Let $(X,d)$ be a connected metric space and let $x\in X$. I was wondering if it can happen that there exists $\delta>0$ such that $B_\epsilon(x)=X$ for any $\epsilon>\delta$ and $B_\epsilon(x)\neq X$ for every $\epsilon<\delta$.
If we remove the hypothesis that $X$ is connected, then the discrete metric shows that this can happen. Is it sufficient to assume $X$ connected to remove this circumstance? Also, in general, is there some standard assumption that prevents this circumstance?
Take $X=[-1,1]$, endowed with the usual metric. Then $B_\varepsilon(0=X$ if $\varepsilon>1$ and $B_\varepsilon(0)\ne X$ if $\varepsilon<1$.