A question of countable product and uncountable product

Question

It is intuitive and easy to show that a countable product of separable spaces is separable. It is tempting to think that an uncountable product of separable spaces will not be separable (a) Show that R^R is separable. Can the same be said about (R^R,Tbox)? Hint: Consider the collection of all step functions with finitely many steps, rational step heights, and whose steps are all on rational intervals. (b) This might help you partially regain your sanity. Show that there exists a subspace of R^R that is not separable. Hint: Forf∈RR,definethesetA:={g∈R^R |g(x)=f(x)for all but countably many x ∈ R}