Exercise 13, § 9

Let X be a topological space, $\mathfrak{F}(X)$ the set of non-empty closed parts of X, $\mathscr{R}(X)$ the set of non-empty quasi-compact parts of X; these sets are equipped with the topology induced by the topology $\mathfrak{T}_\Omega$ on $\mathfrak{P}_0(X)$ (I, p. 91, exer. 7).

a) Prove that if $\mathfrak{B} \subset \mathfrak{F}(X)$ is quasi-compact for $\mathfrak{T}_\Omega$ and if X is regular, the union in X of the sets $M \in \mathfrak{B}$ is closed.

b) Prove that if $\mathfrak{B} \subset \mathscr{R}(X)$ is quasi-compact for $\mathfrak{T}_\Omega$, the union in X of the sets $M \in \mathfrak{B}$ is quasi-compact.