b5614c00693af78f_77' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>p</mi></mrow><annotation encoding='application/x-tex'>p</annotation></semantics></math> (hence also the connected component since we’re in a compact Hausdorff space). But the argument <a href='/nlab/show/compact+Hausdorff+rings+are+profinite#compopen'>here</a> shows that for each <math class='maruku-mathml' display='inline' id='mathml_2db5d3e9193c4a4cbce879f4b5614c00693af78f_78' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>x</mi><mo>∈</mo><mi>K</mi></mrow><annotation encoding='application/x-tex'>x \in K</annotation></semantics></math> there is a clopen neighborhood of <math class='maruku-mathml' display='inline' id='mathml_2db5d3e9193c4a4cbce879f4b5614c00693af78f_79' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>x</mi></mrow><annotation encoding='application/x-tex'>x</annotation></semantics></math> contained in <math class='maruku-mathml' display='inline' id='mathml_2db5d3e9193c4a4cbce879f4b5614c00693af78f_80' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo>¬</mo><mo stretchy='false'>{</mo><mi>p</mi><mo stretchy='false'>}</mo></mrow><annotation encoding='application/x-tex'>\neg \{p\}</annotation></semantics></math>; finitely many of these clopens cover <math class='maruku-mathml' display='inline' id='mathml_2db5d3e9193c4a4cbce879f4b5614c00693af78f_81' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>K</mi></mrow><annotation encoding='application/x-tex'>K</annotation></semantics></math>, and the claim follows by considering their union.</p> </div> <p>It follows from this result that all such spaces <math class='maruku-mathml' display='inline' id='mathml_2db5d3e9193c4a4cbce879f4b5614c00693af78f_82' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> are homeomorphic: they all have Cantor space as their one-point compactifications, and so they are all homeomorphic to the space obtained obtained by removing a single point from Cantor space. This applies for example to spaces obtained by removing a finite number <math class='maruku-mathml' display='inline' id='mathml_2db5d3e9193c4a4cbce879f4b5614c00693af78f_83' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>n</mi><mo>≥</mo><mn>1</mn></mrow><annotation encoding='application/x-tex'>n \geq 1</annotation></semantics></math> of points from Cantor space.</p> <p>Cantor space is also a “universal” compact metric space in the following sense.</p> <div class='num_theorem' id='HA'> <h6 id='theorem_2'>Theorem</h6> <p><strong>(Hausdorff-Alexandroff)</strong> Every compact metric space is a continuous image of Cantor space.</p> </div> <p>This implies that every compact metric space is a <a class='existingWikiWord' href='/nlab/show/diff/quotient+space'>quotient space</a> of Cantor space, since a surjective map between <a class='existingWikiWord' href='/nlab/show/diff/compactum'>compact Hausdorff spaces</a> is a closed surjection, and closed surjections are quotient maps.</p> <div class='proof'> <h6 id='proof_2'>Proof</h6> <p>First, every compact metric space <math class='maruku-mathml' display='inline' id='mathml_2db5d3e9193c4a4cbce879f4b5614c00693af78f_84' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> is <a class='existingWikiWord' href='/nlab/show/diff/separable+space'>separable</a>: has a <a class='existingWikiWord' href='/nlab/show/diff/countable+set'>countable</a> <a class='existingWikiWord' href='/nlab/show/diff/dense+subspace'>dense set</a> <math class='maruku-mathml' display='inline' id='mathml_2db5d3e9193c4a4cbce879f4b5614c00693af78f_85' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>{</mo><msub><mi>x</mi> <mn>0</mn></msub><mo>,</mo><msub><mi>x</mi> <mn>1</mn></msub><mo>,</mo><mi>…</mi><mo stretchy='false'>}</mo></mrow><annotation encoding='application/x-tex'>\{x_0, x_1, \ldots\}</annotation></semantics></math>. Assume, as we may, that the metric <math class='maruku-mathml' display='inline' id='mathml_2db5d3e9193c4a4cbce879f4b5614c00693af78f_86' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>d</mi><mo>:</mo><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: X \times X \to [0, \infty)</annotation></semantics></math> is valued in <math class='maruku-mathml' display='inline' id='mathml_2db5d3e9193c4a4cbce879f4b5614c00693af78f_87' xmlns='http://www.w3.org/1998/Math/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>. Then the map <math class='maruku-mathml' display='inline' id='mathml_2db5d3e9193c4a4cbce879f4b5614c00693af78f_88' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>y</mi><mo>:</mo><mi>X</mi><mo>→</mo><mo stretchy='false'>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><msup><mo stretchy='false'>]</mo> <mi>ℕ</mi></msup></mrow><annotation encoding='application/x-tex'>y: X \to [0, 1]^\mathbb{N}</annotation></semantics></math> to the <a class='existingWikiWord' href='/nlab/show/diff/Hilbert+cube'>Hilbert cube</a>, defined by</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_2db5d3e9193c4a4cbce879f4b5614c00693af78f_89' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>y</mi><mo stretchy='false'>(</mo><mi>x</mi><mo stretchy='false'>)</mo><mo>=</mo><mo stretchy='false'>(</mo><mi>d</mi><mo stretchy='false'>(</mo><mi>x</mi><mo>,</mo><msub><mi>x</mi> <mi>n</mi></msub><mo stretchy='false'>)</mo><msub><mo stretchy='false'>)</mo> <mrow><mi>n</mi><mo>∈</mo><mi>ℕ</mi></mrow></msub></mrow><annotation encoding='application/x-tex'>y(x) = (d(x, x_n))_{n \in \mathbb{N}}</annotation></semantics></math></div> <p>(a type of restricted “<a class='existingWikiWord' href='/nlab/show/diff/Yoneda+embedding'>Yoneda embedding</a>”, regarding <a class='existingWikiWord' href='/nlab/show/diff/metric+space'>metric spaces</a> as <a class='existingWikiWord' href='/nlab/show/diff/enriched+category'>enriched categories</a>), is continuous and maps <math class='maruku-mathml' display='inline' id='mathml_2db5d3e9193c4a4cbce879f4b5614c00693af78f_90' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> onto a <a class='existingWikiWord' href='/nlab/show/diff/closed+subspace'>closed subspace</a> of <math class='maruku-mathml' display='inline' id='mathml_2db5d3e9193c4a4cbce879f4b5614c00693af78f_91' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>I</mi> <mi>ℕ</mi></msup></mrow><annotation encoding='application/x-tex'>I^\mathbb{N}</annotation></semantics></math>. As mentioned at <a href='https://ncatlab.org/nlab/show/Peano+curve#construction'>Peano curve</a>, there is a continuous surjection <math class='maruku-mathml' display='inline' id='mathml_2db5d3e9193c4a4cbce879f4b5614c00693af78f_92' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi><mo>→</mo><msup><mi>I</mi> <mi>ℕ</mi></msup></mrow><annotation encoding='application/x-tex'>C \to I^\mathbb{N}</annotation></semantics></math>. Taking the <a class='existingWikiWord' href='/nlab/show/diff/pullback'>pullback</a> in <a class='existingWikiWord' href='/nlab/show/diff/Top'>Top</a></p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_2db5d3e9193c4a4cbce879f4b5614c00693af78f_93' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mrow><mtable><mtr><mtd><mi>K</mi></mtd> <mtd><mo>↪</mo></mtd> <mtd><mi>C</mi></mtd></mtr> <mtr><mtd><mpadded lspace='-100%width' width='0'><mi>surj</mi></mpadded><mo stretchy='false'>↓</mo></mtd> <mtd /> <mtd><mo stretchy='false'>↓</mo><mpadded width='0'><mi>surj</mi></mpadded></mtd></mtr> <mtr><mtd><mi>X</mi></mtd> <mtd><mover><mo>↪</mo><mi>y</mi></mover></mtd> <mtd><msup><mi>I</mi> <mi>ℕ</mi></msup></mtd></mtr></mtable></mrow></mrow><annotation encoding='application/x-tex'>\array{ K & \hookrightarrow & C \\ \mathllap{surj} \downarrow & & \downarrow \mathrlap{surj} \\ X & \stackrel{y}{\hookrightarrow} & I^\mathbb{N} } </annotation></semantics></math></div> <p>we see that to produce a continuous surjection <math class='maruku-mathml' display='inline' id='mathml_2db5d3e9193c4a4cbce879f4b5614c00693af78f_94' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding='application/x-tex'>C \to X</annotation></semantics></math>, it suffices to exhibit a continuous surjection <math class='maruku-mathml' display='inline' id='mathml_2db5d3e9193c4a4cbce879f4b5614c00693af78f_95' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi><mo>→</mo><mi>K</mi></mrow><annotation encoding='application/x-tex'>C \to K</annotation></semantics></math>.</p> <p>In fact, every closed subspace <math class='maruku-mathml' display='inline' id='mathml_2db5d3e9193c4a4cbce879f4b5614c00693af78f_96' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>K</mi><mo>↪</mo><mi>C</mi></mrow><annotation encoding='application/x-tex'>K \hookrightarrow C</annotation></semantics></math> admits a retraction. There is a clever trick for seeing this: represent Cantor space <math class='maruku-mathml' display='inline' id='mathml_2db5d3e9193c4a4cbce879f4b5614c00693af78f_97' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi></mrow><annotation encoding='application/x-tex'>C</annotation></semantics></math> instead as the subspace of <math class='maruku-mathml' display='inline' id='mathml_2db5d3e9193c4a4cbce879f4b5614c00693af78f_98' xmlns='http://www.w3.org/1998/Math/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> whose points, when written in base <math class='maruku-mathml' display='inline' id='mathml_2db5d3e9193c4a4cbce879f4b5614c00693af78f_99' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>6</mn></mrow><annotation encoding='application/x-tex'>6</annotation></semantics></math>, have just <math class='maruku-mathml' display='inline' id='mathml_2db5d3e9193c4a4cbce879f4b5614c00693af78f_100' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>0</mn></mrow><annotation encoding='application/x-tex'>0</annotation></semantics></math>'s and <math class='maruku-mathml' display='inline' id='mathml_2db5d3e9193c4a4cbce879f4b5614c00693af78f_101' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>5</mn></mrow><annotation encoding='application/x-tex'>5</annotation></semantics></math>'s in their representation. This subspace has the geometric property that if <math class='maruku-mathml' display='inline' id='mathml_2db5d3e9193c4a4cbce879f4b5614c00693af78f_102' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>x</mi><mo>,</mo><mi>y</mi><mo>∈</mo><mi>C</mi></mrow><annotation encoding='application/x-tex'>x, y \in C</annotation></semantics></math>, then <math class='maruku-mathml' display='inline' id='mathml_2db5d3e9193c4a4cbce879f4b5614c00693af78f_103' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mfrac><mrow><mi>x</mi><mo>+</mo><mi>y</mi></mrow><mn>2</mn></mfrac><mo>∉</mo><mi>C</mi></mrow><annotation encoding='application/x-tex'>\frac{x+y}{2} \notin C</annotation></semantics></math>. As a result, for <math class='maruku-mathml' display='inline' id='mathml_2db5d3e9193c4a4cbce879f4b5614c00693af78f_104' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>z</mi><mo>∈</mo><mi>C</mi></mrow><annotation encoding='application/x-tex'>x, y, z \in C</annotation></semantics></math> we have <math class='maruku-mathml' display='inline' id='mathml_2db5d3e9193c4a4cbce879f4b5614c00693af78f_105' xmlns='http://www.w3.org/1998/Math/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>x</mi><mo>,</mo><mi>z</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>d(x, y) = d(x, z)</annotation></semantics></math> only if <math class='maruku-mathml' display='inline' id='mathml_2db5d3e9193c4a4cbce879f4b5614c00693af78f_106' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>y</mi><mo>=</mo><mi>z</mi></mrow><annotation encoding='application/x-tex'>y = z</annotation></semantics></math> and so: given a closed subspace <math class='maruku-mathml' display='inline' id='mathml_2db5d3e9193c4a4cbce879f4b5614c00693af78f_107' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>K</mi></mrow><annotation encoding='application/x-tex'>K</annotation></semantics></math> of <math class='maruku-mathml' display='inline' id='mathml_2db5d3e9193c4a4cbce879f4b5614c00693af78f_108' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi></mrow><annotation encoding='application/x-tex'>C</annotation></semantics></math>, there is for each <math class='maruku-mathml' display='inline' id='mathml_2db5d3e9193c4a4cbce879f4b5614c00693af78f_109' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>x</mi><mo>∈</mo><mi>C</mi></mrow><annotation encoding='application/x-tex'>x \in C</annotation></semantics></math> a <em>unique</em> element <math class='maruku-mathml' display='inline' id='mathml_2db5d3e9193c4a4cbce879f4b5614c00693af78f_110' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>k</mi> <mi>x</mi></msub><mo>∈</mo><mi>K</mi></mrow><annotation encoding='application/x-tex'>k_x \in K</annotation></semantics></math> such that <math class='maruku-mathml' display='inline' id='mathml_2db5d3e9193c4a4cbce879f4b5614c00693af78f_111' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>d</mi><mo stretchy='false'>(</mo><mi>x</mi><mo>,</mo><msub><mi>k</mi> <mi>x</mi></msub><mo stretchy='false'>)</mo><mo>=</mo><mi>d</mi><mo stretchy='false'>(</mo><mi>x</mi><mo>,</mo><mi>K</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>d(x, k_x) = d(x, K)</annotation></semantics></math>. The assignment <math class='maruku-mathml' display='inline' id='mathml_2db5d3e9193c4a4cbce879f4b5614c00693af78f_112' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>x</mi><mo>↦</mo><msub><mi>k</mi> <mi>x</mi></msub></mrow><annotation encoding='application/x-tex'>x \mapsto k_x</annotation></semantics></math> is continuous (in fact a <a class='existingWikiWord' href='/nlab/show/diff/locally+constant+function'>locally constant function</a> on <math class='maruku-mathml' display='inline' id='mathml_2db5d3e9193c4a4cbce879f4b5614c00693af78f_113' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi><mo>∖</mo><mi>K</mi></mrow><annotation encoding='application/x-tex'>C \setminus K</annotation></semantics></math>, and continuous on <math class='maruku-mathml' display='inline' id='mathml_2db5d3e9193c4a4cbce879f4b5614c00693af78f_114' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>K</mi></mrow><annotation encoding='application/x-tex'>K</annotation></semantics></math> as is easily seen) and provides the desired retraction.</p> </div> <h2 id='related_concepts'>Related concepts</h2> <ul> <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/Sierpinski+space'>Sierpinski space</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/long+line'>long line</a>, <a class='existingWikiWord' href='/nlab/show/diff/Warsaw+circle'>Warsaw circle</a>,</p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/line+with+two+origins'>line with two origins</a></p> </li> </ul> <h2 id='references'>References</h2><ins class='diffins'> </ins><ins class='diffins'><p>Named after <a class='existingWikiWord' href='/nlab/show/diff/Georg+Cantor'>Georg Cantor</a>.</p></ins> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Friedhelm+Waldhausen'>Friedhelm Waldhausen</a>, p. 3-4 in <em>Topologie</em> (<a href='https://www.math.uni-bielefeld.de/~fw/ein.pdf'>pdf</a>)</p> </li> <li> <p>Proof Wiki, <em><a href='https://proofwiki.org/wiki/Cantor_Space_as_Countably_Infinite_Product'>Cantor Space as Countably Infinite Product</a></em></p> </li> </ul> <p> </p> </div> <div class="revisedby"> <p> Last revised on June 24, 2024 at 11:01:25. 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