Let $X$ be a compact , metric space.
Assume that the closure of every each open ball it the closed ball with same center and radius.
Prove that any ball in this space is connected.
[Math] Any ball is connected
compactnessconnectednessgeneral-topologymetric-spaces
Best Answer
I will use $B(x,r)$ for the open ball centered at $x$ with radius $r$, $\bar B(x,r)$ for the corresponding closed ball and $\overline{B(x,r)}$ for the closure of the open ball in $X$.
First, notice that it is sufficient to prove that every closed ball is connected, because of $$B(x,r) = \bigcup_{s<r}\bar B(x,s).$$ (Remember that a union of a family of connected sets with non-empty intersection is connected.)
Suppose some closed ball $\bar B(x,r)$ is disconnected. Then $\bar B(x,r)=U+V$ for some clopen subsets $U,V$ of $\bar B(x,r)$. But $\bar B(x,r)$ is closed in $X$, so $U$ and $V$ are also closed in $X$. Since $X$ is compact, this implies that $U$ and $V$ are compact. Assume without loss of generality, that $x\in U$. Since $V$ is compact, $\min_{y\in V} d(x,y)=:q$ exists. Note that $0<q\leq r$. Fix a point $y\in V$ such that $d(x,y)=q$.
Observe that $B(x,q)\cap V=\emptyset$. Therefore, $B(x,q)\subseteq U$. But $U$ is closed in $X$, so $\overline{B(x,q)}\subseteq U$. On the other hand, $\bar B(x,q)$ contains $y$, so $\bar B(x,q)\nsubseteq U$. So, $\bar B(x,q)\neq\overline{B(x,q)}$.
We conclude that if $\bar B(x,r)=\overline{B(x,r)}$ holds for all $x\in X, r\in[0,\infty)$, all balls must be connected.