Subgroups of products of strongly metrizable semitopological groups (Q1632742)

A family \(\mathcal U\) of subsets of a subset \(X\) is \textit{star-finite} if each \(U\in\mathcal U\) intersects only finitely many members of \(\mathcal U\). If \(\mathcal U\) can be decomposed into a countable union of star-finite covers of \(X\), it is called a \(\sigma\)-\textit{star-finite cover}. A regular space \(X\) is called \textit{strongly metrizable} if it has a base that is a \(\sigma\)-star-finite cover. Every separable metrizable space is strongly metrizable and every strongly metrizable space is metrizable. Let \(G\) be a group with a topology. If the multiplication of the group is jointly (separately) continuous, then \(G\) is called a \textit{paratopological (semitopological) group}. If in addition the inversion in a paratopological group is continuous, then it is called a \textit{topological group}. Let \(G\) be a semitopological group. Denote by \(\mathcal{N}(e)\) the family of open neighborhoods of the identity \(e\) in \(G\). The group \(G\) is called \(\omega\)-\textit{balanced} if for every \(U\in\mathcal{N}(e)\) there exists a countable family \(\gamma\subseteq\mathcal N(e)\) such that for every \(x\in G\) there is \(V\in\gamma\) such that \(xVx^{-1}\subseteq U\). The group \(G\) is called \textit{strictly \(\omega\)-balanced} if there is a local base \(\mathcal{B}\) at the identity \(e\) that satisfies: for every \(U\in\mathcal{B}\), the family \(\{Ux:x\in G\}\) has a cover which is a \(\sigma\)-star-finite basic cover and it is dominated by a countable family \(\gamma\subseteq\mathcal{N}(e)\). The property of being strictly \(\omega\)-balanced implies \(\omega\)-balanced and it is closed under taking subgroups and direct products. Now let \(\mathcal{P}\) be a topological (algebraic) property. \(G\) is \textit{range-}\(\mathcal{P}\) if for every \(U\in\mathcal{N}(e)\) there exists a continuous homomorphism \(p\) from \(G\) to some semitopological group \(H\) with property \(\mathcal{P}\) such that \(p^{-1}(V)\subseteq U\), for some neighborhood \(V\) of the identity in \(H\). A regular semitopological group \(G\) has \textit{countable index of regularity} if for every \(U\in\mathcal{N}(e)\), one can find \(V\in\mathcal{N}(e)\) and a countable family \(\gamma\subseteq\mathcal{N}(e)\) such that \(\bigcap\limits_{W\in\gamma}VW^{-1}\subseteq U\). The first main result of the paper is the equivalence among the following properties for a regular paratopological (semitopological) group \(G\): (1) \(G\) is strictly \(\omega\)-balanced with a countable index of regularity; (2) \(G\) is range-strongly metrizable; (3) \(G\) is topologically isomorphic to a subgroup of a topological product of strongly metrizable paratopological (semitopological) groups. The authors obtain three other similar equivalences for a regular topological group \(G\): (1') \(G\) is strictly \(\omega\)-balanced; (2') \(G\) is range-strongly metrizable; (3') \(G\) is topologically isomorphic to a subgroup of a topological product of strongly metrizable topological groups. Applying this result, a characterization of a closed subgroup of a product of strongly metrizable topological groups is given. Finally, some open questions are listed.