Exercise 12, § 9

Let $X$ be a topological space, $\mathfrak{F}(X)$ the set of non-empty closed subsets of $X$, $\mathfrak{K}(X)$ the set of non-empty quasi-compact subsets of $X$;

endow each of these sets with the topology induced by the topology $\mathfrak{T}_\Omega$ on $\mathfrak{B}_0(X)$ (§ 2, Exercise 7).

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

b) Show that if $\mathfrak{B} \subset \mathfrak{K}(X)$ is quasi-compact with respect to $\mathfrak{T}_\Omega$ then the union of the sets $M \in \mathfrak{B}$ is quasi-compact.