Corollary

For a subset of a product of topological spaces to be relatively quasi-compact it is necessary and sufficient that each of its projections should be relatively quasi-compact in the corresponding factor.

Necessity follows from Theorem 2 of § 4. To prove sufficiency, let $A$ be a subset of $\prod_i X_i$ such that, for each index $i$, $\operatorname{pr}_i(A)$ is contained in a quasi-compact subset $K_i$ of $X_i$; then $A$ is contained in the quasi-compact subset $\prod_i K_i$ of $\prod_i X_i$.