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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.

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
Full Text: DOI
[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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