The Algebraic Sum of Sets of Real Numbers with Strong Measure Zero Sets

Nowik, Andrej ; Scheepers, Marion ; Weiss, Tomasz

J. Symbolic Logic, Tome 63 (1998) no. 1, p. 301-324 / Harvested from Project Euclid

Résumé

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 strong measure zero and Hurewicz's covering property then its algebraic sum with any set in $\mathcal{APC}$ ' is a set in $\mathcal{APC}$ '. ($\mathcal{APC}$ ' 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 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.

Détail

Publié le : 1998-03-14
Classification:  Strong Measure Zero Set,  Strong First Category Set,  Always First Category Set,  Hurewicz's Property,  Rothberger's Property,  s$_0$-Set,  $\gamma$-set,  Lusin Set,  $\lambda$-Set,  (*)-Set,  Add($\mathcal{M}$)-Small Set,  03E20,  28E15,  54F65,  54G99