The Topology of Ultrafilters as Subspaces of the Cantor Set and Other Topics

In the first part of this thesis (Chapter 1), we will identify ultrafilters on ω with subspaces of 2 through characteristic functions, and study their topological properties. More precisely, let P be one of the following topological properties. • P = being completely Baire. • P = countable dense homogeneity. • P = every closed subset has the perfect set property. We will show that, under Martin’s Axiom for countable posets, there exist non-principal ultrafilters U ,V ⊆ 2 such that U has property P and V does not have property P . The case ‘P = being completely Baire’ actually follows from a result obtained independently by Marciszewski, of which we were not aware (see Theorem 1.37 and the remarks following it). Using the same methods, still under Martin’s Axiom for countable posets, we will construct a non-principal ultrafilter U ⊆ 2 such that U is countable dense homogeneous. This consistently answers a question of Hrušák and Zamora Avilés. All of Chapter 1 is joint work with David Milovich. In the second part of the thesis (Chapter 2 and Chapter 3), we will study CLPcompactness and h-homogeneity, with an emphasis on products (especially infinite powers). Along the way, we will investigate the behaviour of clopen sets in products (see Section 2.1 and Section 3.2). In Chapter 2, we will construct a Hausdorff space X such that X is CLP-compact if and only if κ is finite. This answers a question of Steprāns and Šostak.