CINXE.COM

<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.1 plus MathML 2.0 plus SVG 1.1//EN" "http://www.w3.org/2002/04/xhtml-math-svg/xhtml-math-svg-flat.dtd" > <html xmlns="http://www.w3.org/1999/xhtml"> <head> <title> cartesian space (changes) in nLab </title> <meta http-equiv="Content-Type" content="text/html; charset=UTF-8" /> <meta name="robots" content="noindex,nofollow" /> <meta name="viewport" content="width=device-width, initial-scale=1" /> <link href="/stylesheets/instiki.css?1676280126" media="all" rel="stylesheet" type="text/css" /> <link href="/stylesheets/mathematics.css?1660229990" media="all" rel="stylesheet" type="text/css" /> <link href="/stylesheets/syntax.css?1660229990" media="all" rel="stylesheet" type="text/css" /> <link href="/stylesheets/nlab.css?1676280126" media="all" rel="stylesheet" type="text/css" /> <link rel="stylesheet" type="text/css" href="https://cdn.jsdelivr.net/gh/dreampulse/computer-modern-web-font@master/fonts.css"/> <style type="text/css"> h1#pageName, div.info, .newWikiWord a, a.existingWikiWord, .newWikiWord a:hover, [actiontype="toggle"]:hover, #TextileHelp h3 { color: #226622; } a:visited.existingWikiWord { color: #164416; } </style> <style type="text/css"><![CDATA[/*></style> <script src="/javascripts/prototype.js?1660229990" type="text/javascript"></script> <script src="/javascripts/effects.js?1660229990" type="text/javascript"></script> <script src="/javascripts/dragdrop.js?1660229990" type="text/javascript"></script> <script src="/javascripts/controls.js?1660229990" type="text/javascript"></script> <script src="/javascripts/application.js?1660229990" type="text/javascript"></script> <script src="/javascripts/page_helper.js?1660229990" type="text/javascript"></script> <script src="/javascripts/thm_numbering.js?1660229990" type="text/javascript"></script> <script type="text/x-mathjax-config"> <![CDATA[//><!]]> </script> <script type="text/javascript"> <![CDATA[//><!]]> </script> <link href="https://ncatlab.org/nlab/atom_with_headlines" rel="alternate" title="Atom with headlines" type="application/atom+xml" /> <link href="https://ncatlab.org/nlab/atom_with_content" rel="alternate" title="Atom with full content" type="application/atom+xml" /> <script type="text/javascript"> document.observe("dom:loaded", function() { generateThmNumbers(); }); </script> </head> <body> <div id="Container"> <div id="Content"> <h1 id="pageName"> <svg xmlns="http://www.w3.org/2000/svg" width="1.872em" height="1.8em" viewBox="0 0 190 181"> <path fill="#226622" d="M72.8 145c-1.6 17.3-15.7 10-23.6 20.2-5.6 7.3 4.8 15 11.4 15 11.5-.2 19-13.4 26.4-20.3 3.3-3 8.2-4 11.2-7.2a14 14 0 0 0 2.9-11.1c-1.4-9.6-12.4-18.6-16.9-27.2-5-9.6-10.7-27.4-24.1-27.7-17.4-.3-.4 26 4.7 30.7 2.4 2.3 5.4 4.1 7.3 6.9 1.6 2.3 2.1 5.8-1 7.2-5.9 2.6-12.4-6.3-15.5-10-8.8-10.6-15.5-23-26.2-31.8-5.2-4.3-11.8-8-18-3.7-7.3 4.9-4.2 12.9.2 18.5a81 81 0 0 0 30.7 23c3.3 1.5 12.8 5.6 10 10.7-2.5 5.2-11.7 3-15.6 1.1-8.4-3.8-24.3-21.3-34.4-13.7-3.5 2.6-2.3 7.6-1.2 11.1 2.8 9 12.2 17.2 20.9 20.5 17.3 6.7 34.3-8 50.8-12.1z"/> <path fill="#a41e32" d="M145.9 121.3c-.2-7.5 0-19.6-4.5-26-5.4-7.5-12.9-1-14.1 5.8-1.4 7.8 2.7 14.1 4.8 21.3 3.4 12 5.8 29-.8 40.1-3.6-6.7-5.2-13-7-20.4-2.1-8.2-12.8-13.2-15.1-1.9-2 9.7 9 21.2 12 30.1 1.2 4 2 8.8 6.4 10.3 6.9 2.3 13.3-4.7 17.7-8.8 12.2-11.5 36.6-20.7 43.4-36.4 6.7-15.7-13.7-14-21.3-7.2-9.1 8-11.9 20.5-23.6 25.1 7.5-23.7 31.8-37.6 38.4-61.4 2-7.3-.8-29.6-13-19.8-14.5 11.6-6.6 37.6-23.3 49.2z"/> <path fill="#193c78" d="M86.3 47.5c0-13-10.2-27.6-5.8-40.4 2.8-8.4 14.1-10.1 17-1 3.8 11.6-.3 26.3-1.8 38 11.7-.7 10.5-16 14.8-24.3 2.1-4.2 5.7-9.1 11-6.7 6 2.7 7.4 9.2 6.6 15.1-2.2 14-12.2 18.8-22.4 27-3.4 2.7-8 6.6-5.9 11.6 2 4.4 7 4.5 10.7 2.8 7.4-3.3 13.4-16.5 21.7-16 14.6.7 12 21.9.9 26.2-5 1.9-10.2 2.3-15.2 3.9-5.8 1.8-9.4 8.7-15.7 8.9-6.1.1-9-6.9-14.3-9-14.4-6-33.3-2-44.7-14.7-3.7-4.2-9.6-12-4.9-17.4 9.3-10.7 28 7.2 35.7 12 2 1.1 11 6.9 11.4 1.1.4-5.2-10-8.2-13.5-10-11.1-5.2-30-15.3-35-27.3-2.5-6 2.8-13.8 9.4-13.6 6.9.2 13.4 7 17.5 12C70.9 34 75 43.8 86.3 47.4z"/> </svg> nLab cartesian space (changes) </h1> <div class="navigation"> <a href='#navEnd'>Skip the Navigation Links</a> | <a href="/nlab/show/diff/HomePage" accesskey="H" title="Home page">Home Page</a> | <a href="/nlab/all_pages" accesskey="A" title="List of all pages">All Pages</a> | <a href="/nlab/latest_revisions" accesskey="U" title="Latest edits and page creations">Latest Revisions</a> | <a href="https://nforum.ncatlab.org/discussion/14785/#Item_1" title="Discuss this page in its dedicated thread on the nForum" style="color: black">Discuss this page</a> | <form accept-charset="utf-8" action="/nlab/search" id="navigationSearchForm" method="get"> <fieldset class="search"><input type="text" id="searchField" name="query" value="Search" style="display:inline-block; float: left;" onfocus="this.value == 'Search' ? this.value = '' : true" onblur="this.value == '' ? this.value = 'Search' : true" /></fieldset> </form> </div> <div id="revision"> Showing changes from revision #20 to #21: <ins class="diffins">Added</ins> | <del class="diffdel">Removed</del> | <del class="diffmod">Chan</del><ins class="diffmod">ged</ins> <div class='rightHandSide'> <div class='toc clickDown' tabindex='0'> <h3 id='context'>Context</h3> <h4 id='topology'>Topology</h4> <div class='hide'> <a class='existingWikiWord' href='/nlab/show/diff/topology'>topology</a> (<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>) see also <a class='existingWikiWord' href='/nlab/show/diff/differential+topology'>differential topology</a>, <a class='existingWikiWord' href='/nlab/show/diff/algebraic+topology'>algebraic topology</a>, <a class='existingWikiWord' href='/nlab/show/diff/functional+analysis'>functional analysis</a> and <a class='existingWikiWord' href='/nlab/show/diff/topological+homotopy+theory'>topological</a> <a class='existingWikiWord' href='/nlab/show/diff/homotopy+theory'>homotopy theory</a> <a class='existingWikiWord' href='/nlab/show/diff/Introduction+to+Topology'>Introduction</a> Basic concepts <ul> <li> <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> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/topological+space'>topological space</a>, <a class='existingWikiWord' href='/nlab/show/diff/locale'>locale</a> </li> <li> <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> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/finer+topology'>finer/coarser topology</a> </li> <li> <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> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/separation+axioms'>separation</a>, <a class='existingWikiWord' href='/nlab/show/diff/sober+topological+space'>sobriety</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/continuous+map'>continuous function</a>, <a class='existingWikiWord' href='/nlab/show/diff/homeomorphism'>homeomorphism</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/uniformly+continuous+map'>uniformly continuous function</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/embedding+of+topological+spaces'>embedding</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/open+map'>open map</a>, <a class='existingWikiWord' href='/nlab/show/diff/closed+map'>closed map</a> </li> <li> <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> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/convergence'>convergence</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/category'>category</a> <a class='existingWikiWord' href='/nlab/show/diff/Top'>Top</a> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/convenient+category+of+topological+spaces'>convenient category of topological spaces</a></li> </ul> </li> </ul> <a href='Top#UniversalConstructions'>Universal constructions</a> <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/weak+topology'>initial topology</a>, <a class='existingWikiWord' href='/nlab/show/diff/weak+topology'>final topology</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/subspace'>subspace</a>, <a class='existingWikiWord' href='/nlab/show/diff/quotient+space'>quotient space</a>, </li> <li> fiber space, <a class='existingWikiWord' href='/nlab/show/diff/space+attachment'>space attachment</a> </li> <li> <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> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/mapping+cylinder'>mapping cylinder</a>, <a class='existingWikiWord' href='/nlab/show/diff/cocylinder'>mapping cocylinder</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/mapping+cone'>mapping cone</a>, <a class='existingWikiWord' href='/nlab/show/diff/mapping+cocone'>mapping cocone</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/mapping+telescope'>mapping telescope</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/colimits+of+normal+spaces'>colimits of normal spaces</a> </li> </ul> <a class='existingWikiWord' href='/nlab/show/diff/stuff%2C+structure%2C+property'>Extra stuff, structure, properties</a> <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/nice+topological+space'>nice topological space</a> </li> <li> <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> </li> <li> <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> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/sober+topological+space'>sober space</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/compact+space'>compact space</a>, <a class='existingWikiWord' href='/nlab/show/diff/proper+map'>proper map</a> <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> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/compactly+generated+topological+space'>compactly generated space</a> </li> <li> <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> </li> <li> <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> </li> <li> <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> </li> <li> <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> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/cell+complex'>cell complex</a>, <a class='existingWikiWord' href='/nlab/show/diff/CW+complex'>CW-complex</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/pointed+topological+space'>pointed space</a> </li> <li> <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> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/topological+group'>topological group</a> </li> <li> <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> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/topological+manifold'>topological manifold</a> </li> </ul> Examples <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/empty+space'>empty space</a>, <a class='existingWikiWord' href='/nlab/show/diff/point+space'>point space</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/discrete+object'>discrete space</a>, <a class='existingWikiWord' href='/nlab/show/diff/codiscrete+space'>codiscrete space</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/Sierpinski+space'>Sierpinski space</a> </li> <li> <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> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/Euclidean+space'>Euclidean space</a> <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> <a class='existingWikiWord' href='/nlab/show/diff/cylinder+object'>cylinder</a>, <a class='existingWikiWord' href='/nlab/show/diff/cone'>cone</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/sphere'>sphere</a>, <a class='existingWikiWord' href='/nlab/show/diff/ball'>ball</a> </li> <li> <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> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/polytope'>polytope</a>, <a class='existingWikiWord' href='/nlab/show/diff/polyhedron'>polyhedron</a> </li> <li> <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>) </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/classifying+space'>classifying space</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/configuration+space+of+points'>configuration space</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/path'>path</a>, <a class='existingWikiWord' href='/nlab/show/diff/loop'>loop</a> </li> <li> <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> <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> <a class='existingWikiWord' href='/nlab/show/diff/Zariski+topology'>Zariski topology</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/Cantor+space'>Cantor space</a>, <a class='existingWikiWord' href='/nlab/show/diff/Mandelbrot+set'>Mandelbrot space</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/Peano+curve'>Peano curve</a> </li> <li> <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> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/K-topology'>K-topology</a>, <a class='existingWikiWord' href='/nlab/show/diff/Dowker+space'>Dowker space</a> </li> <li> <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> </li> </ul> Basic statements <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/Hausdorff+implies+sober'>Hausdorff spaces are sober</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/schemes+are+sober'>schemes are sober</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/continuous+images+of+compact+spaces+are+compact'>continuous images of compact spaces are compact</a> </li> <li> <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> </li> <li> <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> </li> <li> <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> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/compact+spaces+equivalently+have+converging+subnets'>compact spaces equivalently have converging subnet of every net</a> <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/Lebesgue+number+lemma'>Lebesgue number lemma</a> </li> <li> <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> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/compact+spaces+equivalently+have+converging+subnets'>compact spaces equivalently have converging subnet of every net</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/sequentially+compact+metric+spaces+are+totally+bounded'>sequentially compact metric spaces are totally bounded</a> </li> </ul> </li> <li> <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> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/paracompact+Hausdorff+spaces+are+normal'>paracompact Hausdorff spaces are normal</a> </li> <li> <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> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/closed+injections+are+embeddings'>closed injections are embeddings</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/proper+maps+to+locally+compact+spaces+are+closed'>proper maps to locally compact spaces are closed</a> </li> <li> <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> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/locally+compact+and+sigma-compact+spaces+are+paracompact'>locally compact and sigma-compact spaces are paracompact</a> </li> <li> <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> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/second-countable+regular+spaces+are+paracompact'>second-countable regular spaces are paracompact</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/CW-complexes+are+paracompact+Hausdorff+spaces'>CW-complexes are paracompact Hausdorff spaces</a> </li> </ul> Theorems <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/Urysohn%27s+lemma'>Urysohn's lemma</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/Tietze+extension+theorem'>Tietze extension theorem</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/Tychonoff+theorem'>Tychonoff theorem</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/tube+lemma'>tube lemma</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/Michael%27s+theorem'>Michael's theorem</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/Brouwer%27s+fixed+point+theorem'>Brouwer's fixed point theorem</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/topological+invariance+of+dimension'>topological invariance of dimension</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/Jordan+curve+theorem'>Jordan curve theorem</a> </li> </ul> Analysis Theorems <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/Heine-Borel+theorem'>Heine-Borel theorem</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/intermediate+value+theorem'>intermediate value theorem</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/extreme+value+theorem'>extreme value theorem</a> </li> </ul> <a class='existingWikiWord' href='/nlab/show/diff/topological+homotopy+theory'>topological homotopy theory</a> <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/homotopy'>left homotopy</a>, <a class='existingWikiWord' href='/nlab/show/diff/homotopy'>right homotopy</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/homotopy+equivalence'>homotopy equivalence</a>, <a class='existingWikiWord' href='/nlab/show/diff/deformation+retract'>deformation retract</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/fundamental+group'>fundamental group</a>, <a class='existingWikiWord' href='/nlab/show/diff/covering+space'>covering space</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/fundamental+theorem+of+covering+spaces'>fundamental theorem of covering spaces</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/homotopy+group'>homotopy group</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/weak+homotopy+equivalence'>weak homotopy equivalence</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/Whitehead+theorem'>Whitehead's theorem</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/Freudenthal+suspension+theorem'>Freudenthal suspension theorem</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/nerve+theorem'>nerve theorem</a> </li> <li> <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> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/topological+cofiber+sequence'>cofiber sequence</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/Str%C3%B8m+model+structure'>Strøm model category</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/classical+model+structure+on+topological+spaces'>classical model structure on topological spaces</a> </li> </ul> </div> <h4 id='differential_geometry'>Differential geometry</h4> <div class='hide'> <a class='existingWikiWord' href='/nlab/show/diff/synthetic+differential+geometry'>synthetic</a> <a class='existingWikiWord' href='/nlab/show/diff/differential+geometry'>differential geometry</a> Introductions <a class='existingWikiWord' href='/nlab/show/diff/Introduction+to+Topology+--+1'>from point-set topology to differentiable manifolds</a> <a class='existingWikiWord' href='/nlab/show/diff/geometry+of+physics'>geometry of physics</a>: <a class='existingWikiWord' href='/nlab/show/diff/geometry+of+physics+--+coordinate+systems'>coordinate systems</a>, <a class='existingWikiWord' href='/nlab/show/diff/geometry+of+physics+--+smooth+sets'>smooth spaces</a>, <a class='existingWikiWord' href='/nlab/show/diff/geometry+of+physics+--+manifolds+and+orbifolds'>manifolds</a>, <a class='existingWikiWord' href='/nlab/show/diff/geometry+of+physics+--+smooth+homotopy+types'>smooth homotopy types</a>, <a class='existingWikiWord' href='/nlab/show/diff/geometry+of+physics+--+supergeometry'>supergeometry</a> Differentials <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/differentiation'>differentiation</a>, <a class='existingWikiWord' href='/nlab/show/diff/chain+rule'>chain rule</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/differentiable+map'>differentiable function</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/infinitesimal+object'>infinitesimal space</a>, <a class='existingWikiWord' href='/nlab/show/diff/infinitesimally+thickened+point'>infinitesimally thickened point</a>, <a class='existingWikiWord' href='/nlab/show/diff/amazing+right+adjoint'>amazing right adjoint</a> </li> </ul> <a class='existingWikiWord' href='/nlab/show/diff/V-manifold'>V-manifolds</a> <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/differentiable+manifold'>differentiable manifold</a>, <a class='existingWikiWord' href='/nlab/show/diff/coordinate+system'>coordinate chart</a>, <a class='existingWikiWord' href='/nlab/show/diff/atlas'>atlas</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/smooth+manifold'>smooth manifold</a>, <a class='existingWikiWord' href='/nlab/show/diff/smooth+structure'>smooth structure</a>, <a class='existingWikiWord' href='/nlab/show/diff/exotic+smooth+structure'>exotic smooth structure</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/analytic+manifold'>analytic manifold</a>, <a class='existingWikiWord' href='/nlab/show/diff/complex+manifold'>complex manifold</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/formal+smooth+manifold'>formal smooth manifold</a>, <a class='existingWikiWord' href='/nlab/show/diff/derived+smooth+manifold'>derived smooth manifold</a> </li> </ul> <a class='existingWikiWord' href='/nlab/show/diff/smooth+set'>smooth space</a> <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/diffeological+space'>diffeological space</a>, <a class='existingWikiWord' href='/nlab/show/diff/Fr%C3%B6licher+space'>Frölicher space</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/manifold+structure+of+mapping+spaces'>manifold structure of mapping spaces</a> </li> </ul> Tangency <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/tangent+bundle'>tangent bundle</a>, <a class='existingWikiWord' href='/nlab/show/diff/frame+bundle'>frame bundle</a> <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/vector+field'>vector field</a>, <a class='existingWikiWord' href='/nlab/show/diff/multivector+field'>multivector field</a>, <a class='existingWikiWord' href='/nlab/show/diff/tangent+Lie+algebroid'>tangent Lie algebroid</a>; </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/differential+forms+in+synthetic+differential+geometry'>differential forms</a>, <a class='existingWikiWord' href='/nlab/show/diff/de+Rham+complex'>de Rham complex</a>, <a class='existingWikiWord' href='/nlab/show/diff/Dolbeault+complex'>Dolbeault complex</a> <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/pullback+of+a+differential+form'>pullback of differential forms</a>, <a class='existingWikiWord' href='/nlab/show/diff/invariant+differential+form'>invariant differential form</a>, <a class='existingWikiWord' href='/nlab/show/diff/Maurer-Cartan+form'>Maurer-Cartan form</a>, <a class='existingWikiWord' href='/nlab/show/diff/horizontal+differential+form'>horizontal differential form</a>, </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/cogerm+differential+form'>cogerm differential form</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/integration+of+differential+forms'>integration of differential forms</a> </li> </ul> </li> </ul> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/local+diffeomorphism'>local diffeomorphism</a>, <a class='existingWikiWord' href='/nlab/show/diff/formally+%C3%A9tale+morphism'>formally étale morphism</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/submersion'>submersion</a>, <a class='existingWikiWord' href='/nlab/show/diff/formally+smooth+morphism'>formally smooth morphism</a>, </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/immersion'>immersion</a>, <a class='existingWikiWord' href='/nlab/show/diff/formally+unramified+morphism'>formally unramified morphism</a>, </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/de+Rham+space'>de Rham space</a>, <a class='existingWikiWord' href='/nlab/show/diff/crystal'>crystal</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/infinitesimal+disk+bundle'>infinitesimal disk bundle</a> </li> </ul> The magic algebraic facts <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/embedding+of+smooth+manifolds+into+formal+duals+of+R-algebras'>embedding of smooth manifolds into formal duals of R-algebras</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/smooth+Serre-Swan+theorem'>smooth Serre-Swan theorem</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/derivations+of+smooth+functions+are+vector+fields'>derivations of smooth functions are vector fields</a> </li> </ul> Theorems <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/Hadamard+lemma'>Hadamard lemma</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/Borel%27s+theorem'>Borel's theorem</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/Boman%27s+theorem'>Boman's theorem</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/Whitney+extension+theorem'>Whitney extension theorem</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/Steenrod+approximation+theorem'>Steenrod-Wockel approximation theorem</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/Whitney+embedding+theorem'>Whitney embedding theorem</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/Poincar%C3%A9+lemma'>Poincare lemma</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/Stokes+theorem'>Stokes theorem</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/de+Rham+theorem'>de Rham theorem</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/Hochschild-Kostant-Rosenberg+theorem'>Hochschild-Kostant-Rosenberg theorem</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/differential+cohomology+diagram'>differential cohomology hexagon</a> </li> </ul> Axiomatics <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/Kock-Lawvere+axiom'>Kock-Lawvere axiom</a> <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/smooth+topos'>smooth topos</a>, <a class='existingWikiWord' href='/nlab/show/diff/super+smooth+topos'>super smooth topos</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/microlinear+space'>microlinear space</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/integration+axiom'>integration axiom</a> </li> </ul> </li> </ul> <a class='existingWikiWord' href='/nlab/show/diff/cohesive'>cohesion</a> <ul> <li> (<a class='existingWikiWord' href='/nlab/show/diff/shape+modality'>shape modality</a> <math class='maruku-mathml' display='inline' id='mathml_5f0572437a3ae214a10493832748988197295afb_1' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo>⊣</mo></mrow><annotation encoding='application/x-tex'>\dashv</annotation></semantics></math> <a class='existingWikiWord' href='/nlab/show/diff/flat+modality'>flat modality</a> <math class='maruku-mathml' display='inline' id='mathml_5f0572437a3ae214a10493832748988197295afb_2' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo>⊣</mo></mrow><annotation encoding='application/x-tex'>\dashv</annotation></semantics></math> <a class='existingWikiWord' href='/nlab/show/diff/sharp+modality'>sharp modality</a>) <math class='maruku-mathml' display='inline' id='mathml_5f0572437a3ae214a10493832748988197295afb_3' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mo lspace='0em' rspace='thinmathspace'>esh</mo><mo>⊣</mo><mo>♭</mo><mo>⊣</mo><mo>♯</mo><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(\esh \dashv \flat \dashv \sharp )</annotation></semantics></math> <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/discrete+object'>discrete object</a>, <a class='existingWikiWord' href='/nlab/show/diff/codiscrete+object'>codiscrete object</a>, <a class='existingWikiWord' href='/nlab/show/diff/concrete+object'>concrete object</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/points-to-pieces+transform'>points-to-pieces transform</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/cohesive+%28infinity%2C1%29-topos+--+structures'>structures in cohesion</a> </li> </ul> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/dR-shape+modality'>dR-shape modality</a> <math class='maruku-mathml' display='inline' id='mathml_5f0572437a3ae214a10493832748988197295afb_4' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo>⊣</mo></mrow><annotation encoding='application/x-tex'>\dashv</annotation></semantics></math> <a class='existingWikiWord' href='/nlab/show/diff/dR-flat+modality'>dR-flat modality</a> <math class='maruku-mathml' display='inline' id='mathml_5f0572437a3ae214a10493832748988197295afb_5' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mo lspace='0em' rspace='thinmathspace'>esh</mo> <mi>dR</mi></msub><mo>⊣</mo><msub><mo>♭</mo> <mi>dR</mi></msub></mrow><annotation encoding='application/x-tex'>\esh_{dR} \dashv \flat_{dR}</annotation></semantics></math> </li> </ul> <a class='existingWikiWord' href='/nlab/show/diff/infinitesimal+cohesive+%28infinity%2C1%29-topos'>infinitesimal cohesion</a> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/classical+modality'>classical modality</a></li> </ul> <a class='existingWikiWord' href='/nlab/show/diff/tangent+cohesive+%28%E2%88%9E%2C1%29-topos'>tangent cohesion</a> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/differential+cohomology+diagram'>differential cohomology diagram</a></li> </ul> <a class='existingWikiWord' href='/nlab/show/diff/differential+cohesive+%28infinity%2C1%29-topos'>differential cohesion</a> <ul> <li> (<a class='existingWikiWord' href='/nlab/show/diff/reduction+modality'>reduction modality</a> <math class='maruku-mathml' display='inline' id='mathml_5f0572437a3ae214a10493832748988197295afb_6' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo>⊣</mo></mrow><annotation encoding='application/x-tex'>\dashv</annotation></semantics></math> <a class='existingWikiWord' href='/nlab/show/diff/infinitesimal+shape+modality'>infinitesimal shape modality</a> <math class='maruku-mathml' display='inline' id='mathml_5f0572437a3ae214a10493832748988197295afb_7' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo>⊣</mo></mrow><annotation encoding='application/x-tex'>\dashv</annotation></semantics></math> <a class='existingWikiWord' href='/nlab/show/diff/infinitesimal+flat+modality'>infinitesimal flat modality</a>) <math class='maruku-mathml' display='inline' id='mathml_5f0572437a3ae214a10493832748988197295afb_8' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mi>ℜ</mi><mo>⊣</mo><mi>ℑ</mi><mo>⊣</mo><mi>&</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(\Re \dashv \Im \dashv \&)</annotation></semantics></math> <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/reduced+object'>reduced object</a>, <a class='existingWikiWord' href='/nlab/show/diff/coreduced+object'>coreduced object</a>, <a class='existingWikiWord' href='/nlab/show/diff/formally+smooth+object'>formally smooth object</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/formally+%C3%A9tale+morphism'>formally étale map</a> </li> <li> <a href='cohesive+%28infinity%2C1%29-topos+--+infinitesimal+cohesion#StructuresInDifferentialCohesion'>structures in differential cohesion</a> </li> </ul> </li> </ul> <a class='existingWikiWord' href='/nlab/show/diff/super+formal+smooth+infinity-groupoid'>graded differential cohesion</a> <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/bosonic+modality'>fermionic modality</a> <math class='maruku-mathml' display='inline' id='mathml_5f0572437a3ae214a10493832748988197295afb_9' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo>⊣</mo></mrow><annotation encoding='application/x-tex'>\dashv</annotation></semantics></math> <a class='existingWikiWord' href='/nlab/show/diff/bosonic+modality'>bosonic modality</a> <math class='maruku-mathml' display='inline' id='mathml_5f0572437a3ae214a10493832748988197295afb_10' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo>⊣</mo></mrow><annotation encoding='application/x-tex'>\dashv</annotation></semantics></math> <a class='existingWikiWord' href='/nlab/show/diff/rheonomy+modality'>rheonomy modality</a> <math class='maruku-mathml' display='inline' id='mathml_5f0572437a3ae214a10493832748988197295afb_11' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mo>⇉</mo><mo>⊣</mo><mo>⇝</mo><mo>⊣</mo><mi>Rh</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(\rightrightarrows \dashv \rightsquigarrow \dashv Rh)</annotation></semantics></math> </li> </ul> <a class='existingWikiWord' href='/nlab/show/diff/orbifold+cohomology'>singular cohesion</a> <div class='maruku-equation' id='Diagram'><math class='maruku-mathml' display='block' id='mathml_5f0572437a3ae214a10493832748988197295afb_12' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mrow><mtable><mtr><mtd /> <mtd /> <mtd><mi>id</mi></mtd> <mtd><mo>⊣</mo></mtd> <mtd><mi>id</mi></mtd></mtr> <mtr><mtd /> <mtd /> <mtd><mo>∨</mo></mtd> <mtd /> <mtd><mo>∨</mo></mtd></mtr> <mtr><mtd /> <mtd><mover><mrow /><mi>fermionic</mi></mover></mtd> <mtd><mo>⇉</mo></mtd> <mtd><mo>⊣</mo></mtd> <mtd><mo>⇝</mo></mtd> <mtd><mover><mrow /><mi>bosonic</mi></mover></mtd></mtr> <mtr><mtd /> <mtd /> <mtd><mo>⊥</mo></mtd> <mtd /> <mtd><mo>⊥</mo></mtd></mtr> <mtr><mtd /> <mtd><mover><mrow /><mi>bosonic</mi></mover></mtd> <mtd><mo>⇝</mo></mtd> <mtd><mo>⊣</mo></mtd> <mtd><mi mathvariant='normal'>R</mi><mspace width='negativethinmathspace' /><mspace width='negativethinmathspace' /><mi mathvariant='normal'>h</mi></mtd> <mtd><mover><mrow /><mi>rheonomic</mi></mover></mtd></mtr> <mtr><mtd /> <mtd /> <mtd><mo>∨</mo></mtd> <mtd /> <mtd><mo>∨</mo></mtd></mtr> <mtr><mtd /> <mtd><mover><mrow /><mi>reduced</mi></mover></mtd> <mtd><mi>ℜ</mi></mtd> <mtd><mo>⊣</mo></mtd> <mtd><mi>ℑ</mi></mtd> <mtd><mover><mrow /><mi>infinitesimal</mi></mover></mtd></mtr> <mtr><mtd /> <mtd /> <mtd><mo>⊥</mo></mtd> <mtd /> <mtd><mo>⊥</mo></mtd></mtr> <mtr><mtd /> <mtd><mover><mrow /><mi>infinitesimal</mi></mover></mtd> <mtd><mi>ℑ</mi></mtd> <mtd><mo>⊣</mo></mtd> <mtd><mi>&</mi></mtd> <mtd><mover><mrow /><mtext>étale</mtext></mover></mtd></mtr> <mtr><mtd /> <mtd /> <mtd><mo>∨</mo></mtd> <mtd /> <mtd><mo>∨</mo></mtd></mtr> <mtr><mtd /> <mtd><mover><mrow /><mi>cohesive</mi></mover></mtd> <mtd><mo lspace='0em' rspace='thinmathspace'>esh</mo></mtd> <mtd><mo>⊣</mo></mtd> <mtd><mo>♭</mo></mtd> <mtd><mover><mrow /><mi>discrete</mi></mover></mtd></mtr> <mtr><mtd /> <mtd /> <mtd><mo>⊥</mo></mtd> <mtd /> <mtd><mo>⊥</mo></mtd></mtr> <mtr><mtd /> <mtd><mover><mrow /><mi>discrete</mi></mover></mtd> <mtd><mo>♭</mo></mtd> <mtd><mo>⊣</mo></mtd> <mtd><mo>♯</mo></mtd> <mtd><mover><mrow /><mi>continuous</mi></mover></mtd></mtr> <mtr><mtd /> <mtd /> <mtd><mo>∨</mo></mtd> <mtd /> <mtd><mo>∨</mo></mtd></mtr> <mtr><mtd /> <mtd /> <mtd><mi>∅</mi></mtd> <mtd><mo>⊣</mo></mtd> <mtd><mo>*</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding='application/x-tex'> \array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& \mathrm{R}\!\!\mathrm{h} & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{&#233;tale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& \esh &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast } </annotation></semantics></math></div> Models <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/Models+for+Smooth+Infinitesimal+Analysis'>Models for Smooth Infinitesimal Analysis</a> <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/C%5E%E2%88%9E-ring'>smooth algebra</a> (<math class='maruku-mathml' display='inline' id='mathml_5f0572437a3ae214a10493832748988197295afb_13' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>C</mi> <mn>∞</mn></msup></mrow><annotation encoding='application/x-tex'>C^\infty</annotation></semantics></math>-ring) </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/smooth+locus'>smooth locus</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/Fermat+theory'>Fermat theory</a> </li> </ul> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/Cahiers+topos'>Cahiers topos</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/smooth+infinity-groupoid'>smooth ∞-groupoid</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/formal+smooth+infinity-groupoid'>formal smooth ∞-groupoid</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/super+formal+smooth+infinity-groupoid'>super formal smooth ∞-groupoid</a> </li> </ul> <a class='existingWikiWord' href='/nlab/show/diff/Lie+theory'>Lie theory</a>, <a class='existingWikiWord' href='/nlab/show/diff/infinity-Lie+theory'>∞-Lie theory</a> <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/Lie+algebra'>Lie algebra</a>, <a class='existingWikiWord' href='/nlab/show/diff/L-infinity-algebra'>Lie n-algebra</a>, <a class='existingWikiWord' href='/nlab/show/diff/L-infinity-algebra'>L-∞ algebra</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/Lie+group'>Lie group</a>, <a class='existingWikiWord' href='/nlab/show/diff/Lie+2-group'>Lie 2-group</a>, <a class='existingWikiWord' href='/nlab/show/diff/smooth+infinity-groupoid'>smooth ∞-group</a> </li> </ul> <a class='existingWikiWord' href='/nlab/show/diff/differential+equation'>differential equations</a>, <a class='existingWikiWord' href='/nlab/show/diff/variational+calculus'>variational calculus</a> <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/D-geometry'>D-geometry</a>, <a class='existingWikiWord' href='/nlab/show/diff/D-module'>D-module</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/jet+bundle'>jet bundle</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/variational+bicomplex'>variational bicomplex</a>, <a class='existingWikiWord' href='/nlab/show/diff/Euler-Lagrange+complex'>Euler-Lagrange complex</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/Euler-Lagrange+equation'>Euler-Lagrange equation</a>, <a class='existingWikiWord' href='/nlab/show/diff/De+Donder-Weyl-Hamilton+equation'>de Donder-Weyl formalism</a>, </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/phase+space'>phase space</a> </li> </ul> <a class='existingWikiWord' href='/nlab/show/diff/Chern-Weil+theory'>Chern-Weil theory</a>, <a class='existingWikiWord' href='/nlab/show/diff/Chern-Weil+theory+in+Smooth%E2%88%9EGrpd'>∞-Chern-Weil theory</a> <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/connection+on+a+bundle'>connection on a bundle</a>, <a class='existingWikiWord' href='/nlab/show/diff/connection+on+a+smooth+principal+infinity-bundle'>connection on an ∞-bundle</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/differential+cohomology'>differential cohomology</a> <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/ordinary+differential+cohomology'>ordinary differential cohomology</a>, <a class='existingWikiWord' href='/nlab/show/diff/Deligne+cohomology'>Deligne complex</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/differential+K-theory'>differential K-theory</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/differential+cobordism+cohomology'>differential cobordism cohomology</a> </li> </ul> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/parallel+transport'>parallel transport</a>, <a class='existingWikiWord' href='/nlab/show/diff/higher+parallel+transport'>higher parallel transport</a>, <a class='existingWikiWord' href='/nlab/show/diff/fiber+integration+in+differential+cohomology'>fiber integration in differential cohomology</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/holonomy'>holonomy</a>, <a class='existingWikiWord' href='/nlab/show/diff/higher+parallel+transport'>higher holonomy</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/gauge+theory'>gauge theory</a>, <a class='existingWikiWord' href='/nlab/show/diff/higher+gauge+field'>higher gauge theory</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/Wilson+loop'>Wilson line</a>, <a class='existingWikiWord' href='/nlab/show/diff/Wilson+surface'>Wilson surface</a> </li> </ul> <a class='existingWikiWord' href='/nlab/show/diff/Cartan+geometry'>Cartan geometry</a> (<a class='existingWikiWord' href='/nlab/show/diff/super-Cartan+geometry'>super</a>, <a class='existingWikiWord' href='/nlab/show/diff/higher+Cartan+geometry'>higher</a>) <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/Klein+geometry'>Klein geometry</a>, (<a class='existingWikiWord' href='/nlab/show/diff/higher+Klein+geometry'>higher</a>) </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/G-structure'>G-structure</a>, <a class='existingWikiWord' href='/nlab/show/diff/torsion+of+a+G-structure'>torsion of a G-structure</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/Euclidean+geometry'>Euclidean geometry</a>, <a class='existingWikiWord' href='/nlab/show/diff/hyperbolic+geometry'>hyperbolic geometry</a>, <a class='existingWikiWord' href='/nlab/show/diff/elliptic+geometry'>elliptic geometry</a> </li> <li> (<a class='existingWikiWord' href='/nlab/show/diff/pseudo-Riemannian+metric'>pseudo</a>-)<a class='existingWikiWord' href='/nlab/show/diff/Riemannian+geometry'>Riemannian geometry</a> <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/orthogonal+structure'>orthogonal structure</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/isometry'>isometry</a>, <a class='existingWikiWord' href='/nlab/show/diff/Killing+vector+field'>Killing vector field</a>, <a class='existingWikiWord' href='/nlab/show/diff/Killing+spinor'>Killing spinor</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/spacetime'>spacetime</a>, <a class='existingWikiWord' href='/nlab/show/diff/super+spacetime'>super-spacetime</a> </li> </ul> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/complex+geometry'>complex geometry</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/symplectic+geometry'>symplectic geometry</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/conformal+geometry'>conformal geometry</a> </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='#properties'>Properties</a><ul><li><a href='#topological_structure'>Topological structure</a></li><li><a href='#smooth_structures'>Smooth structures</a></li></ul></li><li><a href='#the_category_of_cartesian_spaces'>The category of Cartesian spaces</a></li><li><a href='#related_concepts'>Related concepts</a></li><li><a href='#references'>References</a></li></ul></div> <h2 id='Idea'>Idea</h2> A Cartesian space is a <a class='existingWikiWord' href='/nlab/show/diff/finite+product'>finite</a> <a class='existingWikiWord' href='/nlab/show/diff/cartesian+product'>Cartesian product</a> of the <a class='existingWikiWord' href='/nlab/show/diff/real+number'>real line</a> <math class='maruku-mathml' display='inline' id='mathml_5f0572437a3ae214a10493832748988197295afb_14' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>ℝ</mi></mrow><annotation encoding='application/x-tex'>\mathbb{R}</annotation></semantics></math> with itself. Hence, a Cartesian space has the form <math class='maruku-mathml' display='inline' id='mathml_5f0572437a3ae214a10493832748988197295afb_15' 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> where <math class='maruku-mathml' display='inline' id='mathml_5f0572437a3ae214a10493832748988197295afb_16' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>n</mi></mrow><annotation encoding='application/x-tex'>n</annotation></semantics></math> is some <a class='existingWikiWord' href='/nlab/show/diff/natural+number'>natural number</a> (possibly <a class='existingWikiWord' href='/nlab/show/diff/zero'>zero</a>). What exactly this means depends on which <a class='existingWikiWord' href='/nlab/show/diff/category'>category</a> the <a class='existingWikiWord' href='/nlab/show/diff/real+number'>real line</a> <math class='maruku-mathml' display='inline' id='mathml_5f0572437a3ae214a10493832748988197295afb_17' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>ℝ</mi></mrow><annotation encoding='application/x-tex'>\mathbb{R}</annotation></semantics></math> is being considered as an <a class='existingWikiWord' href='/nlab/show/diff/object'>object</a> of. For instance, if <math class='maruku-mathml' display='inline' id='mathml_5f0572437a3ae214a10493832748988197295afb_18' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>ℝ</mi></mrow><annotation encoding='application/x-tex'>\mathbb{R}</annotation></semantics></math> is regarded as a <a class='existingWikiWord' href='/nlab/show/diff/topological+space'>topological space</a> (hence an object in the category <a class='existingWikiWord' href='/nlab/show/diff/Top'>Top</a>), then <math class='maruku-mathml' display='inline' id='mathml_5f0572437a3ae214a10493832748988197295afb_19' 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> is a <a class='existingWikiWord' href='/nlab/show/diff/Euclidean+space'>Euclidean space</a>. Another possibility is to regard <math class='maruku-mathml' display='inline' id='mathml_5f0572437a3ae214a10493832748988197295afb_20' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>ℝ</mi></mrow><annotation encoding='application/x-tex'>\mathbb{R}</annotation></semantics></math> as a <a class='existingWikiWord' href='/nlab/show/diff/topological+manifold'>topological manifold</a>, <a class='existingWikiWord' href='/nlab/show/diff/differentiable+manifold'>differentiable manifold</a> or <a class='existingWikiWord' href='/nlab/show/diff/smooth+manifold'>smooth manifold</a> (hence an object in the categories <a class='existingWikiWord' href='/nlab/show/diff/TopMfd'>TopMfd</a> or <a class='existingWikiWord' href='/nlab/show/diff/Diff'>Diff</a>), in which case <math class='maruku-mathml' display='inline' id='mathml_5f0572437a3ae214a10493832748988197295afb_21' 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> is the archetypical such <a class='existingWikiWord' href='/nlab/show/diff/manifold'>manifold</a> of <a class='existingWikiWord' href='/nlab/show/diff/dimension+of+a+manifold'>dimension</a> <math class='maruku-mathml' display='inline' id='mathml_5f0572437a3ae214a10493832748988197295afb_22' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>n</mi></mrow><annotation encoding='application/x-tex'>n</annotation></semantics></math> (the basic <a class='existingWikiWord' href='/nlab/show/diff/coordinate+system'>coordinate chart</a>). The Cartesian space <math class='maruku-mathml' display='inline' id='mathml_5f0572437a3ae214a10493832748988197295afb_23' 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> with its standard <a class='existingWikiWord' href='/nlab/show/diff/topological+space'>topology</a>/<a class='existingWikiWord' href='/nlab/show/diff/smooth+structure'>smooth structure</a> is also called real <math class='maruku-mathml' display='inline' id='mathml_5f0572437a3ae214a10493832748988197295afb_24' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>n</mi></mrow><annotation encoding='application/x-tex'>n</annotation></semantics></math>-dimensional space. One could also regard <math class='maruku-mathml' display='inline' id='mathml_5f0572437a3ae214a10493832748988197295afb_25' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>ℝ</mi></mrow><annotation encoding='application/x-tex'>\mathbb{R}</annotation></semantics></math> as the <a class='existingWikiWord' href='/nlab/show/diff/real+vector+space'>real vector space</a> of <a class='existingWikiWord' href='/nlab/show/diff/dimension'>dimension</a> 1, in which case <math class='maruku-mathml' display='inline' id='mathml_5f0572437a3ae214a10493832748988197295afb_26' 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> would be the real vector space of dimension <math class='maruku-mathml' display='inline' id='mathml_5f0572437a3ae214a10493832748988197295afb_27' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>n</mi></mrow><annotation encoding='application/x-tex'>n</annotation></semantics></math> (maybe understood as equipped with the <a class='existingWikiWord' href='/nlab/show/diff/basis+of+a+vector+space'>linear basis</a> canonically induced by this presentation), hence <a class='existingWikiWord' href='/nlab/show/diff/generalized+the'>the</a> “real <math class='maruku-mathml' display='inline' id='mathml_5f0572437a3ae214a10493832748988197295afb_28' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>n</mi></mrow><annotation encoding='application/x-tex'>n</annotation></semantics></math>-dimensional vector space”. However, a vector space would typically not be referred to as a “Cartesian space”. But in between the last two examples, regarding <math class='maruku-mathml' display='inline' id='mathml_5f0572437a3ae214a10493832748988197295afb_29' 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> as an <a class='existingWikiWord' href='/nlab/show/diff/affine+space'>affine space</a> makes it the<del class='diffdel'> the</del> basis of “<a href='https://en.m.wikipedia.org/wiki/Analytic_geometry'>analytic geometry</a>” in the sense originally due to <a class='existingWikiWord' href='/nlab/show/diff/Ren%C3%A9+Descartes'>René Descartes</a> (Cartesian geometry<a href='/nlab/new/Cartesian+geometry'>?</a>), and this is where the term “Cartesian space” originates from. One might also speak of the <a class='existingWikiWord' href='/nlab/show/diff/complexification'>complexified</a> cartesian space <math class='maruku-mathml' display='inline' id='mathml_5f0572437a3ae214a10493832748988197295afb_30' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>ℂ</mi> <mi>n</mi></msup></mrow><annotation encoding='application/x-tex'>\mathbb{C}^n</annotation></semantics></math> as a <a class='existingWikiWord' href='/nlab/show/diff/complex+manifold'>complex manifold</a>, or indeed of the cartesian space <math class='maruku-mathml' display='inline' id='mathml_5f0572437a3ae214a10493832748988197295afb_31' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>K</mi> <mi>n</mi></msup></mrow><annotation encoding='application/x-tex'>K^n</annotation></semantics></math> for any <a class='existingWikiWord' href='/nlab/show/diff/field'>field</a> <math class='maruku-mathml' display='inline' id='mathml_5f0572437a3ae214a10493832748988197295afb_32' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>K</mi></mrow><annotation encoding='application/x-tex'>K</annotation></semantics></math>, maybe regarded as an <a class='existingWikiWord' href='/nlab/show/diff/analytic+space'>analytic space</a>, though this is not common terminology. <div class='un_example'> <h6 id='example'>Example</h6> In particular: <ul> <li><math class='maruku-mathml' display='inline' id='mathml_5f0572437a3ae214a10493832748988197295afb_33' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>ℝ</mi> <mn>0</mn></msup></mrow><annotation encoding='application/x-tex'>\mathbb{R}^0</annotation></semantics></math> is the <a class='existingWikiWord' href='/nlab/show/diff/point'>point</a>,</li> <li><math class='maruku-mathml' display='inline' id='mathml_5f0572437a3ae214a10493832748988197295afb_34' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>ℝ</mi> <mn>1</mn></msup></mrow><annotation encoding='application/x-tex'>\mathbb{R}^1</annotation></semantics></math> is the <a class='existingWikiWord' href='/nlab/show/diff/real+number'>real line</a>,</li> <li><math class='maruku-mathml' display='inline' id='mathml_5f0572437a3ae214a10493832748988197295afb_35' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>ℝ</mi> <mn>2</mn></msup></mrow><annotation encoding='application/x-tex'>\mathbb{R}^2</annotation></semantics></math> is the real <a class='existingWikiWord' href='/nlab/show/diff/plane'>plane</a>, which may be identified (in two canonical ways) with the <a class='existingWikiWord' href='/nlab/show/diff/complex+number'>complex plane</a> <math class='maruku-mathml' display='inline' id='mathml_5f0572437a3ae214a10493832748988197295afb_36' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>ℂ</mi></mrow><annotation encoding='application/x-tex'>\mathbb{C}</annotation></semantics></math>.</li> </ul> </div> <div class='un_remark'> <h6 id='remark'>Remark</h6> Cartesian spaces carry plenty of further canonical structure: <ul> <li> It is canonically a <a class='existingWikiWord' href='/nlab/show/diff/metric+space'>metric space</a> and the <a class='existingWikiWord' href='/nlab/show/diff/Euclidean+topology'>Euclidean topology</a> is the corresponding <a class='existingWikiWord' href='/nlab/show/diff/metric+space'>metric topology</a>. </li> <li> There is a canonical <a class='existingWikiWord' href='/nlab/show/diff/smooth+structure'>smooth structure</a> on <math class='maruku-mathml' display='inline' id='mathml_5f0572437a3ae214a10493832748988197295afb_37' 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> that makes it a <a class='existingWikiWord' href='/nlab/show/diff/smooth+manifold'>smooth manifold</a>. </li> <li> A Cartesian space is canonically a<a class='existingWikiWord' href='/nlab/show/diff/vector+space'>vector space</a> over the <a class='existingWikiWord' href='/nlab/show/diff/field'>field</a> of <a class='existingWikiWord' href='/nlab/show/diff/real+number'>real number</a>s. </li> </ul> Sometimes one is interested in allowing <math class='maruku-mathml' display='inline' id='mathml_5f0572437a3ae214a10493832748988197295afb_38' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>n</mi></mrow><annotation encoding='application/x-tex'>n</annotation></semantics></math> to take other values, in which case one wants a <a class='existingWikiWord' href='/nlab/show/diff/cartesian+product'>product</a> in some category that might not be the Cartesian product on underlying sets. For example, if one is studying Cartesian spaces as <a class='existingWikiWord' href='/nlab/show/diff/inner+product+space'>inner product space</a>s, then one might want an <math class='maruku-mathml' display='inline' id='mathml_5f0572437a3ae214a10493832748988197295afb_39' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>ℵ</mi> <mn>0</mn></msub></mrow><annotation encoding='application/x-tex'>\aleph_0</annotation></semantics></math>-dimensional Cartesian space to be the <math class='maruku-mathml' display='inline' id='mathml_5f0572437a3ae214a10493832748988197295afb_40' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>ℵ</mi> <mn>0</mn></msub></mrow><annotation encoding='application/x-tex'>\aleph_0</annotation></semantics></math>-dimensional <a class='existingWikiWord' href='/nlab/show/diff/Hilbert+space'>Hilbert space</a> <math class='maruku-mathml' display='inline' id='mathml_5f0572437a3ae214a10493832748988197295afb_41' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>l</mi> <mn>2</mn></msup></mrow><annotation encoding='application/x-tex'>l^2</annotation></semantics></math>, which is a proper subset of the cartesian product <math class='maruku-mathml' display='inline' id='mathml_5f0572437a3ae214a10493832748988197295afb_42' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>ℝ</mi> <mrow><msub><mi>ℵ</mi> <mn>0</mn></msub></mrow></msup></mrow><annotation encoding='application/x-tex'>\mathbb{R}^{\aleph_0}</annotation></semantics></math>. </div> <h2 id='properties'>Properties</h2> <h3 id='topological_structure'>Topological structure</h3> The <a class='existingWikiWord' href='/nlab/show/diff/ball'>open n-ball</a> is <a class='existingWikiWord' href='/nlab/show/diff/homeomorphism'>homeomorphic</a> <a class='existingWikiWord' href='/nlab/show/diff/cartesian+space'>Cartesian space</a> <math class='maruku-mathml' display='inline' id='mathml_5f0572437a3ae214a10493832748988197295afb_43' 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> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_5f0572437a3ae214a10493832748988197295afb_44' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>𝔹</mi> <mi>n</mi></msup><mo>≃</mo><msup><mi>ℝ</mi> <mi>n</mi></msup><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> \mathbb{B}^n \simeq \mathbb{R}^n \,. </annotation></semantics></math></div> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/topological+invariance+of+dimension'>topological invariance of dimension</a></li> </ul> <h3 id='smooth_structures'>Smooth structures</h3> For all <math class='maruku-mathml' display='inline' id='mathml_5f0572437a3ae214a10493832748988197295afb_45' 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/ball'>open n-ball</a> with its standard <a class='existingWikiWord' href='/nlab/show/diff/smooth+structure'>smooth structure</a> is <a class='existingWikiWord' href='/nlab/show/diff/diffeomorphism'>diffeomorphic</a> to the <a class='existingWikiWord' href='/nlab/show/diff/cartesian+space'>Cartesian space</a> <math class='maruku-mathml' display='inline' id='mathml_5f0572437a3ae214a10493832748988197295afb_46' 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> with its standard smooth structure <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_5f0572437a3ae214a10493832748988197295afb_47' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>𝔹</mi> <mi>n</mi></msup><mo>≃</mo><msup><mi>ℝ</mi> <mi>n</mi></msup><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> \mathbb{B}^n \simeq \mathbb{R}^n \,. </annotation></semantics></math></div> In fact, in <math class='maruku-mathml' display='inline' id='mathml_5f0572437a3ae214a10493832748988197295afb_48' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>d</mi><mo>≠</mo><mn>4</mn></mrow><annotation encoding='application/x-tex'>d \neq 4</annotation></semantics></math> there is no choice: <div class='un_theorem'> <h6 id='theorem'>Theorem</h6> For <math class='maruku-mathml' display='inline' id='mathml_5f0572437a3ae214a10493832748988197295afb_49' 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> a <a class='existingWikiWord' href='/nlab/show/diff/natural+number'>natural number</a> with <math class='maruku-mathml' display='inline' id='mathml_5f0572437a3ae214a10493832748988197295afb_50' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>n</mi><mo>≠</mo><mn>4</mn></mrow><annotation encoding='application/x-tex'>n \neq 4</annotation></semantics></math>, there is a unique (up to <a class='existingWikiWord' href='/nlab/show/diff/isomorphism'>isomorphism</a>) smooth structure on the <a class='existingWikiWord' href='/nlab/show/diff/cartesian+space'>Cartesian space</a> <math class='maruku-mathml' display='inline' id='mathml_5f0572437a3ae214a10493832748988197295afb_51' 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>. </div> This was shown in (<a href='#Stallings'>Stallings</a>). <div class='un_theorem'> <h6 id='theorem_2'>Theorem</h6> In <math class='maruku-mathml' display='inline' id='mathml_5f0572437a3ae214a10493832748988197295afb_52' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>d</mi><mo>=</mo><mn>4</mn></mrow><annotation encoding='application/x-tex'>d = 4</annotation></semantics></math> the analog of this statement is false. One says that on <math class='maruku-mathml' display='inline' id='mathml_5f0572437a3ae214a10493832748988197295afb_53' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>ℝ</mi> <mn>4</mn></msup></mrow><annotation encoding='application/x-tex'>\mathbb{R}^4</annotation></semantics></math> there exist <a class='existingWikiWord' href='/nlab/show/diff/exotic+smooth+structure'>exotic smooth structure</a>s. </div> <div class='un_theorem'> <h6 id='theorem_3'>Theorem</h6> In <a class='existingWikiWord' href='/nlab/show/diff/dimension'>dimension</a> <math class='maruku-mathml' display='inline' id='mathml_5f0572437a3ae214a10493832748988197295afb_54' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>d</mi><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding='application/x-tex'>d \in \mathbb{N}</annotation></semantics></math> for <math class='maruku-mathml' display='inline' id='mathml_5f0572437a3ae214a10493832748988197295afb_55' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>d</mi><mo>≠</mo><mn>4</mn></mrow><annotation encoding='application/x-tex'>d \neq 4</annotation></semantics></math> we have: every open subset of <math class='maruku-mathml' display='inline' id='mathml_5f0572437a3ae214a10493832748988197295afb_56' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>ℝ</mi> <mi>d</mi></msup></mrow><annotation encoding='application/x-tex'>\mathbb{R}^d</annotation></semantics></math> which is <a class='existingWikiWord' href='/nlab/show/diff/homeomorphism'>homeomorphic</a> to <math class='maruku-mathml' display='inline' id='mathml_5f0572437a3ae214a10493832748988197295afb_57' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>𝔹</mi> <mi>d</mi></msup></mrow><annotation encoding='application/x-tex'>\mathbb{B}^d</annotation></semantics></math> is also <a class='existingWikiWord' href='/nlab/show/diff/diffeomorphism'>diffeomorphic</a> to it. </div> See the first page of (<a href='#Ozols'>Ozols</a>) for a list of references. <div class='un_remark'> <h6 id='remark_2'>Remark</h6> In dimension 4 the analog statement fails due to the existence of <a class='existingWikiWord' href='/nlab/show/diff/exotic+smooth+structure'>exotic smooth structure</a>s on <math class='maruku-mathml' display='inline' id='mathml_5f0572437a3ae214a10493832748988197295afb_58' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>ℝ</mi> <mn>4</mn></msup></mrow><annotation encoding='application/x-tex'>\mathbb{R}^4</annotation></semantics></math>. </div> <h2 id='the_category_of_cartesian_spaces'>The category of Cartesian spaces</h2> See <a class='existingWikiWord' href='/nlab/show/diff/CartSp'>CartSp</a>. <h2 id='related_concepts'>Related concepts</h2> <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/super+Cartesian+space'>super Cartesian space</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/polydisc'>polydisk</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/affine+space'>affine space</a> </li> </ul> In <a class='existingWikiWord' href='/nlab/show/diff/complex+geometry'>complex geometry</a> for purposes of <a class='existingWikiWord' href='/nlab/show/diff/%C4%8Cech+cohomology'>Cech cohomology</a> the role of Cartesian spaces is played by <a class='existingWikiWord' href='/nlab/show/diff/Stein+manifold'>Stein manifolds</a>. <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/topological+invariance+of+dimension'>topological invariance of dimension</a></li> </ul> <h2 id='references'>References</h2> Named after <a class='existingWikiWord' href='/nlab/show/diff/Ren%C3%A9+Descartes'>René Descartes</a>. <ul> <li id='Stallings'> <a class='existingWikiWord' href='/nlab/show/diff/John+Stallings'>John Stallings</a>, The piecewise linear structure of Euclidean space , Proc. Cambridge Philos. Soc. 58 (1962), 481-488. (<a href='http://journals.cambridge.org/production/action/cjoGetFulltext?fulltextid=2118140'>pdf</a>) </li> <li id='Ozols'> V. Ozols, Largest normal neighbourhoods , Proceedings of the American Mathematical Society Vol. 61, No. 1 (Nov., 1976), pp. 99-101 (<a href='http://www.jstor.org/stable/2041672'>jstor</a>) </li> </ul> There are various slight variations of the category <math class='maruku-mathml' display='inline' id='mathml_5f0572437a3ae214a10493832748988197295afb_59' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>CartSp</mi></mrow><annotation encoding='application/x-tex'>CartSp</annotation></semantics></math> that one can consider without changing its basic properties as a category of test spaces for <a class='existingWikiWord' href='/nlab/show/diff/generalized+smooth+space'>generalized smooth space</a>s. A different choice that enjoys some popularity in the literature is the category of open (contractible) subsets of Euclidean spaces. For more references on this see <a class='existingWikiWord' href='/nlab/show/diff/diffeological+space'>diffeological space</a>. The <a class='existingWikiWord' href='/nlab/show/diff/site'>site</a> <math class='maruku-mathml' display='inline' id='mathml_5f0572437a3ae214a10493832748988197295afb_60' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>ThCartSp</mi></mrow><annotation encoding='application/x-tex'>ThCartSp</annotation></semantics></math> of infinitesimally thickened Cartesian spaces is known as the site for the <a class='existingWikiWord' href='/nlab/show/diff/Cahiers+topos'>Cahiers topos</a>. It is considered in detal in section 5 of <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/Anders+Kock'>Anders Kock</a>, Convenient vector spaces embed into the Cahiers topos (<a href='http://www.numdam.org/item?id=CTGDC_1986__27_1_3_0'>numdam</a>)</li> </ul> and briefly mentioned in example 2) on p. 191 of <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/Anders+Kock'>Anders Kock</a>, Synthetic differential geometry (<a href='http://home.imf.au.dk/kock/sdg99.pdf'>pdf</a>)</li> </ul> following the original article <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/Eduardo+Dubuc'>Eduardo Dubuc</a>, Sur les modeles de la geometrie differentielle synthetique (<a href='http://www.numdam.org/item?id=CTGDC_1979__20_3_231_0'>numdam</a>).</li> </ul> With an eye towards <a class='existingWikiWord' href='/nlab/show/diff/Fr%C3%B6licher+space'>Frölicher space</a>s the site is also considered in section 5 of <ul> <li>Hirokazu Nishimura, Beyond the Regnant Philosophy of Manifolds (<a href='http://arxiv.org/abs/0912.0827'>arXiv:0912.0827</a>)</li> </ul> </div> <div class="revisedby"> Last revised on August 8, 2022 at 08:10:11. See the <a href="/nlab/history/cartesian+space" style="color: #005c19">history</a> of this page for a list of all contributions to it. </div> <div class="navigation navfoot"> <a href="/nlab/edit/cartesian+space" accesskey="E" class="navlink" id="edit" rel="nofollow">Edit</a><a href="https://nforum.ncatlab.org/discussion/14785/#Item_1">Discuss</a><a href="/nlab/revision/diff/cartesian+space/20" accesskey="B" class="navlinkbackintime" id="to_previous_revision" rel="nofollow">Previous revision</a><a href="/nlab/show/cartesian+space" accesskey="C" class="navlink" id="see_changes" rel="nofollow">Hide changes</a><a href="/nlab/history/cartesian+space" accesskey="S" class="navlink" id="history" rel="nofollow">History (20 revisions)</a> <a href="/nlab/show/cartesian+space/cite" style="color: black">Cite</a> <a href="/nlab/print/cartesian+space" accesskey="p" id="view_print" rel="nofollow">Print</a> <a href="/nlab/source/cartesian+space" id="view_source" rel="nofollow">Source</a> </div> </div>  </div>  </body> </html>