If the sum of sets is open then one of them is open?

Question

I have managed to prove that if one of two subsets $A$ and $B$ is open, then $A+B$ is open. The next question is:

If $A+B$ is open, is it (always) true that either $A$ or $B$ is open (or both)?

For clarity:

$$ A+B = \{ a+b \mid a\in A, b\in B \} $$

Answer 1

Take $A=[0,\infty)$ and $B=(-\infty,0]$. Then $A+B = \mathbb{R}$ which is open (and closed), but neither $A$ nor $B$ is open.

Answer 2

No,

consider $A=\mathbb Q$ and $B=(\mathbb R\setminus \mathbb Q)\cup\{0\}$. Then $A+B=\mathbb R$ but none of $A,B$ is open.