Metrizability of CHART groups

TL;DR

This paper investigates the conditions under which compact Hausdorff admissible right topological (CHART) groups are metrizable, extending known results from topological groups and establishing new criteria involving countability and compactness.

Contribution

The paper proves that for CHART groups, the weight equals the pi-character, and it extends metrizability criteria to CHART groups under various conditions, including the continuum hypothesis.

Findings

w(G)=πχ(G) for CHART groups

Metrizability criteria extend to CHART groups with countability conditions

Sequentially compact CHART groups are metrizable under CH

Abstract

For compact Hausdorff admissible right topological (CHART) group $G$, we prove $w(G)=\pi\chi(G)$. This equality is well known for compact topological groups. This implies the criteria for the metrizability of CHART groups: if $G$ is first-countable (2013, Moors, Namioka) or $G$ is Fr\'echet (2013, Glasner, Megrelishvili), or $G$ has countable $\pi$-character (2022, Reznichenko) then $G$ is metrizable. Under the continuum hypothesis (CH) assumption, a sequentially compact CHART group is metrizable. Namioka's theorem that metrizable CHART groups are topological groups extends to CHART groups with small weight.