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.

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