On 2-swelling topological groups

Abstract

A topological group $G$ is called 2-swelling if for any compact subsets $A,B\subset G$ and elements $a,b,c\in G$ the inclusions $aA\cup bB\subset A\cup B$ and $aA\cap bB\subset c(A\cap B)$ are equivalent to the equalities $aA\cup bB=A\cup B$ and $aA\cap bB=c(A\cap B)$. We prove that an (abelian) topological group $G$ is 2-swelling if each 3-generated (resp. 2-generated) subgroup of $G$ is discrete. This implies that the additive group $\mathbb Q$ of rationals is 2-swelling and each locally finite topological group is 2-swelling.