I'm struggling with the following question (from an introductory analysis course):
A metric space $E$ is said to be locally connected if for all $x \in E$, there exists $r > 0$ such that $B_r(x)$ is connected (this is the open ball with radius $r$ centered at $x$). Show that if $E$ is locally connected, then $E$ is the disjoint union of open connected sets.
(For reference, we have learned that a space $E$ is connected if $E$ and $\emptyset$ are the only sets that are both open and closed in $E$, and that $E$ is not connected $\iff$ there exists nonempty, disjoint open sets $A$ and $B$ such that $A \bigcup B = E$ ).
I get the feeling that I need to show that $E$ is a union of certain (disjoint) connected balls in $E$, but (if this were the case) I haven't been able to set up a way to determine which balls/sets I need to form the union.
I appreciate any and all help; thanks in advance.