Strongly meager sets and their uniformly continuous images

Abstract

We prove the following theorems: (1) Suppose that f; 2W 2W is a continuous function and X is a Sierpiiiski set. Then (A) for any strongly set Y, the image f[X Y] is an so-set, (B) f [X] is a perfectly meager set in the transitive sense. (2) Every strongly meager set is completely Ramsey null. This paper is a continuation of earlier works by the authors and by M. Scheepers (see [N], [NSW], [S]) in which properties (mainly, the algebraic sum) of certain singular subsets of the real line R and of the Cantor set 2' were investigated. Throughout the paper, by a set of real numbers we mean a subset of 2' and by + we denote the standard modulo 2 coordinatewise addition in 2W. Let us also assume that a measure zero (or negligible) set always denotes a Lebesgue set. We apply the following definition of sets of real numbers. Definition 1. An uncountable set X is said to be a Luzin (respectively, Sierpin'ski) set iff for each meager (respectively, zero) set Y, XnY is at most countable. We say that a set X is of strong (respectively, strongly meager) iff for each meager (respectively, zero) set Y, X Y 7& 2'. Remark 1. It is well known (see [M] for example) that every Luzin set is strongly zero. Quite recently J. Pawlikowski proved that each Sierpin'ski set must be strongly meager as well (see [P]). Let us recall that a set X is called an so-set (or Marczewski set) iff for each perfect set P one can find a perfect set QCP that is disjoint from X. M. Scheepers showed in [S] that for a Sierpin'ski set X and a strong set Y, X Y is an so-set. Later, in [NSW] it was proven that this also holds when X is strongly meager. We have the following functional version of the M. Scheepers' result. Theorem 1. Let X be a Sierpin'ski set and let Y be a strong set. Assume also that f: 2' -* 2W is a continuous function. Then the image f[X Y] is an so-set. Received by the editors July 16, 1998 and, in revised form, September 9, 1998 and March 10, 1999. 2000 Mathematics Subject Classification. Primary 03E15, 03E20, 28E15.