TITLE: Every open ball is connected
QUESTION [9 upvotes]: 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!

REPLY [19 votes]: 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.