Show that sum of the two closed sets in $\mathbb{R^2}$ is not always closed.
In $\mathbb{R^2}$, consider the sets $A =\{(x,y)\,\,:\,\, x>0,\,\,xy=1\}$ and $B =\{(x,y)\,\,:\,\, x>0,\,\,xy=-1\}$ which are closed, show that the sum A + B is not closed. $$ A+B:=\{a+b : a \in A, b \in B\} $$
To have better intuition, you can visualize the two sets as the following picture
Consider the sequence $$(x_n)=\Big\{\Big(\frac{2}{n},0\Big)\Big\}_{1}^\infty=\Big\{\Big(\frac{1}{n},n\Big)+\Big(\frac{1}{n},-n\Big)\Big\}_{1}^\infty \in A+B$$ Here $$x_n \longrightarrow (0,0) \notin A+B$$ since the first coordinate of the element in $A+B$ is always positive .
So, $A+B$ is not closed!