Let $(X,d)$ be a metric space, let $B(x,r)$ be the open ball of radius $r$ about $x$ and $N(x,r)$ be the set of elements $x\in X$ such that $d(x,r)\leq r$. It is well-known that it is not alway true that $N(x,r)$ is the closure of $B(x,r)$.

I need, for some research, to rescrict my attention to metric spaces for which that property is true, i.e. $N(x,r)$ is the closure of $B(x,r)$. Do they have a particular name in literature?

Thanks in advance,

Valerio