# Locally compact, complete, bounded spaces [closed]

Is every locally compact, complete, bounded space compact?

It seems that this fact is implicitly used in the definition of Gromov-Hausdorff convergence for pointed spaces on wikipedia, but I'm not able to prove it.

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## closed as too localized by Stefan Geschke, Yemon Choi, Dmitri Pavlov, Andreas Blass, Bill JohnsonAug 19 '11 at 0:01

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Do you mean Metric space? – Mahdi Majidi-Zolbanin Aug 18 '11 at 18:43
You talk about metric spaces, I assume? Otherwise I wouldn't know what completeness and boundedness mean. Moreover, by boundedness, do you just mean that the metric is bounded? Discrete spaces are locally compact and carry complete bounded metrics without being compact. – Stefan Geschke Aug 18 '11 at 18:44
By bounded, do you mean finite diameter, or totally bounded? This affects whether or not your question is true. – Richard Rast Aug 18 '11 at 18:45

I don't believe this is true. For example, take an infinite set and put discrete metric on it, that is, $d(x,y)=0$ if $x=y$ and $d(x,y)=1$ if $x\not=y$. Then I believe this is locally compact, complete and bounded, but it is not compact. Did I understand your question correctly?