# Characterizations of relative compactness for a subset of a complete space

I need this information, but could not find it. This has something to do with that fact that if $A\subset X$, with $(X,d)$ being a complete metric space, then every sequence in $A$ admits a Cauchy subsequence. It also has something to do with total boundedness. I am wondering if someone has a link or if someone knows what exactly the formal characterizations of relative compactness for a subset of a complete space are.

In that case, $A$ is relatively compact if and only if $\overline{A}$ is totally bounded.
The theorem is, in a metric space, a set $A$ is compact if and only if $A$ is complete and totally bounded.
A similar theorem is, in a complete metric space, a set $A$ is compact if and only if $A$ is closed and totally bounded.