TITLE: Question about closed sets
QUESTION [5 upvotes]: Let $A$ and $B$ be subsets of $\mathbb{R}^n$ (where $\mathbb{R}^n$ is Euclidean n-space).
Define $A + B = \{ x + y : x \in A , y \in B \}.$
Now If $A$ and $B$ are closed sets, is $A+B$ also a closed set?

REPLY [2 votes]: \begin{align}
A + B &= \{ a + b \mid a \in A, b \in B\} \\
&= \bigcup_{b \in B} \{ a + b \mid a \in A\} \\
&= \bigcup_{b \in B} (A+b)\,.
\end{align}
If one of the sets $A,B$ is finite, then $A+B$ is closed, because $A+b$ is closed (I think this holds in all normed spaces). But since infinite unions of closed sets are not closed in general, it seems that there should be plenty of counterexamples (two already posted here).