A Hofmann-Mislove Theorem for Scott open sets

Abstract

We consider the intersection map on the family of non-empty $\omega$-Scott-open sets of the lattice of opens of a topological space. We prove that in a certain class of topological spaces the intersection map forms a continuous retraction onto the space of countably compact subsets of the space equipped with (the sequentialisation of) the upper Vietoris topology. This class consists of all sequential spaces which are sequentially Hausdorff.