TITLE: When does boundedness imply totally boundedness in a metric space
QUESTION [8 upvotes]: For a subset of a metric space, quoted from Wikipedia:

Total boundedness implies boundedness.
  For subsets of $\mathbb{R}^n$ the two are
  equivalent.

I was wondering what are some more general conditions for a metric space than being $\mathbb{R}^n$, so that boundedness can imply totally boundedness?
Thanks and regards!

REPLY [9 votes]: A metric space is totally bounded if and only if its completion is compact.  A subset of a complete metric space is totally bounded if and only if its closure is compact.  A metric space $X$ has the property that its bounded subsets are totally bounded if and only if the completion of $X$ has the property that its closed and bounded subsets are compact, a property sometimes called the Heine-Borel property.  
Montel spaces are examples of these.
Here's an open access article by Williamson and Janos you may find interesting.  For example, Theorem 1 (which they credit to a 1937 paper of Vaughan) says that a metrizable, $\sigma$-compact, locally compact topological space has a compatible metric with the Heine-Borel property.