# Connectedness of Metric Spaces [duplicate]

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!!!

## marked as duplicate by YuiTo Cheng, Jyrki Lahtonen, Cesareo, John B, José Carlos Santos general-topology StackExchange.ready(function() { if (StackExchange.options.isMobile) return; $('.dupe-hammer-message-hover:not(.hover-bound)').each(function() { var$hover = $(this).addClass('hover-bound'),$msg = $hover.siblings('.dupe-hammer-message');$hover.hover( function() { $hover.showInfoMessage('', { messageElement:$msg.clone().show(), transient: false, position: { my: 'bottom left', at: 'top center', offsetTop: -7 }, dismissable: false, relativeToBody: true }); }, function() { StackExchange.helpers.removeMessages(); } ); }); }); May 19 at 16:54

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...