Non-Hausdorff Topology and Domain Theory

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.

We will see a relatively recent theorem of Jimmie Lawson’s: consider a locally compact space X with a semiclosed ordering ≤, and a closed connected chain C inside X; then every closed interval inside C is compact in X. The proof is perhaps surprising, as it relies on Koch’s arc theorem. Read the full post.

Last time, we have seen that every arc chain is an arc. This month, we will see that every arc is an arc chain. The proof proceeds by inventing a suitable partial ordering, the cut-point ordering. We will also see that all second-countable arcs (or arc-chains) are isomorphic, in particular they are all isomorphic to [0, 1], and that there are non-second-countable arcs (or arc-chains) of the same cardinality that are not isomorphic. Read the full post.

Last month, we have seen that ℒ(X) is locally compact in its Scott topology (equivalently, core-compact, equivalently, stably compact) if and only if OX is locally compact in its Scott topology (equivalently, core-compact, equivalently, stably compact), and that those properties are equivalent to X being core-compact, provided that X is a poset with its Scott topology or a ⊙-consonant second-countable space. Matthew de Brecht observed that the latter condition can be relaxed. We will see that we only need X to be second-countable or consonant for the equivalence to hold. Read the full post.