Real Analysis Exchange

Marczewski Fields and Ideals

M. Balcerzak and A. Bartoszewicz

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For an $X\neq\emptyset$ and a given family $\mathcal{F}\subset \mathcal {P}(X)\setminus \{ \emptyset \} $, we consider the \mf \sodf 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 $S^0(\mathcal{F})$ 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.

Article information

Real Anal. Exchange, Volume 26, Number 2 (2000), 703-716.

First available in Project Euclid: 27 June 2008

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 28A05: Classes of sets (Borel fields, $\sigma$-rings, etc.), measurable sets, Suslin sets, analytic sets [See also 03E15, 26A21, 54H05] 03E75: Applications of set theory 06E25: Boolean algebras with additional operations (diagonalizable algebras, etc.) [See also 03G25, 03F45] 54A10: Several topologies on one set (change of topology, comparison of topologies, lattices of topologies) 54E52: Baire category, Baire spaces

Marczewski sets field of sets Baire category Lebesgue measure density topology


Balcerzak, M.; Bartoszewicz, A. Marczewski Fields and Ideals. Real Anal. Exchange 26 (2000), no. 2, 703--716.

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