In a metric space, if $A$ is open and $B$ is closed, is $A + B$ open or closed?

Question

Let $A, B \in E^n$, and consider their sum $A + B = \{x+y \mid x \in A, y \in B\}$. Suppose that $A$ is open and $B$ is closed.

• Is it always true that $A+B$ is open?

• Is it always true that $A+B$ is closed?

Answer 1

Try simple examples. The most basic closed set, for instance, is a point. What happens in this case?