finite intersection property

A collection $𝒜={\{A_{\alpha }\}}_{\alpha \in I}$ of subsets of a set $X$ is said to have the finite intersection property, abbreviated f.i.p., if every finite subcollection $\{A_1,A_2,\mathrm{\dots },A_n\}$ of $𝒜$ satisifes ${\bigcap }_{i=1}^n{A}_i\ne \mathrm{\varnothing }$.

The finite intersection property is most often used to give the following http://planetmath.org/node/3769equivalent condition for the http://planetmath.org/node/503compactness of a topological space (a proof of which may be found http://planetmath.org/node/4181here):

Proposition.

A topological space $X$ is compact if and only if for every collection $\mathrm{C}\mathrm{=}{\mathrm{\{}{C}_{\alpha }\mathrm{\}}}_{\alpha \mathrm{\in }J}$ of closed subsets of $X$ having the finite intersection property, ${\mathrm{\bigcap }}_{\alpha \mathrm{\in }J}{C}_{\alpha }\mathrm{\ne }\mathrm{\varnothing }$.

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

$$C_1\supset C_2\supset C_3\supset \mathrm{\cdots }\text{.}$$

In this case, $𝒞$ automatically has the finite intersection property, and if each $C_i$ is a closed subset of a compact topological space, then, by the proposition, ${\bigcap }_{i=1}^{\mathrm{\infty }}{C}_i\ne \mathrm{\varnothing }$.

The f.i.p. characterization of 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.