# finite intersection property

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 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 $\mathcal{C}=\{C_{\alpha}\}_{\alpha\in J}$ of closed subsets of $X$ having the finite intersection property, $\bigcap_{\alpha\in J}C_{\alpha}\neq\emptyset$.

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

 $C_{1}\supset C_{2}\supset C_{3}\supset\cdots\text{.}$

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 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.

## References

• 1 J. Munkres, Topology, 2nd ed. Prentice Hall, 1975.
 Title finite intersection property Canonical name FiniteIntersectionProperty Date of creation 2013-03-22 13:34:05 Last modified on 2013-03-22 13:34:05 Owner azdbacks4234 (14155) Last modified by azdbacks4234 (14155) Numerical id 17 Author azdbacks4234 (14155) Entry type Definition Classification msc 54D30 Synonym finite intersection condition Synonym f.i.c. Synonym f.i.p. Related topic Compact Related topic Intersection Related topic Finite Defines finite intersection property