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.

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?

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.
• Yes, assuming $A_1\cup A_2$ is not connected we get those $C,D$ – user123 Feb 28 at 11:25
• @hopefully thanks :) note that i didn't use the fact that the space is $\Bbb R^n$ – user123 Feb 28 at 11:27
• but disconnectedness say that $A_{1} \cup A_{2} \subset C \cup D$ not equal – hopefully Feb 28 at 11:28