On the Sierpinski-erdös and the Oxtoby-ulam Theorems for Some New Sigma-ideals of Sets12

Let $(*) denote the family of subsets of the unit square defined to be of first category (Lebesgue measure zero) in almost every vertical line in the sense of measure (category). Theorem 1. There is a homeomorphism of the unit square onto itself mapping a given set in tyji?) onto a set of Lebesgue measure zero. Theorem 2. There is a set belonging to both Í» and * that cannot be mapped onto a set of first category by a homeomorphism of the unit square onto itself. Let C denote the Cantor set, regarded as the product of a sequence of 2-element groups, and let A denote one of the o-ideals of subsets of C studied by Schmidt and Mycielski. Theorem 3. Assuming the continuum hypothesis, the Sierpinski-Eraos theorem holds for A and the class of subsets of C of Haar measure zero (or of first category). Theorem 4. The Oxtoby-Ulam theorem holds for the image of A under the Cantor mapping of C onto the unit interval. 1. On the Sierpinskl-Erdos theorem. This section is inspired by the work of Schmidt [8] and Mycielski [4] on a-ideals of sets on C (the Cantor set), or for that matter on I (the unit interval). In correspondence with Mycielski's definition we have: Given a set S c C and a set K of natural numbers we define a positional game T(S, K) with perfect information between two players I and II. The players choose consecutive terms of a sequence (x0, xx, x2, ■ ■ ■) E C (where C is hereby regarded as the topological space (LI" xXt, IT~ XT¡), X¡ = (0, 1}, and T¡ is the discrete topology on X¡), the choice x¡ is made by player I if i G K, and by player II if i £ K. The player choosing x¡ knows S, K, and x0, xx, . . . , x¡_,. Player I wins if (x0, xx, x2,. . .) E S, and player II wins in the other case. Let WXX(K) be the class of sets S c C for which player II has a winning strategy in the game T(S, K). Let M = {Ks¡Si Si¡: s¡ = 0, 1; 1 < / < n, n E J) be a system of sets of natural numbers such that KS¡S:¡ ,. is Presented to the Society, April 16, 1978; received by the editors October 2, 1976 and, in revised form, December 14, 1977. AMS (MOS) subject classifications (1970). Primary 54A05; Secondary 28A65, 90D05.