[Math] Prove that if the closure of each open ball in compact metric space is the closed ball with the same radius, then any ball in this space is connected

Question

I'm having some difficulty with the following problem in general topology:

Prove that if the closure of each open ball in compact metric space is the closed ball with the same center and radius, then any ball in this space is connected.

I've tried many things – looking at the components of the open ball, the closed ball, assuming there is a non constant function from the open\closed ball to $\{0,1\}$ & more – But I always ended up with more mess then I can handle.

Any ideas?

Answer 1

Assume $X$ is disconnected, i.e. the disjoint union of two closed sets $A$ and $B$. Note that $B$ is compact. For any point $a\in A$, there is thus a point $b\in B$ such that $d(a,b)=r$ is the largest radius of a ball $B_r(a)$ contained in $A$. Can you finish from here?