The Hawaiian earring group is topologically incomplete

TL;DR

This paper investigates the topological properties of the fundamental group of Hawaiian earring-like continua, revealing that these groups are uncountable, regular, but not Baire spaces, thus lacking compatible complete metrics.

Contribution

It demonstrates that the fundamental groups of certain Peano continua have two natural topological structures that are not Baire spaces, highlighting their topological incompleteness.

Findings

The fundamental group admits two natural topological group structures.

Neither of these structures is a Baire space.

The groups are uncountable and regular but not complete.

Abstract

The premier exhibition of the following phenomenon: The fundamental group of any Peano continuum constructed in similar fashion to the Hawaiian earring admits two natural distinct topological group structures. However despite being uncountable and regular, neither group is a Baire space and hence neither group admits a compatible complete metric.