Derived set

Let $(X,d)$ be a metric space and let $A\subseteq X$. The derived set of $A$, denoted $A'$, is the set of all limit points of $A$:

The derived set isolates where a set “accumulates.” It is useful in studying closed sets, perfect sets, and in iterative constructions like the Cantor–Bendixson process.

Examples:

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