If union is disconnected and open then the sets are open

Question

Let $A$ and $B$ be subsets of $\mathbb{R}^{2}$, if $A \cup B$ is disconnected and open, then is it necessary that both $A$ and $B $ are open?

I feel that it is true, as disconnectedness and openness together restrict from taking semi-open/closed sets here, but don't know how to prove.

Answer 1

No. Let $$X=\{(x,y)\in\mathbb R^2:x>1\}\cup\{(x,y)\in\mathbb R^2:x<-1\}.$$ Then $X$ is open and disconnected. Now let $$A=X\cap\{(x,y)\in\mathbb R^2:y\geq 0\},$$ $$B=X\cap\{(x,y)\in\mathbb R^2:y<0\}.$$ Then $X=A\cup B$, but $A$ is not open.