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