Abstract

We prove the following theorems: If X has strong measure zero and if Y has strong first category, then their algebraic sum has property s$_0$. If X has Hurewicz's covering property, then it has strong measure zero if, and only if, its algebraic sum with any first category set is a first category set. If X has strong measure zero and Hurewicz's covering property then its algebraic sum with any set in $\mathcal{APC}$ ' is a set in $\mathcal{APC}$ '. These results extend: Fremlin and Miller's theorem that strong measure zero sets having Hurewicz's property have Rothberger's property, Galvin and Miller's theorem that the algebraic sum of a set with the $\gamma$-property and of a first category set is a first category set, and Bartoszynski and Judah's characterization of SR$^\mathcal{M}$-sets. They also characterize the property introduced by Gerlits and Nagy in terms of older concepts.