On closed discrete subsets of normal dense subspaces of products

We consider the question: Can a normal dense subspace of a product of separable metric spaces contain a closed discrete subset whose cardinality is greater than c? We prove that if Y is a normal dense convex subspace of a product of real lines, then e(Y)≤c, but every product ∏{Mα:α<2τ} of non-trivial separable metric spaces contains a perfectly normal dense subspace Z, so the answer is “yes” since for such a subspace we have 2τ≤2|Z| and, by Hajnal-Juhász inequality, |Z|≤2e(Z). (The subspace Z which we construct actually has a closed discrete subset of cardinality τ.)