The question is "Let $X$ be a metric space, find a closed subset $A$ such that $A'\neq \emptyset$ and $(A')' = \emptyset $"

Well in $\mathbb R$ with the standard topology, $\mathbb Q$ is close and open simultaneously (hope I'm not wrong with that) therefore it answers the demands. Yet I can't find an example in which $A$ is closed and not clopen. Are there any?

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    $\begingroup$ $\mathbb{Q}$ is neither closed nor open. $\endgroup$ – John Griffin Oct 16 '17 at 3:53

Consider $A=\left\{ \displaystyle\frac{1}{n} : n\in\mathbb{N}\right\}\cup\{0\}\subseteq\mathbb{R}$. $A$ is closed (in fact, $A$ is compact). We have that $A'=\{0\}$ but $(A')'=\emptyset$

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