Publication Date

Summer 2019

Advisor(s) - Committee Chair

Tom Richmond (Director), Melanie Autin, and Dominic Lanphier

Degree Program

Department of Mathematics

Degree Type

Master of Science

Abstract

This thesis explores functionally Alexandroff topologies and the order theory asso- ciated when considering the collection of such topologies on some set X. We present several theorems about the properties of these topologies as well as their partially ordered set.

The first chapter introduces functionally Alexandroff topologies and motivates why this work is of interest to topologists. This chapter explains the historical context of this relatively new type of topology and how this work relates to previous work in topology. Chapter 2 presents several theorems describing properties of functionally Alexandroff topologies ad presents a characterization for the functionally Alexandroff topologies on a finite set X. The third and fourth chapters present facts about the lattice of functionally Alexandroff topologies, with Chapter 4 being dedicated to an algorithm which generates a complement in this lattice.

Disciplines

Discrete Mathematics and Combinatorics | Geometry and Topology | Set Theory

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