Let $Acup B$ be open, disconnected in $Bbb{R}^2$ where $A,B$ are non-empty, disjoint. Are both $A,B$ open in $Bbb{R}^2$?

I have tried it in the following manner-
Assume $A$ is not open. Then $exists xin A$ such that $xnotin A^circ$ i.e. $forall epsilon>0, B(x,epsilon)notsubset A$ .
Now $xin Acup B$, open in $Bbb{R}^2$. Hence $exists r>0$ such that $B(x,r)subset Acup B$. But $B(x,r)notsubset A$, so we must have $B(x,r)cap Bneqemptyset$. Again for any $0<epsilonle r, B(x,epsilon)subset Acup B$ and $B(x,epsilon)notsubset A$, so we must have $B(x,epsilon)cap Bneq emptyset$ for all $0<epsilonle r$. Hence $xin overline{B}$.
$therefore xin Acapoverline{B}implies Acap overline{B}neq emptyset$.
Now if both $A$ and $B$ are connected then $Acup B$ will be connected (but here it is given that $Acup B $ is disconnected). So we must have either $A$ or $B$ is disconnected.
Now from this stage I cannot proceed further to get a contradiction. Can anyone guide me to conclude? Thanks for help in advance.