Distance Between Subsets in Connected Spaces Suppose $\langle X, d \rangle$ is a metric space.  For any two sets $F,G \subseteq X$, by abuse of notation define $d(F,G) = \inf \{ d(f,g): f \in F, g \in G \}$. 
Let $\rho > 0$, $x \in X$, and $E \subseteq X$ be such that the open ball of radius $\rho$ centered at $x$ has non-empty intersection with both $E$ and its complement.  If $X$ is connected, does it follow that $d(B_{\rho}(x) \cap E, B_{\rho}(x) \cap E^c) = 0$, where $E^c$ is the complement of $E$ in $X$?
If the statement is false, are there obvious natural conditions that one could place on the metric space in question (rather than on the open ball centered at $x$) that guarantee that $d(B_{\rho}(x) \cap E, B_{\rho}(x) \cap E^c) = 0$?

Counterexamples for connected spaces have already been given by Daniel Fischer and Stefan H.
It turns out that connectedness is somewhat tangential to the issue. The property you are after is inherited by dense subspaces, so it also applies to  $\mathbb{Q}^n$ for example. That means it makes sense to look for conditions
on the completion of $X$. A reasonable sufficient condition is that
the completion is a length space, because in a length space all open balls
are path-connected.
