Closure - 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 .

The closure of a subset $A\subseteq X$ in a topological space $(X,\mathcal{T})$ is

\overline{A}=\bigcap\{F\subseteq X : F \text{ is a closed set and } A\subseteq F\}.

Equivalent characterizations

Equivalently, a point $x\in X$ lies in $\overline{A}$ if and only if every neighborhood of $x$ intersects $A$ .

Remarks

A set $F$ is closed exactly when $F=\overline{F}$ . Closure is closely tied to limit points and the derived set.

Examples