Derived set

The derived set of a subset $A\subseteq X$ in a topological space is the set

where “limit point” is as in limit point .

The derived set records the accumulation behavior of $A$, and it satisfies the basic relationship with closure :

Examples:

• In $\mathbb{R}$, if $A=\{1/n : n\in\mathbb{N}\}$ then $A'=\{0\}$.
• In $\mathbb{R}$, if $A=\mathbb{Z}$ then $A'=\varnothing$.
• In $\mathbb{R}$, if $A=(0,1)$ then $A'=[0,1]$.