work of Corson [@Corson] provide a wide range of topological groups that are such examples. Our analysis of these examples strengthen some results of Hernandez [@CH] on the theory of $\aleph_0$-bounded groups.

Some terminology and notation
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For collections $\mathcal{A}$ and $\mathcal{B}$ the symbol ${{\sf S}_1}(\mathcal{A},\mathcal{B})$ denotes the statement that

> For each sequence $(A_n:n<\omega)$ of elements of $\mathcal{A}$ there is a sequence $(b_n:n<\omega)$ such that for each $n$ $b_n\in A_n$, and $\{b_n:n<\omega\}\in\mathcal{B}$.

Let $Y$ be a topological space. Then ${\mathcal{O}}$ denotes the collection of all open covers of $Y$. If the selection principle ${{\sf S}_1}({\mathcal{O}},{\mathcal{O}})$ holds for $Y$ we say $Y$ is a *Rothberger space* (or has the *Rothberger* property). This property is named after F. Rothberger who introduced it in his study of Borel’s notion of strong measure zero - [@rothberger38].

A subspace $X$ of the metric space $(Y,d)$ is a *strong measure zero* subspace if there is for each sequence $(\epsilon_n:n<\omega)$ of positive real numbers a partition $X=\cup_{n<\omega}X_n$ such that for each $n$ the $d$-diameter of $X_n$ is less than $\epsilon_n$. In [@smzpow] we characterized the strong measure zero subspaces of $\sigma$-compact metric spaces in terms of Ramseyan partition relations.

For a topological group $(G,*)$ and a 