The Two-Faced God Janus or What Does n-Hausdorfness Have to Do With Dynamics and Topology?

This thesis is centered around the study of topological dynamics and analytic topology, as well as an unexpected intersection between the two, which revolves around the notion of an n-Hausdorff space. In the Dynamics part of this thesis, we discuss the author's two main results in topological dynamics. The first is about the Ellis semigroup of substitution systems, which extends previous results in this area. It states that the Ellis semigroup of a certain type of constant-length substitution dynamical systems has two minimal ideals, and further calculates the number of idempotents in these ideals. This requires a novel approach towards considering the factor maps to the maximal equicontinuous factor of these dynamical systems - a reworking of an old theorem which takes up a chapter in the thesis. The second result is about the Furstenberg topology of a point-distal dynamical system. Since the constantlength substitution systems we had considered in the previous sections are also point-distal, it can be considered a rather general result. It shows that if a pointdistal system is an almost k-to-1 extension of its maximal equicontinuous factor, the Furstenberg topology restricted to a (in some sense canonical) subspace is at most k + 1-Hausdorff. In the Analytic Topology part of the thesis, we discuss the n-Hausdorff property in its original context, as a natural part of a series of combinatorial generalisations of separation axioms. These combinatorial generalisations were introduced by several authors throughout the past 20 years. However, n-Hausdorfness in particular is interesting in light of a couple of still-open questions of Arhangelskii. The more easily stateable of the two is whether the cardinality of a T1 first countable Lindel of space exceeds continuum. The main work of the author in this part involves the many examples of spaces which satisfy a combinatorial separation axiom and also have (or lack) various other properties, such as being Lindel of, first countable, compact, or being T1. The author has contributed towards the proofs of the theorems given in this part.