## Electronic Theses and Dissertations

#### Award/Availability

Open Access Thesis

#### 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 metalindel¨of, 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.

#### Date of Award

2017

#### Degree Name

Master of Arts

#### Department

Department of Mathematics

#### First Advisor

Adrienne Stanley

#### Date Original

2017

#### Object Description

1 PDF file (v + 33 pages)

#### Copyright

©2017 - Eric Scheidecker

#### Language

EN

#### Recommended Citation

Scheidecker, Eric, "Properties of left-separated spaces and their unions" (2017). *Electronic Theses and Dissertations*. 461.

https://scholarworks.uni.edu/etd/461