Proposition 14

a) Let $(X_i)_{i \in I}$ be a family of locally compact spaces such that $X_i$ is compact for all but a finite number of indices. Then the product space $X = \prod_{i \in I} X_i$ is locally compact.

b) Conversely, if the product of a family $(X_i)_{i \in I}$ of non-empty topological spaces is locally compact, then the factors $X_i$ are compact for all but a finite number of indices, and the factors which are not compact are locally compact.

a) Let $x = (x_i)$ be a point of $X$. For each index $i \in I$ such that $X_i$ is locally compact but not compact, let $V_i$ be a compact neighbourhood of $x_i$ in $X_i$, and for all other indices $i$ put $V_i = X_i$. Then $\prod_{i \in I} V_i$ is a compact neighbourhood of $x$ in $X$ (no. 5, Theorem 3). Also $X$ is Hausdorff by , no. 2, Proposition 7, and is therefore locally compact

b) If $X = \prod_{i \in I} X_i$ is locally compact and the $X_i$ non-empty, then each of the $X_i$ is homeomorphic to a closed subspace of $X$ ($§ 4$, no. 2, Proposition 4 and $§ 4$, no. 3, Corollary to Proposition 7), hence locally compact by Proposition 13. Let $a = (a_i)$ be a point of $X$ and let $V$ be a compact neighbourhood of $a$; since we have $\operatorname{pr}_i V = X_i$ for all but a finite number of indices ($§ 4$, no. 1), it follows from no. 4, Corollary 1 to Theorem 2, that the $X_i$ are compact except for a finite number of indices.