For many of the spaces we work in the closure of open balls are the closed balls, however, I had an example in a class recently that there are some spaces in which this is not true. This is easily proved using either $\mathbb{Z}$ with the euclidean metric or a non-trivial set with the discrete metric.
I was wondering if there is a condition on metric spaces which is equivalent to this intuitive statement being true. I define "the closed balls" as follows:
The closed ball around $z$ of radius $r$, $\hat{B}_r(z) = \{x\in X:d(x,z)\le r\}$
I thought convexity might be a sufficient condition but that's just postulation, don't know how I'd prove it.