Metrizability of CHART groups

Abstract

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.