Question about closed sets

Question

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?

Answer 1

Take $n = 2$, take $A$ to be the $y$-axis and take $B$ to be the positive quadrant of the hyperbola $y = \frac{1}x$. Then $A$ and $B$ are both closed, but $A + B$ is the set of $(x, y)$ such that $x > 0$, which is not closed.

If $A$ and $B$ are both closed and one of them is compact, then $A + B$ is closed. See Closed sum of sets for a proof.

Answer 2

Not always. Consider in $\mathbb{R}$: $$A = \mathbb{Z}, \quad B = \left\{ n + \frac{1}{n} : n \geqslant 2 \right\}$$ so $\frac{1}{n} \in A + B$ but $0 \not \in A+B$.

Answer 3

\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).