Exercise 14, § 8

Show that the axiom $(O_{III})$ is equivalent to each of the following axioms:

(O''''III) If $F$ is any closed subset of $X$ then the intersection of all the closed neighbourhoods of $F$ is equal to $F$.

(O'''''III) If $\mathfrak{F}$ is any filter on $X$ which converges to a point $a$, then the filter $\mathfrak{F}$ on $X$ which has the closures of the sets of $\mathfrak{F}$ as a base also converges to $a$.