Some properties of Pre-uniform spaces

TL;DR

This paper introduces and explores properties of pre-uniform spaces and pre-proximities, highlighting differences from existing definitions, and demonstrates their implications in topology and group theory.

Contribution

It defines a new version of pre-uniformity, proves regularity of pre-uniform pre-topologies, and introduces methods to generate pre-uniformities and pre-proximities.

Findings

Pre-uniform pre-topologies are regular.

Existence of non-discrete pre-uniform structures on finite sets.

Strongly pre-topological groups are completely regular.

Abstract

In this paper, we introduce the notions of pre-uniform spaces and pre-proximities and investigate some basic properties about them, where the definition of pre-uniformity here is different with the pre-uniformities which are studied in \cite{BR2016}, \cite{GM2007} and \cite{K2016} respectively. First, we prove that each pre-uniform pre-topology is regular, and give an example to show that there exists a pre-uniform structure on a finite set such that the pre-uniform pre-topology is not discrete. Moreover, we give three methods of generating (strongly) pre-uniformities, that is, the definition of a pre-base, a family of strongly pre-uniform covers, or a family of strongly pre-uniform pseudometrics. As an application, we show that each strongly pre-topological group is completely regular. Finally, we pose the concept of the pre-proximity on a set and discuss some properties of the pre-proximity.