In 1958, Andrew Gleason proved that for every compact Hausdorff space X there exists an extremally disconnected compact Hausdorff space X' and a continuous surjection p: X' --> X with the property that every other continuous surjection from an extremally disconnected compact Hausdorff space onto X factors via surjection through p. Later, several authors have extended this construction to wider contexts, including the Gleason cover for an elementary topos introduced by Johnstone in 1980.
We investigate properties of the Gleason cover for not necessarily sober T_0 Alexandroff spaces, i. e. spaces determined by partially ordered sets. First, we introduce the notion of co-local homeomorphism for such spaces, and prove that for every finite T_0 topological space X there exists a unique irreducible co-local homeomorphism p: X' --> X from finite extremally disconnected space X' onto X. Next, we extend this approach to arbitrary Alexandroff topological spaces. We finish with several characterizations of Alexandroff spaces with Alexandroff Gleason covers.
Keywords: Gleason cover, Alexandroff space, co-local homeomorphism
2020 MSC: 06F30, 18F60, 18F70, 54B30, 54G05, 54D80
Theory and Applications of Categories, Vol. 39, 2023, No. 1, pp 1-20.