The quantitative difference between countable compactness and compactness ✩

We establish here some inequalities between distances of pointwise bounded subsets H of RX 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 RX. 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 Cp(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. © 2008 Elsevier Inc. All rights reserved