Precalibers, monolithic spaces, first countability, and homogeneity in the class of compact spaces (Q952617)

The importance of some cardinal invariants in compact spaces is studied. A compact \(T_2\)-space \(X\) is either separable or can be continuously mapped onto a compact space of weight \(\omega_1\) which has a dense non-separable subspace. A strong form of homogeneity, called shell-homogeneity, is introduced and studied. For example: a shell-homogeneous monolithic compactum is first countable; for any first countable zero-dimensional space \(X\), the space \(X^{\omega}\) is shell-homogeneous. Several open problems are formulated.