Marczewski fields and ideals. (English) Zbl 1009.28001

Balcerzak, M.; Bartoszewicz, A.; Rzepecka, J.; Wroński, S.

Real Anal. Exch. 26(2000-2001), No. 2, 703-715 (2001).

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

Keywords:

Marczewski sets; field of sets; Baire category; Lebesgue measure; density topology; Marczewski field; interval algebra; perfect sets