On Topologized Fundamental Groups with Small Loop Transfer Viewpoints

TL;DR

This paper investigates the properties of topologized fundamental groups with small loop transfer conditions, analyzing when they form topological groups and providing examples of their behaviors.

Contribution

It introduces small loop transfer spaces to study the topological structures of fundamental groups and establishes conditions for their topological group properties.

Findings

Conditions for $ ext{pi}_1^{qtop}$ and $ ext{pi}_1^{wh}$ to coincide

Criteria for these groups to be topological groups

Examples showing the failure of reverse implications

Abstract

In this paper, by introducing some kind of small loop transfer spaces at a point, we study the behavior of topologized fundamental groups with the compact-open topology and the whisker topology, $\pi_{1}^{qtop}(X,x_{0})$ and $\pi_{1}^{wh}(X,x_{0})$, respectively. In particular, we give necessary or sufficient conditions for coincidence and being topological group of these two topologized fundamental groups. Finally, we give some examples to show that the reverse of some of these implications do not hold, in general.