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I was trying to find an example of a bounded metric space which is not totally bounded. The only example I could come up whith was the natural numbers with the discrete metric.

However, like any other space with the discrete metric, I find this example very artificial (not to say disappointing), so I was wondering if there is any non-trivial example for that.

(by non-trivial I mean not envolving the discrete metric or, if possible, envolving non discrete sets)

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Take the open unit ball $B$ in an infinite dimensional Banach space. If $B$ is totally bounded so is its closure (exercise!). Hence $\textrm{cl}(B)$ is complete and totally bounded, thus compact. But this is false.

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Start with an arbitrary metric space (with metric $d$) that is not totally bounded, and take the new metric $\overline{d}(x,y) = \min(1, d(x,y))$.

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