Topological fundamental groups and small generated coverings

TL;DR

This paper investigates the properties of the small generated subgroup of the quasitopological fundamental group, its impact on universal coverings, and introduces conditions for their existence and topological group structure.

Contribution

It introduces small generated spaces and coverings, provides a criterion for their existence, and establishes when the fundamental group forms a topological group.

Findings

Small generated spaces have indiscrete fundamental groups.

Necessary and sufficient conditions for small generated coverings are provided.

Fundamental groups of semi-locally small generated spaces are topological groups.

Abstract

This paper is devoted to study some topological properties of the SG subgroup, $\pi_1^{sg}(X,x)$, of the quasitopological fundamental group of a based space $(X,x)$, $\pt$, its topological properties as a subgroup of the topological fundamental group $\pi_1^{\tau}(X,x)$ and its influence on the existence of universal covering of $X$. First, we introduce small generated spaces which have indiscrete topological fundamental groups and also small generated coverings which are universal coverings in the categorical sense. Second, we give a necessary and sufficient condition for the existence of the small generated coverings. Finally, by introducing the notion of semi-locally small generatedness we show that the quasitopological fundamental groups of semi-locally small generated spaces are topological groups.