# 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 $d(x,y)\leq r$ ?

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?

Valerio

• Fix your typo "d(x,r) \leq r" since r is a real number, not a point in X. Sep 6 '11 at 15:35
• I've fixed the typo and improved the title. Feel free to revert either change if you don't think they are appropriate.
– j.c.
Sep 6 '11 at 15:39
• Note that every metric space that is not a single point has a uniformly equivalent distance for which that property does not hold, that is, a truncated distance $(x,y)\mapsto\min(d(x,y),r)$. Just to point out that it is really a property of the distance function. Sep 6 '11 at 15:48
• @Pietro: Out of curiosity: is it known which metric spaces have an equivalent distance function for which the property does hold? Sep 6 '11 at 15:59
• Thanks for the correctio of the typo and also for improving the title. Sep 6 '11 at 16:59

I'm not sure what they're called, but according to this 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.
• This equivalent condition is interesting. At the beginning it looks some kind of convexity, but it is not: also concave subspace of $\mathbb R^2$ can have that property, if the concaveness is not too strong. This property says that I can move from a point towards another point without getting back. How can I call this space? Let's try.. "spaces with no pits". How does that sound? Sep 7 '11 at 9:13