On The Whisker Topology

TL;DR

This paper investigates the properties of the whisker topology on the fundamental group, demonstrating its behavior in product spaces, resolving an open question about its structure, and analyzing its separability in specific cases.

Contribution

It establishes new properties of the whisker topology, including product preservation, existence of a non-discrete non-abelian Hausdorff example, and non-separability on the earring group.

Findings

Whisker topology preserves products.

Existence of a space with non-discrete, non-abelian, Hausdorff whisker topology.

Whisker topology is not separable on the earring group.

Abstract

The purpose of this paper is to explore properties of the whisker topology, which is a topology endowed on the fundamental group and whose utility is to detect locally complicated phenomena in pathological topological spaces. We show that the whisker topology preserves products, resolve an open question regarding the existence of a space which makes $\pi_1^{wh}(X,x_0)$ a non-discrete, non-abelian, and Hausdorff topological group, and show the whisker topology is not separable on the earring group $\pi_1(\Er^1,x_0)$.