On the preservation of topological properties under group multiplication in topological groups

The Lindelöf property, cellularity, countable compactness, countable pracompactness, and pseudocompactness are not finitely productive properties. Multiplying subsets of a topological group does not preserve these properties either.

We continue the study started by A.V. Arhangel'skii a few years ago and show that if U is an open Lindelöf (countably cellular, or countably compact) subset of a topological group G and a subset F of G is Lindelöf (countably cellular, countably compact or countably pracompact), then the group products UF and FU are also Lindelöf (countably cellular, countably compact or countably pracompact) subspaces of G. Therefore, the open subgroup of G algebraically generated by $U\cup F$ is Lindelöf (countably cellular, or is the union of a countable family of open countably compact or countably pracompact subsets). Similarly, if U is an open pseudocompact subset of G and a set $F\subseteq G$ is pseudocompact, then the group products UF and FU are pseudocompact subspaces of G.

It is also established that if B and C are bounded subsets of a locally feebly compact paratopological group G, then the sets $B^{-1}$, $C^{-1}$ and BC are bounded in G. Hence, every bounded subset of G is contained in an open σ-bounded subgroup of G.