Closed sets are complements of open sets

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 .

In a topological space $X$ , a subset $F\subseteq X$ is closed if and only if its complement $X\setminus F$ is open.

Consequently:

This duality between open and closed sets is a basic tool in topology and analysis, especially for closure, limit points, and compactness arguments.