Marczewski Fields and Ideals

Abstract

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.