---
tag: 8ZAPAPYJFG
kind: concept
name: Derived set
msc: [54A05]
sources:
  - url: https://en.wikipedia.org/wiki/Derived_set_(mathematics)
same_as:
  wikidata: Q322340
  wikipedia: Derived set (mathematics)
---

## Summary

The derived set of a subset $A$ of a [[topological space|0CQHFPGJ45]] is the set of all [[limit points|0E3X7TXS67]] of $A$.
It is written $A'$.
The [[closure|0QEAJB91GZ]] of $A$ is $A \cup A'$, and repeating the operation $A \mapsto A'$ leads to the Cantor–Bendixson analysis of closed sets.

## Explanations

### L1 human

A limit point of $A$ is a point that has points of $A$, other than itself, arbitrarily close to it.
The derived set collects all such points, whether they lie in $A$ or not.
For the set $A = \set{1, 1/2, 1/3, \dots}$ in $\R$, every point of $A$ is alone in a small interval, but the points pile up at $0$.
So the derived set of $A$ is $\set{0}$.