{"id":1457,"date":"2018-03-30T15:53:39","date_gmt":"2018-03-30T13:53:39","guid":{"rendered":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/?page_id=1457"},"modified":"2022-11-19T15:17:16","modified_gmt":"2022-11-19T14:17:16","slug":"g-delta-subsets-of-locally-compact-sober-spaces","status":"publish","type":"page","link":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/?page_id=1457","title":{"rendered":"G-delta subsets of locally compact sober spaces"},"content":{"rendered":"

A kind of space that has good properties and that\u00a0I regularly bump into are the G<\/em>\u03b4<\/sub> subspaces of locally compact sober spaces. \u00a0For example, every continuous valuation on such a space automatically extends to a measure on its Borel\u00a0\u03c3-algebra. \u00a0Or: every such space is consonant<\/a>. \u00a0Or: projective limits of directed systems of locally finite continuous valuations (where the index set has a countable cofinal subfamily) on such spaces exist and are unique [1].<\/p>\n

How should I call those spaces? \u00a0“G<\/em>\u03b4<\/sub> subspace of a locally compact sober space” is longish…<\/p>\n

I recently asked the question to Matthew de Brecht<\/a>, because those spaces include his own quasi-Polish spaces, without the separability requirement. \u00a0His angle was rather to see whether that kind of space was already known under another name, and he had the intuition that that should be some notion of completeness. \u00a0Hence call them “X-complete”, for some unknown X.<\/p>\n

Let me see.<\/p>\n

    \n
  1. Every locally compact sober space is X-complete (sure).<\/li>\n
  2. Every complete metric space, in its open ball topology, is X-complete. \u00a0This is because the space embeds in its space of formal balls, which is then a continuous dcpo [2]. \u00a0That includes all Polish spaces.<\/li>\n
  3. Every continuous complete quasi-metric space, in its\u00a0d<\/em>-Scott topology, is\u00a0X-complete. \u00a0The reason is similar [3]. \u00a0That includes all quasi-Polish<\/a> spaces.<\/li>\n
  4. Every T0<\/sub>\u00a0completely regular\u00a0\u010cech-complete space is X-complete. \u00a0This is because, by Frol\u00edk’s Theorem (Exercise 7.6.22 in the book<\/a>) those spaces are exactly those that embed as a (dense) G<\/em>\u03b4<\/sub> subspace of a compact T2<\/sub>\u00a0space (its Stone-\u010cech compactification).<\/li>\n<\/ol>\n

    On the other hand:<\/p>\n

      \n
    1. Every X-complete space is Choquet-complete. \u00a0This is because every locally compact sober space is Choquet-complete (Proposition 8.3.24 in the book<\/a>; \u03b1 even has a stationary winning strategy) and a\u00a0G<\/em>\u03b4<\/sub> subspace of a Choquet-complete space in which\u00a0\u03b1 has a stationary winning strategy is Choquet-complete. \u00a0(I cannot find the latter in the book<\/a>, damn. \u00a0But that has an easy proof. \u00a0Let X<\/em> be the intersection of a decreasing sequence Wn<\/sub><\/em> of opens in Y.\u00a0<\/em>\u00a0At the n<\/em>th turn played in X<\/em>,\u00a0\u03b1 looks at the open subset V<\/em>\u00a0of X<\/em> that\u00a0\u03b2 has just played. \u00a0This is the intersection with\u00a0X<\/em> of an open subset\u00a0V’<\/em> of\u00a0Y<\/em>. \u00a0Then\u00a0\u03b1 simulates the stationary winning strategy it has on\u00a0Y<\/em> to find a smaller open subset\u00a0U<\/em> of\u00a0Y<\/em>, and plays U<\/i>\u00a0\u2229 Wn<\/sub><\/em>. \u00a0Note that this new strategy is not stationary, but it is Markov<\/em>, in the sense that it only depends on what\u00a0\u03b2 has just played and\u00a0n<\/em>, only. \u00a0It is also true that\u00a0G<\/em>\u03b4<\/sub> subspace of a Choquet-complete space is Choquet-complete, without the need to rely on stationary strategies, but the argument is a bit more complicated.)<\/li>\n<\/ol>\n

      Matthew de Brecht thought that X-completeness sounded a lot like\u00a0\u010cech-completeness, and that maybe we could replace the Stone-\u010cech compactification in the proof of item 4 by some non-Hausdorff “compactification”. \u00a0Proposition 18 of [4] seems to be exactly what we need… but there is a gap in the proof.<\/p>\n

      This leads to a whole set of questions:<\/p>\n

        \n
      1. Is every X-complete space \u010cech-complete, in the weak sense that I am using in Exercise 7.6.21 of the book<\/a>? \u00a0That notion implies Choquet-completeness.<\/li>\n
      2. Is every\u00a0\u010cech-complete space X-complete?<\/li>\n
      3. Is every X-complete space also a\u00a0G<\/em>\u03b4<\/sub> subspace of a stably (locally?) compact space? \u00a0Embedding a space\u00a0X<\/em> into some space of filters of open subsets of\u00a0X<\/em> may be a good idea, following [4] (see also Exercise 9.3.10 in the book<\/a>).<\/li>\n
      4. Can we characterize X-complete spaces by some form of (Choquet, Banach-Mazur) game?<\/li>\n<\/ol>\n

        Update (March 20, 2019).\u00a0 With M. de Brecht, X. Jia, and Z. Lyu, we have put everything we know about those spaces in a paper<\/a> submitted to ISDT 2019<\/a>.\u00a0 We have decided to call these spaces LCS-complete\u2014LCS stands for “locally compact sober”.<\/p>\n

        Question 8 has a negative answer, by the way: every subspace of stably locally compact space is coherent, and there are non-coherent LCS-complete spaces, for example any non-coherent continuous dcpo, such as the set of negative integers with two incomparable elements added below all of them.<\/p>\n

          \n
        1. Jean Goubault-Larrecq. \u00a0Products and projective limits of continuous valuations on T0<\/sub>\u00a0spaces. \u00a0arXiv:1803.05259<\/a> [math.PR]<\/li>\n
        2. Abbas Edalat and Reinhold Heckmann.\u00a0\u00a0A computational model for metric spaces.\u00a0\u00a0Theoretical Computer Science Vol. 193, 1998, pages 53-73.<\/li>\n
        3. Jean Goubault-Larrecq and Kok Min Ng. A few notes on formal balls<\/a>. \u00a0Logical Methods in Computer Science Vol. 13(4:18), 2017, pages. 1\u201334.<\/li>\n
        4. Michael B. Smyth. Stable compactification I<\/a>. Journal of the London Mathematical Society\u00a0s2-45<\/span>, Issue\u00a02<\/span><\/a>. \u00a0April 1992, pages 321-340.<\/li>\n<\/ol>\n

          \u2014 Jean Goubault-Larrecq<\/a>\u00a0(March 30th, 2018)\"jgl-2011\"<\/p>\n","protected":false},"excerpt":{"rendered":"

          A kind of space that has good properties and that\u00a0I regularly bump into are the G\u03b4 subspaces of locally compact sober spaces. \u00a0For example, every continuous valuation on such a space automatically extends to a measure on its Borel\u00a0\u03c3-algebra. \u00a0Or: … 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":{"_crdt_document":"","footnotes":""},"class_list":["post-1457","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=\/wp\/v2\/pages\/1457","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=1457"}],"version-history":[{"count":5,"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=\/wp\/v2\/pages\/1457\/revisions"}],"predecessor-version":[{"id":5929,"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=\/wp\/v2\/pages\/1457\/revisions\/5929"}],"wp:attachment":[{"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=1457"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}