Non-Hausdorff Topology and Domain Theory

In 1993, Knijnenburg studied lenses, and came up with a simple dcpo that shows that the topological and the ordinary Egli-Milner orderings on lenses can differ. This dcpo settles more than this question. For example, it is an algebraic dcpo that is neither coherent nor weakly Hausdorff. It has a lens whose closure differs from its downward closure, and it is a projective limit of weakly Hausdorff spaces that is not weakly Hausdorff, as we will see in the full post.

We continue our study of the class of compactly Choquet-complete spaces. We will show two things. First, that this class is closed under finite and countable products, but not under arbitrary products. While the technique for finite products consists in simulating plays in every space in parallel, the technique for countable products is a bit more subtle, and requires us to start the simulation of each individual game with a variable delay. Second, we will see that images of compactly Choquet-complete spaces under continuous, almost open maps, are still compactly Choquet-complete spaces, similarly to the cases of Choquet-complete and convergence Choquet-complete spaces, as dealt with in the book. Read the full post.

A completely Baire space is a space that is not only Baire, but whose closed subspaces are all Baire. We will continue our exploration of compactly Choquet-complete spaces, and we will see that they are all completely Baire. In fact, every compactly Choquet-complete space is Baire, and every closed subspace of a compactly Choquet-complete space is still compactly Choquet-complete. Read the full post.

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.