Maybe.
Hope I can find a counterexample that A' is not closed in the future. Thank you Plato, for your patient replies.
Consider $\displaystyle \mathbb{R}$ with the topology generated by $\displaystyle \{(-\infty,a) \mid a \in \mathbb{R} \}$, which is a T0 space. The derived set of $\displaystyle \{0\}$ is $\displaystyle (0, \infty)$, which is open.
If this is Lynn Steen’s Left Order Topology, then isn’t it true that $\displaystyle 0$ is a limit point of $\displaystyle \{0\}?$
If so, then is $\displaystyle 0$ in the derived set of $\displaystyle \{0\}?$
If that is so, what is its complement?