Closed subsets $A,B\subset\mathbb{R}^2$ so that $A+B$ is not closed I am looking for closed subsets $A,B\subset\mathbb{R}^2$ so that $A+B$ is not closed.
I define
$A+B=\{a+b:a\in A,b\in B\}$
I thought of this example, but it is only in $\mathbb{R}$. Take:
$A=\{\frac{1}{n}:n\in\mathbb{Z^+}\}\cup\{0\}$ and $B=\mathbb{Z}$
both of these are closed (is this correct?). But their sum $A+B=\mathbb{Q}$ which is not closed.

Let $F=\{(x,y)\in \mathbb{R}^2 : xy\geq 1\}$ be the set of points $(x,y)$ which lie  on and above the arm of hyperbola $xy=1$ in the first quadrant and C be the Y-axis. then
$F+C=\{ (x,y):x>0\}$ which is not closed.

See the above figure in first picture $F$ and $C$ both are closed  but in second picture $F+C$ is not closed.
