$G_ \kappa$ subspaces of hyadic spaces

Abstract:

A hyadic space is a continuous image of a hyperspace of a compact space. For an infinite cardinal $\kappa$, an intersection of at most $\kappa$ many open subsets of $X$ is called a ${G_\kappa }$ subset of $X$. We construct, in ZFC, a compact separable space of uncountable $\pi$-weight and of cardinality continuum. This space is a ${G_\omega }$ subset of a hyadic space. We show that a compact space that does not contain any convergent sequences and which contains the Stone-Čech compactification of the countable discrete space cannot be imbedded as a ${G_\kappa }$ subset, where $\kappa$ is less than the continuum, of any hyadic space.