Metrizability of CHART groups

For compact Hausdorff admissible right topological (CHART) group $G$, we prove $w(G)=πχ(G)$. 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 Fréchet (2013, Glasner, Megrelishvili), or $G$ has countable $π$-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.

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