I'm reading about separation axioms weaker than $T_1$ on topological spaces. One article makes the following claim:
In an arbitrary topological space $(X,\tau)$ (Not necessarily $T_1$)
"If for all $x\in X$ it satisfies {$x$}' is a closed set then for all $x\in X$ it satisfies {$x$}' is the union of disjoint closed sets."
Could you please give me some observation by which is true?
Remark: {$x$}' means the derived set of the point $x$.