Wikipedia [article](https://en.wikipedia.org/wiki/Totally_bounded_space) on totally bounded spaces states "... the completion of a totally bounded space might not be compact in the absence of choice." Where is the axiom of choice used, and do you need it for metric spaces or only for general uniform spaces?



Totally bounded spaces and axiom of choice
The issue here is that a metric space might not have non-trivial (read: not eventually constant) Cauchy sequences. For example, if the underlying space is a Dedekind finite set.


Indeed it is consistent that there is a dense subset of $[0,1]$ which is Dedekind finite. As a space with the inherited metric it is complete already and totally bounded, but it is not compact as it is not closed on $[0,1]$. 


