Localic Products of Spaces

Abstract

The subject is spatiality of localic products of topological spaces, in particular metrizable spaces, or equivalently, preservation of products under the embedding of the category of sober spaces into the category of locales. A key theorem characterizes spatiality of (localic) products in terms of winning strategies of a strictly determined topological game. Further results concern metrizable spaces. The product of two metrizable spaces X1 and X2 is non‐spatial if and only if they have closed subspaces F1 and F2 respectively, which can be embedded into the Cantor set as dense subspaces with disjoint images. The locale X×loc Y is spatial for all Y if and only if X is complete and has no closed subspace homeomorphic to the irrationals. These spaces are called spatial multipliers. Using W. Hurewicz's characterization of strongly Baire spaces (every non‐empty closed subspace is of second category in itself) as spaces no closed subspace of which is homeomorphic to the rationals, we find that the characterization of spatial multipliers implies that a space which is not strongly Baire has spatial product only with the spatial multipliers. Accordingly for metrizable spaces the interesting questions lie within the strongly Baire spaces. The product of a complete space and a strongly Baire space is spatial. Spatiality of the localic product of two strongly Baire metrizable spaces implies that the topological product is strongly Baire. There is a Galois connection induced by the relation ‘X ∼ Y if and only if X×loc Y is spatial’. The poset of Galois‐closed classes contains subsets order isomorphic to the powerset of C−(C = 2N0). Finally, the question of absoluteness of Borel locales is considered, and it is shown that a metrizable locale which is Borel in every completion is spatial.

Some structure of Borel locales

“…This paper is very closely connected to my previous paper [2] and to Till Plewe's paper [4]. It is probably not advisable to try to read this paper without reading at least the introduction of [2].…”

“…The joins and meets are taken in the lattice of sublocales. It turns out-this is a major result of Plewe's [4]-that an F σδ Borel set which is not G δσ , in R, is never an F σδ sublocale.…”

“…It is probably not advisable to try to read this paper without reading at least the introduction of [2]. Nothing here depends on Plewe's work; but much of the interest of this paper does depend on [4].…”

Directed inverse limits of spatial locales

“…Since finite products of locally compact locales are locally compact [5,7] and hence spatial, we have the following corollary: Corollary 1. Arbitrary products of compact locally compact locales are spatial.…”

“…seems to be rather difficult. Results concerning preservation of products (and some applications) can be found in [2,7,8]. Here we consider preservation of directed inverse limits; in particular Isbell's question whether a directed inverse limit of (quasi-)compact spatial locales is again spatial [1, p. 24].…”