{"id":1737,"date":"2019-03-24T19:36:21","date_gmt":"2019-03-24T18:36:21","guid":{"rendered":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/?page_id=1737"},"modified":"2023-06-16T18:55:01","modified_gmt":"2023-06-16T16:55:01","slug":"countably-presented-locales","status":"publish","type":"page","link":"https:\/\/projects.lsv.ens-paris-saclay.fr\/topology\/?page_id=1737","title":{"rendered":"Countably presented locales"},"content":{"rendered":"\n

Reinhold Heckmann has a very nice paper [1] in which he proves the following: every countably presented locale is spatial. As he himself mentions, the result was first proved by Fourman and Grayson in 1982. The new thing is that he shows how tightly this is connected with the Baire property. This also gives a localic description of Matthew de Brecht’s quasi-Polish spaces, as those whose frame of open sets is countably presented.<\/p>\n\n\n\n

Completely Baire spaces<\/h2>\n\n\n\n

A topological space is Baire<\/em> if and only if every countable intersection of dense open sets is dense. All complete metric spaces are Baire, but there are more. The following was first proved by John Isbell in a localic context [2]. That is Proposition 8.3.24 in the book<\/a>.<\/p>\n\n\n\n

Lemma 1.<\/strong> Every locally compact sober space is Baire.<\/p>\n\n\n\n

One can say more. A space is completely Baire<\/em> if and only if all its closed subspaces are Baire. There are Baire spaces that are not completely Baire. For example, take the set Q<\/strong> of rational numbers with the usual metric topology. This is not Baire, since the intersection of the open sets Q<\/strong>\u2013{q<\/em>},<\/em> q<\/em> \u2208<\/em> Q<\/strong>, is empty. However, Q <\/strong>with a new top element added above all rational numbers, and with the topology whose open sets are the open sets of Q<\/strong> union top, plus the empty set, is Baire… because every non-empty open set contains the top element. And Q <\/strong>is a closed subspace of that, which is not Baire. (In a previous version, I used N<\/strong> instead of Q<\/strong>, but that is silly, as Zhenchao Lyu pointed out to me.)<\/p>\n\n\n\n

Lemma 2.<\/strong> Every locally compact sober space is completely Baire.<\/p>\n\n\n\n

Indeed, it is easy to see that every closed subspace of a locally compact sober space is locally compact and sober.<\/p>\n\n\n\n

Heckmann calls the completely Baire spaces spaces with the local Baire<\/em> property. I am using the name that Matthew de Brecht uses in his papers.<\/p>\n\n\n\n

\u03a0<\/strong>0<\/sup>2<\/sub> subsets<\/h2>\n\n\n\n

Matthew de Brecht does a lot with so-called \u03a0<\/strong>0<\/sup>2<\/sub> subsets [3], and so does Heckmann, under a different name.<\/p>\n\n\n\n

Let us fix a topological space X<\/em>. Following Heckmann, let us call UCO subset<\/em> of X<\/em> any space that is the union of a closed subset C<\/em> and of an open subset V<\/em> of X<\/em>. (UCO stands for “union of closed and open”.)<\/p>\n\n\n\n

If we write C<\/em> as the complement of an open set U<\/em>, then our UCO set can be written as the set of points x<\/em> such that x<\/em> \u2208 U<\/em> implies x<\/em> \u2208 V<\/em>. (Recall that “A<\/em> implies B<\/em>” means the same thing as “not A<\/em> or B<\/em>“.) Then we can write this set under the convenient notation U<\/em> imp<\/strong> V<\/em>, which I will also use even if U<\/em> is not open. (I will not use \u21d2 for implication imp<\/strong>, because that would be in conflict with the residuation operation on frames which we will use below.)<\/p>\n\n\n\n

A \u03a0<\/strong>0<\/sup>2<\/sub> subset of X<\/em> is then a countable intersection of UCO sets, namely the subsets of all points satisfying all implications x<\/em> \u2208 Un<\/sub><\/em> implies x<\/em> \u2208 Vn<\/sub><\/em>, where Un<\/sub><\/em> and Vn<\/sub><\/em> are fixed open subsets of X<\/em>, n<\/em> \u2208 N<\/strong>.<\/p>\n\n\n\n

Note that all open subsets, all closed subsets, and all G\u03b4<\/sub> subsets, are \u03a0<\/strong>0<\/sup>2<\/sub>.<\/p>\n\n\n\n

Heckmann notices that the Baire property extends to \u03a0<\/strong>0<\/sup>2<\/sub> subsets.<\/p>\n\n\n\n

Proposition 3.<\/strong> In a Baire space, every countable intersection of dense \u03a0<\/strong>0<\/sup>2<\/sub> subsets is dense.<\/p>\n\n\n\n

Proof. Let X<\/em> be a Baire space. If A<\/em> is dense, and is a countable intersection of subsets, then each of these subsets is dense. Moreover, a countable intersection of countable intersections of UCO sets is a countable intersection of UCO sets. With these remarks in mind, we see that it is enough to show that every countable intersection of dense UCO subsets Un<\/sub><\/em> imp<\/strong> Vn<\/sub><\/em>, is dense.<\/p>\n\n\n\n

Let On<\/sub><\/em> be the set cl(Un<\/sub><\/em>) imp<\/strong> Vn<\/sub><\/em>, where cl(Un<\/sub><\/em>) is the closure of Un<\/sub><\/em>. Note that On<\/sub><\/em> is open. We claim that On<\/sub><\/em> is dense. For every non-empty open set U<\/em>, we must show that U<\/em> intersects On<\/sub><\/em>. We consider two cases:<\/p>\n\n\n\n