The quantitative difference between countable compactness and compactness

Abstract

We establish here some inequalities between distances of pointwise bounded subsets H of R X to the space of real-valued continuous functions C ( X ) that allow us to examine the quantitative difference between (pointwise) countable compactness and compactness of H relative to C ( X ) . We prove, amongst other things, that if X is a countably K-determined space the worst distance of the pointwise closure H ¯ of H to C ( X ) is at most 5 times the worst distance of the sets of cluster points of sequences in H to C ( X ) : here distance refers to the metric of uniform convergence in R X . We study the quantitative behavior of sequences in H approximating points in H ¯ . As a particular case we obtain the results known about angelicity for these C p ( X ) spaces obtained by Orihuela. We indeed prove our results for spaces C ( X , Z ) (hence for Banach-valued functions) and we give examples that show when our estimates are sharp.