Alex Simpson: Point-free Descriptive Set Theory (Part 1)
Descriptive set theory classifies subsets of a topological space according to their descriptive complexity: how difficult it is to describe a set in terms of structurally simple sets such as open and closed sets. The aim of this two-part series of talks is to develop a similar classification programme for locales: a generalisation of the notion of topological space to "point-free" spaces.
In Part 1, I shall review the notion of locale and how it "generalises" the notion of topological space to incorporate a new realm of point-free spaces alongside traditional set-theoretic spaces. Many concepts from topology have natural generalisations to the point-free setting. I shall focus in particular on the notion of sublocale, which is the point-free analogue of the notion of subspace in topology. It is a common theme in point-free topology that set-theoretic constructions from topology can be replaced by corresponding free-algebraic constructions in locale theory. In particular, a theorem of Joyal and Tierney achieves this for the notion of sublocale.
In Part 2, I shall develop a point-free analogue of the Borel hierarchy for locales. Such a programme was initiated by Isbell, who adapted the usual definition of the Borel hierarchy from subspaces to sublocales. I shall advocate a different approach, namely that the point-free Borel hierarchy should be construed in terms of free-algebraic constructions. This approach is justified by the theorem that, in the special case of Polish topological spaces (considered as the basis of most investigations in ordinary descriptive set theory), the induced point-free Borel hierarchy coincides with the standard set-theoretic one. Indeed this result generalises also to the broader class of quasi Polish spaces in the sense of de Brecht.