Marczewski fields and ideals


M. Balcerzak, A. Bartoszewicz, J. Rzepecka, S. Wronski

Real Anal. Exchange, to appear; 10 pages

For a non-empty set X and a given family F of subsets of X such that F does not contain the empty set, we consider the Marczewski field S(F) which consists of subsets A of X such that each set U in F contains a set V in F with such that V is either disjoint with of contained in A. We also study the respective ideal S0(F). We show general properties of S(F) and certain representation theorems. For instance we prove that the interval algebra in [0,1) is a Marczewski field. We are also interested in situations where S(F)=S(\tau \ {emptyset}) for a topology \tau on X. We propose a general method which establishes S(F) and S0(F) provided that F is the family of perfect sets with respect to \tau, and \tau is a certain ideal topology on R connected with measure or category.

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Last modified December 16, 2000.