Abstract

It is well known that any Vitali set on the real line $\Bbb{R}$ does not possess the Baire property. In this article we prove the following: Let $S$ be a Vitali set, $S_r$ be the image of $S$ under the translation of $\Bbb {R}$ by a rational number $r$ and $\Cal F = \{S_r: r \text{ is rational}\}$. Then for each non-empty proper subfamily $\Cal F'$ of $\Cal F$ the union $\bigcup \Cal F'$ does not possess the Baire property.

Keywords: Vitali set; Baire property.

MSC: 03E15, 03E20

Pages:  87$-$92     

Volume  63 ,  Issue  2 ,  2011