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Title: Topics in general and set-theoretic topology : slice sets, rigid subsets of the reals, Toronto spaces, cleavability, and 'neight'
Author: Brian, William R.
ISNI:       0000 0004 2747 5436
Awarding Body: University of Oxford
Current Institution: University of Oxford
Date of Award: 2013
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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.
Supervisor: Knight, Robin W. Sponsor: Clarendon Fund
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID:  DOI: Not available
Keywords: Mathematics ; Analytic Topology or Topology ; Mathematical logic and foundations ; 2-point set ; slice set ; forcing ; homogeneity ; rigid ; real line ; cleavability ; cleavage ; Toronto space ; Toronto problem ; neight