Some properties of Pre-uniform spaces

Fucai Lin; Yufan Xie; Ting Wu; Meng Bao

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arXiv · 2203.10725

Some properties of Pre-uniform spaces

Fucai Lin·Yufan Xie·Ting Wu·Meng Bao

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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.

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Fucai Lin, Yufan Xie, Ting Wu, Meng Bao. 2022-03-21. Some properties of Pre-uniform spaces. https://arxiv.org/abs/2203.10725

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