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A topological space $(X, \tau)$ is compact iff there exists a complete and totally bounded metric (see here).

Do you know an example of a non-compact metrizable topological space for which we have a metric that is complete, but then not totally bounded, and a metric that is not complete, but totally bounded?

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    $\begingroup$ Take $\mathbb{R}$ with the usual metric for the first example, and $(0,1)$ with the usual metric for the second. You can assume they are the same space since they are bijective. $\endgroup$ – Prahlad Vaidyanathan May 4 '17 at 15:59
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Take $\mathbb{R}$ with the usual metric for the first example, and $(0,1)$ with the usual metric for the second. You can assume they are the same space since they are bijective.

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