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

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- Feb 4th 2010, 11:12 AMLCopper2010Complex Variables
Prove that if a set contains each of its accumulation points, then it must be a closed set.

- Feb 4th 2010, 11:57 AMPlato
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. - Feb 5th 2010, 04:12 AMHallsofIvy
What does this have to do with complex variables?