On endomorphisms of abelian topological groups

Abstract:

A family $\Phi$ of continuous endomorphisms of a topological group $G$ is said to be small if for every subgroup $H$ of $G$ of cardinality ${\text {card}}(H) < {\text {card}}(G)$ there exists an element $g \in G$ such that $\Phi g \cap H = \emptyset$. M. I. Kabenjuk [5] proved that if $G$ is a compact connected Hausdorff abelian group of countable weight then every countable family $\Phi$ of nontrivial endomorphisms of $G$ is small. He asked if "compact" can be replaced by "complete". In this note the answer is given in the negative, but it is shown that "compact" can be replaced by "locally compact".