TITLE: A question about the derived set
QUESTION [5 upvotes]: Let $A$ be a topological set and $A'=\{x: x \text{ is an accumulation point of A}\}$, which is called the derived set of $A$. So, we can define $A''$ of $A'$.
Then do we have $A''\subset A'$?
Thanks for your help.

REPLY [6 votes]: It’s true if $X$ is $T_1$.
If $x\in A''\setminus A'$, and $U$ is any open nbhd of $x$, then $U\cap A'\ne\varnothing$, but there is an open nbhd $U_0$ of $x$ such that $U_0\cap A\subseteq\{x\}$. Let $y\in U_0\cap A'$; then $U_0$ is an open nbhd of $y$, so $U_0\cap(A\setminus\{y\})\ne\varnothing$. It follows that $U_0\cap A=\{x\}$. But then $U_0\setminus\{x\}$ is an open nbhd of $y$ whose intersection with $A\setminus\{y\}$ is empty, which is impossible. Thus, $A''\subseteq A'$.