Show that the X is disconnected

Question

Let $(X,d)$ be a metric space. We assume $X=A\cup B$, with both of the sets non empty, $A \cap B=\emptyset$. A contains non of B's limit points and B contains non of A's limit points. Show that X is disconnected.

I thought I wanted to show that the sets A and B are both open but I have some theorem that says if a set contains all it´s limit points it´s closed. But I dont know how to show that X is disconnected if A and B are closed.

So if anyone knows how to do that or if I´m missing somethig it would help alot.

Answer 1

Since $A$ and $B$ are closed and they are complementary $A$ and $B$ are also open sets and you have what do you look for.

Answer 2

Hint: The sets $X\setminus \overline A$ and $X\setminus \overline B$ are open.

Answer 3

You can see that $\overline{B}\subseteq B$, where $\overline{B}$ is the closure of $B$. Similarly we have $\overline{A}\subseteq A$. Therefore both $A$ and $B$ are closed.