Let $A , B \subset \mathbb { R } ^ { n }$ and define $A + B = \{ a + b ; a \in A , b \in B \}$. If $A$ and $B$ are open, is $A + B$ open? If $A$ and $B$ are closed, is $A + B$ closed? Justify your answers.
Let $A , B \subset \mathbb { R } ^ { n }$ and define $A + B = \{ a + b ; a \in A , b \in B \}$. If $A$ and $B$ are open, is $A + B$ open? If $A$ and $B$ are closed, is $A + B$ closed? Justify your answers.