= Local compactness of a product with finitely many noncompact factors
A <product topology> on locally <compact> factors is locally <compact> if all but finitely many factors are <compact>. At a <point>, choose a <compact> <neighborhood> in each of the finitely many exceptional factors and take the full <compact space> in every other coordinate. <Tychonoff's theorem> makes their product <compact>, and the product contains an open basic <neighborhood> because only finitely many coordinates were restricted. In contrast $\mathbb R^{\mathbb N}$ is not locally <compact>: a basic <neighborhood> has a free coordinate projecting onto all of the noncompact real line, so no <compact set> can contain it.
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