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The algebraic sum of sets of real numbers with strong measure zero sets. (English) Zbl 0901.03036
Authors’ summary: We prove the following theorems:
(1) If \(X\) has strong measure zero and if \(Y\) has strong first category, then their algebraic sum has property \(s_0\).
(2) 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.
(3) if \(X\) has srong measure zero and Hurewicz’s covering property then its algebraic sum with any set in \({\mathcal A} {\mathcal F} {\mathcal C}'\) is a set in \({\mathcal A} {\mathcal F} {\mathcal C}'\). \(({\mathcal A} {\mathcal F} {\mathcal C}'\) is included in the class of sets always of first category, and includes the class of strong first category sets.)
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 Bartoszyński and Judah’s characterization of \(\text{SR}^{\mathcal M}\)-sets. They also characterize the property \((*)\) introduced by Gerlits and Nagy in terms of older concepts.

MSC:
03E20 Other classical set theory (including functions, relations, and set algebra)
28E15 Other connections with logic and set theory
54F65 Topological characterizations of particular spaces
54G99 Peculiar topological spaces
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