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Marczewski fields and ideals. (English) Zbl 1009.28001

For an \(X\neq\emptyset\) and a given family \({\mathcal F}\subset{\mathcal P}(X)\setminus\{\emptyset\}\), we consider the Marczewski field \(S({\mathcal F})\) which consists of sets \(A\subset X\) such that each set \(U\in{\mathcal F}\) contains a set \(V\in{\mathcal F}\) with \(V\subset A\) or \(V\cap A=\emptyset\). We also study the respective ideal \(S^0({\mathcal F})\). We show general properties of \(S({\mathcal F})\) 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 \(S({\mathcal F})= S(\tau\setminus\{\emptyset\})\) for a topology \(\tau\) on \(X\). We propose a general method which establishes \(S({\mathcal F})\) and \(S^0({\mathcal F})\) provided that \({\mathcal F}\) is the family of perfect sets with respect to \(\tau\), and \(\tau\) is a certain ideal topology on \(\mathbb{R}\) connected with measure or category.

MSC:

28A05 Classes of sets (Borel fields, \(\sigma\)-rings, etc.), measurable sets, Suslin sets, analytic sets
03E15 Descriptive set theory
54A10 Several topologies on one set (change of topology, comparison of topologies, lattices of topologies)
54E52 Baire category, Baire spaces
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