Is $\displaystyle \overline{B((1,0),1)} \cup B((-1,0),1)$ a connected subset of $\displaystyle \Re^2$? I'm sure it is, although I can't find a proof. (Note the "overline" denotes the closure).

Let A,B be two connected subsets of a topological space X such that $\displaystyle A \cap \overline{B} \neq \emptyset$. Prove that $\displaystyle A \cup B$ is connected.