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name="query" value="Search" style="display:inline-block; float: left;" onfocus="this.value == 'Search' ? this.value = '' : true" onblur="this.value == '' ? this.value = 'Search' : true" /></fieldset> </form> <span id='navEnd'></span> </div> <div id="revision"> <p class="show_diff"> Showing changes from revision #48 to #49: <ins class="diffins">Added</ins> | <del class="diffdel">Removed</del> | <del class="diffmod">Chan</del><ins class="diffmod">ged</ins> </p> <p>.</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='#for_topological_spaces'>For topological spaces</a></li><li><a href='#for_noncommutative_topological_spaces_algebras'>For non-commutative topological spaces (<math class='maruku-mathml' display='inline' id='mathml_36d2b9ac71eb02eda287e1afa8662e532c98741a_1' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>C</mi> <mo>*</mo></msup></mrow><annotation encoding='application/x-tex'>C^\ast</annotation></semantics></math>-algebras)</a></li></ul></li><li><a href='#properties'>Properties</a><ul><li><a href='#BasicProperties'>Basic properties</a></li><li><a href='#UniversalProperty'>Universal property</a></li><li><a href='#MonoidalFunctoriality'>Monoidal functoriality</a></li></ul></li><li><a href='#examples'>Examples</a><ul><li><a href='#general'>General</a></li><li><a href='#compactification_of_discrete_spaces'>Compactification of discrete spaces</a></li><li><a href='#ExamplesSpheres'><span> Euclidean spaces compactify to<del class='diffmod'> Spheres</del><ins class='diffmod'> spheres</ins></span></a></li><li><a href='#linear_representations_compactify_to_representation_spheres'>Linear representations compactify to representation spheres</a></li><li><a href='#thom_spaces'>Thom spaces</a></li><li><a href='#locally_compact_hausdorff_spaces'>Locally compact Hausdorff spaces</a></li></ul></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>The <em>one-point compactification</em> of a <a class='existingWikiWord' href='/nlab/show/diff/topological+space'>topological space</a> <math class='maruku-mathml' display='inline' id='mathml_36d2b9ac71eb02eda287e1afa8662e532c98741a_2' 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 new <a class='existingWikiWord' href='/nlab/show/diff/compact+space'>compact space</a> <math class='maruku-mathml' display='inline' id='mathml_36d2b9ac71eb02eda287e1afa8662e532c98741a_3' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>X</mi> <mo>*</mo></msup><mo>=</mo><mi>X</mi><mo>∪</mo><mo stretchy='false'>{</mo><mn>∞</mn><mo stretchy='false'>}</mo></mrow><annotation encoding='application/x-tex'>X^* = X \cup \{\infty\}</annotation></semantics></math> obtained by adding a single new point “<math class='maruku-mathml' display='inline' id='mathml_36d2b9ac71eb02eda287e1afa8662e532c98741a_4' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>∞</mn></mrow><annotation encoding='application/x-tex'>\infty</annotation></semantics></math>” to the original space and declaring in <math class='maruku-mathml' display='inline' id='mathml_36d2b9ac71eb02eda287e1afa8662e532c98741a_5' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>X</mi> <mo>*</mo></msup></mrow><annotation encoding='application/x-tex'>X^*</annotation></semantics></math> the <a class='existingWikiWord' href='/nlab/show/diff/complement'>complements</a> of the original <a class='existingWikiWord' href='/nlab/show/diff/closed+subspace'>closed</a> <a class='existingWikiWord' href='/nlab/show/diff/compact+space'>compact</a> <a class='existingWikiWord' href='/nlab/show/diff/subspace'>subspaces</a> to be <a class='existingWikiWord' href='/nlab/show/diff/open+subspace'>open</a>.</p> <p>One may think of the new point added as the “point at infinity” of the original space. A <a class='existingWikiWord' href='/nlab/show/diff/continuous+map'>continuous function</a> on <math class='maruku-mathml' display='inline' id='mathml_36d2b9ac71eb02eda287e1afa8662e532c98741a_6' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> <em><a class='existingWikiWord' href='/nlab/show/diff/vanishing+at+infinity'>vanishes at infinity</a></em> precisely if it extends to a continuous function on <math class='maruku-mathml' display='inline' id='mathml_36d2b9ac71eb02eda287e1afa8662e532c98741a_7' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>X</mi> <mo>*</mo></msup></mrow><annotation encoding='application/x-tex'>X^*</annotation></semantics></math> and literally takes the value zero at the point “<math class='maruku-mathml' display='inline' id='mathml_36d2b9ac71eb02eda287e1afa8662e532c98741a_8' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>∞</mn></mrow><annotation encoding='application/x-tex'>\infty</annotation></semantics></math>”.</p> <p>This one-point compactification is also known as the <em>Alexandroff compactification</em> after a 1924 paper by <a class='existingWikiWord' href='/nlab/show/diff/Pavel+Aleksandrov'>Павел Сергеевич Александров</a> (then transliterated ‘P.S. Aleksandroff’).</p> <p>The one-point compactification is usually applied to a non-<a class='existingWikiWord' href='/nlab/show/diff/compact+space'>compact</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> space. In the more general situation, it may not really be a <a class='existingWikiWord' href='/nlab/show/diff/compactification'>compactification</a> and hence is called the <em>one-point extension</em> or <em>Alexandroff extension</em>.</p> <h2 id='definition'>Definition</h2> <h3 id='for_topological_spaces'>For topological spaces</h3> <div class='num_defn' id='OnePointExtension'> <h6 id='definition_2'>Definition</h6> <p><strong>(one-point extension)</strong></p> <p>Let <math class='maruku-mathml' display='inline' id='mathml_36d2b9ac71eb02eda287e1afa8662e532c98741a_9' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> be any <a class='existingWikiWord' href='/nlab/show/diff/topological+space'>topological space</a>. Its <strong>one-point extension</strong> <math class='maruku-mathml' display='inline' id='mathml_36d2b9ac71eb02eda287e1afa8662e532c98741a_10' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>X</mi> <mo>*</mo></msup></mrow><annotation encoding='application/x-tex'>X^*</annotation></semantics></math> is the topological space</p> <ul> <li> <p>whose underlying <a class='existingWikiWord' href='/nlab/show/diff/set'>set</a> is the <a class='existingWikiWord' href='/nlab/show/diff/disjoint+union'>disjoint union</a> <math class='maruku-mathml' display='inline' id='mathml_36d2b9ac71eb02eda287e1afa8662e532c98741a_11' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi><mo>∪</mo><mo stretchy='false'>{</mo><mn>∞</mn><mo stretchy='false'>}</mo></mrow><annotation encoding='application/x-tex'>X \cup \{\infty\}</annotation></semantics></math> of that of <math class='maruku-mathml' display='inline' id='mathml_36d2b9ac71eb02eda287e1afa8662e532c98741a_12' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> with a <a class='existingWikiWord' href='/nlab/show/diff/singleton'>singleton set</a> <math class='maruku-mathml' display='inline' id='mathml_36d2b9ac71eb02eda287e1afa8662e532c98741a_13' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>{</mo><mn>∞</mn><mo stretchy='false'>}</mo></mrow><annotation encoding='application/x-tex'>\{\infty\}</annotation></semantics></math>;</p> </li> <li> <p>and whose <a class='existingWikiWord' href='/nlab/show/diff/open+subspace'>open subsets</a> are</p> <ol> <li> <p>the <a class='existingWikiWord' href='/nlab/show/diff/open+subspace'>open subsets</a> of <math class='maruku-mathml' display='inline' id='mathml_36d2b9ac71eb02eda287e1afa8662e532c98741a_14' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi><mspace width='thinmathspace' /><mo>⊂</mo><mspace width='thinmathspace' /><msup><mi>X</mi> <mo>*</mo></msup></mrow><annotation encoding='application/x-tex'>X \,\subset\, X^*</annotation></semantics></math> ;</p> </li> <li> <p>the <a class='existingWikiWord' href='/nlab/show/diff/complement'>complements</a> <math class='maruku-mathml' display='inline' id='mathml_36d2b9ac71eb02eda287e1afa8662e532c98741a_15' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>X</mi> <mo>*</mo></msup><mo>∖</mo><mi>CK</mi><mo>=</mo><mo stretchy='false'>(</mo><mi>X</mi><mo>∖</mo><mi>CK</mi><mo stretchy='false'>)</mo><mo>∪</mo><mo stretchy='false'>{</mo><mn>∞</mn><mo stretchy='false'>}</mo></mrow><annotation encoding='application/x-tex'>X^\ast \setminus CK = (X \setminus CK) \cup \{\infty\}</annotation></semantics></math> of the <a class='existingWikiWord' href='/nlab/show/diff/closed+subspace'>closed</a> <a class='existingWikiWord' href='/nlab/show/diff/compact+space'>compact</a> subsets <math class='maruku-mathml' display='inline' id='mathml_36d2b9ac71eb02eda287e1afa8662e532c98741a_16' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>CK</mi><mo>⊂</mo><mi>X</mi></mrow><annotation encoding='application/x-tex'>CK \subset X</annotation></semantics></math>.</p> </li> </ol> </li> </ul> </div> <p>(<a href='#Aleksandrov24'>Aleksandrov 24</a>, see <a href='#Kelley75'>Kelley 75, p. 150</a>)</p> <div class='num_remark'> <h6 id='remark'>Remark</h6> <p>If <math class='maruku-mathml' display='inline' id='mathml_36d2b9ac71eb02eda287e1afa8662e532c98741a_17' 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/Hausdorff+space'>Hausdorff</a>, then it is sufficient to speak of compact subsets in def. <a class='maruku-ref' href='#OnePointExtension'>1</a>, since <a class='existingWikiWord' href='/nlab/show/diff/compact+subspaces+of+Hausdorff+spaces+are+closed'>compact subspaces of Hausdorff spaces are closed</a>.</p> </div> <div class='num_lemma' id='OnePointExtensionWellDefined'> <h6 id='lemma'>Lemma</h6> <p><strong>(one-point extension is well-defined)</strong></p> <p>The <a class='existingWikiWord' href='/nlab/show/diff/topological+space'>topology</a> on the one-point extension in def. <a class='maruku-ref' href='#OnePointExtension'>1</a> is indeed well defined in that the given set of subsets is indeed closed under arbitrary unions and finite intersections.</p> </div> <div class='proof'> <h6 id='proof'>Proof</h6> <p>The unions and finite intersections of the open subsets inherited from <math class='maruku-mathml' display='inline' id='mathml_36d2b9ac71eb02eda287e1afa8662e532c98741a_18' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> are closed among themselves by the assumption that <math class='maruku-mathml' display='inline' id='mathml_36d2b9ac71eb02eda287e1afa8662e532c98741a_19' 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 topological space.</p> <p>It is hence sufficient to see that</p> <ol> <li> <p>the unions and finite intersection of the <math class='maruku-mathml' display='inline' id='mathml_36d2b9ac71eb02eda287e1afa8662e532c98741a_20' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mi>X</mi><mo>\</mo><mi>CK</mi><mo stretchy='false'>)</mo><mo>∪</mo><mo stretchy='false'>{</mo><mn>∞</mn><mo stretchy='false'>}</mo></mrow><annotation encoding='application/x-tex'>(X \backslash CK) \cup \{\infty\}</annotation></semantics></math> are closed among themselves,</p> </li> <li> <p>the union and intersection of a subset of the form <math class='maruku-mathml' display='inline' id='mathml_36d2b9ac71eb02eda287e1afa8662e532c98741a_21' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>U</mi><munder><mo>⊂</mo><mtext>open</mtext></munder><mi>X</mi><mo>⊂</mo><msup><mi>X</mi> <mo>*</mo></msup></mrow><annotation encoding='application/x-tex'>U \underset{\text{open}}{\subset} X \subset X^\ast</annotation></semantics></math> with one of the form <math class='maruku-mathml' display='inline' id='mathml_36d2b9ac71eb02eda287e1afa8662e532c98741a_22' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mi>X</mi><mo>\</mo><mi>CK</mi><mo stretchy='false'>)</mo><mo>∪</mo><mo stretchy='false'>{</mo><mn>∞</mn><mo stretchy='false'>}</mo></mrow><annotation encoding='application/x-tex'>(X \backslash CK) \cup \{\infty\}</annotation></semantics></math> is again of one of the two kinds.</p> </li> </ol> <p>Regarding the first statement: Under <a class='existingWikiWord' href='/nlab/show/diff/De+Morgan+duality'>de Morgan duality</a></p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_36d2b9ac71eb02eda287e1afa8662e532c98741a_23' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><munder><mo lspace='thinmathspace' rspace='thinmathspace'>⋂</mo><mrow><mi>i</mi><mo>∈</mo><munder><mi>J</mi><mtext>finite</mtext></munder></mrow></munder><mo stretchy='false'>(</mo><mi>X</mi><mo>\</mo><msub><mi>CK</mi> <mi>i</mi></msub><mo>∪</mo><mo stretchy='false'>{</mo><mn>∞</mn><mo stretchy='false'>}</mo><mo stretchy='false'>)</mo><mo>=</mo><mrow><mo>(</mo><mi>X</mi><mo>\</mo><mrow><mo>(</mo><munder><mo lspace='thinmathspace' rspace='thinmathspace'>⋃</mo><mrow><mi>i</mi><mo>∈</mo><munder><mi>J</mi><mtext>finite</mtext></munder></mrow></munder><msub><mi>CK</mi> <mi>i</mi></msub><mo>)</mo></mrow><mo>)</mo></mrow><mo>∪</mo><mo stretchy='false'>{</mo><mn>∞</mn><mo stretchy='false'>}</mo></mrow><annotation encoding='application/x-tex'> \underset{i \in \underset{\text{finite}}{J}}{\bigcap} (X \backslash CK_i \cup \{\infty\}) = \left( X \backslash \left(\underset{i \in \underset{\text{finite}}{J}}{\bigcup} CK_i \right)\right) \cup \{\infty\} </annotation></semantics></math></div> <p>and</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_36d2b9ac71eb02eda287e1afa8662e532c98741a_24' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><munder><mo lspace='thinmathspace' rspace='thinmathspace'>⋃</mo><mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></munder><mo stretchy='false'>(</mo><mi>X</mi><mo>\</mo><msub><mi>CK</mi> <mi>i</mi></msub><mo>∪</mo><mo stretchy='false'>{</mo><mn>∞</mn><mo stretchy='false'>}</mo><mo stretchy='false'>)</mo><mo>=</mo><mrow><mo>(</mo><mi>X</mi><mo>\</mo><mrow><mo>(</mo><munder><mo lspace='thinmathspace' rspace='thinmathspace'>⋂</mo><mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></munder><msub><mi>CK</mi> <mi>i</mi></msub><mo>)</mo></mrow><mo>)</mo></mrow><mo>∪</mo><mo stretchy='false'>{</mo><mn>∞</mn><mo stretchy='false'>}</mo></mrow><annotation encoding='application/x-tex'> \underset{i \in I}{\bigcup} ( X \backslash CK_i \cup \{\infty\} ) = \left(X \backslash \left(\underset{i \in I}{\bigcap} CK_i \right)\right) \cup \{\infty\} </annotation></semantics></math></div> <p>and so the first statement follows from the fact that finite unions of compact subspaces and arbitrary intersections of closed compact subspaces are themselves again compact (<a href='compact+space#UnionsAndIntersectionOfCompactSubspaces'>this prop.</a>).</p> <p>Regarding the second statement: That <math class='maruku-mathml' display='inline' id='mathml_36d2b9ac71eb02eda287e1afa8662e532c98741a_25' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>U</mi><mo>⊂</mo><mi>X</mi></mrow><annotation encoding='application/x-tex'>U \subset X</annotation></semantics></math> is open means that there exists a closed subset <math class='maruku-mathml' display='inline' id='mathml_36d2b9ac71eb02eda287e1afa8662e532c98741a_26' 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 \subset X</annotation></semantics></math> with <math class='maruku-mathml' display='inline' id='mathml_36d2b9ac71eb02eda287e1afa8662e532c98741a_27' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>U</mi><mo>=</mo><mi>X</mi><mo>\</mo><mi>C</mi></mrow><annotation encoding='application/x-tex'>U = X\backslash C</annotation></semantics></math>. Now using <a class='existingWikiWord' href='/nlab/show/diff/De+Morgan+duality'>de Morgan duality</a> we find</p> <ol> <li> <p>for intersections:</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_36d2b9ac71eb02eda287e1afa8662e532c98741a_28' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mrow><mtable columnalign='right left right left right left right left right left' columnspacing='0em' displaystyle='true'><mtr><mtd><mi>U</mi><mo>∩</mo><mo stretchy='false'>(</mo><mo stretchy='false'>(</mo><mi>X</mi><mo>\</mo><mi>CK</mi><mo stretchy='false'>)</mo><mo>∪</mo><mo stretchy='false'>{</mo><mn>∞</mn><mo stretchy='false'>}</mo><mo stretchy='false'>)</mo></mtd> <mtd><mo>=</mo><mo stretchy='false'>(</mo><mi>X</mi><mo>\</mo><mi>C</mi><mo stretchy='false'>)</mo><mo>∩</mo><mo stretchy='false'>(</mo><mi>X</mi><mo>\</mo><mi>CK</mi><mo stretchy='false'>)</mo></mtd></mtr> <mtr><mtd /> <mtd><mo>=</mo><mi>X</mi><mo>\</mo><mo stretchy='false'>(</mo><mi>C</mi><mo>∪</mo><mi>CK</mi><mo stretchy='false'>)</mo><mo>.</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding='application/x-tex'> \begin{aligned} U \cap ( (X\backslash CK) \cup \{\infty\} ) & = (X \backslash C) \cap (X \backslash CK) \\ & = X \backslash (C \cup CK). \end{aligned} </annotation></semantics></math></div> <p>Since finite unions of closed subsets are closed, this is again an open subset of <math class='maruku-mathml' display='inline' id='mathml_36d2b9ac71eb02eda287e1afa8662e532c98741a_29' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math>;</p> </li> <li> <p>for unions:</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_36d2b9ac71eb02eda287e1afa8662e532c98741a_30' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mrow><mtable columnalign='right left right left right left right left right left' columnspacing='0em' displaystyle='true'><mtr><mtd><mi>U</mi><mo>∪</mo><mo stretchy='false'>(</mo><mi>X</mi><mo>\</mo><mi>CK</mi><mo stretchy='false'>)</mo><mo>∪</mo><mo stretchy='false'>{</mo><mn>∞</mn><mo stretchy='false'>}</mo></mtd> <mtd><mo>=</mo><mo stretchy='false'>(</mo><mi>X</mi><mo>\</mo><mi>C</mi><mo stretchy='false'>)</mo><mo>∪</mo><mo stretchy='false'>(</mo><mi>X</mi><mo>\</mo><mi>CK</mi><mo stretchy='false'>)</mo><mo>∪</mo><mo stretchy='false'>{</mo><mn>∞</mn><mo stretchy='false'>}</mo></mtd></mtr> <mtr><mtd /> <mtd><mo>=</mo><mo stretchy='false'>(</mo><mi>X</mi><mo>\</mo><mo stretchy='false'>(</mo><mi>C</mi><mo>∩</mo><mi>CK</mi><mo stretchy='false'>)</mo><mo stretchy='false'>)</mo><mo>∪</mo><mo stretchy='false'>{</mo><mn>∞</mn><mo stretchy='false'>}</mo><mo>.</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding='application/x-tex'> \begin{aligned} U \cup (X \backslash CK) \cup \{\infty\} & = (X \backslash C) \cup (X \backslash CK) \cup \{\infty\} \\ & = (X \backslash (C \cap CK)) \cup \{\infty\} . \end{aligned} </annotation></semantics></math></div> <p>For this to be open in <math class='maruku-mathml' display='inline' id='mathml_36d2b9ac71eb02eda287e1afa8662e532c98741a_31' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>X</mi> <mo>*</mo></msup></mrow><annotation encoding='application/x-tex'>X^\ast</annotation></semantics></math> we need that <math class='maruku-mathml' display='inline' id='mathml_36d2b9ac71eb02eda287e1afa8662e532c98741a_32' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi><mo>∩</mo><mi>CK</mi></mrow><annotation encoding='application/x-tex'>C \cap CK</annotation></semantics></math> is again compact. This follows because <a class='existingWikiWord' href='/nlab/show/diff/subsets+are+closed+in+a+closed+subspace+precisely+if+they+are+closed+in+the+ambient+space'>subsets are closed in a closed subspace precisely if they are closed in the ambient space</a> and because <a class='existingWikiWord' href='/nlab/show/diff/closed+subspaces+of+compact+spaces+are+compact'>closed subsets of compact spaces are compact</a>.</p> </li> </ol> </div> <h3 id='for_noncommutative_topological_spaces_algebras'>For non-commutative topological spaces (<math class='maruku-mathml' display='inline' id='mathml_36d2b9ac71eb02eda287e1afa8662e532c98741a_33' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>C</mi> <mo>*</mo></msup></mrow><annotation encoding='application/x-tex'>C^\ast</annotation></semantics></math>-algebras)</h3> <p>Dually in <a class='existingWikiWord' href='/nlab/show/diff/noncommutative+topology'>non-commutative topology</a> the one-point compactification corresponds to the <a class='existingWikiWord' href='/nlab/show/diff/unitization+of+a+C-star-algebra'>unitisation of C*-algebras</a>.</p> <h2 id='properties'>Properties</h2> <h3 id='BasicProperties'>Basic properties</h3> <p>We discuss the basic properties of the construction <math class='maruku-mathml' display='inline' id='mathml_36d2b9ac71eb02eda287e1afa8662e532c98741a_34' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>X</mi> <mo>*</mo></msup></mrow><annotation encoding='application/x-tex'>X^\ast</annotation></semantics></math> in def. <a class='maruku-ref' href='#OnePointExtension'>1</a>, in particular that it always yields a <a class='existingWikiWord' href='/nlab/show/diff/compact+space'>compact topological space</a> (prop. <a class='maruku-ref' href='#OnePointExtensionIsCompact'>1</a> below) and the ingredients needed to see its <a class='existingWikiWord' href='/nlab/show/diff/universal+construction'>universal property</a> in the Hausdorff case <a href='#OnePointExtensionIsCompact'>below</a>.</p> <div class='num_prop' id='OnePointExtensionIsCompact'> <h6 id='proposition'>Proposition</h6> <p><strong>(one-point extension is compact)</strong></p> <p>For <math class='maruku-mathml' display='inline' id='mathml_36d2b9ac71eb02eda287e1afa8662e532c98741a_35' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> any <a class='existingWikiWord' href='/nlab/show/diff/topological+space'>topological space</a>, we have that its one-point extension <math class='maruku-mathml' display='inline' id='mathml_36d2b9ac71eb02eda287e1afa8662e532c98741a_36' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>X</mi> <mo>*</mo></msup></mrow><annotation encoding='application/x-tex'>X^\ast</annotation></semantics></math> (def. <a class='maruku-ref' href='#OnePointExtension'>1</a>) is a <a class='existingWikiWord' href='/nlab/show/diff/compact+space'>compact topological space</a>.</p> </div> <div class='proof'> <h6 id='proof_2'>Proof</h6> <p>Let <math class='maruku-mathml' display='inline' id='mathml_36d2b9ac71eb02eda287e1afa8662e532c98741a_37' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>{</mo><msub><mi>U</mi> <mi>i</mi></msub><mo>⊂</mo><msup><mi>X</mi> <mo>*</mo></msup><msub><mo stretchy='false'>}</mo> <mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></msub></mrow><annotation encoding='application/x-tex'>\{U_i \subset X^\ast\}_{i \in I}</annotation></semantics></math> be an <a class='existingWikiWord' href='/nlab/show/diff/open+cover'>open cover</a>. We need to show that this has a finite subcover.</p> <p>That we have a cover means that</p> <ol> <li> <p>there must exist <math class='maruku-mathml' display='inline' id='mathml_36d2b9ac71eb02eda287e1afa8662e532c98741a_38' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>i</mi> <mn>∞</mn></msub><mo>∈</mo><mi>I</mi></mrow><annotation encoding='application/x-tex'>i_\infty \in I</annotation></semantics></math> such that <math class='maruku-mathml' display='inline' id='mathml_36d2b9ac71eb02eda287e1afa8662e532c98741a_39' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>U</mi> <mrow><mi>i</mi><mn>∞</mn></mrow></msub><mo>⊃</mo><mo stretchy='false'>{</mo><mn>∞</mn><mo stretchy='false'>}</mo></mrow><annotation encoding='application/x-tex'>U_{i \infty} \supset \{\infty\}</annotation></semantics></math> is an <a class='existingWikiWord' href='/nlab/show/diff/neighborhood'>open neighbourhood</a> of the extra point. But since, by construction, the only open subsets containing that point are of the form <math class='maruku-mathml' display='inline' id='mathml_36d2b9ac71eb02eda287e1afa8662e532c98741a_40' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mi>X</mi><mo>\</mo><mi>CK</mi><mo stretchy='false'>)</mo><mo>∪</mo><mo stretchy='false'>{</mo><mn>∞</mn><mo stretchy='false'>}</mo></mrow><annotation encoding='application/x-tex'>(X \backslash CK) \cup \{\infty\}</annotation></semantics></math>, it follows that there is a compact closed subset <math class='maruku-mathml' display='inline' id='mathml_36d2b9ac71eb02eda287e1afa8662e532c98741a_41' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>CK</mi><mo>⊂</mo><mi>X</mi></mrow><annotation encoding='application/x-tex'>CK \subset X</annotation></semantics></math> with <math class='maruku-mathml' display='inline' id='mathml_36d2b9ac71eb02eda287e1afa8662e532c98741a_42' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi><mo>\</mo><mi>CK</mi><mo>⊂</mo><msub><mi>U</mi> <mrow><mi>i</mi><mn>∞</mn></mrow></msub></mrow><annotation encoding='application/x-tex'>X \backslash CK \subset U_{i \infty}</annotation></semantics></math>.</p> </li> <li> <p><math class='maruku-mathml' display='inline' id='mathml_36d2b9ac71eb02eda287e1afa8662e532c98741a_43' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>{</mo><msub><mi>U</mi> <mi>i</mi></msub><mo>⊂</mo><mi>X</mi><msub><mo stretchy='false'>}</mo> <mrow><mi>i</mi><mo>∈</mo><mi>i</mi></mrow></msub></mrow><annotation encoding='application/x-tex'>\{U_i \subset X\}_{i \in i}</annotation></semantics></math> is in particular an open cover of that closed compact subset <math class='maruku-mathml' display='inline' id='mathml_36d2b9ac71eb02eda287e1afa8662e532c98741a_44' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>CK</mi><mo>⊂</mo><mi>X</mi></mrow><annotation encoding='application/x-tex'>CK \subset X</annotation></semantics></math>. This being compact means that there is a finite subset <math class='maruku-mathml' display='inline' id='mathml_36d2b9ac71eb02eda287e1afa8662e532c98741a_45' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>J</mi><mo>⊂</mo><mi>I</mi></mrow><annotation encoding='application/x-tex'>J \subset I</annotation></semantics></math> so that <math class='maruku-mathml' display='inline' id='mathml_36d2b9ac71eb02eda287e1afa8662e532c98741a_46' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>{</mo><msub><mi>U</mi> <mi>i</mi></msub><mo>⊂</mo><mi>X</mi><msub><mo stretchy='false'>}</mo> <mrow><mi>i</mi><mo>∈</mo><mo lspace='0em' rspace='thinmathspace'>J</mo><mo>⊂</mo><mi>X</mi></mrow></msub></mrow><annotation encoding='application/x-tex'>\{U_i \subset X\}_{i \in \J \subset X}</annotation></semantics></math> is still a cover of <math class='maruku-mathml' display='inline' id='mathml_36d2b9ac71eb02eda287e1afa8662e532c98741a_47' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>CK</mi></mrow><annotation encoding='application/x-tex'>CK</annotation></semantics></math>.</p> </li> </ol> <p>Together this implies that</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_36d2b9ac71eb02eda287e1afa8662e532c98741a_48' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>{</mo><msub><mi>U</mi> <mi>i</mi></msub><mo>⊂</mo><mi>X</mi><msub><mo stretchy='false'>}</mo> <mrow><mi>i</mi><mo>∈</mo><mi>J</mi><mo>⊂</mo><mi>I</mi></mrow></msub><mo>∪</mo><mo stretchy='false'>{</mo><msub><mi>U</mi> <mrow><msub><mi>i</mi> <mn>∞</mn></msub></mrow></msub><mo stretchy='false'>}</mo></mrow><annotation encoding='application/x-tex'> \{U_i \subset X\}_{i \in J \subset I} \cup \{ U_{i_\infty} \} </annotation></semantics></math></div> <p>is a finite subcover of the original cover.</p> </div> <div class='num_prop' id='HausdorffOnePointCompactification'> <h6 id='proposition_2'>Proposition</h6> <p><strong>(one-point extension of locally compact space is Hausdorff precisely if original space is)</strong></p> <p>Let <math class='maruku-mathml' display='inline' id='mathml_36d2b9ac71eb02eda287e1afa8662e532c98741a_49' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> be a <a class='existingWikiWord' href='/nlab/show/diff/locally+compact+topological+space'>locally compact topological space</a>. Then its one-point extension <math class='maruku-mathml' display='inline' id='mathml_36d2b9ac71eb02eda287e1afa8662e532c98741a_50' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>X</mi> <mo>*</mo></msup></mrow><annotation encoding='application/x-tex'>X^\ast</annotation></semantics></math> (def. <a class='maruku-ref' href='#OnePointExtension'>1</a>) is a <a class='existingWikiWord' href='/nlab/show/diff/Hausdorff+space'>Hausdorff topological space</a> precisely if <math class='maruku-mathml' display='inline' id='mathml_36d2b9ac71eb02eda287e1afa8662e532c98741a_51' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> is.</p> </div> <div class='proof'> <h6 id='proof_3'>Proof</h6> <p>It is clear that if <math class='maruku-mathml' display='inline' id='mathml_36d2b9ac71eb02eda287e1afa8662e532c98741a_52' 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 Hausdorff then <math class='maruku-mathml' display='inline' id='mathml_36d2b9ac71eb02eda287e1afa8662e532c98741a_53' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>X</mi> <mo>*</mo></msup></mrow><annotation encoding='application/x-tex'>X^\ast</annotation></semantics></math> is not.</p> <p>For the converse, assume that <math class='maruku-mathml' display='inline' id='mathml_36d2b9ac71eb02eda287e1afa8662e532c98741a_54' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> is Hausdorff.</p> <p>Since <math class='maruku-mathml' display='inline' id='mathml_36d2b9ac71eb02eda287e1afa8662e532c98741a_55' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>X</mi> <mo>*</mo></msup><mo>=</mo><mi>X</mi><mo>∪</mo><mo stretchy='false'>{</mo><mn>∞</mn><mo stretchy='false'>}</mo></mrow><annotation encoding='application/x-tex'>X^\ast = X \cup \{\infty\}</annotation></semantics></math> as underlying sets, we only need to check that for <math class='maruku-mathml' display='inline' id='mathml_36d2b9ac71eb02eda287e1afa8662e532c98741a_56' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>x</mi><mo>∈</mo><mi>X</mi></mrow><annotation encoding='application/x-tex'>x \in X</annotation></semantics></math> any point, then there is an open neighbourhood <math class='maruku-mathml' display='inline' id='mathml_36d2b9ac71eb02eda287e1afa8662e532c98741a_57' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>U</mi> <mi>x</mi></msub><mo>⊂</mo><mi>X</mi><mo>⊂</mo><msup><mi>X</mi> <mo>*</mo></msup></mrow><annotation encoding='application/x-tex'>U_x \subset X \subset X^\ast</annotation></semantics></math> and an open neighbourhood <math class='maruku-mathml' display='inline' id='mathml_36d2b9ac71eb02eda287e1afa8662e532c98741a_58' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>V</mi> <mn>∞</mn></msub><mo>⊂</mo><msup><mi>X</mi> <mo>*</mo></msup></mrow><annotation encoding='application/x-tex'>V_\infty \subset X^\ast</annotation></semantics></math> of the extra point which are disjoint.</p> <p>That <math class='maruku-mathml' display='inline' id='mathml_36d2b9ac71eb02eda287e1afa8662e532c98741a_59' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> is locally compact implies by definition that there exists an open neighbourhood <math class='maruku-mathml' display='inline' id='mathml_36d2b9ac71eb02eda287e1afa8662e532c98741a_60' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>U</mi> <mi>k</mi></msub><mo>⊃</mo><mo stretchy='false'>{</mo><mi>x</mi><mo stretchy='false'>}</mo></mrow><annotation encoding='application/x-tex'>U_k \supset \{x\}</annotation></semantics></math> whose <a class='existingWikiWord' href='/nlab/show/diff/closed+subspace'>topological closure</a> <math class='maruku-mathml' display='inline' id='mathml_36d2b9ac71eb02eda287e1afa8662e532c98741a_61' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>CK</mi><mo>≔</mo><mi>Cl</mi><mo stretchy='false'>(</mo><msub><mi>U</mi> <mi>x</mi></msub><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>CK \coloneqq Cl(U_x)</annotation></semantics></math> is a closed compact neighbourhood <math class='maruku-mathml' display='inline' id='mathml_36d2b9ac71eb02eda287e1afa8662e532c98741a_62' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>CK</mi><mo>⊃</mo><mo stretchy='false'>{</mo><mi>x</mi><mo stretchy='false'>}</mo></mrow><annotation encoding='application/x-tex'>CK \supset \{x\}</annotation></semantics></math>. Hence</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_36d2b9ac71eb02eda287e1afa8662e532c98741a_63' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>V</mi> <mn>∞</mn></msub><mo>≔</mo><mo stretchy='false'>(</mo><mi>X</mi><mo>\</mo><mi>CK</mi><mo stretchy='false'>)</mo><mo>∪</mo><mo stretchy='false'>{</mo><mn>∞</mn><mo stretchy='false'>}</mo><mo>⊂</mo><msup><mi>X</mi> <mo>*</mo></msup></mrow><annotation encoding='application/x-tex'> V_\infty \coloneqq (X \backslash CK ) \cup \{\infty\} \subset X^\ast </annotation></semantics></math></div> <p>is an open neighbourhood of <math class='maruku-mathml' display='inline' id='mathml_36d2b9ac71eb02eda287e1afa8662e532c98741a_64' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>{</mo><mn>∞</mn><mo stretchy='false'>}</mo></mrow><annotation encoding='application/x-tex'>\{\infty\}</annotation></semantics></math> and the two are disjoint</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_36d2b9ac71eb02eda287e1afa8662e532c98741a_65' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>U</mi> <mi>x</mi></msub><mo>∩</mo><msub><mi>V</mi> <mn>∞</mn></msub><mo>=</mo><mi>∅</mi></mrow><annotation encoding='application/x-tex'> U_x \cap V_\infty = \emptyset </annotation></semantics></math></div> <p>by construction.</p> </div> <div class='num_prop' id='InclusionIntoOnePointExtensionIsOpenEmbedding'> <h6 id='proposition_3'>Proposition</h6> <p><strong>(inclusion into one-point extension is open embedding)</strong></p> <p>Let <math class='maruku-mathml' display='inline' id='mathml_36d2b9ac71eb02eda287e1afa8662e532c98741a_66' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> be a <a class='existingWikiWord' href='/nlab/show/diff/topological+space'>topological space</a>. Then the evident inclusion function</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_36d2b9ac71eb02eda287e1afa8662e532c98741a_67' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>i</mi><mspace width='thickmathspace' /><mo lspace='verythinmathspace'>:</mo><mspace width='thickmathspace' /><mi>X</mi><mo>⟶</mo><msup><mi>X</mi> <mo>*</mo></msup></mrow><annotation encoding='application/x-tex'> i \;\colon\; X \longrightarrow X^\ast </annotation></semantics></math></div> <p>into its one-point extension (def. <a class='maruku-ref' href='#OnePointExtension'>1</a>) is</p> <ol> <li> <p>a <a class='existingWikiWord' href='/nlab/show/diff/continuous+map'>continuous function</a></p> </li> <li> <p>an <a class='existingWikiWord' href='/nlab/show/diff/open+map'>open map</a></p> </li> <li> <p>an <a class='existingWikiWord' href='/nlab/show/diff/embedding+of+topological+spaces'>embedding of topological spaces</a>.</p> </li> </ol> </div> <div class='proof'> <h6 id='proof_4'>Proof</h6> <p>Regarding the first point: For <math class='maruku-mathml' display='inline' id='mathml_36d2b9ac71eb02eda287e1afa8662e532c98741a_68' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>U</mi><mo>⊂</mo><mi>X</mi></mrow><annotation encoding='application/x-tex'>U \subset X</annotation></semantics></math> open and <math class='maruku-mathml' display='inline' id='mathml_36d2b9ac71eb02eda287e1afa8662e532c98741a_69' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>CK</mi><mo>⊂</mo><mi>X</mi></mrow><annotation encoding='application/x-tex'>CK \subset X</annotation></semantics></math> closed and compact, the preimages of the corresponding open subsets in <math class='maruku-mathml' display='inline' id='mathml_36d2b9ac71eb02eda287e1afa8662e532c98741a_70' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>X</mi> <mo>*</mo></msup></mrow><annotation encoding='application/x-tex'>X^\ast</annotation></semantics></math> are</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_36d2b9ac71eb02eda287e1afa8662e532c98741a_71' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>i</mi> <mrow><mo lspace='verythinmathspace' rspace='0em'>−</mo><mn>1</mn></mrow></msup><mo stretchy='false'>(</mo><mi>U</mi><mo stretchy='false'>)</mo><mo>=</mo><mi>U</mi><mphantom><mi>AAAA</mi></mphantom><msup><mi>i</mi> <mrow><mo lspace='verythinmathspace' rspace='0em'>−</mo><mn>1</mn></mrow></msup><mo stretchy='false'>(</mo><mo stretchy='false'>(</mo><mi>X</mi><mo>\</mo><mi>CK</mi><mo stretchy='false'>)</mo><mo>∪</mo><mn>∞</mn><mo stretchy='false'>)</mo><mo>=</mo><mi>X</mi><mo>\</mo><mi>CK</mi></mrow><annotation encoding='application/x-tex'> i^{-1}(U) = U \phantom{AAAA} i^{-1}( (X \backslash CK) \cup \infty ) = X \backslash CK </annotation></semantics></math></div> <p>which are open in <math class='maruku-mathml' display='inline' id='mathml_36d2b9ac71eb02eda287e1afa8662e532c98741a_72' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math>.</p> <p>Regarding the second point: The image of an open subset <math class='maruku-mathml' display='inline' id='mathml_36d2b9ac71eb02eda287e1afa8662e532c98741a_73' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>U</mi><mo>⊂</mo><mi>X</mi></mrow><annotation encoding='application/x-tex'>U \subset X</annotation></semantics></math> is <math class='maruku-mathml' display='inline' id='mathml_36d2b9ac71eb02eda287e1afa8662e532c98741a_74' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>i</mi><mo stretchy='false'>(</mo><mi>U</mi><mo stretchy='false'>)</mo><mo>=</mo><mi>U</mi><mo>⊂</mo><msup><mi>X</mi> <mo>*</mo></msup></mrow><annotation encoding='application/x-tex'>i(U) = U \subset X^\ast</annotation></semantics></math>, which is open by definition.</p> <p>Regarding the third point: We need to show that <math class='maruku-mathml' display='inline' id='mathml_36d2b9ac71eb02eda287e1afa8662e532c98741a_75' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>i</mi><mo lspace='verythinmathspace'>:</mo><mi>X</mi><mo>→</mo><mi>i</mi><mo stretchy='false'>(</mo><mi>X</mi><mo stretchy='false'>)</mo><mo>⊂</mo><msup><mi>X</mi> <mo>*</mo></msup></mrow><annotation encoding='application/x-tex'>i \colon X \to i(X) \subset X^\ast</annotation></semantics></math> is a <a class='existingWikiWord' href='/nlab/show/diff/homeomorphism'>homeomorphism</a>. This is immediate from the definition of <math class='maruku-mathml' display='inline' id='mathml_36d2b9ac71eb02eda287e1afa8662e532c98741a_76' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>X</mi> <mo>*</mo></msup></mrow><annotation encoding='application/x-tex'>X^\ast</annotation></semantics></math>.</p> </div> <div class='num_prop' id='CompactHausdorffSpaceIsCompactificationOfComplementOfAnyPoint'> <h6 id='proposition_4'>Proposition</h6> <p>If <math class='maruku-mathml' display='inline' id='mathml_36d2b9ac71eb02eda287e1afa8662e532c98741a_77' 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 <a class='existingWikiWord' href='/nlab/show/diff/compactum'>compact Hausdorff space</a> and <math class='maruku-mathml' display='inline' id='mathml_36d2b9ac71eb02eda287e1afa8662e532c98741a_78' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>x</mi> <mn>0</mn></msub><mo>∈</mo><mi>X</mi></mrow><annotation encoding='application/x-tex'>x_0 \in X</annotation></semantics></math> any point, then <math class='maruku-mathml' display='inline' id='mathml_36d2b9ac71eb02eda287e1afa8662e532c98741a_79' 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/homeomorphism'>homeomorphic</a> to the one-point compactification (Def. <a class='maruku-ref' href='#OnePointExtension'>1</a>) of its <a class='existingWikiWord' href='/nlab/show/diff/complement'>complement</a> <a class='existingWikiWord' href='/nlab/show/diff/subspace'>subspace</a> <math class='maruku-mathml' display='inline' id='mathml_36d2b9ac71eb02eda287e1afa8662e532c98741a_80' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi><mo>∖</mo><mo stretchy='false'>{</mo><msub><mi>x</mi> <mn>0</mn></msub><mo stretchy='false'>}</mo><mo>⊂</mo><mi>X</mi></mrow><annotation encoding='application/x-tex'>X \setminus \{x_0\} \subset X</annotation></semantics></math>:</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_36d2b9ac71eb02eda287e1afa8662e532c98741a_81' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi><mo>≃</mo><mo stretchy='false'>(</mo><mi>X</mi><mo>∖</mo><mo stretchy='false'>{</mo><msub><mi>x</mi> <mn>0</mn></msub><mo stretchy='false'>}</mo><msup><mo stretchy='false'>)</mo> <mo>*</mo></msup><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> X \simeq (X \setminus \{x_0\})^\ast \,. </annotation></semantics></math></div> <p>Observe also that <math class='maruku-mathml' display='inline' id='mathml_36d2b9ac71eb02eda287e1afa8662e532c98741a_82' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi><mo>∖</mo><mo stretchy='false'>{</mo><msub><mi>x</mi> <mn>0</mn></msub><mo stretchy='false'>}</mo></mrow><annotation encoding='application/x-tex'>X \setminus \{x_0\}</annotation></semantics></math>, being an <a class='existingWikiWord' href='/nlab/show/diff/open+subspace'>open subspace</a> of a <a class='existingWikiWord' href='/nlab/show/diff/compactum'>compact Hausdorff space</a>, is a <a class='existingWikiWord' href='/nlab/show/diff/locally+compact+topological+space'>locally compact topological space</a>, since <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>, and of course it is Hausdorff, since <math class='maruku-mathml' display='inline' id='mathml_36d2b9ac71eb02eda287e1afa8662e532c98741a_83' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> is.</p> </div> <div class='proof'> <h6 id='proof_5'>Proof</h6> <p>Since <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>, the open neighbourhoods of <math class='maruku-mathml' display='inline' id='mathml_36d2b9ac71eb02eda287e1afa8662e532c98741a_84' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>x</mi><mo>∈</mo><mi>X</mi></mrow><annotation encoding='application/x-tex'>x \in X</annotation></semantics></math> are equivalently the complements of closed, and hence compact closed, subsets in <math class='maruku-mathml' display='inline' id='mathml_36d2b9ac71eb02eda287e1afa8662e532c98741a_85' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi><mo>∖</mo><mo stretchy='false'>{</mo><mi>x</mi><mo stretchy='false'>}</mo></mrow><annotation encoding='application/x-tex'>X \setminus \{x\}</annotation></semantics></math>. By def. <a class='maruku-ref' href='#OnePointExtension'>1</a> this means that the function</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_36d2b9ac71eb02eda287e1afa8662e532c98741a_86' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mrow><mtable><mtr><mtd><mi>X</mi></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mo stretchy='false'>(</mo><mi>X</mi><mo>∖</mo><mo stretchy='false'>{</mo><msub><mi>x</mi> <mn>0</mn></msub><mo stretchy='false'>}</mo><msup><mo stretchy='false'>)</mo> <mo>*</mo></msup></mtd></mtr></mtable></mrow></mrow><annotation encoding='application/x-tex'> \array{ X &\longrightarrow& (X \setminus \{x_0\})^\ast } </annotation></semantics></math></div> <p>which is the identity on <math class='maruku-mathml' display='inline' id='mathml_36d2b9ac71eb02eda287e1afa8662e532c98741a_87' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi><mo>∖</mo><mo stretchy='false'>{</mo><msub><mi>x</mi> <mn>0</mn></msub><mo stretchy='false'>}</mo></mrow><annotation encoding='application/x-tex'>X \setminus \{x_0\}</annotation></semantics></math> and sends <math class='maruku-mathml' display='inline' id='mathml_36d2b9ac71eb02eda287e1afa8662e532c98741a_88' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>x</mi> <mn>0</mn></msub><mo>↦</mo><mn>∞</mn></mrow><annotation encoding='application/x-tex'>x_0 \mapsto \infty</annotation></semantics></math> (hence which is just the identity on the underlying sets) is a <a class='existingWikiWord' href='/nlab/show/diff/homeomorphism'>homeomorphism</a>.</p> </div> <h3 id='UniversalProperty'>Universal property</h3> <p>As a <a class='existingWikiWord' href='/nlab/show/diff/pointed+topological+space'>pointed</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 space</a>, the one-point compactification of <math class='maruku-mathml' display='inline' id='mathml_36d2b9ac71eb02eda287e1afa8662e532c98741a_89' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> may be described by a <a class='existingWikiWord' href='/nlab/show/diff/universal+construction'>universal property</a>:</p> <p>For every <a class='existingWikiWord' href='/nlab/show/diff/pointed+topological+space'>pointed</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 space</a> <math class='maruku-mathml' display='inline' id='mathml_36d2b9ac71eb02eda287e1afa8662e532c98741a_90' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mi>Y</mi><mo>,</mo><msub><mi>y</mi> <mn>0</mn></msub><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(Y, y_0)</annotation></semantics></math> and every <a class='existingWikiWord' href='/nlab/show/diff/continuous+map'>continuous map</a> <math class='maruku-mathml' display='inline' id='mathml_36d2b9ac71eb02eda287e1afa8662e532c98741a_91' xmlns='http://www.w3.org/1998/Math/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> such that the <a class='existingWikiWord' href='/nlab/show/diff/preimage'>pre-image</a> <math class='maruku-mathml' display='inline' id='mathml_36d2b9ac71eb02eda287e1afa8662e532c98741a_92' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>f</mi> <mrow><mo lspace='verythinmathspace' rspace='0em'>−</mo><mn>1</mn></mrow></msup><mo stretchy='false'>(</mo><mi>K</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>f^{-1}(K)</annotation></semantics></math> is compact for all closed sets <math class='maruku-mathml' display='inline' id='mathml_36d2b9ac71eb02eda287e1afa8662e532c98741a_93' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>K</mi></mrow><annotation encoding='application/x-tex'>K</annotation></semantics></math> not containing <math class='maruku-mathml' display='inline' id='mathml_36d2b9ac71eb02eda287e1afa8662e532c98741a_94' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>y</mi> <mn>0</mn></msub></mrow><annotation encoding='application/x-tex'>y_0</annotation></semantics></math>, there is a unique basepoint-preserving continuous map <math class='maruku-mathml' display='inline' id='mathml_36d2b9ac71eb02eda287e1afa8662e532c98741a_95' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>X</mi> <mo>*</mo></msup><mo>→</mo><mi>Y</mi></mrow><annotation encoding='application/x-tex'>X^\ast \to Y</annotation></semantics></math> that extends <math class='maruku-mathml' display='inline' id='mathml_36d2b9ac71eb02eda287e1afa8662e532c98741a_96' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>f</mi></mrow><annotation encoding='application/x-tex'>f</annotation></semantics></math>.</p> <p>To see this, note that such a map is necessarily unique. It suffices to show existence. Extend <math class='maruku-mathml' display='inline' id='mathml_36d2b9ac71eb02eda287e1afa8662e532c98741a_97' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>f</mi></mrow><annotation encoding='application/x-tex'>f</annotation></semantics></math> to a map <math class='maruku-mathml' display='inline' id='mathml_36d2b9ac71eb02eda287e1afa8662e532c98741a_98' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>f</mi> <mo>*</mo></msup><mo>:</mo><msup><mi>X</mi> <mo>*</mo></msup><mo>→</mo><mi>Y</mi></mrow><annotation encoding='application/x-tex'>f^\ast: X^\ast \to Y</annotation></semantics></math> such that <math class='maruku-mathml' display='inline' id='mathml_36d2b9ac71eb02eda287e1afa8662e532c98741a_99' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>∞</mn><mo>↦</mo><msub><mi>y</mi> <mn>0</mn></msub></mrow><annotation encoding='application/x-tex'>\infty\mapsto y_0</annotation></semantics></math>. If <math class='maruku-mathml' display='inline' id='mathml_36d2b9ac71eb02eda287e1afa8662e532c98741a_100' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>U</mi><mo>⊂</mo><mi>Y</mi></mrow><annotation encoding='application/x-tex'>U\subset Y</annotation></semantics></math> is open, and <math class='maruku-mathml' display='inline' id='mathml_36d2b9ac71eb02eda287e1afa8662e532c98741a_101' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>y</mi> <mn>0</mn></msub><mo>∉</mo><mi>U</mi></mrow><annotation encoding='application/x-tex'>y_0\notin U</annotation></semantics></math> then <math class='maruku-mathml' display='inline' id='mathml_36d2b9ac71eb02eda287e1afa8662e532c98741a_102' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><msup><mi>f</mi> <mo>*</mo></msup><msup><mo stretchy='false'>)</mo> <mrow><mo lspace='verythinmathspace' rspace='0em'>−</mo><mn>1</mn></mrow></msup><mo stretchy='false'>(</mo><mi>U</mi><mo stretchy='false'>)</mo><mo>=</mo><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^\ast)^{-1}(U) = f^{-1}(U)</annotation></semantics></math> is open. If <math class='maruku-mathml' display='inline' id='mathml_36d2b9ac71eb02eda287e1afa8662e532c98741a_103' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>y</mi> <mn>0</mn></msub><mo>∈</mo><mi>U</mi></mrow><annotation encoding='application/x-tex'>y_0\in U</annotation></semantics></math> then <math class='maruku-mathml' display='inline' id='mathml_36d2b9ac71eb02eda287e1afa8662e532c98741a_104' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Y</mi><mo>∖</mo><mi>U</mi></mrow><annotation encoding='application/x-tex'>Y\setminus U</annotation></semantics></math> is closed and <math class='maruku-mathml' display='inline' id='mathml_36d2b9ac71eb02eda287e1afa8662e532c98741a_105' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>f</mi> <mrow><mo lspace='verythinmathspace' rspace='0em'>−</mo><mn>1</mn></mrow></msup><mo stretchy='false'>(</mo><mi>Y</mi><mo>∖</mo><mi>U</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>f^{-1}(Y\setminus U)</annotation></semantics></math> is compact, by the hypothesis. Put <math class='maruku-mathml' display='inline' id='mathml_36d2b9ac71eb02eda287e1afa8662e532c98741a_106' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>K</mi><mo>=</mo><msup><mi>f</mi> <mrow><mo lspace='verythinmathspace' rspace='0em'>−</mo><mn>1</mn></mrow></msup><mo stretchy='false'>(</mo><mi>Y</mi><mo>∖</mo><mi>U</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>K = f^{-1}(Y\setminus U)</annotation></semantics></math>; then <math class='maruku-mathml' display='inline' id='mathml_36d2b9ac71eb02eda287e1afa8662e532c98741a_107' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>K</mi></mrow><annotation encoding='application/x-tex'>K</annotation></semantics></math> is closed and compact and so <math class='maruku-mathml' display='inline' id='mathml_36d2b9ac71eb02eda287e1afa8662e532c98741a_108' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><msup><mi>f</mi> <mo>*</mo></msup><msup><mo stretchy='false'>)</mo> <mrow><mo lspace='verythinmathspace' rspace='0em'>−</mo><mn>1</mn></mrow></msup><mo stretchy='false'>(</mo><mi>U</mi><mo stretchy='false'>)</mo><mo>=</mo><msup><mi>X</mi> <mo>*</mo></msup><mo>∖</mo><mi>K</mi></mrow><annotation encoding='application/x-tex'>(f^\ast)^{-1}(U) = X^\ast\setminus K</annotation></semantics></math> is open. It follows that <math class='maruku-mathml' display='inline' id='mathml_36d2b9ac71eb02eda287e1afa8662e532c98741a_109' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>f</mi> <mo>*</mo></msup></mrow><annotation encoding='application/x-tex'>f^\ast</annotation></semantics></math> is continuous.</p> <p>This property characterizes <math class='maruku-mathml' display='inline' id='mathml_36d2b9ac71eb02eda287e1afa8662e532c98741a_110' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>X</mi> <mo>*</mo></msup></mrow><annotation encoding='application/x-tex'>X^\ast</annotation></semantics></math> in an <a class='existingWikiWord' href='/nlab/show/diff/essentially+unique'>essentially unique</a> manner.</p> <p><math class='maruku-mathml' display='inline' id='mathml_36d2b9ac71eb02eda287e1afa8662e532c98741a_111' 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/dense+subspace'>dense</a> in <math class='maruku-mathml' display='inline' id='mathml_36d2b9ac71eb02eda287e1afa8662e532c98741a_112' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>X</mi> <mo>*</mo></msup></mrow><annotation encoding='application/x-tex'>X^*</annotation></semantics></math> precisely if <math class='maruku-mathml' display='inline' id='mathml_36d2b9ac71eb02eda287e1afa8662e532c98741a_113' 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 already compact. Note that <math class='maruku-mathml' display='inline' id='mathml_36d2b9ac71eb02eda287e1afa8662e532c98741a_114' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>X</mi> <mo>*</mo></msup></mrow><annotation encoding='application/x-tex'>X^*</annotation></semantics></math> is technically a <a class='existingWikiWord' href='/nlab/show/diff/compactification'>compactification</a> of <math class='maruku-mathml' display='inline' id='mathml_36d2b9ac71eb02eda287e1afa8662e532c98741a_115' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> only in this case.</p> <p><math class='maruku-mathml' display='inline' id='mathml_36d2b9ac71eb02eda287e1afa8662e532c98741a_116' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>X</mi> <mo>*</mo></msup></mrow><annotation encoding='application/x-tex'>X^*</annotation></semantics></math> is <a class='existingWikiWord' href='/nlab/show/diff/Hausdorff+space'>Hausdorff</a> (hence a <a class='existingWikiWord' href='/nlab/show/diff/compactum'>compactum</a>) if and only if <math class='maruku-mathml' display='inline' id='mathml_36d2b9ac71eb02eda287e1afa8662e532c98741a_117' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> is already both Hausdorff and <a class='existingWikiWord' href='/nlab/show/diff/locally+compact+topological+space'>locally compact</a> (see prop. <a class='maruku-ref' href='#HausdorffOnePointCompactification'>2</a>).</p> <h3 id='MonoidalFunctoriality'>Monoidal functoriality</h3> <div class='num_prop' id='OnePointCompactificationFunctor'> <h6 id='proposition_5'>Proposition</h6> <p>The operation of one-point compactification (Def. <a class='maruku-ref' href='#OnePointExtension'>1</a>) does not extend to a <a class='existingWikiWord' href='/nlab/show/diff/functor'>functor</a> on the whole <a class='existingWikiWord' href='/nlab/show/diff/category'>category</a> of <a class='existingWikiWord' href='/nlab/show/diff/topological+space'>topological spaces</a>. But it does extend to a <a class='existingWikiWord' href='/nlab/show/diff/functor'>functor</a> on <a class='existingWikiWord' href='/nlab/show/diff/locally+compact+topological+space'>locally compact Hausdorff spaces</a> with <a class='existingWikiWord' href='/nlab/show/diff/proper+map'>proper maps</a> between them.</p> </div> <p>(e.g. <a href='#Cutler20'>Cutler 20, Prop. 1.6</a>)</p> <div class='num_prop' id='OnePointCompactificationAndSmashProduct'> <h6 id='proposition_6'>Proposition</h6> <p><strong>(<a class='existingWikiWord' href='/nlab/show/diff/one-point+compactification+intertwines+Cartesian+product+with+smash+product'>one-point compactification intertwines Cartesian product with smash product</a>)</strong></p> <p>On the <a class='existingWikiWord' href='/nlab/show/diff/subcategory'>subcategory</a> <math class='maruku-mathml' display='inline' id='mathml_36d2b9ac71eb02eda287e1afa8662e532c98741a_118' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>Top</mi> <mi>LCHaus</mi></msub></mrow><annotation encoding='application/x-tex'>Top_{LCHaus}</annotation></semantics></math> in <a class='existingWikiWord' href='/nlab/show/diff/Top'>Top</a> of <a class='existingWikiWord' href='/nlab/show/diff/locally+compact+topological+space'>locally compact Hausdorff spaces</a> with <a class='existingWikiWord' href='/nlab/show/diff/proper+map'>proper maps</a> between them, the <a class='existingWikiWord' href='/nlab/show/diff/functor'>functor</a> of <a class='existingWikiWord' href='/nlab/show/diff/one-point+compactification'>one-point compactification</a> (Prop. <a class='maruku-ref' href='#OnePointCompactificationFunctor'>5</a>)</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_36d2b9ac71eb02eda287e1afa8662e532c98741a_119' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><msup><mo stretchy='false'>)</mo> <mi>cpt</mi></msup><mspace width='thickmathspace' /><mo lspace='verythinmathspace'>:</mo><mspace width='thickmathspace' /><msub><mi>Top</mi> <mi>LCHaus</mi></msub><mo>⟶</mo><msup><mi>Top</mi> <mrow><mo>*</mo><mo stretchy='false'>/</mo></mrow></msup></mrow><annotation encoding='application/x-tex'> (-)^{cpt} \;\colon\; Top_{LCHaus} \longrightarrow Top^{\ast/} </annotation></semantics></math></div> <ol> <li> <p>sends <a class='existingWikiWord' href='/nlab/show/diff/coproduct'>coproducts</a>, hence <a class='existingWikiWord' href='/nlab/show/diff/disjoint+union+topological+space'>disjoint union topological spaces</a>, to <a class='existingWikiWord' href='/nlab/show/diff/wedge+sum'>wedge sums</a> of <a class='existingWikiWord' href='/nlab/show/diff/pointed+topological+space'>pointed topological spaces</a>;</p> </li> <li> <p>sends <a class='existingWikiWord' href='/nlab/show/diff/cartesian+product'>Cartesian products</a>, hence <a class='existingWikiWord' href='/nlab/show/diff/product+topological+space'>product topological spaces</a>, to <a class='existingWikiWord' href='/nlab/show/diff/smash+product'>smash products</a> of <a class='existingWikiWord' href='/nlab/show/diff/pointed+topological+space'>pointed topological spaces</a>;</p> </li> </ol> <p>hence constitutes a <a class='existingWikiWord' href='/nlab/show/diff/monoidal+functor'>strong monoidal functor</a> for both <a class='existingWikiWord' href='/nlab/show/diff/monoidal+category'>monoidal</a> structures of these <a class='existingWikiWord' href='/nlab/show/diff/distributive+monoidal+category'>distributive monoidal categories</a> in that there are <a class='existingWikiWord' href='/nlab/show/diff/natural+transformation'>natural</a> <a class='existingWikiWord' href='/nlab/show/diff/homeomorphism'>homeomorphisms</a></p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_36d2b9ac71eb02eda287e1afa8662e532c98741a_120' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo maxsize='1.2em' minsize='1.2em'>(</mo><mi>X</mi><mo>⊔</mo><mi>Y</mi><msup><mo maxsize='1.2em' minsize='1.2em'>)</mo> <mi>cpt</mi></msup><mspace width='thickmathspace' /><mo>≃</mo><mspace width='thickmathspace' /><msup><mi>X</mi> <mi>cpt</mi></msup><mo>∨</mo><msup><mi>Y</mi> <mi>cpt</mi></msup><mspace width='thinmathspace' /><mo>,</mo></mrow><annotation encoding='application/x-tex'> \big( X \sqcup Y \big)^{cpt} \;\simeq\; X^{cpt} \vee Y^{cpt} \,, </annotation></semantics></math></div> <p>and</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_36d2b9ac71eb02eda287e1afa8662e532c98741a_121' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo maxsize='1.2em' minsize='1.2em'>(</mo><mi>X</mi><mo>×</mo><mi>Y</mi><msup><mo maxsize='1.2em' minsize='1.2em'>)</mo> <mi>cpt</mi></msup><mspace width='thickmathspace' /><mo>≃</mo><mspace width='thickmathspace' /><msup><mi>X</mi> <mi>cpt</mi></msup><mo>∧</mo><msup><mi>Y</mi> <mi>cpt</mi></msup><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> \big( X \times Y \big)^{cpt} \;\simeq\; X^{cpt} \wedge Y^{cpt} \,. </annotation></semantics></math></div></div> <p>This is briefly mentioned in <a href='#Bredon93'>Bredon 93, p. 199</a>. The argument is spelled out in: <a href='https://math.stackexchange.com/a/1645794/58526'>MO:a/1645794</a>, <a href='#Cutler20'>Cutler 20, Prop. 1.6</a>.</p> <h2 id='examples'>Examples</h2> <h3 id='general'>General</h3> <p>\begin{example} \label{OnePointExtensionOfCompactSpace} If <math class='maruku-mathml' display='inline' id='mathml_36d2b9ac71eb02eda287e1afa8662e532c98741a_122' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> is already itself <a class='existingWikiWord' href='/nlab/show/diff/compact+space'>compact</a>, then its one-point extension in the sense of Def. <a class='maruku-ref' href='#OnePointExtension'>1</a> is the <a class='existingWikiWord' href='/nlab/show/diff/disjoint+union'>disjoint union</a> of <math class='maruku-mathml' display='inline' id='mathml_36d2b9ac71eb02eda287e1afa8662e532c98741a_123' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> with a <a class='existingWikiWord' href='/nlab/show/diff/singleton'>singleton</a> <math class='maruku-mathml' display='inline' id='mathml_36d2b9ac71eb02eda287e1afa8662e532c98741a_124' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>{</mo><mn>∞</mn><mo stretchy='false'>}</mo></mrow><annotation encoding='application/x-tex'>\{ \infty \}</annotation></semantics></math>.</p> <p>Namely, in this case the open neighbourhood <math class='maruku-mathml' display='inline' id='mathml_36d2b9ac71eb02eda287e1afa8662e532c98741a_125' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mi>X</mi><mo>∖</mo><mi>X</mi><mo stretchy='false'>)</mo><mo>∪</mo><mo stretchy='false'>{</mo><mn>∞</mn><mo stretchy='false'>}</mo></mrow><annotation encoding='application/x-tex'>(X \setminus X) \cup \{\infty\}</annotation></semantics></math> of <math class='maruku-mathml' display='inline' id='mathml_36d2b9ac71eb02eda287e1afa8662e532c98741a_126' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>∞</mn></mrow><annotation encoding='application/x-tex'>\infty</annotation></semantics></math> consists of just <math class='maruku-mathml' display='inline' id='mathml_36d2b9ac71eb02eda287e1afa8662e532c98741a_127' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>∞</mn></mrow><annotation encoding='application/x-tex'>\infty</annotation></semantics></math> itself, which is hence an <a class='existingWikiWord' href='/nlab/show/diff/open+point'>open point</a>. But it is also a <a class='existingWikiWord' href='/nlab/show/diff/closed+point'>closed point</a>, being the <a class='existingWikiWord' href='/nlab/show/diff/complement'>complement</a> of <math class='maruku-mathml' display='inline' id='mathml_36d2b9ac71eb02eda287e1afa8662e532c98741a_128' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math>. \end{example}</p> <h3 id='compactification_of_discrete_spaces'>Compactification of discrete spaces</h3> <p>\begin{example}\label{CompactifOfCountableSet} <strong>(one-point compactification of <a class='existingWikiWord' href='/nlab/show/diff/countable+set'>countable</a> <a class='existingWikiWord' href='/nlab/show/diff/discrete+object'>discrete space</a>)</strong> \linebreak Consider the <a class='existingWikiWord' href='/nlab/show/diff/natural+number'>natural numbers</a> <math class='maruku-mathml' display='inline' id='mathml_36d2b9ac71eb02eda287e1afa8662e532c98741a_129' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>ℕ</mi></mrow><annotation encoding='application/x-tex'>\mathbb{N}</annotation></semantics></math> regarded as a <a class='existingWikiWord' href='/nlab/show/diff/discrete+object'>discrete space</a>. This is not <a class='existingWikiWord' href='/nlab/show/diff/compact+space'>compact</a>. Its one-point compactification has as <a class='existingWikiWord' href='/nlab/show/diff/underlying+set'>underlying set</a> the <a class='existingWikiWord' href='/nlab/show/diff/disjoint+union'>disjoint union</a> <math class='maruku-mathml' display='inline' id='mathml_36d2b9ac71eb02eda287e1afa8662e532c98741a_130' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>ℕ</mi><mo>∪</mo><mo stretchy='false'>{</mo><mn>∞</mn><mo stretchy='false'>}</mo></mrow><annotation encoding='application/x-tex'>\mathbb{N}\cup\{\infty\}</annotation></semantics></math> of the natural numbers with an element “at <a class='existingWikiWord' href='/nlab/show/diff/infinity'>infinity</a>”, and its <a class='existingWikiWord' href='/nlab/show/diff/open+subspace'>open subsets</a> are those <a class='existingWikiWord' href='/nlab/show/diff/subset'>subsets</a> that either do not contain <math class='maruku-mathml' display='inline' id='mathml_36d2b9ac71eb02eda287e1afa8662e532c98741a_131' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>∞</mn></mrow><annotation encoding='application/x-tex'>\infty</annotation></semantics></math>, or which contain <math class='maruku-mathml' display='inline' id='mathml_36d2b9ac71eb02eda287e1afa8662e532c98741a_132' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>∞</mn></mrow><annotation encoding='application/x-tex'>\infty</annotation></semantics></math> and are <a class='existingWikiWord' href='/nlab/show/diff/cofinite+subset'>cofinite subsets</a>.</p> <p>This space is actually a <a class='existingWikiWord' href='/nlab/show/diff/Stone+space'>Stone space</a>, and corresponds via <a class='existingWikiWord' href='/nlab/show/diff/Stone+duality'>Stone duality</a> to the <a class='existingWikiWord' href='/nlab/show/diff/Boolean+algebra'>Boolean algebra</a> of the finite and cofinite subsets of <math class='maruku-mathml' display='inline' id='mathml_36d2b9ac71eb02eda287e1afa8662e532c98741a_133' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>ℕ</mi></mrow><annotation encoding='application/x-tex'>\mathbb{N}</annotation></semantics></math>, with the usual <a class='existingWikiWord' href='/nlab/show/diff/Boolean+algebra'>Boolean algebra</a> operations of <a class='existingWikiWord' href='/nlab/show/diff/union'>union</a> and <a class='existingWikiWord' href='/nlab/show/diff/complement'>set-complement</a>.</p> <p>To see this, notice that the <a class='existingWikiWord' href='/nlab/show/diff/clopen+set'>clopen sets</a> in the space are those that are either <a class='existingWikiWord' href='/nlab/show/diff/finite+set'>finite</a> and not containing <math class='maruku-mathml' display='inline' id='mathml_36d2b9ac71eb02eda287e1afa8662e532c98741a_134' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>∞</mn></mrow><annotation encoding='application/x-tex'>\infty</annotation></semantics></math>, or <a class='existingWikiWord' href='/nlab/show/diff/cofinite+subset'>cofinite</a> and containing <math class='maruku-mathml' display='inline' id='mathml_36d2b9ac71eb02eda287e1afa8662e532c98741a_135' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>∞</mn></mrow><annotation encoding='application/x-tex'>\infty</annotation></semantics></math>. So a clopen set is determined by giving either a finite or a cofinite subset of <math class='maruku-mathml' display='inline' id='mathml_36d2b9ac71eb02eda287e1afa8662e532c98741a_136' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>ℕ</mi></mrow><annotation encoding='application/x-tex'>\mathbb{N}</annotation></semantics></math>, and then adding <math class='maruku-mathml' display='inline' id='mathml_36d2b9ac71eb02eda287e1afa8662e532c98741a_137' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>∞</mn></mrow><annotation encoding='application/x-tex'>\infty</annotation></semantics></math> if it is cofinite. Under this correspondence, the Boolean algebra operations on finite and cofinite subsets of <math class='maruku-mathml' display='inline' id='mathml_36d2b9ac71eb02eda287e1afa8662e532c98741a_138' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>ℕ</mi></mrow><annotation encoding='application/x-tex'>\mathbb{N}</annotation></semantics></math> correspond to the Boolean algebra operations on the clopen sets.</p> <p>Another reason that this space is important is because to give a continuous map <math class='maruku-mathml' display='inline' id='mathml_36d2b9ac71eb02eda287e1afa8662e532c98741a_139' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>ℕ</mi><mo>∪</mo><mo stretchy='false'>{</mo><mn>∞</mn><mo stretchy='false'>}</mo><mo>→</mo><mi>X</mi></mrow><annotation encoding='application/x-tex'>\mathbb{N}\cup\{\infty\}\to X</annotation></semantics></math> to a topological space <math class='maruku-mathml' display='inline' id='mathml_36d2b9ac71eb02eda287e1afa8662e532c98741a_140' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> is to give a convergent sequence in <math class='maruku-mathml' display='inline' id='mathml_36d2b9ac71eb02eda287e1afa8662e532c98741a_141' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math>. This can then be used as a foundation: <a class='existingWikiWord' href='/nlab/show/diff/Johnstone%27s+topological+topos'>Johnstone's topological topos</a> is a category of <a class='existingWikiWord' href='/nlab/show/diff/sheaf'>sheaves</a> on the continuous <a class='existingWikiWord' href='/nlab/show/diff/endofunction'>endofunctions</a> of <math class='maruku-mathml' display='inline' id='mathml_36d2b9ac71eb02eda287e1afa8662e532c98741a_142' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mi>ℕ</mi><mo>∪</mo><mo stretchy='false'>{</mo><mn>∞</mn><mo stretchy='false'>}</mo><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(\mathbb{N}\cup\{\infty\})</annotation></semantics></math>, and <a class='existingWikiWord' href='/nlab/show/diff/subsequential+space'>subsequential spaces</a> are a subcategory of <a class='existingWikiWord' href='/nlab/show/diff/concrete+sheaf'>concrete sheaves</a>. \end{example}</p> <h3 id='ExamplesSpheres'><span> Euclidean spaces compactify to<del class='diffmod'> Spheres</del><ins class='diffmod'> spheres</ins></span></h3> <p>We discuss how the <a class='existingWikiWord' href='/nlab/show/diff/one-point+compactification'>one-point compactification</a> of <a class='existingWikiWord' href='/nlab/show/diff/Euclidean+space'>Euclidean space</a> of <a class='existingWikiWord' href='/nlab/show/diff/dimension'>dimension</a> <math class='maruku-mathml' display='inline' id='mathml_36d2b9ac71eb02eda287e1afa8662e532c98741a_143' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>n</mi></mrow><annotation encoding='application/x-tex'>n</annotation></semantics></math> is the <a class='existingWikiWord' href='/nlab/show/diff/sphere'>n-sphere</a>.</p> <div class='num_example' id='nSphereIsOnePointCompactificationOfRn'> <h6 id='example'>Example</h6> <p><strong>(<a class='existingWikiWord' href='/nlab/show/diff/one-point+compactification'>one-point compactification</a> of <a class='existingWikiWord' href='/nlab/show/diff/Euclidean+space'>Euclidean n-space</a> is the <a class='existingWikiWord' href='/nlab/show/diff/sphere'>n-sphere</a>)</strong></p> <p>For <math class='maruku-mathml' display='inline' id='mathml_36d2b9ac71eb02eda287e1afa8662e532c98741a_144' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>n</mi><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding='application/x-tex'>n \in \mathbb{N}</annotation></semantics></math> the <a class='existingWikiWord' href='/nlab/show/diff/sphere'>n-sphere</a> with its standard <a class='existingWikiWord' href='/nlab/show/diff/topological+space'>topology</a> (e.g. as a <a class='existingWikiWord' href='/nlab/show/diff/subspace'>subspace</a> of the <a class='existingWikiWord' href='/nlab/show/diff/Euclidean+space'>Euclidean space</a> <math class='maruku-mathml' display='inline' id='mathml_36d2b9ac71eb02eda287e1afa8662e532c98741a_145' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>ℝ</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup></mrow><annotation encoding='application/x-tex'>\mathbb{R}^{n+1}</annotation></semantics></math> with its <a class='existingWikiWord' href='/nlab/show/diff/metric+topology'>metric topology</a>) is <a class='existingWikiWord' href='/nlab/show/diff/homeomorphism'>homeomorphic</a> to the one-point compactification (def. <a class='maruku-ref' href='#OnePointExtension'>1</a>) of the <a class='existingWikiWord' href='/nlab/show/diff/Euclidean+space'>Euclidean space</a> <math class='maruku-mathml' display='inline' id='mathml_36d2b9ac71eb02eda287e1afa8662e532c98741a_146' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>ℝ</mi> <mi>n</mi></msup></mrow><annotation encoding='application/x-tex'>\mathbb{R}^n</annotation></semantics></math></p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_36d2b9ac71eb02eda287e1afa8662e532c98741a_147' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>S</mi> <mi>n</mi></msup><mo>≃</mo><mo stretchy='false'>(</mo><msup><mi>ℝ</mi> <mi>n</mi></msup><msup><mo stretchy='false'>)</mo> <mo>*</mo></msup><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> S^n \simeq (\mathbb{R}^n)^\ast \,. </annotation></semantics></math></div></div> <div class='proof'> <h6 id='proof_6'>Proof</h6> <p>Pick a point <math class='maruku-mathml' display='inline' id='mathml_36d2b9ac71eb02eda287e1afa8662e532c98741a_148' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>∞</mn><mo>∈</mo><msup><mi>S</mi> <mi>n</mi></msup></mrow><annotation encoding='application/x-tex'>\infty \in S^n</annotation></semantics></math>. By <a class='existingWikiWord' href='/nlab/show/diff/stereographic+projection'>stereographic projection</a> we have a <a class='existingWikiWord' href='/nlab/show/diff/homeomorphism'>homeomorphism</a></p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_36d2b9ac71eb02eda287e1afa8662e532c98741a_149' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>S</mi> <mi>n</mi></msup><mo>∖</mo><mo stretchy='false'>{</mo><mn>∞</mn><mo stretchy='false'>}</mo><mo>≃</mo><msup><mi>ℝ</mi> <mi>n</mi></msup><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> S^n \setminus \{\infty\} \simeq \mathbb{R}^n \,. </annotation></semantics></math></div> <p>With this it only remains to see that for <math class='maruku-mathml' display='inline' id='mathml_36d2b9ac71eb02eda287e1afa8662e532c98741a_150' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>U</mi> <mn>∞</mn></msub><mo>⊃</mo><mo stretchy='false'>{</mo><mn>∞</mn><mo stretchy='false'>}</mo></mrow><annotation encoding='application/x-tex'>U_\infty \supset \{\infty\}</annotation></semantics></math> an open neighbourhood of <math class='maruku-mathml' display='inline' id='mathml_36d2b9ac71eb02eda287e1afa8662e532c98741a_151' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>∞</mn></mrow><annotation encoding='application/x-tex'>\infty</annotation></semantics></math> in <math class='maruku-mathml' display='inline' id='mathml_36d2b9ac71eb02eda287e1afa8662e532c98741a_152' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>S</mi> <mi>n</mi></msup></mrow><annotation encoding='application/x-tex'>S^n</annotation></semantics></math> then the complement <math class='maruku-mathml' display='inline' id='mathml_36d2b9ac71eb02eda287e1afa8662e532c98741a_153' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>S</mi> <mi>n</mi></msup><mo>∖</mo><msub><mi>U</mi> <mn>∞</mn></msub></mrow><annotation encoding='application/x-tex'>S^n \setminus U_\infty</annotation></semantics></math> is compact closed, and conversely that the complement of every compact closed subset of <math class='maruku-mathml' display='inline' id='mathml_36d2b9ac71eb02eda287e1afa8662e532c98741a_154' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>S</mi> <mi>n</mi></msup><mo>∖</mo><mo stretchy='false'>{</mo><mn>∞</mn><mo stretchy='false'>}</mo></mrow><annotation encoding='application/x-tex'>S^n \setminus \{\infty\}</annotation></semantics></math> is an open neighbourhood of <math class='maruku-mathml' display='inline' id='mathml_36d2b9ac71eb02eda287e1afa8662e532c98741a_155' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>{</mo><mn>∞</mn><mo stretchy='false'>}</mo></mrow><annotation encoding='application/x-tex'>\{\infty\}</annotation></semantics></math>.</p> <p>Observe that under <a class='existingWikiWord' href='/nlab/show/diff/stereographic+projection'>stereographic projection</a> the open subspaces <math class='maruku-mathml' display='inline' id='mathml_36d2b9ac71eb02eda287e1afa8662e532c98741a_156' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>U</mi> <mn>∞</mn></msub><mo>∖</mo><mo stretchy='false'>{</mo><mn>∞</mn><mo stretchy='false'>}</mo><mo>⊂</mo><msup><mi>S</mi> <mi>n</mi></msup><mo>∖</mo><mo stretchy='false'>{</mo><mn>∞</mn><mo stretchy='false'>}</mo></mrow><annotation encoding='application/x-tex'>U_\infty \setminus \{\infty\} \subset S^n \setminus \{\infty\}</annotation></semantics></math> are identified precisely with the <a class='existingWikiWord' href='/nlab/show/diff/closed+subspace'>closed</a> and <a class='existingWikiWord' href='/nlab/show/diff/bounded+set'>bounded subsets</a> of <math class='maruku-mathml' display='inline' id='mathml_36d2b9ac71eb02eda287e1afa8662e532c98741a_157' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>ℝ</mi> <mi>n</mi></msup></mrow><annotation encoding='application/x-tex'>\mathbb{R}^n</annotation></semantics></math>. (Closure is immediate, boundedness follows because an open neighbourhood of <math class='maruku-mathml' display='inline' id='mathml_36d2b9ac71eb02eda287e1afa8662e532c98741a_158' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>{</mo><mn>∞</mn><mo stretchy='false'>}</mo><mo>∈</mo><msup><mi>S</mi> <mi>n</mi></msup></mrow><annotation encoding='application/x-tex'>\{\infty\} \in S^n</annotation></semantics></math> needs to contain an open ball around <math class='maruku-mathml' display='inline' id='mathml_36d2b9ac71eb02eda287e1afa8662e532c98741a_159' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>0</mn><mo>∈</mo><msup><mi>ℝ</mi> <mi>n</mi></msup><mo>≃</mo><msup><mi>S</mi> <mi>n</mi></msup><mo>∖</mo><mo stretchy='false'>{</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mn>∞</mn><mo stretchy='false'>}</mo></mrow><annotation encoding='application/x-tex'>0 \in \mathbb{R}^n \simeq S^n \setminus \{-\infty\}</annotation></semantics></math> in the <em>other</em> stereographic projection, which under change of chart gives a bounded subset. )</p> <p>By the <a class='existingWikiWord' href='/nlab/show/diff/Heine-Borel+theorem'>Heine-Borel theorem</a> the closed and bounded subsets of <math class='maruku-mathml' display='inline' id='mathml_36d2b9ac71eb02eda287e1afa8662e532c98741a_160' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>ℝ</mi> <mi>n</mi></msup></mrow><annotation encoding='application/x-tex'>\mathbb{R}^n</annotation></semantics></math> are precisely the compact, and hence the compact closed, subsets of <math class='maruku-mathml' display='inline' id='mathml_36d2b9ac71eb02eda287e1afa8662e532c98741a_161' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>ℝ</mi> <mi>n</mi></msup><mo>≃</mo><msup><mi>S</mi> <mi>n</mi></msup><mo>∖</mo><mo stretchy='false'>{</mo><mn>∞</mn><mo stretchy='false'>}</mo></mrow><annotation encoding='application/x-tex'>\mathbb{R}^n \simeq S^n \setminus \{\infty\}</annotation></semantics></math>.</p> </div> <div class='num_remark' id='RelevanceForMonopolesAndInstantons'> <h6 id='remark_2'>Remark</h6> <p><strong>(relevance for <a class='existingWikiWord' href='/nlab/show/diff/monopole'>monopoles</a> and <a class='existingWikiWord' href='/nlab/show/diff/instanton'>instantons</a> in <a class='existingWikiWord' href='/nlab/show/diff/gauge+theory'>gauge theory</a>)</strong></p> <p>In <a class='existingWikiWord' href='/nlab/show/diff/physics'>physics</a>, Example <a class='maruku-ref' href='#nSphereIsOnePointCompactificationOfRn'>1</a> governs the phenomenon of <a class='existingWikiWord' href='/nlab/show/diff/monopole'>monopoles</a> and <a class='existingWikiWord' href='/nlab/show/diff/instanton'>instantons</a> for <a class='existingWikiWord' href='/nlab/show/diff/gauge+theory'>gauge theory</a> on <a class='existingWikiWord' href='/nlab/show/diff/Minkowski+space'>Minkowski spacetime</a> or <a class='existingWikiWord' href='/nlab/show/diff/Euclidean+space'>Euclidean space</a>: While such spaces themselves are not <a class='existingWikiWord' href='/nlab/show/diff/compact+space'>compact</a>, the consistency condition that any <a class='existingWikiWord' href='/nlab/show/diff/field+history'>field configuration</a> carries a <a class='existingWikiWord' href='/nlab/show/diff/finite+number'>finite</a> <a class='existingWikiWord' href='/nlab/show/diff/energy'>energy</a> requires that <a class='existingWikiWord' href='/nlab/show/diff/gauge+field'>gauge fields</a> <em><a class='existingWikiWord' href='/nlab/show/diff/vanishing+at+infinity'>vanish at infinity</a></em>.</p> <p>This means that if <math class='maruku-mathml' display='inline' id='mathml_36d2b9ac71eb02eda287e1afa8662e532c98741a_162' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>A</mi></mrow><annotation encoding='application/x-tex'>A</annotation></semantics></math> is the <a class='existingWikiWord' href='/nlab/show/diff/classifying+space'>classifying space</a> for the corresponding gauge field – e.g. <math class='maruku-mathml' display='inline' id='mathml_36d2b9ac71eb02eda287e1afa8662e532c98741a_163' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>A</mi><mo>=</mo><mi>B</mi><mi>G</mi></mrow><annotation encoding='application/x-tex'>A = B G</annotation></semantics></math> for <a class='existingWikiWord' href='/nlab/show/diff/Yang-Mills+theory'>Yang-Mills theory</a> with <a class='existingWikiWord' href='/nlab/show/diff/gauge+group'>gauge group</a> <math class='maruku-mathml' display='inline' id='mathml_36d2b9ac71eb02eda287e1afa8662e532c98741a_164' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>G</mi></mrow><annotation encoding='application/x-tex'>G</annotation></semantics></math> – and if <math class='maruku-mathml' display='inline' id='mathml_36d2b9ac71eb02eda287e1afa8662e532c98741a_165' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>a</mi> <mn>∞</mn></msub><mo>∈</mo><mi>A</mi></mrow><annotation encoding='application/x-tex'>a_\infty \in A</annotation></semantics></math> denotes the <a class='existingWikiWord' href='/nlab/show/diff/pointed+topological+space'>base point</a> witnessing vanishing fields, then a <a class='existingWikiWord' href='/nlab/show/diff/field+history'>field configuration</a>/<a class='existingWikiWord' href='/nlab/show/diff/cocycle'>cocycle</a></p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_36d2b9ac71eb02eda287e1afa8662e532c98741a_166' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>ℝ</mi> <mi>n</mi></msup><mover><mo>⟶</mo><mi>c</mi></mover><mi>A</mi></mrow><annotation encoding='application/x-tex'> \mathbb{R}^n \overset{c}{\longrightarrow} A </annotation></semantics></math></div> <p>on <math class='maruku-mathml' display='inline' id='mathml_36d2b9ac71eb02eda287e1afa8662e532c98741a_167' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>ℝ</mi> <mi>n</mi></msup></mrow><annotation encoding='application/x-tex'>\mathbb{R}^n</annotation></semantics></math> in <math class='maruku-mathml' display='inline' id='mathml_36d2b9ac71eb02eda287e1afa8662e532c98741a_168' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>A</mi></mrow><annotation encoding='application/x-tex'>A</annotation></semantics></math>-<a class='existingWikiWord' href='/nlab/show/diff/generalized+cohomology'>cohomology</a> <em><a class='existingWikiWord' href='/nlab/show/diff/vanishing+at+infinity'>vanishes at infinity</a></em> if outside any <a class='existingWikiWord' href='/nlab/show/diff/compact+space'>compact</a> <a class='existingWikiWord' href='/nlab/show/diff/subset'>subset</a> its value is the vanishing field configuration <math class='maruku-mathml' display='inline' id='mathml_36d2b9ac71eb02eda287e1afa8662e532c98741a_169' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>a</mi> <mn>∞</mn></msub></mrow><annotation encoding='application/x-tex'>a_\infty</annotation></semantics></math>.</p> <p>But by Def. <a class='maruku-ref' href='#OnePointExtension'>1</a> this is equivalent to the cocycle <a class='existingWikiWord' href='/nlab/show/diff/extension'>extends</a> to the <a class='existingWikiWord' href='/nlab/show/diff/one-point+compactification'>one-point compactification</a> as a <a class='existingWikiWord' href='/nlab/show/diff/morphism'>morphism</a> of <a class='existingWikiWord' href='/nlab/show/diff/pointed+topological+space'>pointed topological space</a>:</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_36d2b9ac71eb02eda287e1afa8662e532c98741a_170' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mrow><mtable><mtr><mtd><mo maxsize='1.2em' minsize='1.2em'>(</mo><msup><mi>ℝ</mi> <mi>n</mi></msup><mo maxsize='1.2em' minsize='1.2em'>)</mo><mo>≃</mo><msup><mi>S</mi> <mi>n</mi></msup></mtd> <mtd><mover><mo>⟶</mo><mi>c</mi></mover></mtd> <mtd><mi>A</mi></mtd></mtr> <mtr><mtd><mn>∞</mn></mtd> <mtd><mo>↦</mo></mtd> <mtd><msub><mi>a</mi> <mn>∞</mn></msub></mtd></mtr></mtable></mrow></mrow><annotation encoding='application/x-tex'> \array{ \big( \mathbb{R}^n \big) \simeq S^n & \overset{c}{\longrightarrow} & A \\ \infty &\mapsto& a_\infty } </annotation></semantics></math></div> <p>The following graphics illustrates this for <math class='maruku-mathml' display='inline' id='mathml_36d2b9ac71eb02eda287e1afa8662e532c98741a_171' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>A</mi><mo>=</mo><msup><mi>S</mi> <mi>n</mi></msup></mrow><annotation encoding='application/x-tex'>A = S^n</annotation></semantics></math> an <a class='existingWikiWord' href='/nlab/show/diff/sphere'>n-sphere</a> itself, hence for charges in <a class='existingWikiWord' href='/nlab/show/diff/cohomotopy'>Cohomotopy cohomology theory</a>:</p> <center> <a href='https://arxiv.org/pdf/1909.12277.pdf#page=9'> <img src='https://ncatlab.org/schreiber/files/CohomotopyOfEuclideanNSpaceVanishingAtInfinity.jpg' width='640' /> </a> </center> <blockquote> <p>graphics grabbed from <a href='cohomotopy+charge#SatiSchreiber19'>SS 19</a></p> </blockquote> <p>For more see at <em><a href='Yang-Mills+instanton#FromTheMathsToThePhysicsStory'>Yang-Mills instanton – SU(2)-instantons from the correct maths to the traditional physics story</a></em>.</p> </div> <h3 id='linear_representations_compactify_to_representation_spheres'>Linear representations compactify to representation spheres</h3> <p>Via the presentation of example <a class='maruku-ref' href='#nSphereIsOnePointCompactificationOfRn'>1</a>, the canonical <a class='existingWikiWord' href='/nlab/show/diff/action'>action</a> of the <a class='existingWikiWord' href='/nlab/show/diff/orthogonal+group'>orthogonal group</a> <math class='maruku-mathml' display='inline' id='mathml_36d2b9ac71eb02eda287e1afa8662e532c98741a_172' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>O</mi><mo stretchy='false'>(</mo><mi>N</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>O(N)</annotation></semantics></math> on <math class='maruku-mathml' display='inline' id='mathml_36d2b9ac71eb02eda287e1afa8662e532c98741a_173' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>ℝ</mi> <mi>n</mi></msup></mrow><annotation encoding='application/x-tex'>\mathbb{R}^n</annotation></semantics></math> induces an action of <math class='maruku-mathml' display='inline' id='mathml_36d2b9ac71eb02eda287e1afa8662e532c98741a_174' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>O</mi><mo stretchy='false'>(</mo><mi>n</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>O(n)</annotation></semantics></math> on <math class='maruku-mathml' display='inline' id='mathml_36d2b9ac71eb02eda287e1afa8662e532c98741a_175' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>S</mi> <mi>n</mi></msup></mrow><annotation encoding='application/x-tex'>S^n</annotation></semantics></math>, which preserves the basepoint <math class='maruku-mathml' display='inline' id='mathml_36d2b9ac71eb02eda287e1afa8662e532c98741a_176' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>∞</mn></mrow><annotation encoding='application/x-tex'>\infty</annotation></semantics></math> (the “point at infinity”).</p> <p>This construction presents the <a class='existingWikiWord' href='/nlab/show/diff/J-homomorphism'>J-homomorphism</a> in <a class='existingWikiWord' href='/nlab/show/diff/stable+homotopy+theory'>stable homotopy theory</a> and is encoded for instance in the definition of <a class='existingWikiWord' href='/nlab/show/diff/orthogonal+spectrum'>orthogonal spectra</a>.</p> <p>Slightly more generally, for <math class='maruku-mathml' display='inline' id='mathml_36d2b9ac71eb02eda287e1afa8662e532c98741a_177' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>V</mi></mrow><annotation encoding='application/x-tex'>V</annotation></semantics></math> any real <a class='existingWikiWord' href='/nlab/show/diff/vector+space'>vector space</a> of <a class='existingWikiWord' href='/nlab/show/diff/dimension'>dimension</a> <math class='maruku-mathml' display='inline' id='mathml_36d2b9ac71eb02eda287e1afa8662e532c98741a_178' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>n</mi></mrow><annotation encoding='application/x-tex'>n</annotation></semantics></math> one has <math class='maruku-mathml' display='inline' id='mathml_36d2b9ac71eb02eda287e1afa8662e532c98741a_179' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>S</mi> <mi>n</mi></msup><mo>≃</mo><mo stretchy='false'>(</mo><mi>V</mi><msup><mo stretchy='false'>)</mo> <mo>*</mo></msup></mrow><annotation encoding='application/x-tex'>S^n \simeq (V)^\ast</annotation></semantics></math>. In this context and in view of the previous case, one usually writes</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_36d2b9ac71eb02eda287e1afa8662e532c98741a_180' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>S</mi> <mi>V</mi></msup><mo>≔</mo><mo stretchy='false'>(</mo><mi>V</mi><msup><mo stretchy='false'>)</mo> <mo>*</mo></msup></mrow><annotation encoding='application/x-tex'> S^V \coloneqq (V)^\ast </annotation></semantics></math></div> <p>for the <math class='maruku-mathml' display='inline' id='mathml_36d2b9ac71eb02eda287e1afa8662e532c98741a_181' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>n</mi></mrow><annotation encoding='application/x-tex'>n</annotation></semantics></math>-<a class='existingWikiWord' href='/nlab/show/diff/sphere'>sphere</a> obtained as the one-point compactification of the vector space <math class='maruku-mathml' display='inline' id='mathml_36d2b9ac71eb02eda287e1afa8662e532c98741a_182' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>V</mi></mrow><annotation encoding='application/x-tex'>V</annotation></semantics></math>.</p> <p>As a special case of Prop. <a class='maruku-ref' href='#OnePointCompactificationAndSmashProduct'>6</a> we have:</p> <div class='num_prop'> <h6 id='proposition_7'>Proposition</h6> <p>For <math class='maruku-mathml' display='inline' id='mathml_36d2b9ac71eb02eda287e1afa8662e532c98741a_183' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>V</mi><mo>,</mo><mi>W</mi><mo>∈</mo><msub><mi>Vect</mi> <mi>ℝ</mi></msub></mrow><annotation encoding='application/x-tex'>V,W \in Vect_{\mathbb{R}}</annotation></semantics></math> two real <a class='existingWikiWord' href='/nlab/show/diff/vector+space'>vector spaces</a>, there is a <a class='existingWikiWord' href='/nlab/show/diff/natural+transformation'>natural</a> <a class='existingWikiWord' href='/nlab/show/diff/homeomorphism'>homeomorphism</a></p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_36d2b9ac71eb02eda287e1afa8662e532c98741a_184' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>S</mi> <mi>V</mi></msup><mo>∧</mo><msup><mi>S</mi> <mi>W</mi></msup><mo>≃</mo><msup><mi>S</mi> <mrow><mi>V</mi><mo>⊕</mo><mi>W</mi></mrow></msup></mrow><annotation encoding='application/x-tex'> S^V \wedge S^W \simeq S^{V\oplus W} </annotation></semantics></math></div> <p>between the <a class='existingWikiWord' href='/nlab/show/diff/smash+product'>smash product</a> of their one-point compactifications and the one-point compactification of the <a class='existingWikiWord' href='/nlab/show/diff/direct+sum'>direct sum</a>.</p> </div> <div class='num_remark'> <h6 id='remark_3'>Remark</h6> <p>In particular, it follows directly from this that the <a class='existingWikiWord' href='/nlab/show/diff/suspension'>suspension</a> <math class='maruku-mathml' display='inline' id='mathml_36d2b9ac71eb02eda287e1afa8662e532c98741a_185' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Σ</mi><mo stretchy='false'>(</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mo stretchy='false'>)</mo><mo>≃</mo><msup><mi>S</mi> <mn>1</mn></msup><mo>∧</mo><mo stretchy='false'>(</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>\Sigma(-) \simeq S^1 \wedge (-)</annotation></semantics></math> of the <math class='maruku-mathml' display='inline' id='mathml_36d2b9ac71eb02eda287e1afa8662e532c98741a_186' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>n</mi></mrow><annotation encoding='application/x-tex'>n</annotation></semantics></math>-sphere is the <math class='maruku-mathml' display='inline' id='mathml_36d2b9ac71eb02eda287e1afa8662e532c98741a_187' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(n+1)</annotation></semantics></math>-sphere, up to <a class='existingWikiWord' href='/nlab/show/diff/homeomorphism'>homeomorphism</a>:</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_36d2b9ac71eb02eda287e1afa8662e532c98741a_188' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mrow><mtable columnalign='right left right left right left right left right left' columnspacing='0em' displaystyle='true'><mtr><mtd><mi>Σ</mi><msup><mi>S</mi> <mi>n</mi></msup></mtd> <mtd><mo>≃</mo><msup><mi>S</mi> <mrow><msup><mi>ℝ</mi> <mn>1</mn></msup></mrow></msup><mo>∧</mo><msup><mi>S</mi> <mrow><msup><mi>ℝ</mi> <mi>n</mi></msup></mrow></msup></mtd></mtr> <mtr><mtd /> <mtd><mo>≃</mo><msup><mi>S</mi> <mrow><msup><mi>ℝ</mi> <mn>1</mn></msup><mo>⊕</mo><msup><mi>ℝ</mi> <mi>n</mi></msup></mrow></msup></mtd></mtr> <mtr><mtd /> <mtd><mo>≃</mo><msup><mi>S</mi> <mrow><msup><mi>ℝ</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup></mrow></msup></mtd></mtr> <mtr><mtd /> <mtd><mo>≃</mo><msup><mi>S</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup></mtd></mtr></mtable></mrow><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> \begin{aligned} \Sigma S^n & \simeq S^{\mathbb{R}^1} \wedge S^{\mathbb{R}^n} \\ & \simeq S^{\mathbb{R}^1 \oplus \mathbb{R}^n} \\ & \simeq S^{\mathbb{R}^{n+1}} \\ & \simeq S^{n+1} \end{aligned} \,. </annotation></semantics></math></div></div> <p>\linebreak</p> <h3 id='thom_spaces'>Thom spaces</h3> <p>For <math class='maruku-mathml' display='inline' id='mathml_36d2b9ac71eb02eda287e1afa8662e532c98741a_189' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> a <a class='existingWikiWord' href='/nlab/show/diff/compact+space'>compact topological space</a> and <math class='maruku-mathml' display='inline' id='mathml_36d2b9ac71eb02eda287e1afa8662e532c98741a_190' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>V</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding='application/x-tex'>V \to X</annotation></semantics></math> a <a class='existingWikiWord' href='/nlab/show/diff/vector+bundle'>vector bundle</a>, then the (<a class='existingWikiWord' href='/nlab/show/diff/homotopy+type'>homotopy type</a> of the) one-point compactification of the total space <math class='maruku-mathml' display='inline' id='mathml_36d2b9ac71eb02eda287e1afa8662e532c98741a_191' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>V</mi></mrow><annotation encoding='application/x-tex'>V</annotation></semantics></math> is the <a class='existingWikiWord' href='/nlab/show/diff/Thom+space'>Thom space</a> of <math class='maruku-mathml' display='inline' id='mathml_36d2b9ac71eb02eda287e1afa8662e532c98741a_192' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>V</mi></mrow><annotation encoding='application/x-tex'>V</annotation></semantics></math>, equivalent to <math class='maruku-mathml' display='inline' id='mathml_36d2b9ac71eb02eda287e1afa8662e532c98741a_193' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>D</mi><mo stretchy='false'>(</mo><mi>V</mi><mo stretchy='false'>)</mo><mo stretchy='false'>/</mo><mi>S</mi><mo stretchy='false'>(</mo><mi>V</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>D(V)/S(V)</annotation></semantics></math>.</p> <p>For a simple example: the real <a class='existingWikiWord' href='/nlab/show/diff/projective+plane'>projective plane</a> <math class='maruku-mathml' display='inline' id='mathml_36d2b9ac71eb02eda287e1afa8662e532c98741a_194' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>ℝℙ</mi> <mn>2</mn></msup></mrow><annotation encoding='application/x-tex'>\mathbb{RP}^2</annotation></semantics></math> is the one-point compactification of the ‘open’ <a class='existingWikiWord' href='/nlab/show/diff/M%C3%B6bius+strip'>Möbius strip</a>, as line bundle over <math class='maruku-mathml' display='inline' id='mathml_36d2b9ac71eb02eda287e1afa8662e532c98741a_195' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>S</mi> <mn>1</mn></msup></mrow><annotation encoding='application/x-tex'>S^1</annotation></semantics></math>. This is a special case of the more general observation that <math class='maruku-mathml' display='inline' id='mathml_36d2b9ac71eb02eda287e1afa8662e532c98741a_196' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>ℝℙ</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup></mrow><annotation encoding='application/x-tex'>\mathbb{RP}^{n+1}</annotation></semantics></math> is the Thom space of the <a class='existingWikiWord' href='/nlab/show/diff/tautological+line+bundle'>tautological line bundle</a> over <math class='maruku-mathml' display='inline' id='mathml_36d2b9ac71eb02eda287e1afa8662e532c98741a_197' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>ℝℙ</mi> <mi>n</mi></msup></mrow><annotation encoding='application/x-tex'>\mathbb{RP}^n</annotation></semantics></math>.</p> <h3 id='locally_compact_hausdorff_spaces'>Locally compact Hausdorff spaces</h3> <div class='num_example' id='LocallyCompatcHausdorffSpaceIsOpenSubspaceOfCompactHausdorffSpace'> <h6 id='example_2'>Example</h6> <p><strong>(every <a class='existingWikiWord' href='/nlab/show/diff/locally+compact+topological+space'>locally compact</a> <a class='existingWikiWord' href='/nlab/show/diff/Hausdorff+space'>Hausdorff space</a> is an <a class='existingWikiWord' href='/nlab/show/diff/open+subspace'>open</a> <a class='existingWikiWord' href='/nlab/show/diff/subspace'>subspace</a> of a <a class='existingWikiWord' href='/nlab/show/diff/compactum'>compact Hausdorff space</a>)</strong></p> <p>Every <a class='existingWikiWord' href='/nlab/show/diff/locally+compact+topological+space'>locally compact</a> <a class='existingWikiWord' href='/nlab/show/diff/Hausdorff+space'>Hausdorff space</a> is <a class='existingWikiWord' href='/nlab/show/diff/homeomorphism'>homeomorphic</a> to a <a class='existingWikiWord' href='/nlab/show/diff/open+subspace'>open</a> <a class='existingWikiWord' href='/nlab/show/diff/subspace+topology'>topological subspace</a> of a <a class='existingWikiWord' href='/nlab/show/diff/compact+space'>compact topological space</a>.</p> </div> <div class='proof'> <h6 id='proof_7'>Proof</h6> <p>In one direction the statement is that <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> (see there for the proof). What we need to show is that every locally compact Hausdorff spaces arises this way.</p> <p>So let <math class='maruku-mathml' display='inline' id='mathml_36d2b9ac71eb02eda287e1afa8662e532c98741a_198' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> be a locally compact Hausdorff space. By prop. <a class='maruku-ref' href='#OnePointExtensionIsCompact'>1</a> and prop. <a class='maruku-ref' href='#HausdorffOnePointCompactification'>2</a> its one-point extension <math class='maruku-mathml' display='inline' id='mathml_36d2b9ac71eb02eda287e1afa8662e532c98741a_199' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>X</mi> <mo>*</mo></msup></mrow><annotation encoding='application/x-tex'>X^\ast</annotation></semantics></math> (def. <a class='maruku-ref' href='#OnePointExtension'>1</a>) is a <a class='existingWikiWord' href='/nlab/show/diff/compactum'>compact Hausdorff space</a>. By prop. <a class='maruku-ref' href='#InclusionIntoOnePointExtensionIsOpenEmbedding'>3</a> the canonical inclusion <math class='maruku-mathml' display='inline' id='mathml_36d2b9ac71eb02eda287e1afa8662e532c98741a_200' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi><mo>→</mo><msup><mi>X</mi> <mo>*</mo></msup></mrow><annotation encoding='application/x-tex'>X \to X^\ast</annotation></semantics></math> is an <a class='existingWikiWord' href='/nlab/show/diff/open+map'>open</a> <a class='existingWikiWord' href='/nlab/show/diff/embedding+of+topological+spaces'>embedding of topological spaces</a>.</p> </div> <h2 id='related_concepts'>Related concepts</h2> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Thom+space'>Thom space</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/compactification'>compactification</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Stone-%C4%8Cech+compactification'>Stone-Cech compactification</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/end+compactification'>end compactification</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Bohr+compactification'>Bohr compactification</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Kaluza-Klein+mechanism'>Kaluza-Klein compactification</a></p> </li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/configuration+space+of+points'>configuration space (mathematics)</a></p> </li> <li> <p><a href='Yang-Mills+instanton#FromTheMathsToThePhysicsStory'>Yang-Mills instanton – SU(2)-instantons from the correct maths to the traditional physics story</a>_</p> </li> </ul> <h2 id='references'>References</h2> <p>The concept goes back to</p> <ul> <li id='Aleksandrov24'><a class='existingWikiWord' href='/nlab/show/diff/Pavel+Aleksandrov'>Pavel Aleksandrov</a>, <em>Über die Metrisation der im Kleinen kompakten topologischen Räume</em>, Mathematische Annalen (1924) Volume: 92, page 294-301 (<a href='https://eudml.org/doc/159072'>dml:159072</a>)</li> </ul> <p>Textbook accounts:</p> <ul> <li id='Kelley75'> <p><a class='existingWikiWord' href='/nlab/show/diff/John+L.+Kelley'>John Kelley</a>, p. 150 of: <em>General topology</em>, D. van Nostrand, New York (1955), reprinted as: Graduate Texts in Mathematics, Springer (1975) [[ISBN:978-0-387-90125-1](https://www.springer.com/gp/book/9780387901251)]</p> </li> <li id='Bredon93'> <p><a class='existingWikiWord' href='/nlab/show/diff/Glen+Bredon'>Glen Bredon</a>, p. 199 of: <em>Topology and Geometry</em>, Graduate Texts in Mathematics 139, Springer 1993 (<a href='https://link.springer.com/book/10.1007/978-1-4757-6848-0'>doi:10.1007/978-1-4757-6848-0</a>, <a href='http://virtualmath1.stanford.edu/~ralph/math215b/Bredon.pdf'>pdf</a>)</p> </li> </ul> <p>Review:</p> <ul> <li id='Cutler20'><a class='existingWikiWord' href='/nlab/show/diff/Tyrone+Cutler'>Tyrone Cutler</a>, <em>The category of pointed topological spaces</em> (2020) [[pdf](https://www.math.uni-bielefeld.de/~tcutler/pdf/Elementary%20Homotopy%20Theory%20II%20-%20The%20Pointed%20Category.pdf), <a class='existingWikiWord' href='/nlab/files/CutlerPointedTopologicalSpaces.pdf' title='pdf'>pdf</a>]</li> </ul> <p>See also</p> <ul> <li>Wikipedia, <em><a href='https://en.wikipedia.org/wiki/Alexandroff_extension'>Alexandroff extension</a></em></li> </ul> <p> </p> <p> </p> <p> </p> </div> <div class="revisedby"> <p> Last revised on May 11, 2024 at 08:27:02. 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