Open Access
2000/2001 Marczewski Fields and Ideals
M. Balcerzak, A. Bartoszewicz
Real Anal. Exchange 26(2): 703-716 (2000/2001).


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.


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M. Balcerzak. A. Bartoszewicz. "Marczewski Fields and Ideals." Real Anal. Exchange 26 (2) 703 - 716, 2000/2001.


Published: 2000/2001
First available in Project Euclid: 27 June 2008

zbMATH: 1009.28001
MathSciNet: MR1844387

Primary: 03E75 , 06E25 , 28A05 , 54A10 , 54E52

Keywords: Baire category , density topology , field of sets , Lebesgue measure , Marczewski sets

Rights: Copyright © 2000 Michigan State University Press

Vol.26 • No. 2 • 2000/2001
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