Special points in compact spaces

Given a collection ', of cardinality K, of subsets of a compact space X, we prove the existence of a point x such that whenever C E ' and X E C, there exists a GA-set Z with A < K and x E Z C C . We investigate the case when ' is the collection of all cozerosets of X and also when X is a dyadic space. We apply this result to homogeneous compact spaces. Another application is a characterization of 20' among dyadic spaces.