I'm looking for a topologic space $X$ and a $A\subset X$ closed such that $A'$ is not closed. Where $A'$ is the derived set of $A$. I'm trying with a $T_0$ space, like Sierpinski topology but I couldn't find the example.
If you can help me with any ideas it would be of great help.
You can't find an example because the derived set of a closed set is always closed. You can find the proof in the answer to this question.