Skula spaces I: clopen selectors

A Skula space is a space that is obtained from another space X by giving it the Skula topology instead, which is generated by the open sets and the closed sets of the original space X. In 1990, Alan Dow and Stephen Watson gave a simple necessary and sufficient condition for a compact Hausdorff space to be Skula. I will show what it is—it involves the notion of clopen selectors in the title—, how it relates to the theory of Noetherian spaces, and I will end with a corollary of theirs about compact Hausdorff scattered spaces of scattering height at most 3 being Skula. That will be an opportunity to talk about scattered spaces, totally disconnected spaces, connected and quasi-connected components. I would also like start the full post with a tribute to Maurice Pouzet, a nice human fellow and a great mathematician, who passed away on December 31st, 2023.