{"id":1089,"date":"2016-10-14T17:24:10","date_gmt":"2016-10-14T15:24:10","guid":{"rendered":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/?page_id=1089"},"modified":"2022-11-19T15:23:08","modified_gmt":"2022-11-19T14:23:08","slug":"some-easy-new-facts-on-consonant-spaces","status":"publish","type":"page","link":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/?page_id=1089","title":{"rendered":"Some new, easy facts on consonant spaces"},"content":{"rendered":"

Last time, we had stated and proved the Dolecki-Greco-Lechicki theorem<\/a>: every regular \u010cech-complete space is consonant.\u00a0 I would like to show that there are some other classes of consonant spaces, among T0<\/sub> spaces.\u00a0 The results are going to be easy consequences of results from the book<\/a>.<\/p>\n

We start with the following.<\/p>\n

Theorem.<\/strong> Every G<\/em>\u03b4<\/sub> subspace of a well-filtered locally compact space is consonant.<\/p>\n

Recall that a space is well-filtered if and only if, for every filtered family of compact saturated sets whose intersection is contained in some open set U<\/em>, one member of the family must be included in U<\/em> (Section 8.3.1 in the book<\/a>).\u00a0 Every sober space, notably, is well-filtered.\u00a0 For a T0<\/sub> locally compact space, in fact, sobriety and well-filteredness are equivalent properties (Propositions 8.3.5 and 8.3.8 in the book<\/a>).<\/p>\n

Before we start the proof, let us recall that every locally compact space is consonant.\u00a0 The value of the theorem lies in the fact that G<\/em>\u03b4<\/sub> subspaces of consonant spaces need not<\/em> be consonant [1, Proposition 7.3].<\/p>\n

Proof.\u00a0 Let Y<\/em> be a G<\/em>\u03b4<\/sub> subset of X<\/em>, where X<\/em> is well-filtered and locally compact.\u00a0 We can write V<\/em> as the intersection of an antitone sequence of open subsets Vn<\/sub><\/em> of X<\/em>.\u00a0 By antitone, I mean that V<\/em>0<\/sub> \u2287 V<\/em>1<\/sub> \u2287 … \u2287\u00a0Vn<\/sub><\/em> \u2287 …\u00a0 Let U<\/strong><\/em> be a Scott-open family of open subsets of Y<\/em>, and U<\/em> be in U<\/strong><\/em>.<\/p>\n

Write int<\/em> for the interior operator in X<\/em>.\u00a0 By the definition of the subspace topology, there is an open subset U’<\/em> of X<\/em> such that U’<\/em> \u22c2 Y<\/em> = U<\/em>.\u00a0 By local compactness, U’<\/em> \u22c2\u00a0V<\/em>0<\/sub> is the union of the directed family of sets int<\/em>(Q<\/em>), where Q<\/em> ranges over the family Q<\/strong><\/em>0<\/sub><\/strong> of compact saturated subsets of U’<\/em> \u22c2\u00a0V<\/em>0<\/sub>.\u00a0 (That family is clearly directed.\u00a0 Moreover, local compactness tells us that every point of U’<\/em> \u22c2\u00a0V<\/em>0<\/sub> lies in int<\/em>(Q<\/em>) for some Q<\/em> in Q<\/strong><\/em>0<\/sub><\/strong>.)\u00a0 The union over all\u00a0Q<\/em> in Q<\/strong><\/em>0<\/sub><\/strong> of the sets int<\/em>(Q<\/em>) \u22c2 Y<\/em> is then equal to U’<\/em> \u22c2\u00a0V<\/em>0<\/sub> \u22c2 Y<\/em>, that is, to\u00a0U<\/em> \u22c2\u00a0V<\/em>0<\/sub>, i.e., U<\/em>.<\/p>\n

Since U<\/em> is in U<\/strong><\/em> and U<\/strong><\/em> is Scott-open, int<\/em>(Q<\/em>) \u22c2 Y<\/em> is in U<\/strong><\/em> for some Q<\/em> in Q<\/strong><\/em>0<\/sub><\/strong>.\u00a0 Let\u00a0Q<\/em>0<\/sub> be this compact saturated set Q<\/em>, U’<\/em>0<\/sub> be int<\/em>(Q<\/em>0<\/sub>), and U<\/em>0<\/sub> be equal to\u00a0U’<\/em>0<\/sub> \u22c2 Y<\/em>.\u00a0 Note that\u00a0U<\/em>0<\/sub> is in U<\/strong><\/em>, and that U’<\/em>0<\/sub> \u2286 Q<\/em>0<\/sub> \u2286 U’<\/em> \u22c2\u00a0V<\/em>0<\/sub>.<\/p>\n

Do the same thing with\u00a0U’<\/em>0<\/sub> \u22c2\u00a0V<\/em>1<\/sub> instead of U’<\/em> \u22c2\u00a0V<\/em>0<\/sub>.\u00a0 There is a compact saturated subset Q<\/em>1<\/sub> of\u00a0U’<\/em>0<\/sub> \u22c2\u00a0V<\/em>1<\/sub> such that int<\/em>(Q<\/em>1<\/sub>) \u22c2 Y<\/em> is in U<\/strong><\/em>.\u00a0 Let U’<\/em>1<\/sub> be int<\/em>(Q<\/em>1<\/sub>), and U<\/em>1<\/sub> be equal to\u00a0U’<\/em>1<\/sub> \u22c2 Y<\/em>.\u00a0 Note that\u00a0U<\/em>1<\/sub> is in U<\/strong><\/em>, and that U’<\/em>1<\/sub> \u2286 Q<\/em>1<\/sub> \u2286 U’<\/em>0<\/sub> \u22c2\u00a0V<\/em>1<\/sub>.<\/p>\n

Iterating this construction, we obtain for each natural number n<\/em> a compact saturated subset Qn<\/sub><\/em> of\u00a0X<\/em>, an open subset U’n<\/sub><\/em> of X<\/em>, and an open subset Un<\/sub><\/em> of Y<\/em>, such that Un<\/sub><\/em> is in U<\/strong><\/em>, and U’n+1<\/sub><\/em> \u2286 Qn+1<\/sub><\/em> \u2286 U’n<\/sub><\/em> \u22c2\u00a0Vn<\/sub><\/em>, for every n<\/em> in N<\/strong>.<\/p>\n

Let Q<\/em> be the intersection of all Qn<\/sub><\/em>, n<\/em> in N<\/strong>.\u00a0 Since X<\/em> is well-filtered, Q<\/em> is compact saturated in X<\/em> (Proposition 8.3.6 in the book<\/a>).<\/p>\n

Since every Qn<\/sub><\/em> is included in Vn<\/sub><\/em>, Q<\/em> is included in the intersection of the Vn<\/sub><\/em>s, which happens to be Y<\/em>.\u00a0 We check that Q<\/em> is also compact saturated in Y<\/em>.\u00a0 The specialization quasi-ordering of\u00a0Y<\/em> is the restriction of that of X<\/em> (Proposition 4.9.5 in the book<\/a>), so Q<\/em> is upwards-closed, namely, saturated, in Y<\/em>.\u00a0 For every open cover (Wj<\/sub><\/em>)j \u2208 J<\/sub><\/em> of Q<\/em> in Y<\/em>, we write each\u00a0Wj<\/sub><\/em> as the intersection of some open subset\u00a0W’j<\/sub><\/em> of X<\/em> with Y<\/em>.\u00a0 Then (W’j<\/sub><\/em>)j \u2208 J<\/sub><\/em> is an open cover of Q<\/em> in X<\/em>, from which we can extract a finite subcover (W’j<\/sub><\/em>)j \u2208 K<\/sub><\/em> (K<\/em> finite).\u00a0 It is then clear that (Wj<\/sub><\/em>)j \u2208 K<\/sub><\/em> is a finite subcover of Q<\/em>, since Q<\/em> is included in Y<\/em>.
\n<\/em><\/p>\n

We observe that Q<\/em> is included in Q<\/em>0<\/sub>, which is included in U’<\/em> \u22c2\u00a0V<\/em>0<\/sub>,.\u00a0 Since it is also included in\u00a0Y<\/em>, it is included in\u00a0U’<\/em> \u22c2 Y = U<\/em>.\u00a0 Therefore U<\/em> is in \uffedQ.<\/p>\n

It remains to show that every W<\/em> in\u00a0\uffedQ is in U<\/strong><\/em>.\u00a0 Write W<\/em> as the intersection of some open subset W’<\/em> of X<\/em> with Y<\/em>.\u00a0 Since Q<\/em>, which is equal to the filtered intersection of the compact saturated subsets Qn<\/sub><\/em>, is included in W<\/em>, hence in W’<\/em>, some\u00a0Qn<\/sub><\/em> is included in W’<\/em> by well-filteredness.\u00a0 It follows that\u00a0U’n<\/sub><\/em> is included in W’<\/em>.\u00a0 Taking intersections with Y<\/em>, Un<\/sub><\/em> is included in W<\/em>.\u00a0 Since Un<\/sub><\/em> is in U<\/strong><\/em>, so is W<\/em>.\u00a0 \u2610<\/p>\n

That theorem has the following nice consequence.<\/p>\n

Corollary.<\/strong> Every continuous Yoneda-complete quasi-metric space X<\/em>, d<\/em> is consonant in its d<\/em>-Scott topology.<\/p>\n

Proof.\u00a0 Every continuous dcpo is sober in its Scott topology (Proposition 8.2.12 (b) in the book<\/a>), hence well-filtered, and also locally compact (Corollary 5.1.36).\u00a0 The definition of the d<\/em>-Scott topology means that the map x<\/em> \u27fc (x<\/em>, 0) is a topological embedding of X<\/em> into its dcpo of formal balls B<\/strong>(X<\/em>, d<\/em>), and we equate X<\/em> with a subspace of B<\/strong>(X<\/em>, d<\/em>) through this map.<\/p>\n

For every \u03b5 > 0, let V<\/em>\u03b5<\/sub> be the set of formal balls (x<\/em>, r<\/em>) whose radius is < \u03b5.\u00a0 This is Scott-open in B<\/strong>(X<\/em>, d<\/em>): by the Kostanek-Waszkewicz theorem (Theorem 7.4.27), the supremum of a directed family of formal balls (xi<\/sub><\/em>, ri<\/sub><\/em>) is of the form (x<\/em>, r<\/em>) where r<\/em> = inf ri<\/sub><\/em> (and x<\/em> is the d<\/em>-limit of the net consisting of the points xi<\/sub><\/em>); if r<\/em> < \u03b5, then some\u00a0ri<\/sub><\/em> is also < \u03b5.<\/p>\n

The family of open sets V<\/em>1\/2n<\/sup><\/em><\/sub>, n<\/em> in N<\/strong>, is then an antitone sequence of open sets of B<\/strong>(X<\/em>, d<\/em>), whose intersection equals X<\/em>.\u00a0 In other words, X<\/em> is a G<\/em>\u03b4<\/sub> subspace of B<\/strong>(X<\/em>, d<\/em>), and we conclude by applying the previous Theorem.\u00a0 \u2610<\/p>\n

We retrieve, notably, that every complete metric space is consonant, because every complete metric space is continuous Yoneda-complete, and its open ball topology coincides with the d<\/em>-Scott topology.\u00a0 This yields another proof of that result, which is classically obtained by noticing that every complete metric space is (completely) regular and \u010cech-complete.<\/p>\n

What we gain is that the above corollary applies to a whole family of\u00a0T0<\/sub> spaces that are far from being regular: recall that a\u00a0T0<\/sub> regular space is automatically T2<\/sub>, whereas most continuous Yoneda-complete quasi-metric spaces are not.<\/p>\n

\u2014 Jean Goubault-Larrecq<\/a>\u00a0(October 14th, 2016)\"jgl-2011\"<\/p>\n

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  1. Dolecki, Szymon, Greco, Gabriele H., and Lechicki, Alojzy, 1995. When Do the Upper Kuratowski Topology (Homeomorphically, Scott Topology) and the Co-Compact Topology Coincide?<\/em><\/a> Transactions of the American Mathematical Society, 347(8), 2869\u20132884.<\/li>\n<\/ol>\n","protected":false},"excerpt":{"rendered":"

    Last time, we had stated and proved the Dolecki-Greco-Lechicki theorem: every regular \u010cech-complete space is consonant.\u00a0 I would like to show that there are some other classes of consonant spaces, among T0 spaces.\u00a0 The results are going to be easy … Continue reading →<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"class_list":["post-1089","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=\/wp\/v2\/pages\/1089","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=1089"}],"version-history":[{"count":5,"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=\/wp\/v2\/pages\/1089\/revisions"}],"predecessor-version":[{"id":5944,"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=\/wp\/v2\/pages\/1089\/revisions\/5944"}],"wp:attachment":[{"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=1089"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}