= Solution
\b[False.] The additive group $\mathbb Q$ with its usual metric is a <Hausdorff space> but not a <locally compact space>. If a compact set were a neighbourhood of zero, it would contain all rationals in some interval $(-\epsilon,\epsilon)$. Choose an irrational $a$ in that interval and rational numbers $q_n\to a$ in the real line. All sufficiently late $q_n$ would lie in the compact neighbourhood, so a subsequence would converge in $\mathbb Q$. Its real limit would have to be $a$, a contradiction.
The other suggested example is the additive <Hilbert space> $\ell^2$. A compact neighbourhood of zero would contain a closed ball of some positive radius $r$. The vectors $re_n$ in that ball have pairwise distance $r\sqrt2$, and thus no convergent subsequence. This again contradicts metric compactness. Both examples are <topological groups> whose operations are continuous, so Hausdorffness cannot supply local compactness.
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