A collection $\mathcal{A}=\{A_{\alpha}\}_{{\alpha\in I}}$ of subsets of a set $X$ is said to have the finite intersection property, abbreviated f.i.p., if every finite subcollection $\{A_{1},A_{2},\ldots,A_{n}\}$ of $\mathcal{A}$ satisifes $\bigcap_{{i=1}}^{n}A_{i}\neq\emptyset$.

The finite intersection property is most often used to give the following equivalent condition for the compactness of a topological space (a proof of which may be found here):

An important special case of the preceding is that in which $\mathcal{C}$ is a countable collection of non-empty nested sets, i.e., when we have

In this case, $\mathcal{C}$ automatically has the finite intersection property, and if each $C_{i}$ is a closed subset of a compact topological space, then, by the proposition, $\bigcap_{{i=1}}^{\infty}C_{i}\neq\emptyset$.

The f.i.p. characterization of compactness may be used to prove a general result on the uncountability of certain compact Hausdorff spaces, and is also used in a proof of Tychonoff’s Theorem.