## Semester

Summer

## Date of Graduation

2023

## Document Type

Dissertation

## Degree Type

PhD

## College

Eberly College of Arts and Sciences

## Department

Mathematics

## Committee Chair

Jerzy Wojciechowski

## Committee Member

Hong-Jian Lai

## Committee Member

Rong Luo

## Committee Member

John Goldwasser

## Committee Member

Elaine Eschen

## Abstract

The purpose of this thesis is to present new different spaces as attempts to generalize the concept of topological vector spaces. A topological vector space, a well-known concept in mathematics, is a vector space over a field \mathbb{F} with a topology that makes the addition and scalar multiplication operations of the vector space continuous functions. The field \mathbb{F} is usually \mathbb{R} or \mathbb{C} with their standard topologies. Since every vector space is a finitary matroid, we define two spaces called finite matroidal spaces and matrological spaces by replacing the linear structure of the topological vector space with a finitary matroidal structure. The idea is to combine a finitary matroidal closure operator like the linear closure operator with a topological closure operator into a single closure operator called a common closure operator. Therefore, one may take a set with a finitary matroidal closure operator and a topological closure operator like the topological vector space. The study starts with basic definitions, some fundamental properties and a collection of examples. The finite matroidal spaces and matrological spaces are then presented. Furthermore, the idea of a common closure operator is introduced and then a discussion is given of when to obtain from a set and a common closure operator a finite matroidal space or a matrological space. Finally, relationships of topological vector spaces with both finite matroidal spaces and topological vector spaces are presented.

## Recommended Citation

Hamad, Ziyad M., "Finite Matroidal Spaces and Matrological Spaces" (2023). *Graduate Theses, Dissertations, and Problem Reports*. 12136.

https://researchrepository.wvu.edu/etd/12136

#### Included in

Analysis Commons, Discrete Mathematics and Combinatorics Commons, Geometry and Topology Commons