A topological space $X$ is said to be $e$-separable if $X$ has a $\sigma$-closed-discrete dense subset. Recently, G. Gruenhage and D. Lutzer showed that $e$-separable PIGO spaces are perfect and asked if $e$-separable monotonically normal spaces are perfect in general. The main purpose of this article is to provide examples of $e$-separable monotonically normal spaces which are not perfect. Extremely normal $e$-separable spaces are shown to be stratifiable.

We show that there exists an Abelian topological group $G$ such that the operations in $G$ cannot be extended to the Dieudonné completion $\mu G$ of the space $G$ in such a way that $G$ becomes a topological subgroup of the topological group $\mu G$. This provides a complete answer to a question of V.G. Pestov and M.G. Tkačenko, dating back to 1985. We also identify new large classes of topological groups for which such an extension is possible. The technique developed also allows to find many new solutions to the...