Compactly Choquet-complete spaces I: LCS-complete and Gδ subspaces

A Choquet-complete space is one in which player α has a winning strategy in the strong Choquet game. By winning, it is meant that the intersection of all the open sets played by α is non-empty. If you change the winning condition to “the open sets played by α form a neighborhood base of a point”, then you get the convergence Choquet-complete spaces of Dorais and Mummert. I will embark on a study of compactly Choquet-complete spaces, where the winning condition is “the open sets played by α form a neighborhood base of a non-empty compact set”. These were introduced by M. de Brecht, X. Jia, Zh. Lyu, and me a few years ago. In this first part, I will show that all locally compact sober space are compactly Choquet-complete, and that the class of compactly Choquet-complete spaces is closed under taking G_δ subspaces. I will take the opportunity to fill in a few gaps in the original paper, and to make precise one or two techniques in topological game theory. Read the full post.