Exercise 1, § 4

a) Construct a topological space consisting of four points which satisfies axiom $(O_v)$ but not axiom $(O_{III})$. b) A topological space $X$ which satisfies axiom $(O_v)$ also satisfies axiom $(O_{III})$ if and only if it has the following property: every closed subset of $X$ is the intersection of its neighbourhoods. $X$ is then uniformizable and the completely regular space associated with $X$ ($§ 1$, Exercise 4) is normal.

c) If a topological space satisfies axioms (C) and $(O_{III})$, it satisfies $(O_V)$, and the associated completely regular space is compact.