Subgroups of products of certain paratopological (semitopological) groups

In the first part of this note, we give some sufficient conditions under which a paratopological group is topologically isomorphic to a subgroup of a product of strongly metrizable paratopological groups. In the second part of this note, we show that a regular (Hausdorff, T1) semitopological group G admits a homeomorphic embedding as a subgroup into a product of regular (Hausdorff, T1) first-countable semitopological groups which are σ-spaces if and only if G is locally ω-good, ω-balanced, Ir(G)≤ω (Hs(G)≤ω, Sm(G)≤ω) and with the property that for every open neighborhood U of the identity e of G the cover {xU:x∈G} has a basic refinement F which is σ-discrete with respect to a countable family V of open neighborhoods of e. In the last part of this note, we give an internal characterization of projectively Ti second-countable semitopological groups, for i=0,1,2.