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.
| Identifer | oai:union.ndltd.org:bl.uk/oai:ethos.bl.uk:581400 |
| Date | January 2013 |
| Creators | Brian, William R. |
| Contributors | Knight, Robin W. |
| Publisher | University of Oxford |
| Source Sets | Ethos UK |
| Detected Language | English |
| Type | Electronic Thesis or Dissertation |
| Source | http://ora.ox.ac.uk/objects/uuid:d80b6234-91e4-4c80-8696-a6de7aabc985 |
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