Exercise 7, § 4

A topological space $X$ is said to be perfectly normal if it is normal and if each closed subset of $X$ is a countable intersection of open sets (or, equivalently, if each open subset of $X$ is a countable union of closed sets).

a) A Hausdorff space $X$ is perfectly normal if and only if, given any closed subset $A$ of $X$, there exists a real-valued continuous function $f$ on $X$ such that $f^{-1}(0) = A$ [use the method of Exercise 16 a) of § 1].

b) Show that every perfectly normal space $X$ is completely normal and that every subspace of $X$ is perfectly normal [use Exercises 3 b) and 6].

c) Every lower semi-continuous real-valued function on a perfectly normal space is the upper envelope of a sequence of continuous functions (use the method of § 2, no. 7, Proposition 11).

d) The compact space $X$ obtained by adjoining a point at infinity to an uncountable discrete space is completely normal but not perfectly normal, and every subspace of $X$ is paracompact.