Title:

Topics in general and settheoretic topology : slice sets, rigid subsets of the reals, Toronto spaces, cleavability, and 'neight'

I explore five topics in topology using settheoretic techniques. The first of these is a generalization of 2point sets called slice sets. I show that, for any smallincardinality 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, nondiscrete Toronto space that is not regular falls into one of two strictlydefined classes. I also prove that for every infinite cardinality there are precisely 3 nonT1 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.
