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 $y\in X$ such that $d(x,y)\leq r$. It is well-known that it is not always true that $N(x,r)$ is the closure of $B(x,r)$.


I need, for some research, to restrict 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



Is there a name for the class of metric spaces such that the closure of the open ball of radius $r$ around each point $x$ is the set of elements $y$ such that $d(x,y)\leq r$ ?
I'm not sure what they're called, but according to [this](http://www.mathkb.com/Uwe/Forum.aspx/math/4535/Closure-of-open-balls) site an equivalent characterization of spaces $X$ where $\overline{B(x,r)} = N(x,r)$ is: for all $p\in X$, the
only local minimum of the function $x \rightarrow d(x,p)$ is at $x=p$. The proof is also there.


