# Every open ball is connected

Let $(X,d)$ be a metric space such that for all $x \in X$ and all $r>0$, $\overline{B(x,r)} = \{y \in X \mid d(x,y)\leqslant r\}$ Show that every open ball of $X$ is connected.

Note- I was trying to move with contradiction, but failed!

• It would be interesting to see your proof attempt. You may be on the right track! – Umberto P. Oct 22 '13 at 14:37
• Which notion of "connected" are you trying to prove it with? – rschwieb Oct 22 '13 at 14:39
• I agree, especially if it's a homework related question, you should post your attempted efforts as well. – user2566092 Oct 22 '13 at 14:39
• @DanielRust What about $[0,2) \cup (3,5]$ though? – Arthur Oct 22 '13 at 14:48
• The answer to that is no. Consider the solid balls $B((0,0),1)$ and $B((2,0),1)$ in $\mathbb{R}^2$. Their union is connected but the interior of their union is disconnected. – Dan Rust Oct 22 '13 at 15:17

## 1 Answer

The statement is not true. Take for example $X = [0,2) \cup (3,5]$. All properties are satisfied, but the open ball $B(1.5,2) = [0,2) \cup (3,3.5)$ is not connected.