Sigma-fragmentability of mappings into Cp(K)

We prove the main theorem concerning the σ-fragmentability of a continuous map ϑ : T → Cp(K), where T is a topological space and K is a compact Hausdorff space, and we obtain from it some new and old results about the σ-fragmentability of subsets of Cp(K) and property N∗f. Here are a few of them: A subset T of Cp(K) is σ-fragmented by the norm metric if and only if it is cover-semicomplete, i.e., T is fragmented by a pseudo-metric d such that each d-convergent sequence in T has a cluster point in T. The class of cover-semicomplete spaces contains spaces with countable separation in the sense of Kenderov and Moors. Assume that a compact space K has a family A of compact subsets such that (a) for each A ε A, Cp(A) is σ-fragmented by the norm, (b) if An ε A for each n ε N, then ∪n = 1∞ An is compact and (c) ∪A is dense in K. Then Cp(K) is σ-fragmented by the norm. Finally, suppose K and L are compact spaces and L has property N∗. If Cp(K) embeds in Cp(L), then K has property N∗. The last result can be used to show that Cp(βN) does not embed in Cp(0, 1Γ) for any set Γ. The method of proving the main theorem also yields the following. If K and L are compact and Cp(K) and Cp(L) are σ-fragmented by the norm, so is Cp(K × L).