Past exam of the mathematics course of the University of Cambridge / 2008 / iii / Paper 9 / 1 / d / Solution

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 $(-ϵ,ϵ)$. 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 $r{e}_n$ in that ball have pairwise distance $r\sqrt{2}$, 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.