Example of a bounded space which is not totally bounded 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 example with the discrete metric, I find this one artificial (not to say disappointing), so I was wondering if there is any non-trivial example of this.
(by non-trivial I mean not envolving the discrete metric or, if possible, envolving non discrete sets)

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