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 o with subspaces of 2o 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 subsets of 2o 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 such that Uw is countable dense homogeneous. This consistently answers a question of Hrusak and Zamora Aviles. 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 CLP-compactness 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 Xkappa is CLP-compact if and only if kappa is finite. This answers a question of Steprans and Sostak.;In Chapter 3, we will resolve an issue left open by Terada, by showing that h-homogeneity is productive in the class of zero-dimensional spaces (see Corollary 3.27). Further positive results are Theorem 3.17 (based on a result of Kunen) and Corollary 3.15 (based on a result of Steprans). Corollary 3.29 and Theorem 3.31 generalize results of Motorov and Terada. Finally, we will show that a question of Terada (whether Xo is h-homogeneous for every zero-dimensional first-countable X) is equivalent to a question of Motorov (whether such an infinite power is always divisible by 2) and give some partial answers (see Proposition 3.38, Proposition 3.42, and Corollary 3.43). A positive answer would give a strengthening of a remarkable result by Dow and Pearl (see Theorem 3.34).