Suppose that the sets $A_{1},A_{2} \subset \mathbb{R}^n$ are connected and that they are not disjoint. Prove that $A_{1} \cup A_{2}$ is connected.
The section including this question contains this theorem:
And the following definition:
A is disconnected if there exists open sets B,C such that $A\cap B$ and $A \cap C$ are nonempty, disjoint sets and $A \subset B\cup C.$ A is connected if it is not disconnected.
but I do not know how to use this information to prove the required, could anyone help me please?
Best Answer
Suppose by contradiction that $A_1 \cup A_2 \subset C\cup D$ where $C,D$ are open and $C\cap (A_1 \cup A_2) \ne \emptyset,D\cap (A_1 \cup A_2) \ne \emptyset$ and $(C\cap D) \cap (A_1 \cup A_2) = \emptyset$.
As $A_1,A_2$ are connected each are either in $C$ or $D$ , if ,say , both in $C$ then their union $A_1\cup A_2$ would also be in $C$ and then $D\cap (A_1 \cup A_2)= \emptyset$ a contradiction. So we have that W.L.O.G $A_1 \subset C$ and $A_2 \subset D$ but then $A_1\cap A_2 =\emptyset$ , a contradiction.
Note that we didn't use the fact that we are in $\Bbb R^n$ which means we proved that any union of connected sets with nonempty intersection is connected.