Can sequences in two seperated sets have the same limit point?

Question

A set $X$ is said to be seperated $\iff$ $\exists$ two subsets $A $ and $ B $ of $X$ such that $A \cap cl(B) \neq \emptyset $, $B \cap cl(A) \neq \emptyset$ and $A \cup B =X$.

Now,we do not claim $cl(A) \cap cl(B) \neq \emptyset $. Does this mean that, if there are two sequences each of them in two separated sets then they may converge to the same limit point.

Answer 1

This is possible. Consider the sets $ (-1,0) $ and $ (0,1) $. These sets are separated, but we see that the sequences $ \{\frac{1}{n}\}_{n\geq 1} $ and $ \{-\frac{1}{n}\}_{n\geq 1} $ both converge to $ 0 $.