If Cp(X) is strongly dominated by a second countable space, then X is countable

We establish that a Tychonoff space X is countable if and only if Cp(X) is strongly dominated by a second countable space. The same is true for a compact space K such that Cp(K,[0,1]) is strongly dominated by a second countable space. We also prove that strong domination by a second countable space of the complement of the diagonal of a Tychonoff space X implies that X is an ℵ0-space. Our results solve several published open questions.