{"id":681,"date":"2015-10-25T17:25:25","date_gmt":"2015-10-25T16:25:25","guid":{"rendered":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/?page_id=681"},"modified":"2022-11-19T15:26:09","modified_gmt":"2022-11-19T14:26:09","slug":"ideal-models-i","status":"publish","type":"page","link":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/?page_id=681","title":{"rendered":"Ideal models I"},"content":{"rendered":"

A few months ago, Keye Martin<\/a> drew my attention to his results on so-called ideal models of spaces [1]. \u00a0Ideal domains are incredibly specific dcpos: they are defined as dcpos where each non-finite element is maximal. \u00a0Despite this, Keye Martin was able to show that: (1) every space that has an \u03c9-continuous model has an ideal model, that is, a model that is an ideal domain; (2) the metrizable spaces that have an ideal model are exactly the completely metrizable spaces.<\/p>\n

I will try to expose a few of his ideas here. \u00a0I will probably betray him a lot. \u00a0For example, I will not talk about measurements (one of Keye’s inventions), and I will not stress the role of Choquet-completeness to go beyond “Lawson at the top” domains, or the role of G<\/em>\u03b4<\/sub> subsets so much. \u00a0I have also tried to extend whatever I can to the case of quasi-metric, not just metric, spaces… unfortunately, I failed to achieve this.<\/p>\n

A basic example<\/h2>\n

The prime example of an ideal domain is the infinite binary tree. \u00a0Take finite and infinite words over the two-letter alphabet {0, 1}, and order them by prefix. \u00a0So the empty word is the least element, and is below, say, 0, which is below 01, which is below 011, and so on. In that dcpo, the finite elements are the finite words, and the maximal elements are the infinite words. \u00a0This is an ideal domain.<\/p>\n

Ideal \u27f9 algebraic<\/h2>\n

In general, ideal domains have plenty of properties.<\/p>\n

For one, they are all algebraic dcpos<\/em>.<\/p>\n

For that, we show that every element\u00a0x<\/em> is the supremum of a directed family of finite elements. \u00a0If\u00a0x<\/em> is itself finite, this is clear. \u00a0Otherwise, by the definition of “finite”, and since x<\/em> is not finite, there must be a directed family (xi<\/sub><\/em>)i<\/sub><\/em> in <\/sub>I<\/sub><\/em> of elements whose supremum is above x<\/em>, and such that no xi<\/sub><\/em> is above x<\/em>. \u00a0Since we are in an ideal domain,\u00a0x<\/em> is maximal, so the supremum is equal to\u00a0x<\/em>.\u00a0 Since no xi<\/sub><\/em> is equal to x<\/em>, none is maximal, so they are all finite, and we are done.<\/p>\n

Ideal\u00a0\u27f9 first-countable<\/h2>\n

An ideal domain is not just algebraic: every element is the supremum of a countable chain<\/em> of finite elements.<\/p>\n

Consider any point\u00a0x<\/em> in the chosen ideal domain. \u00a0If\u00a0x<\/em> is finite, then we can take the constant chain whose elements are all equal to x<\/em>.<\/p>\n

Otherwise, let D<\/em> be a directed family of finite elements whose supremum is x<\/em>. Pick one element x<\/em>1<\/sub> from D<\/em>. Since x<\/em>1<\/sub> is finite and x<\/em> is not, there is a strictly larger element in D<\/em>, call it x<\/em>2<\/sub> in D<\/em>, and then an even strictly larger element x<\/em>3<\/sub>, etc. The sequence of elements xn<\/sub><\/em>, n<\/em> \u2208 N<\/strong>, has a supremum x<\/em>‘, and x<\/em>‘ cannot be finite, since otherwise x<\/em>‘ would be below some xn<\/sub><\/em>, contradicting the fact that the sequence is strictly increasing. Hence x<\/em>‘ is maximal, and since it is clearly below x<\/em>, it coincides with it.<\/p>\n

It follows that an ideal domain is first-countable<\/em>: every point has a countable basis of open neighborhoods. Indeed, for every point x<\/em>, write x<\/em> as the supremum of a countable chain xn<\/sub><\/em>, n<\/em> \u2208 N<\/strong>: the open subsets \u2191xn<\/sub><\/em> form the desired basis of open neighborhoods of x<\/em>.<\/p>\n

In this post, I will not make use of first-countability. \u00a0But remember that for next time!<\/p>\n

Constructing an ideal domain of formal balls<\/span><\/h2>\n

First note that if\u00a0X<\/em>,\u00a0d<\/em>\u00a0is a metric space, then its space of formal balls\u00a0B<\/strong>(X<\/em>) is not an ideal domain, and in fact is not even algebraic (unless\u00a0X<\/em>\u00a0is empty): if (x<\/em>,\u00a0r<\/em>) were finite in\u00a0B<\/strong>(X<\/em>), since it is the supremum of the chain of formal balls (x<\/em>,\u00a0r<\/em>+\u03b5), \u03b5>0, it would be equal to one of them, which is absurd.<\/p>\n

Keye Martin has a general construction for building an ideal domain from G<\/em>\u03b4<\/sub> subsets of a continuous dcpo. I said I would betray him, and I will now. Instead of showing you the general construction, let me show you how he would build an ideal domain for a complete metric space X<\/em>, by changing the ordering on its space of formal balls B<\/strong>(X<\/em>).<\/p>\n

Every complete metric space is in particular Smyth-complete, in particular continuous Yoneda-complete. \u00a0What that means is that its dcpo of formal balls B<\/strong>(X<\/em>) is a continuous dcpo (Definition 7.4.72). In that case, let\u00a0B<\/strong>D<\/sub><\/em>(X<\/em>) be just the set of formal balls on X<\/em> again, except with a modified ordering: (x<\/em>,\u00a0r<\/em>) \u2291 (y<\/em>,\u00a0s<\/em>) if and only if:<\/p>\n