Metrizable and weakly metrizable coset spaces (Q1998820)

Metrization theorems play an essential role in general topology and analysis. The classical Birkhoff-Kakutani Theorem states that a topological group \(G\) is metrizable if and only if it is \(T_1\) and first-countable. The condition of being first-countable can be weakened with some additional properties which hold automatically for first-countable spaces. A topological space \(X\) is called a \textit{coset space} if it is homeomorphic to a quotient space \(G/H\) of left/right cosets for some topological group \(G\) and its closed subgroup \(H\). One can naturally ask: \textit{Find metrization conditions for coset spaces.} It is not surprising that in general to be first-countable is only a necessary condition but not sufficient. In this interesting article, generalizing some results for topological groups, the authors give several criteria for the coset space \(G/H\) to be a metric space. The following theorem is the main result of the article. \textbf{Theorem:} For a topological group \(G\) and its closed neutral subgroup \(H\) the following assertions are equivalent: (1) \(G/H\) is a metrizable space; (2) \(G/H\) is a bisequential space; (3) \(G/H\) is a weakly first-countable space; (4) \(G/H\) is a Fréchet-Urysohn space with an \(\omega^\omega\)-base. Several conditions of being a first-countable space are obtained for coset spaces \(G/H\), where \(G\) is a semitopological, quasitopological or paratopological group and \(H\) is a closed neutral subgroup.