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The algebraic sum of sets of real numbers with strong measure zero sets

Published online by Cambridge University Press:  12 March 2014

Andrej Nowik ,
Marion Scheepers  and
Tomasz Weiss
Andrej Nowik
Affiliation:
Ul. Kolobrzeska23/F8, Gdansk-Oliwa 80390, Poland, E-mail: andrzand@ghost.mimuw.edu.pl
Marion Scheepers
Affiliation:
Department of Mathematics and Computer Science, Boise State University, Boise, Idaho 83725, USA, E-mail: marion@math.idbsu.edu
Tomasz Weiss*
Affiliation:
Institute of Mathematics, Warsaw University, Banacha 2, 02-097 Warszawa, Poland
*
Institute of Mathematics, WSRP 08-110 Siedlce, Poland, E-mail: weiss@wsrp.siedlce.pl
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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 S0.

(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 is a set in . ( 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 γ-property and of a first category set is a first category set, and Bartoszyfński and Judah's characterization of -sets. They also characterize the property (*) introduced by Gerlits and Nagy in terms of older concepts.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 63 , Issue 1 , March 1998 , pp. 301 - 324
Copyright
Copyright © Association for Symbolic Logic 1998

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