Domination by a Polish space of the complement of the diagonal of X implies that X is cosmic

We will prove that a Tychonoff space X   is analytic if and only if (X×X)\Δ(X×X)\Δ is dominated by a Polish space; here Δ={(x,x):x∈X}Δ={(x,x):x∈X} is the diagonal of X. This solves two published open questions. We will also establish under CH, that a Tychonoff space X   has a countable network whenever (X×X)\Δ(X×X)\Δ is dominated by a second countable space.