{"id":352,"date":"2014-05-22T18:58:38","date_gmt":"2014-05-22T16:58:38","guid":{"rendered":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/?page_id=352"},"modified":"2022-05-17T09:31:48","modified_gmt":"2022-05-17T07:31:48","slug":"filters-iv-compactifications","status":"publish","type":"page","link":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/?page_id=352","title":{"rendered":"Filters IV: compactifications"},"content":{"rendered":"<p><strong>The Stone-\u010cech compactification: a reminder.<\/strong><\/p>\n<p>Remember from Exercise 6.7.23 that every T<sub>3 1\/2<\/sub> topological space <em>X<\/em> can be embedded in a compact T<sub>2<\/sub> space, called its Stone-\u010cech compactification \u00df<em>X<\/em>. This has the universal property of being the <em>free<\/em> compact T<sub>2<\/sub> space over <em>X<\/em> as a T<sub>3 1\/2<\/sub> space.<\/p>\n<p>More elementarily, every subspace of a compact T<sub>2<\/sub> space must be T<sub>3 1\/2<\/sub>, and every T<sub>3 1\/2<\/sub> space embeds into a compact T<sub>2<\/sub> space.\u00a0 \u00df<em>X<\/em> is in a sense the largest such completion, and <em>X<\/em> is dense in \u00df<em>X<\/em>.<\/p>\n<p>As a free object,\u00a0\u00df<em>X<\/em> is necessarily unique up to isomorphism, but can be implemented in several ways, at least in principle. Exercise 6.7.23 gives one possible implementation of \u00df<em>X<\/em>, as a subspace of the compact space obtained as the product of as many copies of [0, 1] as there are continuous maps from <em>X<\/em> to [0, 1].\u00a0 This is natural considering the definition of complete regularity.\u00a0 There are other implementations, based on filters, which we shall explore.<\/p>\n<p><strong>Spaces of ultrafilters.<\/strong><\/p>\n<p>The most well-known implementation of this program is by building the space <strong>U<\/strong><em>X<\/em> of all ultrafilters of subsets of <em>X<\/em>.\u00a0 This will work out as expected, and provide another implementation of \u00df<em>X<\/em>, only when <em>X<\/em> is a <span style=\"text-decoration: underline;\">discrete space<\/span>.\u00a0 But that is a good start, and anyway the case <em>X<\/em>=<strong>N<\/strong> (the set of natural numbers) is already an intriguing beast.\u00a0 I will not even try to explain what \u00df<strong>N<\/strong> looks like; Jan van Mill calls it the `three-headed monster&#8217;, and this should be enough to scare you away from trying to understand it finely.<\/p>\n<p>So let us fix a discrete space <em>X<\/em>; in other words, a set, with the discrete topology.\u00a0 For a subset <em>U<\/em> of <em>X<\/em>, let us write <em>U<\/em><sup>\u266f<\/sup> for the subset of <strong>U<\/strong><em>X<\/em> of all those ultrafilters <em>F<\/em> of subsets that contain <em>U<\/em>.\u00a0 In other words, <em>F<\/em> is in <em>U<\/em><sup>\u266f<\/sup> if and only if <em>U<\/em> is in <em>F<\/em>.\u00a0 The intersection of finitely many sets <em>U<sub>i<\/sub><\/em><sup>\u266f<\/sup> is (\u22c2<em><sub>i<\/sub> U<sub>i<\/sub><\/em>)<sup>\u266f<\/sup>, so the subsets <em>U<\/em><sup>\u266f<\/sup> form a base for a topology.<\/p>\n<p>I claim that with this topology turns\u00a0<strong>U<\/strong><em>X<\/em> into a compact T<sub>2<\/sub> space.\u00a0 This can be checked by hand:<\/p>\n<ul>\n<li>[T<sub>2<\/sub>] If <em>F<\/em> and <em>F&#8217;<\/em> are two distinct ultrafilters, then there is a subset <em>A<\/em> in <em>F<\/em> and outside <em>F&#8217;<\/em>, say.\u00a0 Then <em>F<\/em> is in <em>A<\/em><sup>\u266f<\/sup>, <em>F&#8217;<\/em> is in (<em>X<\/em> \\ <em>A<\/em>)<sup>\u266f<\/sup>, and <em>A<\/em><sup>\u266f<\/sup> and\u00a0(<em>X<\/em> \\ <em>A<\/em>)<sup>\u266f<\/sup> are disjoint open subsets; so <strong>U<\/strong><em>X<\/em> is T<sub>2<\/sub>.<\/li>\n<li>[Compact] we shall use a handy gadget: the <em>Kowalsky sum<\/em> operation.\u00a0 This is a map\u00a0\u266d : <strong>U<\/strong><strong>U<\/strong><em>X<\/em> <strong>\u2192 U<\/strong><em>X<\/em>, defined by <strong><em>F<\/em><\/strong><sup>\u266d<\/sup>={<em>A<\/em> \u2286 <em>X<\/em> | <em>A<\/em><sup>\u266f<\/sup> is in <strong><em>F<\/em><\/strong>}.\u00a0 Yes, I know, this is hard to digest:\u00a0<strong><em>F<\/em><\/strong> is an ultrafilter of subsets of the set <strong>U<\/strong><em>X<\/em> of ultrafilters of subsets (!).<br \/>\nThis map \u266d has plenty of properties, but let us concentrate on one in particular: for every ultrafilter <strong><em>F<\/em><\/strong> of subsets of <strong>U<\/strong><em>X<\/em>,\u00a0<strong><em>F<\/em><\/strong><sup>\u266d<\/sup> is a limit of <strong><em>F<\/em><\/strong>.\u00a0 Indeed, it suffices to show that any basic open neighborhood <em>U<\/em><sup>\u266f<\/sup> of\u00a0<strong><em>F<\/em><\/strong><sup>\u266d<\/sup> is in <strong><em>F<\/em><\/strong>&#8230; but this is completely mechanical: since\u00a0<strong><em>F<\/em><\/strong><sup>\u266d<\/sup> is in <em>U<\/em><sup>\u266f<\/sup>, U is in\u00a0<strong><em>F<\/em><\/strong><sup>\u266d<\/sup> [definition of #], so <em>U<\/em><sup>\u266f<\/sup> is in <strong><em>F<\/em><\/strong> [definition of <strong><em>F<\/em><\/strong><sup>\u266d<\/sup>].<br \/>\nIn particular, every ultrafilter <strong><em>F<\/em><\/strong> of subsets of <strong>U<\/strong><em>X<\/em> has a limit.\u00a0 We have seen in <a title=\"Filters, part I\" href=\"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/?page_id=251\">Filters, part I<\/a> that that meant that <strong>U<\/strong><em>X<\/em> was compact.<\/li>\n<\/ul>\n<p>We can embed <em>X<\/em> into <strong>U<\/strong><em>X<\/em>, too.\u00a0 Define \u03b7 : <em>X<\/em> \u2192 <strong>U<\/strong><em>X<\/em> by: \u03b7(<em>x<\/em>) is the principal ultrafilter at <em>x<\/em>, namely the filter of all the subsets of <em>X<\/em> that contain <em>x<\/em>.\u00a0 (I had written it (<em>x<\/em>) in previous posts, but this would pose some readibility problems here.\u00a0 A standard notation is <em>x<\/em> with a dot above it.)\u00a0 Check that this is a continuous map; in fact, the inverse image of <em>U<\/em><sup>\u266f<\/sup> is just <em>U<\/em>, which immediately shows that \u03b7 is almost open; since \u03b7 is easily seen to be injective, \u03b7 is an embedding.<\/p>\n<p>Before we proceed, we should mention some of the other properties of \u266d.\u00a0 I will use some of them below (exercise!):<\/p>\n<ul>\n<li>\u266d : <strong>U<\/strong><strong>U<\/strong><em>X<\/em> <strong>\u2192 U<\/strong><em>X<\/em> is continuous: the inverse image of <em>U<\/em><sup>\u266f<\/sup> is <em>U<\/em><sup>\u266f\u266f<\/sup>;<\/li>\n<li><strong>U<\/strong> is a functor; on morphisms (functions) <em>f<\/em> : <em>X<\/em> <strong>\u2192<\/strong> <em>Y<\/em>, it acts by: <strong>U<\/strong><em>f<\/em> : <strong>U<\/strong><em>X<\/em> <strong>\u2192<\/strong> <strong>U<\/strong><em>Y<\/em> maps every ultrafilter <em>F<\/em> in to its image filter <em>f<\/em>[<em>F<\/em>];<\/li>\n<li>\u266d is natural in <em>X<\/em>, that is, \u266d o <strong>UU<\/strong><em>f<\/em> = <strong>U<\/strong><em>f<\/em> o \u266d for every map <em>f<\/em>;<\/li>\n<li>in fact\u00a0\u03b7 is also natural in <em>X<\/em>:\u00a0<strong>U<\/strong><em>f<\/em> o \u03b7 = \u03b7 o <em>f<\/em>;<\/li>\n<li>The composite \u266d o\u00a0\u03b7 : <strong>U<\/strong><em>X<\/em> <strong>\u2192 U<\/strong><em>X<\/em> is equal to the identity map;<\/li>\n<li>The composite \u266d o\u00a0<strong>U<\/strong>\u03b7 : <strong>U<\/strong><em>X<\/em> <strong>\u2192 U<\/strong><em>X<\/em> is also equal to the identity map;<\/li>\n<li>The composites \u266d o \u266d and \u266d o <strong>U<\/strong>\u266d : <strong><strong>U<\/strong>U<\/strong><strong>U<\/strong><em>X<\/em> <strong>\u2192 U<\/strong><em>X<\/em> are the same map.<\/li>\n<\/ul>\n<p>All this means that (<strong>U<\/strong>, \u03b7, \u266d) is a <a title=\"Monad\" href=\"https:\/\/en.wikipedia.org\/wiki\/Monad_%28category_theory%29\">monad<\/a> on the category of sets.\u00a0 A very nice structure, but this would carry us away somehow&#8230; apart from the fact that I&#8217;ll use the equations above anyway.<\/p>\n<p>To show that <strong>U<\/strong><em>X<\/em> is also (isomorphic to) the Stone-\u010cech compactification \u00df<em>X<\/em>, it suffices to show that it satisfies the universal property of being the free compact T<sub>2<\/sub> space over <em>X<\/em>, since all solutions to a universal property are isomorphic.\u00a0 Consider any continuous map <em>g<\/em> : <em>X<\/em> \u2192 <em>Y<\/em>, where <em>Y<\/em> is compact T<sub>2<\/sub>.\u00a0\u00a0 We must show that it extends to a unique continuous map <em>g&#8217;<\/em> from <strong>U<\/strong><em>X<\/em> to <em>Y<\/em>, namely that <em>g&#8217;<\/em>(\u03b7(<em>x<\/em>))=g(x) for every <em>x<\/em> in <em>X<\/em>.<\/p>\n<p>Recall that a continuous map has the property that for every filter that converges to a point, the image filter converges to the image of the point.\u00a0 For every ultrafilter <em>F<\/em> in <strong>U<\/strong><em>X<\/em>, let <strong><em>F<\/em><\/strong><em>=<\/em><strong>U<\/strong>\u03b7(<em>F<\/em>); this is an ultrafilter of subsets of <strong>U<\/strong><em>X<\/em>, which converges to <em><strong>F<\/strong><\/em><sup>\u266d<\/sup>=\u266d o <strong>U<\/strong>\u03b7(<em>F<\/em>)=<em>F<\/em>.\u00a0 If g&#8217; exists, and is continuous, it follows that the image filter <em>g&#8217;<\/em>[<em><strong>F<\/strong><\/em>]=<strong>U<\/strong><em>g&#8217;<\/em> (<em><strong>F<\/strong><\/em>) = <strong>U<\/strong><em>g&#8217;<\/em> (<strong>U<\/strong>\u03b7(<em>F<\/em>)) =\u00a0<strong>U<\/strong>(<em>g&#8217;<\/em> o \u03b7) (<em>F<\/em>) = <strong>U<\/strong><em>g<\/em> (<em>F<\/em>)=<em>g<\/em>[<em>F<\/em>] must converge to <em>g&#8217;<\/em>(<strong>F<\/strong><sup>\u266d<\/sup>) = <em>g&#8217;<\/em>(<em>F<\/em>).\u00a0 Since limits are unique in the T<sub>2<\/sub> space <em>Y<\/em>, this shows that, if <em>g&#8217;<\/em> exists, then it is unique.\u00a0 This also suggests to define <em>g&#8217;<\/em>(<em>F<\/em>) as the limit of <em>g<\/em>[<em>F<\/em>], which exists because <em>Y<\/em> is compact.\u00a0 I&#8217;ll let you check that g&#8217; thus defined is continuous, solving the whole question:<\/p>\n<blockquote><p>When <em>X<\/em> is a discrete space, the space of ultrafilters <strong>U<\/strong><em>X<\/em> is a Stone-\u010cech compactification of <em>X<\/em>, i.e., it is isomorphic to \u00df<em>X<\/em>.<\/p><\/blockquote>\n<p>Oh, by the way, you probably cannot imagine any ultrafilter on X that would not be principal.\u00a0 This might lead you to conjecturing that <strong>U<\/strong><em>X<\/em>=<em>X<\/em> (up to \u03b7) when X is discrete.\u00a0 This is true when <em>X<\/em> is finite, but definitely wrong if <em>X<\/em> is infinite.\u00a0 Assuming that<em> X<\/em> is infinite, one can for example consider the cofinite filter Cof(<em>X<\/em>) on <em>X<\/em>, namely the family of all those subsets of <em>X<\/em> whose complement is finite.\u00a0 Using Zorn&#8217;s Lemma, there is an ultrafilter that contains Cof(<em>X<\/em>), and it cannot be principal: if it were equal to \u03b7(<em>x<\/em>) for some <em>x<\/em>, since it also contains the complement of {x} (which is in Cof(<em>X<\/em>)), it would contain the empty set, contradiction.<\/p>\n<p>I&#8217;m stopping there for now.\u00a0 You&#8217;ve probably had enough for today.\u00a0 Next time, I&#8217;ll tell you about <a title=\"Wallman compactification\" href=\"https:\/\/en.wikipedia.org\/wiki\/Wallman_compactification\">Wallman compactifications<\/a>, a variant on this construction which also works (whatever that means) on non-discrete spaces.\u00a0 I claim this is best explained through Stone duality&#8230; You&#8217;ll have to understand Stone duality pretty deeply to understand my next post: sorry&#8230; but I&#8217;ll do my best to remain understandable.<\/p>\n<p style=\"text-align: right;\">\u2014 <a href=\"https:\/\/www.lsv.ens-paris-saclay.fr\/~goubault\/?l=en\" rel=\"attachment wp-att-993\">Jean Goubault-Larrecq<\/a>\u00a0(May 22nd, 2014)<img loading=\"lazy\" decoding=\"async\" class=\"wp-image-993 alignright\" src=\"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/wp-content\/uploads\/2016\/08\/jgl-2011.png\" alt=\"jgl-2011\" width=\"32\" height=\"44\" \/><\/p>\n","protected":false},"excerpt":{"rendered":"<p>The Stone-\u010cech compactification: a reminder. Remember from Exercise 6.7.23 that every T3 1\/2 topological space X can be embedded in a compact T2 space, called its Stone-\u010cech compactification \u00dfX. This has the universal property of being the free compact T2 &hellip; <a href=\"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/?page_id=352\">Continue reading <span class=\"meta-nav\">&rarr;<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"open","ping_status":"open","template":"","meta":{"_crdt_document":"","footnotes":""},"class_list":["post-352","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=\/wp\/v2\/pages\/352","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=352"}],"version-history":[{"count":21,"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=\/wp\/v2\/pages\/352\/revisions"}],"predecessor-version":[{"id":5336,"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=\/wp\/v2\/pages\/352\/revisions\/5336"}],"wp:attachment":[{"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=352"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}