neighborhood $U$ of its identity element we define: ${\mathcal{O}}(U)=\{x*U:x\in G\}$. Then ${\mathcal{O}}(U)$ is an open cover of $G$. ${\mathcal{O}}_{nbd}$ denotes the family of all open covers of $G$ of the form ${\mathcal{O}}(U)$. A topological group is said to be *Rothberger bounded* if it has the property ${{\sf S}_1}({\mathcal{O}}_{nbd},{\mathcal{O}})$. For $X$ be a subset of the topological space $G$ let ${\mathcal{O}}_X$ denote the covers of $X$ by sets open in $G$. Then $X$ is said to be Rothberger bounded in $G$ if ${{\sf S}_1}({\mathcal{O}}_{nbd},{\mathcal{O}}_X)$ holds. In topological groups the property ${{\sf S}_1}({\mathcal{O}},{\mathcal{O}})$ is generally stronger than ${{\sf S}_1}({\mathcal{O}}_{nbd},{\mathcal{O}})$.

The symbol ${{\sf G}_1}(\mathcal{A},\mathcal{B})$ denotes the following game of length $\omega$: Players ONE and TWO play an inning per $n<\omega$. In inning $n$ ONE first selects a member $O_n\in\mathcal{A}$, and then TWO responds by choosing a $T_n\in O_n$. A play $(O_0,\, T_0,\, \cdots,\, O_n,\, T_n,\,\cdots)$ is won by TWO if $\{T_n:n<\omega\}\in\mathcal{B}$; else, it is won by ONE. F. Galvin [@Galvin] introduced the game ${{\sf G}_1}({\mathcal{O}},{\mathcal{O}})$ and proved that it is related as follows to the well-known point-open game[^1]: ONE has a winning strategy in the point-open game if, and only if, TWO has a winning strategy in ${{\sf G}_1}({\mathcal{O}},{\mathcal{O}})$. TWO has a winning 