The Hawaiian earring group and metrizability

TL;DR

This paper investigates the topological properties of the Hawaiian earring group, demonstrating that it and related groups are nonmetrizable when endowed with the quotient topology from based loops.

Contribution

It establishes the nonmetrizability of the Hawaiian earring group and any retracts, highlighting fundamental topological limitations of these groups.

Findings

Hawaiian earring group is nonmetrizable with quotient topology

Fundamental groups of spaces retracting to the Hawaiian earring are also nonmetrizable

Provides insight into the topological complexity of certain fundamental groups

Abstract

Endowed with quotient topology inherited from the space of based loops, the fundamental group of the Hawaiian earring fails to be metrizable. The fundamental group of any space which retracts to the Hawaiian earring is also nonmetrizable.