Compact scattered subsets and a topological game

Showing that Q is not consonant is quite an ordeal. I have finally managed to understand one of the existing proofs of this fact, due to Costantini and Watson. This would be a bit too long to cover entirely in one post, so the bulk of the explanation will be for another time. Instead, I will explain why the compact subsets of Q are all scattered, and what it means, but the important point of this month’s post is that, reading between the lines, the Costantini-Watson argument relies on a property that I will characterize through the use of a topological game G(K), resembling the strong Choquet game, in which we will see that player I has a winning strategy if K is compact and scattered—and that is an if and only if in any T2 space. Read the full post.

This entry was posted in Uncategorized. Bookmark the permalink.