A collection $\mathcal{A}=\set{A_\alpha}_{\alpha\in I}$ of subsets of a set $X$ is said to have the finite intersection property, abbreviated f.i.p., if every finite subcollection $\set{A_1,A_2,\ldots,A_n}$ of $\mathcal{A}$ satisifes $\bigcap_{i=1}^nA_i\neq\emptyset$

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 \begin{equation*} C_1\supset C_2\supset C_3\supset\cdots\text{.} \end{equation*}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.

Bibliography

1

J. Munkres, Topology, 2nd ed. Prentice Hall, 1975.