Sum of locally compact spaces is locally compact

Question

Is this proof correct?

Let $C_\alpha$ be a collection of of locally compact spaces. For each $\alpha$, $x \in C_\alpha$, the corresponding compact nhood basis $N_{x}$ is then embedded as a compact nhood base of $x \in \bigsqcup C_\alpha$.

Answer 1

Yes, the proof is correct. You can say more generally that if a Hausdorff space $X$ has a cover by open locally compact spaces, then it is locally compact. Clearly the sum of Hausdorff spaces is Hausdorff.

Remark: I assume that locally compact spaces are required to be Hausdorff.