Q:Find two disjoint clsed subsets of (R^2) which are zero distance a part?
my answer is A & B where
A={(1/n) : n belong to z+}X{0}
B={[-1,0]XR}
Is it right
No, it is not correct. Notice that $\displaystyle \{1/n:n\in\mathbb{Z}^+\}=\mathbb{R}^+$. So $\displaystyle A=\{(r,0):r\in\mathbb{R}^+\}$, and this set is not closed since $\displaystyle (0,0)$ is a boundary point not in $\displaystyle A$.
However, consider the sets $\displaystyle C=\{(x,1/x):x\in\mathbb{R}^+\}$ and $\displaystyle D=\{(x,0):x\in\mathbb{R}\}$.