Independent-type structures and the number of closed subsets of a space.

Abstract

Three different notions of an independent family of sets are considered, and it is shown that they are all equivalent under certain conditions. In particular it is proved that in a compact space $X$ in which there is a dyadic system of size $\tau$ there exists also an independent matrix of closed subsets of size $\tau\times 2^\tau$. The cardinal function $M(X,\kappa)$ counting the number of disjoint closed subsets of size larger than or equal to $\kappa$ is introduced and some of its basic properties are studied