Theorem 3 (Tychonoff)

Every product of quasi-compact (resp. compact) spaces is quasi-compact (resp. compact). Conversely, if a product of non-empty spaces is quasi-compact (resp. compact) then each of the factors is quasi-compact (resp. compact).

In view of the characterization of Hausdorff product spaces given in § 8, no. 2, Proposition 7 it is enough to prove the assertions for quasi-compact spaces. If $X = \prod_{i \in I} X_i$ is quasi-compact and non-empty, then $X_i = \operatorname{pr}_i(X)$ is quasi-compact by reason of Theorem 2 of no. 4. Conversely, suppose the $X_i$ are quasi-compact and let $\mathfrak{U}$ be an ultrafilter on $X$; then for each $i \in I$, $\operatorname{pr}_i(\mathfrak{U})$ is an ultrafilter base on $X_i$ (§ 6, no. 6, Proposition 10) which therefore converges by reason of axiom (C'); hence $\mathfrak{U}$ is convergent (§ 7, no. 6, Corollary 1 of Proposition 10) and therefore $X$ is quasi-compact.