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

Andrej Nowik, Marion Scheepers, and Tomasz Weiss
Source: J. Symbolic Logic Volume 63, Issue 1 (1998), 301-324.

#### 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.

Primary Subjects: 03E20
Secondary Subjects: 28E15, 54F65, 54G99
Full-text: Remote access
If you are a member of the ASL, log in to Euclid for access.
Full-text is available via JSTOR, for JSTOR subscribers. Go to this article in JSTOR.

Permanent link to this document: http://projecteuclid.org/euclid.jsl/1183745472
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 6, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9077861905097961, "perplexity_flag": "middle"}