Topics in general and set-theoretic topology: slice sets, rigid subsets of the reals, Toronto spaces, cleavability, and neight

Thesis

I explore five topics in topology using set-theoretic techniques. The first of these is a generalization of 2-point sets called slice sets. I show that, for any small-in-cardinality subset A of the real line, there is a subset of the plane meeting every line in a topological copy of A. Under Martin's Axiom, I show how to improve this result to any totally disconnected A. Secondly, I show that it is consistent with and independent of ZFC to have a topologically rigid subset of the real line that is smaller than the continuum. Thirdly, I define and examine a new cardinal function related to cleavability. Fourthly, I explore the Toronto Problem and prove that any uncountable, Hausdorff, non-discrete Toronto space that is not regular falls into one of two strictly-defined classes. I also prove that for every infinite cardinality there are precisely 3 non-T1 Toronto spaces up to homeomorphism. Lastly, I examine a notion of dimension called the "neight", and prove several theorems that give a lower bound for this cardinal function.

Authors: Brian, W

• Publication date: 2013
• Type of award: DPhil
• Level of award: Doctoral
• Awarding institution: Oxford University, UK
• Language: English
• Keywords: rigid; forcing; homogeneity; slice set; real line; cleavage; 2-point set; Toronto problem; neight; cleavability; Toronto space
• Subjects: Mathematics; Mathematical logic and foundations; Analytic Topology or Topology