Condensations of C-p(X) onto sigma-compact spaces

We show, in particular, that if nw(Nt) <= n for any t is an element of T and C is a dense subspace of the product Pi{Nt : t is an element of T} then, for any continuous (not necessarily surjective) map phi : C -> K of C into a compact space K with t(K) <= k, we have Psi(phi(C)) <= k. This result has several applications in C-p-theory. We prove, among other things, that if K is a non-metrizable Corson compact space then C-p(K) cannot be condensed onto a sigma-compact space. This answers two questions published by Arhangel'skii and Pavlov.