Connectedness of Metric Spaces

Question

Let $(X, \rho)$ be a metric space which is compact suppose that for all $x \in X$ and $r>0$ $\overline{B_\rho(x,r)} =\{y \in X : \rho(x,y) \leq r\}$.

Show that $B_{\rho}(x_0,r_0)$ is connected for all $x_0 \in X$ and $r_0 >0.$

I haven't try anything and cannot think of any on how can i show this problem. Any suggestion, help and hint would help thank you very much!!!

Answer 1

What I think will help, is that if too sets $A$ and $B$ in a compact metric $(X,\rho)$ are closed and disjoint, there are $a_0 \in A$ and $b_0 \in B$ such that $\rho(a_0,b_0)= \rho(A,B) = \inf\{\rho(a,b): a \in A,b\in B\}$. This "gap" can be used to contradict the closure of balls property...