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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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##### References:
 [1] Fundamenta Mathematicae 24 pp 17– (1934) [2] Fundamenta Mathematicae 9 pp 193– (1927) [3] DOI: 10.1090/conm/192/02351 · doi:10.1090/conm/192/02351 [4] DOI: 10.1016/0166-8641(82)90065-7 · Zbl 0503.54020 · doi:10.1016/0166-8641(82)90065-7 [5] Notices of the American Mathematical Society 26 (1979) [6] DOI: 10.1016/0166-8641(84)90038-5 · Zbl 0551.54001 · doi:10.1016/0166-8641(84)90038-5 [7] Fundamenta Mathematicae 129 pp 17– (1988) [8] Fundamenta Mathematicae 113 pp 187– (1981) [9] DOI: 10.1090/S0002-9947-1989-0982239-X · doi:10.1090/S0002-9947-1989-0982239-X [10] Bulletin de la Societe Mathematique de France 47 pp 97– (1919) [11] Real Analysis Exchange 19 pp 521– (1993) [12] Real Analysis Exchange 20 pp 536– (1994) [13] DOI: 10.1080/16073606.1993.9631729 · Zbl 0783.90132 · doi:10.1080/16073606.1993.9631729 [14] Fundamenta Mathematicae 30 pp 50– (1938) [15] Colloquium Mathematicum 62 pp 221– (1991) [16] Handbook of set theoretic topology (1984) · Zbl 0546.00022 [17] DOI: 10.1090/S0002-9947-1981-0613787-2 · doi:10.1090/S0002-9947-1981-0613787-2 [18] Fundamenta Mathematicae 21 pp 114– (1933) [19] Fundamenta Mathematicae 2 pp 155– (1921) [20] DOI: 10.1090/S0002-9939-1954-0063389-3 · doi:10.1090/S0002-9939-1954-0063389-3 [21] DOI: 10.1007/BF02392416 · Zbl 0357.28003 · doi:10.1007/BF02392416 [22] Fundamenta Mathematicae 11 pp 301– (1928)
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