Spanier spaces and covering theory of non-homotopically path Hausdorff spaces

TL;DR

This paper explores the concept of Spanier spaces, introduces Spanier coverings as universal coverings, and investigates their properties and conditions for existence in non-homotopically path Hausdorff spaces, linking them to topological fundamental groups.

Contribution

It provides the first example of Spanier spaces, establishes conditions for Spanier coverings, and analyzes their topological properties and implications for fundamental groups.

Findings

Provided an example of Spanier spaces

Established necessary and sufficient conditions for Spanier coverings

Derived criteria for Hausdorffness of topological fundamental groups

Abstract

H. Fischer et al. (Topology and its Application, 158 (2011) 397-408.) introduced the Spanier group of a based space $(X,x)$ which is denoted by $\psp$. By a Spanier space we mean a space $X$ such that $\psp=\pi_1(X,x)$, for every $x\in X$. In this paper, first we give an example of Spanier spaces. Then we study the influence of the Spanier group on covering theory and introduce Spanier coverings which are universal coverings in the categorical sense. Second, we give a necessary and sufficient condition for the existence of Spanier coverings for non-homotopically path Hausdorff spaces. Finally, we study the topological properties of Spanier groups and find out a criteria for the Hausdorffness of topological fundamental groups.