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Open subspace of a locally compact space is locally compact? The definition of locally compact is given in Willard: A space X is locally compact iff each point in X has nhood base consisting of compact set.

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Well, if $X$ is locally compact in the manner you (and Willard) describe, and $U \subseteq X$ is open, then given $x \in U$ and any open neighbourhood $V$ of $x$ in $U$, it must be that $V$ is open in $X$, and so there is a compact neighbourhood $K$ of $x$ with $K \subseteq V$.

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