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2011, vol. 63, iss. 2, pp. 87-92
On Vitali sets and their unions

emailvitja@mai.liu.se
Keywords: Vitali set; Baire property
Abstract
It is well known that any Vitali set on the real line R does not possess the Baire property. In this article we prove the following: Let S be a Vitali set, Sr be the image of S under the translation of R by a rational number r and Ƒ= {Sr : r is rational}. Then for each non-empty proper subfamily Ƒ' of Ƒ the union UƑ' does not possess the Baire property.
 References Engelking, R. (1989) General topology. Berlin: Heldermann Verlag Kuratowski, K. (1966) Topology. New York - London: Academic press, vol. 1 Vitali, G. (1905) Sul problema della mesura dei gruppi di punti di una retta. Bologna