On quasi-uniform space via strong quasi-uniform covers

Abstract

A quasi-uniformity, unlike a uniformity, is not determined by its quasi-uniform covers. It is generally known that covering space theory applies to topological spaces that are connected. In this dissertation, we will define topology induced by strong quasi-uniform cover which will determine the quasi-uniformity of a quasi-uniform space. And then we will expand the discussion on strong quasi-uniform covers, detailing how quasi-uniform spaces via strong quasi-uniform covers give rise to a topological space with some added axioms which will precisely determine the quasi-uniformity of a given quasi-uniform space.. The dissertation will further detail on how uniformizable topological spaces are precisely the completely regular spaces. This erupts from the idea that uniform spaces can be induced by uniform covers, but in this article, we want to ascertain that the same concept can be applied with quasi-uniform spaces via strong quasi-uniform covers.