finite intersection property

A collectionMathworldPlanetmath 𝒜={Aα}αI of subsets of a set X is said to have the finite intersection property, abbreviated f.i.p., if every finite subcollection {A1,A2,,An} of 𝒜 satisifes i=1nAi.

The finite intersection property is most often used to give the following condition for the of a topological spaceMathworldPlanetmath (a proof of which may be found


A topological space X is compactPlanetmathPlanetmath if and only if for every collection C={Cα}αJ of closed subsets of X having the finite intersection property, αJCα.

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


In this case, 𝒞 automatically has the finite intersection property, and if each Ci is a closed subset of a compact topological space, then, by the propositionPlanetmathPlanetmathPlanetmath, i=1Ci.

The f.i.p. characterizationMathworldPlanetmath 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 TychonoffPlanetmathPlanetmath’s Theorem.


  • 1 J. Munkres, TopologyMathworldPlanetmath, 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 IntersectionMathworldPlanetmath
Related topic Finite
Defines finite intersection property