## Journal of Symbolic Logic

- J. Symbolic Logic
- Volume 63, Issue 1 (1998), 301-324.

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

Andrej Nowik, Marion Scheepers, and Tomasz Weiss

#### Abstract

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.

#### Article information

**Source**

J. Symbolic Logic Volume 63, Issue 1 (1998), 301-324.

**Dates**

First available: 6 July 2007

**Permanent link to this document**

http://projecteuclid.org/euclid.jsl/1183745472

**JSTOR**

links.jstor.org

**Mathematical Reviews number (MathSciNet)**

MR1610427

**Zentralblatt MATH identifier**

0901.03036

**Subjects**

Primary: 03E20: Other classical set theory (including functions, relations, and set algebra)

Secondary: 28E15: Other connections with logic and set theory 54F65: Topological characterizations of particular spaces 54G99: None of the above, but in this section

**Keywords**

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

#### Citation

Nowik, Andrej; Scheepers, Marion; Weiss, Tomasz. The Algebraic Sum of Sets of Real Numbers with Strong Measure Zero Sets. Journal of Symbolic Logic 63 (1998), no. 1, 301--324. http://projecteuclid.org/euclid.jsl/1183745472.