1. ## Complex Variables

Prove that if a set contains each of its accumulation points, then it must be a closed set.

2. Originally Posted by LCopper2010
Prove that if a set contains each of its accumulation points, then it must be a closed set.
Suppose that $\mathcal{A}$ is such a set.
If $x\in\mathcal{A}^c$, the complement, then $x$ is not a limit point of $\mathcal{A}$.
Therefore, the is an open set, $x\in\mathcal{O}_x$ that contains no point of $\mathcal{A}$.
Now show the complement is open.

3. What does this have to do with complex variables?