Suppose $\displaystyle A,B $ are connected subsets of some metric space $\displaystyle X $. Prove that if $\displaystyle A \cap B \neq \emptyset $ the $\displaystyle A \cup B $ is connected.

So suppose $\displaystyle A \cup B $ is not connected.

Then $\displaystyle A \cup B = C \cup D $ where $\displaystyle C $ and $\displaystyle D $ are disjoint and clopen. So $\displaystyle C = E_{O} \cap X $ and $\displaystyle C = E_{C} \cap X $. Also $\displaystyle D = F_{O} \cap X $ and $\displaystyle D = F_{C} \cap X $.

So $\displaystyle A \cup B = (E_{O} \cap X) \cup (F_{O} \cap X) $. Now what? How do you conclude that $\displaystyle A \cap B = \emptyset $?