Theorem 2

If $f$ is a continuous mapping of a quasi-compact space $X$ into a topological space $X'$, then the set $f(X)$ is quasi-compact.

Let $\mathscr{R}$ be a covering of $f(X)$ by open sets in $X'$; then $\overline{f}^{-1}(\mathscr{R})$ is an open covering of $X$ (§ 2, no. 1, Theorem 1); hence there is a finite subset $\mathscr{S}$ of $\mathscr{R}$ such that $(\mathscr{S} \overline{f})$ is a covering of $X$; but then $\mathscr{S}$ is a covering of $f(X)$ and the theorem is proved.