Open Access Thesis
Left-separated spaces are topological spaces which can be well ordered such that every initial segment is closed. In this paper, we examine what topological properties imply left-separation, and under what circumstances left-separation is preserved by unions. We also introduce several known theorems regarding elementary submodels as they are one of the primary tools that we use. We prove that for a topological space X;
1. If X has a point-countable base, then X is left-separated if and only if X has closed intersection with any elementary submodel M such that X ∈ M.
2. If every elementary submodel M with X ∈ M and |M| < λ has closed intersection with X, then X has a left-separated subspace of size λ whose initial segments are closed in X.
3. If X is locally countable and metalindelof, then X is left-separated.
4. If X is neat with |X| = κ + such that X is left-separated in order type κ + · α with α < κ, then d(X) < κ+, or X is the union of less than κ + many nowhere dense sets.
5. If X is left-separated in order type κ and Y is a topological space that is left-separated in order type ω1 such that X ∪ Y is locally countable, then X ∪ Y is left-separated in order type less than or equal to κ · 2.
We finish with several open questions that outline the general direction of our future work.
Year of Submission
Master of Arts
Department of Mathematics
1 PDF file (v, 26 pages)
©2017 Eric Scheidecker
Scheidecker, Eric, "Properties of left-separated spaces and their unions" (2017). Electronic Theses and Dissertations. 461.
Available for download on Saturday, December 22, 2018