A metric space is called pre-compact or totally bounded if any sequence has a Cauchy subsequence; this can be generalised to uniform spaces.
Alternatively, pre-compactness and total boundedness can be defined differently for a uniform space (note that a metric space is a uniform space):
Pre-compact subspace is a subset whose closure is compact.
A subset $S$ of a uniform space $X$ is totally bounded if and only if, given any entourage $E$ in $X$, there exists a finite cover of $S$ by subsets of $X$ each of whose Cartesian squares is a subset of $E$.
Let me call a uniform space to be Cauchy sequential compact, if any sequence in it has a Cauchy subsequence.
I was wondering if
- For a metric space, pre-compactness, total boundedness and Cauchy sequential compactness are all equivalent?
- Same question for a uniform space?
Pre-compactness in the first quote is defined differently from the one in the second quote. So now my question is narrowed down to whether total boundedness and Cauchy sequential compactness are equivalent in both metric spaces and uniform spaces.
Pete's reply says yes for metric spaces, and now what can we say about uniform spaces?
Thanks and regards!