Special Points in Compact Spaces

Abstract

Given a collection $\mathcal {C}$, of cardinality $\kappa$, of subsets of a compact space X, we prove the existence of a point x such that whenever $C \in \mathcal {C}$ and $X \in \bar C$, there exists a ${G_\lambda }$-set Z with $\lambda < \kappa$ and $x \in Z \subset \bar C$. We investigate the case when $\mathcal {C}$ 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 ${2^{{\omega _1}}}$ among dyadic spaces.