Sum of two open sets is open?

Question

how to prove that if $A$ and $B$ are open in $(\mathbb{R},|.|)$ then $A+B$ is open ?

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

Thank you

Answer 1

The continuous linear map $T((x,y)) = x+y$ is surjective, hence an open map. Since $A \times B$ is open, we see that $T(A \times B)$ is open.

Answer 2

Hint: $$A+B = \bigcup_{a\in A}(a+B).$$

Answer 3

If $B$ is an open set, and $a \in \mathbb{R}$, let's show that $a + B = \{a + b : b \in B\}$ is also open.

What do open sets in $\mathbb{R}$ look like? They are unions of open intervals. That is, there exist numbers $a_i, b_i$ etc. such that $$B = \bigcup\limits_i (a_i,b_i)$$ Now, it should be obvious that $a + (a_i,b_i) = (a+a_i,a+ b_i)$, which is open. You can then argue that $$a+B = \bigcup\limits_i (a+a_i,a+b_i)$$ so $a+B$ is a union of open sets.