On zero dimensional sequential spaces

TL;DR

This paper develops tools to identify zero-dimensional sequential spaces and demonstrates that the Hawaiian earring group, with a specific topology, is zero-dimensional and embeds into a free topological group, revealing new structural properties.

Contribution

It introduces methods to recognize zero-dimensional sequential spaces and proves the Hawaiian earring group is zero-dimensional and embeds into a free topological group.

Findings

Hawaiian earring group is 0-dimensional with the quotient topology.

The group is $T_4$ and embeds into the free topological group $F_M(G)$.

Tools for recognizing zero-dimensional sequential spaces are developed.

Abstract

We develop tools to recognize sequential spaces with large inductive dimension zero. We show the Hawaiian earring group $G$ is 0 dimensional, when endowed with the quotient topology, inherited from the space of based loops with the compact open topology. In particular $G$ is $T_4$ and hence inclusion $G \rightarrow F_M (G)$ is a topological embedding into the free topological group $F_M (G)$ in the sense of Markov.