Limit point

A point whose every neighborhood meets a set away from that point.

Limit point

A limit point (or accumulation point) of a subset $A\subseteq X$ in a topological space is a point $x\in X$ such that every neighborhood $N$ of $x$ satisfies

(N\cap (A\setminus\{x\}))\neq\varnothing.

The set of all limit points of $A$ is the derived set of $A$ . Limit points also describe closure via the identity $\overline{A}=A\cup A'$ .

Examples:

In $\mathbb{R}$ , $0$ is a limit point of the set $\{1/n : n\in\mathbb{N}\}$ .
In $\mathbb{R}$ , every point of $(0,1)$ is a limit point of $(0,1)$ .
In a discrete topological space, no subset has a limit point (every point has a singleton neighborhood).