Visual representation of separated sets.

Question

Two subsets $A$ and $B$ of a metric space $X$ are said to be separated if both $A\cap \bar{B}$ and $\bar{A}\cap {B}$ are empty.

My question is that if I take two disjoint subsets $A$ and $B$ of a metric space $X$ then are they separated?

And further for disjoint or any arrangement of sets in $\mathbb{R}^{2}$ I can always find a case where $\exists$ an element $e \in (A\cap \bar{B})$ either $e \in (\bar{A}\cap {B})$ . So I'm not able to visualize it in my mind the concept of separated sets, any visual representation in $\mathbb{R}^{2}$ would be really helpful.

Answer 1

Let $A=\{(x,y)\in\mathbb{R}^2\,|\,x\geqslant0\}$ and let $B=\{(x,y)\in\mathbb{R}^2\,|\,x<0\}$. Then $A$ and $B$ are disjoint, but they are not separated: for instance, $(0,0)\in A\cap\overline B$ (but $\overline A\cap B=\emptyset$).