Exercise 11, § 4

One says that a topological space X is perfectly normal if it is normal, and if every closed subset of X is countable intersection of open sets (or, what amounts to the same thing, if every open subset of X is countable union of closed sets). a) For a separated space X to be perfectly normal, it is necessary and sufficient that for every closed subset A of X, there exists a numerical function f continuous in X and such that $f^{-1}(0) = A$ (method of IX, p. 88, exerc. 19 a)). b) Now prove that every perfectly normal space X is completely normal and that every subspace of X is perfectly normal (use exerc. 7, 9) (IX, p. 102). c) Every lower semicontinuous numerical function on a perfectly normal space is the upper envelope of a sequence of continuous functions (method of IX, p. 18, prop. 11). d) The compact space X obtained by adjoining a point at infinity to a non-countable discrete space is completely normal but not perfectly normal, and every subspace of X is paracompact.

¶ 11) Now prove that the compact non-metrisable space X defined in exerc. 12a) of IX, p. 92 is perfectly normal, and that the compact non-metrisable space Y defined in exerc. 12 d) of IX, p. 93 is completely normal but not perfectly normal.