Box Resolvability

We say that a topological group $G$ is partially box $κ$-resolvable if there exist a dense subset $B$ of $G$ and a subset $A$ of $G$, $|A|=κ$ such that the subsets $\{ aB: a\in A\}$ are pairwise disjoint. If $G=AB$ then $G$ is called box $κ$-resolvable. We prove two theorems. If a topological group $G$ contains an injective convergent sequence then $G$ is box $ω$-resolvable. Every infinite totally bounded topological group $G$ is partially box $n$-resolvable for each natural number $n$, and $G$ is box $κ$-resolvable for each infinite cardinal $κ, κ<|G|$.

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