In mathematics, more specifically in point-set topology, the derived set of a subset ${\displaystyle S}$ of a topological space is the set of all limit points of ${\displaystyle S.}$ It is usually denoted by ${\displaystyle S'.}$

The concept was first introduced by Georg Cantor in 1872 and he developed set theory in large part to study derived sets on the real line.

The derived set of a subset ${\displaystyle S}$ of a topological space ${\displaystyle X,}$ denoted by ${\displaystyle S',}$ is the set of all points ${\displaystyle x\in X}$ that are limit points of ${\displaystyle S,}$ that is, points ${\displaystyle x}$ such that every neighbourhood of ${\displaystyle x}$ contains a point of ${\displaystyle S}$ other than ${\displaystyle x}$ itself.

If ${\displaystyle \mathbb {R} }$ is endowed with its usual Euclidean topology then the derived set of the half-open interval ${\displaystyle [0,1)}$ is the closed interval ${\displaystyle [0,1].}$

Consider ${\displaystyle \mathbb {R} }$ with the topology (open sets) consisting of the empty set and any subset of ${\displaystyle \mathbb {R} }$ that contains 1. The derived set of ${\displaystyle A:=\{1\}}$ is ${\displaystyle A'=\mathbb {R} \setminus \{1\}.}$

Let ${\displaystyle X}$ denote a topological space in what follows.

If ${\displaystyle A}$ and ${\displaystyle B}$ are subsets of ${\displaystyle X,}$ the derived set has the following properties:

A set ${\displaystyle S\subseteq X}$ is closed precisely when ${\displaystyle S'\subseteq S,}$ that is, when ${\displaystyle S}$ contains all its limit points. For any ${\displaystyle S\subseteq X,}$ the set ${\displaystyle S\cup S'}$ is closed and is the closure of ${\displaystyle S}$ (that is, the set ${\displaystyle {\overline {S}}}$).