MATEMATIČKI VESNIK
МАТЕМАТИЧКИ ВЕСНИК



MATEMATIČKI VESNIK
On Vitali sets and their unions
Vitalij A. Chatyrko

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.

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Keywords: Vitali set; Baire property.

MSC: 03E15, 03E20

Pages:  87--92     

Volume  63 ,  Issue  2 ,  2011