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.
