I am aware of the facts that every totally bounded metric space is separable and a metric space is compact iff it is totally bounded and complete but I wanted to know, is every totally bounded metric space is locally compact or not. If not, then give an example of a metric space that is totally bounded but not locally compact.
2026-03-27 14:10:25.1774620625
Every totally bounded metric space is locally compact?
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A subset of a (finite dimensional) Euclidean space is totally bounded if and only if it is bounded. So all you have to do is to pick your favourite bounded but not locally compact subset of Euclidean space. E.g. $\mathbb{Q}\cap [0,1]$.