Strong domination by countable and second countable spaces

We show that, for a Lindelöf Σ-space X, if Cp(X,[0,1]) is strongly dominated by a second countable space, then X is countable. Under Martin's Axiom we prove that there exists a countable space Z that strongly dominates the complement of the diagonal of any first countable compact space. In particular, strong domination by a countable space of the complement of the diagonal of a compact space X need not imply metrizability of X. It turns out that the same countable space Z strongly dominates Cp(X) for an uncountable space X. Our results solve several published open problems.