On dissolute spaces (Q810912)

The dissolution of a locale X (the term is due to the reviewer) is the locale \(X_ d\) whose frame of opens is the congruence lattice of the frame of opens of X; equivalently, it is characterized by the existence of a continuous map \(X_ d\to X\) inducing a bijection between arbitrary sublocales of X and closed sublocales of \(X_ d\) (that is, one dissolves a locale by declaring everything to be closed). It is thus something like the discrete modification of a topological space, although the construction \(X\to X_ d\) is not idempotent; in particular, locales of the form \(X_ d\) are highly disconnected. Not much is known in general about which locales can occur as \(X_ d\) for some X; this paper makes a start on the characterization of dissolute spaces, i.e. those which occur as spatial sublocales of some \(X_ d\). Sample result: a compact Hausdorff space whose third Hausdorff derivative is empty is homeomorphic with a dissolution (but if you substitute ``fourth'' for ``third'', he result fails).