Proposition 4.20.10. Let $\seqi{X}$ be a family of LCH spaces. If $X_{i}$ is compact for all but finitely many $i \in I$, then $X = \prod_{i \in I}X_{i}$ is a LCH space.
Proof. By Proposition 4.8.3, $\prod_{i \in I}X_{i}$ is Hausdorff. Let $x \in \prod_{i \in I}X_{i}$ and $i \in I$. If $X_{i}$ is not compact, let $U_{i} \in \cn_{X_i}(\pi_{i}(x))$ be compact. Otherwise, let $U_{i} = X_{i}$. Let $U = \prod_{i \in I}U_{i}$, then since $U_{i} \ne X_{i}$ for only finitely many $i \in I$, $U \in \cn_{X}(x)$. By Tychonoff’s Theorem, $U$ is compact. Therefore $X$ is locally compact.$\square$