example of two disjoint subsets $A$ and $B$ of $\Bbb R$ so that $A \cup B$ and $A$ are both open in $\Bbb R$ but $B$ is not open in $\Bbb R$

Question

In a question I recently encountered is as follows that give an example of two disjoint subsets $A$ and $B$ of $\Bbb R$ so that $A \cup B$ and $A$ are both open in $\Bbb R$ but $B$ is not open in $\Bbb R$.

I'm clueless so far. Thanks in advance for help.

Answer 1

You can use:

$A=(0,4)$ and $B=[4,5)$

Answer 2

Take $A=(-\infty, 0)$ and $B=[0,\infty).$

Answer 3

$$A=(-1,1), B=(-\infty,-1]\cup [1,\infty)$$

Note that $A$ and $A\cup B$ = $\Bbb R$ are open but B is not open.