Georgian Math. J. 19 (2012), 93 – 99 DOI 10.1515 / gmj-2011-0054 © de Gruyter 2012 On some ideal defined by density topology in the Cantor set Marta Frankowska and Andrzej Nowik Abstract. We prove that the ideal .a/ defined in the Cantor space is not G ı generated. This answers a question of Z. Grande and E. Stro ´ nska in the case of the Cantor space. Keywords. Density topology, nowhere dense sets, ideal of sets, G ı generated ideal. 2010 Mathematics Subject Classification. Primary 03E15; secondary 03E20, 28E15. 1 Introduction The aim of this paper is to prove a solution of the problem from [5, p. 311] but in the case of the Cantor set instead of the real line. The ideal .a/ for the case of the real line was defined in [4], [5] and examined in [4] and [7]. In [5], the authors proved that it is F ı generated (for the definition of this notion see below) and they asked whether this ideal is G ı generated. We prove that the answer is no but in the case of the Cantor space 2 ! . 2 Notations and definitions We denote by 2 <! a collection of all finite sequences of elements from 2 D¹0;1º. For any t 2 2 <! denote Œt D¹x 2 2 ! W t  xº. We denote by  the standard product measure on 2 ! , so .Œx  n/ D 2 n for any n 2 ! and x 2 2 ! . We denote by  d the classical density topology on the real line. Notice that the classical density topology was defined for the case of the real line, but it is well known that we can define an analogous notion also in the case of the Cantor set. Namely, if A  2 ! is a measurable set, then define the set of the density points as follows. Definition 2.1 ([6, Exercise 17.9]). x 2 ˆ 2 ! .A/ ” lim n!1 .Œx  n \ A/ .Œx  n/ D 1: Brought to you by | Simon Fraser University Authenticated Download Date | 6/13/15 2:17 AM