Finite intersection property theorem

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 .

Finite intersection property theorem: Let $X$ be a topological space and let $K\subseteq X$ be given the subspace topology. Then $K$ is compact if and only if for every family $\mathcal{F}$ of closed subsets of $K$ with the finite intersection property, one has $\bigcap_{F\in\mathcal{F}} F\neq\varnothing$ .

This is an alternative to the open cover definition of compactness and is often the most convenient form for “compactness by contradiction.”