1. A topological space (S, T ) is called perfectly normal iff for every closed set F there is a continuous real function f on S with F = f -1({0}).
(a) Show that every metric space is perfectly normal.
(b) Show that every perfectly normal space is normal. Hint: Adapt the proof of Theorem 2.6.1.
2. Let (S, T ) be a perfectly normal space and F a closed subset of S. Here is another way to prove that (a) implies (b) in Theorem 2.6.4 in such spaces. Show that there is an h ∈ Cb (S, T ) such that for every continuous function f from F into (-1, 1) and continuous function g from S into [-1, 1] which equals f on F , hg is continuous from S into (-1, 1) and equals f on F. Hint: Make h = 1 on F, 0 ≤ h <>1 elsewhere.