Limit point - Knowlpedia

Empty set and whole space: ∅ , X ∈ T \varnothing,X\in\mathcal T ∅ , X ∈ T .
Arbitrary unions: the union of any collection of members of T \mathcal T T belongs to T \mathcal T T .
Finite intersections: the intersection of finitely many members of T \mathcal T T belongs to T \mathcal T T .

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.

Relation to closure

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 $\{1/n : n\ge1\}$ .
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).