Proposition 5

The union of a finite family of quasi-compact subsets of a topological space is quasi-compact.

It is sufficient to show that if $A$ and $B$ are two quasi-compact subsets of a topological space $X$, then $A \cup B$ is quasi-compact. Let $\mathscr{R}$ be covering of $A \cup B$; then $\mathscr{R}$ is a covering of $A$ and a covering of $B$; hence $\mathscr{R}$ contains a finite covering $\mathscr{R}_1$ of $A$ and a finite covering $\mathscr{R}_2$ of $B$; $\mathscr{R}_1 \cup \mathscr{R}_2$ is thus a finite covering of $A \cup B$ contained in $\mathscr{R}$.