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This question already has an answer here:

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!!!

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marked as duplicate by YuiTo Cheng, Jyrki Lahtonen, Cesareo, John B, José Carlos Santos general-topology May 19 at 16:54

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

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    $\begingroup$ Whenever I am clueless about a problem, I will search harder. $\endgroup$ – YuiTo Cheng May 19 at 6:00
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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...

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