## Theses and Dissertations @ UNI

#### Availability

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)

#### Copyright

©2017 Eric Scheidecker

#### Language

EN

#### File Format

application/pdf

#### Recommended Citation

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

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