Closed set - 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 closed set in a topological space $(X,\mathcal T)$ is a subset $F\subseteq X$ whose complement $X\setminus F$ is a open set.

Examples

In $\mathbb R$ with its usual topology, $[0,1]$ is closed.
In the discrete topology on $X$ , every subset of $X$ is closed.
In the indiscrete topology on $X$ , the only closed sets are $\varnothing$ and $X$ .