I am havig some trouble with topology. If you have a metric space which is totally bounded en locally compact, is it then compact? At first I though that this was not true. I tried some dicrete metric space and it failed. I tried to take the space of rational numbers with the interval [0,1] and it failed. So I could not find a good couterexample. I also could not prove that it is true... I still believe there is a counterexample. Any tips?
2026-03-26 04:30:52.1774499452
totally bounded and locally compact but not compact
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I think $(0,1)$ is totally bounded, locally compact, and not compact.