A generalization of a problem of Fremlin (Axiomatic Set Theory and Set-theoretic Topology)
Sakaé Fuchino · Institutional Repositories DataBase (IRDB) · 2008
The following result is stated in A. Miller [3] as an answer to a question by David Fbremlin: Theorem 1. (Theorem 3.7 in A. Miller [3]) The following holds in the generic extension obtained by adding at least $\aleph_{3}$ Cohen reals to a model of CH: (1.1)For any family $\mathcal{F}$ of Borel sets Utth $|\mathcal{F}|=\aleph_{2}$ such $that\cap \mathcal{F}=\emptyset$ , there is a subfamily $\mathcal{F}'\subseteq \mathcal{F}$ with $|\mathcal{F}'|\leq\aleph_{1}$ such that $\cap \mathcal{F}'=\emptyset$ .Note that by moving to complements of elements of $\mathcal{F}$ , the assertion (1.1) can be also conceived as a covering property resembling,Lindel\"of property of topological spaces.Thus we shall call here the property (1.1) the IFlremlin-Miller Covering Principle.More generally, for cardinals $\kappa\geq\lambda$ , let us denote with FMCP $(\kappa, \lambda)$ the following parametrized Fremlin-Miller Covering Principle: FMCP $(\kappa, \lambda)$ : For any family $\mathcal{F}$ of Borel sets with $|\mathcal{F}|<\kappa$ such $that\cap \mathcal{F}=\emptyset$ there is $\mathcal{F}'\in[\mathcal{F}]<\lambda$ such $that\cap \mathcal{F}'=\emptyset$ .