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

Abstract

Some new results on relationships between cardinal invariants in compacta are obtained. We establish that every non-separable compactum admits a continuous mapping onto a compactum of the weight ω 1 that has a dense non-separable monolithic subspace (Lemma 1). Lemma 1 easily implies Shapirovskij's theorem that every compactum of countable tightness and of precaliber ω 1 is separable. The lemma also opens the road to some generalizations of this statement and to other results. We also obtain new results on the structure of monolithic compacta and of homogeneous compacta. In particular, a new class of shell-homogeneous compacta is introduced and studied. One of the main results here is Theorem 31 which provides a generous sufficient condition for a homogeneous monolithic compactum to be first countable. Many intriguing open questions are formulated.