{"id":70,"date":"2012-10-12T10:42:20","date_gmt":"2012-10-12T08:42:20","guid":{"rendered":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/?page_id=70"},"modified":"2022-11-19T15:34:56","modified_gmt":"2022-11-19T14:34:56","slug":"70-2","status":"publish","type":"page","link":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/?page_id=70","title":{"rendered":"Models improved"},"content":{"rendered":"

A model of a space\u00a0X<\/em>\u00a0is a dcpo\u00a0Y<\/em>\u00a0whose subspace of maximal elements is isomorphic to\u00a0X<\/em>. \u00a0Of particular importance are those spaces that have \u03c9-continuous models. \u00a0Martin [3], and Mummert and Stephan [1] came very close to characterize them exactly.<\/p>\n

Let’s review the notion. This is covered in Section 7.7.2 of the book<\/a>. The prime example of an\u00a0\u03c9-continuous model is\u00a0IR<\/strong>, the dcpo of all non-empty closed intervals of the real line\u00a0R<\/strong>. \u00a0We order the elements of\u00a0IR<\/strong>\u00a0by: [a, b] is below [c, d] if and only if [c, d] is included in [a, b]. \u00a0Not the other way around! \u00a0Higher in the model means a better approximation to a (possibly unknown) real value, hence a smaller interval. \u00a0Maximal elements are one-point intervals [a, a], which we equate with the real number a. \u00a0So, for example, a chain of better and better approximations to a real number may be [-2, 9], [1, 4], [2.9, 3.8], [3.1, 3.2], [3.14, 3.15], [3.141, 3.142], etc. \u00a0(Can you say pi?) \u00a0Models have been used for giving domain-theoretic foundations to computation over the reals, following this intuition, for one: see papers by Edalat, and by Escard\u00f2, in particular.<\/p>\n

If a space\u00a0Y<\/em>\u00a0has a\u00a0\u03c9-continuous model\u00a0X<\/em>, that gives you countable chains of approximations for any element in\u00a0X<\/em>, with good mathematical properties. \u00a0A natural question, first attacked by Lawson [2], is: which spaces\u00a0Y<\/em>\u00a0have\u00a0\u03c9-continuous models\u00a0X<\/em>\u00a0at all?<\/p>\n

Lawson showed that, if you require the model\u00a0Y<\/em>\u00a0to occur as a subspace of\u00a0X<\/em>\u00a0independently of the fact that\u00a0X<\/em>\u00a0comes with its Scott or Lawson topology, then\u00a0Y<\/em>\u00a0must be Polish; and that every Polish space\u00a0Y<\/em>\u00a0has an\u00a0\u03c9-continuous model\u00a0X<\/em>.<\/p>\n

Martin improved this [3] and showed that the\u00a0T<\/em>3<\/sub>\u00a0spaces that have\u00a0\u03c9-continuous models are exactly the Polish spaces. \u00a0In particular, there is no hope of find a non-trivial theory of \u03c9-continuous models\u00a0for, say, analytic (Souslin) spaces.<\/p>\n

The argument behind this is given in Theorem 7.7.23 of the book<\/a>. \u00a0The key ingredient found by Martin is that:<\/p>\n

Any space that has an \u03c9-continuous model must be Choquet-complete.<\/p><\/blockquote>\n

In fact, countability is irrelevant in this remark, and every space with a continuous model (not necessarily\u00a0\u03c9-continuous) is Choquet-complete. \u00a0One can say even more: every space with a continuous model is\u00a0convergence<\/em>\u00a0Choquet-complete, see Proposition 7.7.19 and Exercise 7.7.20 in the book<\/a>.<\/p>\n

Conversely, any space\u00a0X<\/em>\u00a0with an\u00a0\u03c9-continuous model\u00a0Y<\/em>\u00a0must be countably-based. \u00a0Using Norberg’s Lemma (Lemma 7.7.13 in the book<\/a>), indeed,\u00a0X<\/em>\u00a0must be countably-based in its Scott topology, hence also its subspace\u00a0Y<\/em>.<\/p>\n

Also,\u00a0any space\u00a0X<\/em>\u00a0with an\u00a0\u03c9-continuous model\u00a0Y<\/em>\u00a0must be\u00a0T<\/em>1<\/sub>. This is because any two distinct maximal elements of\u00a0Y<\/em>\u00a0must be incomparable.<\/p>\n

Mummert and Stephan [1, Theorem 6.1 and Corollary 6.3] showed that:<\/p>\n

Let\u00a0X<\/em>\u00a0be countably-based and\u00a0T<\/em>1<\/sub>. \u00a0\u00a0X<\/em>\u00a0has\u00a0an\u00a0\u03c9-continuous model\u00a0Y<\/em>\u00a0if and only if\u00a0X<\/em>\u00a0is Choquet-complete.<\/p><\/blockquote>\n

Moreover, we can even take\u00a0Y<\/em>\u00a0to be\u00a0\u03c9-algebraic. \u00a0The latter is relatively easy: if\u00a0Y<\/em>\u00a0is an\u00a0an\u00a0\u03c9-continuous model\u00a0<\/em>of\u00a0X<\/em>, then its ideal completion will be an\u00a0an\u00a0\u03c9-algebraic model of\u00a0X.<\/em>\u00a0The result quoted above is more intricate, and Mummert and Stephan obtain it through a study of so-called countably-based MF spaces, which, as they show, are exactly the countably-based,\u00a0T<\/em>1<\/sub>, Choquet-complete spaces, up to homeomorphism (their Theorem 5.3). \u00a0An MF space is defined as the set of maximal filters of a poset, with the hull kernel topology. \u00a0See [1] for more details.<\/p>\n

This is rather amazing. \u00a0This says that, provided we work with countably-based and\u00a0T<\/em>1\u00a0<\/sub>spaces, Choquet-completeness and having\u00a0an\u00a0\u03c9-continuous model are\u00a0the same thing<\/em>.<\/p>\n

Thanks to Matthew de Brecht for giving me a pointer to the paper [1].<\/p>\n

\u2014 Jean Goubault-Larrecq<\/a>\u00a0(October 12th, 2012)\"jgl-2011\"<\/p>\n

[1] Carl Mummert and Frank Stephan. \u00a0Topological aspects of poset spaces<\/em>. \u00a0Michigan Mathematical Journal, 59, 2010, 3-24.<\/p>\n

[2] Jimmie D. Lawson. \u00a0Spaces of maximal points<\/em>. \u00a0Mathematical Structures in Computer Science, 7, 1997, 543-555.<\/p>\n

[3] Keye Martin. \u00a0The regular spaces with countably based models<\/em>. \u00a0Theoretical Computer Science, 305, vol. 1-3, 2003, 299-310.<\/p>\n","protected":false},"excerpt":{"rendered":"

A model of a space\u00a0X\u00a0is a dcpo\u00a0Y\u00a0whose subspace of maximal elements is isomorphic to\u00a0X. \u00a0Of particular importance are those spaces that have \u03c9-continuous models. \u00a0Martin [3], and Mummert and Stephan [1] came very close to characterize them exactly. Let’s review … Continue reading →<\/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-70","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=\/wp\/v2\/pages\/70","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=70"}],"version-history":[{"count":7,"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=\/wp\/v2\/pages\/70\/revisions"}],"predecessor-version":[{"id":5974,"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=\/wp\/v2\/pages\/70\/revisions\/5974"}],"wp:attachment":[{"href":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=70"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}