Choquet-Wilker spaces

A Choquet-Wilker space is a space in which the collection of compact saturated subsets is rich: one where, given a compact saturated set Q included in a union U₁ ∪ U₂ of two open sets, Q is included in a union Q₁ ∪ Q₂ of two compact saturated sets, with Q₁ ⊆ U₁ and Q₂ ⊆ U₂. This condition is sometimes called Wilker’s condition, but dates back to (at least) Choquet. We will see that many topological spaces are Choquet-Wilker, but not all are: we will see that the Alexandroff compactification of the space Q of rational numbers is not Choquet-Wilker. I will conclude with a neat observation, due to M. de Brecht and K. Keimel, on the continuity of the binary union operation on the lattice of open sets. Read the full post.