Semigroup extensions of abelian topological groups
In the paper it is stated that a topological group is called \textit{absolutely closed} if it is closed in every topological group containing it as a subgroup; and a topological group is absolutely closed if and only if it is complete [\textit{D. Raikov}, Izv. Akad. Nauk SSSR, Ser. Mat. 10, 513--528 (1946; Zbl 0061.04206)]. Given a topological group $G$ and a class $\mathcal{C}$ of topological semigroups, $G$ is called \textit{absolutely closed} in $\mathcal{C}$ if $G$ is closed in every semigroup $S\in \mathcal{C}$ containing $G$ as a subgroup. \par The author proves that an abelian topological group $G$ is absolutely closed in the class of topological semigroups if and only if $G$ is complete and there is an $n\in \mathbb{N}$ such that the subgroup $nG =\{nx : x \in G\}$ is totally bounded. If for every $n\in \mathbb{N}$, the subgroup $nG$ is not totally bounded, then the topology of $G$ can be extended to a semigroup topology $\tau$ on $G\times\mathbb{Z}^+$ in which $G$ is open and dense, and if $G$ is locally compact, so can be chosen $\tau$.
scientific article; zbMATH DE number 7036263