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!
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4$\begingroup$ It would be interesting to see your proof attempt. You may be on the right track! $\endgroup$– Umberto P.Oct 22, 2013 at 14:37
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$\begingroup$ Which notion of "connected" are you trying to prove it with? $\endgroup$– rschwiebOct 22, 2013 at 14:39
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$\begingroup$ I agree, especially if it's a homework related question, you should post your attempted efforts as well. $\endgroup$– user2566092Oct 22, 2013 at 14:39
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1$\begingroup$ @DanielRust What about $[0,2) \cup (3,5]$ though? $\endgroup$– ArthurOct 22, 2013 at 14:48
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3$\begingroup$ 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. $\endgroup$– Dan RustOct 22, 2013 at 15:17
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1 Answer
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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.
