Dissertations and Theses @ UNI

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Open Access Thesis

Keywords

Topological spaces;

Abstract

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

2017

Degree Name

Master of Arts

Department

Department of Mathematics

First Advisor

Adrienne Stanley

Date Original

2017

Object Description

1 PDF file (v, 26 pages)

Language

en

File Format

application/pdf

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