On Marczewski Sets

Using the methods of Brown and Walsh, we get condition guaranteeing that, for an ideal I of sets in a perfect Polish space some (s 0) sets are not in I. A few examples and corollaries are given. 0. Introduction Papers Br], W1], W2] and C] made a signiicant progress in the studying of (s 0) sets introduced by Marczewski in Sz]. One of the main results states that there exists a nonmeasurable (s 0) set without the Baire property. That was proved in Br] under CH and in W1], W2], C] within ZFC. We analyse the schemes from Br] and W1], W2] and get two criteria for an ideal I (of sets in a perfect Polish space X) to satisfy I 0 n I 6 = ; where I 0 denotes the ideal of all (s 0) sets (in X). The original proofs we base on need only a slight modiication. However, we give new versions in full. We describe some applications. Throughout the paper, we x a perfect Polish space X. A set which has no perfect subset is called totally imperfect. A set E X is called an (s 0) set if each perfect set has a perfect subset disjoint from E (see Sz]). Obviously, (s 0) sets are totally imperfect and, moreover, they form an ideal (see Sz]) which will be written as I 0. For any ideal I P(X), we always assume that X = 2 I (here P(X) is the power set of X). The cardinality of continuum is denoted by c. Further, the following lemma will be useful.