Exercise 1, § 9

Let $X$ be a topological space, $\mathfrak{G}$ a subbase of the topology of $X$ ($§ 2$, no. 3). Show that $X$ is quasi-compact if every open covering of $X$ formed by sets belonging to $\mathfrak{G}$ contains a finite covering of $X$. (Observe that if an ultrafilter $U$ on $X$ were not convergent, then every $x \in X$ would belong to a set of $\mathfrak{G}$ not belonging to $U$; and use the corollary to Proposition 5 of $§ 6$).