Countably compact spaces admitting full r-skeletons are proximal

Abstract

We provide simple characterizations of spaces admitting full r-skeletons, c-skeletons and q-skeletons, by using ω-monotone functions. We use this characterization to prove that every countably compact space admitting a full r-skeleton is proximal; furthermore the characterizations are used to show that the class of spaces admitting full c-skeletons is invariant under subspaces, disjoint topological sums and Σ-products, in addition to prove that the class of spaces admitting full q-skeletons is closed under extensions, continuous images and one-point Lindelöf extensions of disjoint topological sums. These characterizations also yield some positive results for products.