Discreteness and Homogeneity of the Topological Fundamental Group

TL;DR

This paper investigates the conditions under which the topological fundamental group of a space is discrete, linking it to the space being semilocally simply-connected, and explores its properties like homogeneity.

Contribution

It establishes a precise equivalence between discreteness of the topological fundamental group and semilocally simple-connectedness for locally path connected spaces.

Findings

Discreteness of the topological fundamental group characterizes semilocally simply-connected spaces.

The topological fundamental group is always a homogeneous space.

Functoriality remains an open question in the general case.

Abstract

For a locally path connected topological space, the topological fundamental group is discrete if and only if the space is semilocally simply-connected. While functoriality of the topological fundamental group for arbitrary topological spaces remains an open question, the topological fundamental group is always a homogeneous space.