Finite intersection property theorem

Finite intersection property (FIP) theorem: Let $X$ be a compact topological space (in particular, a compact metric space ). If $\{F_\alpha\}_{\alpha\in A}$ is a family of closed subsets of $X$ such that every finite subfamily has nonempty intersection, i.e., $\bigcap_{j=1}^n F_{\alpha_j}\neq \varnothing \quad \text{for all finite choices } \alpha_1,\dots,\alpha_n,$ then $\bigcap_{\alpha\in A} F_\alpha\neq \varnothing.$ Conversely, a space $X$ is compact iff this property holds for all families of closed sets.

This theorem reformulates compactness in terms of closed sets and is often convenient in existence proofs.

Proof sketch (optional): If the total intersection were empty, then the complements $\{X\setminus F_\alpha\}$ would form an open cover of $X$. Compactness gives a finite subcover, meaning finitely many complements cover $X$, i.e., finitely many $F_\alpha$ have empty intersection, contradicting the FIP.