Regarding sum of subsets of $\mathbb{R}^n$ Let $A$ , $B$ be open subsets of $\mathbb{R}^n$ . Is $A+B$ a open subset of $\mathbb{R}^n$? Same question replacing open sets by compact sets. I know that for closed sets  the result is not true.

For $A, B$ open, I think the quickest way is to note that $A + B = \bigcup_{a \in A} (a + B)$. If the sets are instead compact, then note that addition is a continuous map $\mathbf R^n \times \mathbf R^n \to \mathbf R^n$, and that the product of two compact spaces is compact.
