a space is compact iff any family of closed sets having fip has non-empty intersection

TheoremMathworldPlanetmath. A topological spaceMathworldPlanetmath is compactPlanetmathPlanetmath if and only if any collectionMathworldPlanetmath of its closed setsPlanetmathPlanetmath having the finite intersection property has non-empty intersectionMathworldPlanetmath.

The above theorem is essentially the definition of a compact space rewritten using de Morgan’s laws. The usual definition of a compact space is based on open sets and unions. The above characterizationMathworldPlanetmath, on the other hand, is written using closed sets and intersections.

Proof. Suppose X is compact, i.e., any collection of open subsets that cover X has a finite collection that also cover X. Further, suppose {Fi}iI is an arbitrary collection of closed subsets with the finite intersection property. We claim that iIFi is non-empty. Suppose otherwise, i.e., suppose iIFi=. Then,

X = (iIFi)c
= iIFic.

(Here, the complement of a set A in X is written as Ac.) Since each Fi is closed, the collection {Fic}iI is an open cover for X. By compactness, there is a finite subset JI such that X=iJFic. But then X=(iJFi)c, so iJFi=, which contradicts the finite intersection property of {Fi}iI.

The proof in the other direction is analogous. Suppose X has the finite intersection property. To prove that X is compact, let {Fi}iI be a collection of open sets in X that cover X. We claim that this collection contains a finite subcollection of sets that also cover X. The proof is by contradictionMathworldPlanetmathPlanetmath. Suppose that XiJFi holds for all finite JI. Let us first show that the collection of closed subsets {Fic}iI has the finite intersection property. If J is a finite subset of I, then

iJFic = (iJFi)c,

where the last assertion follows since J was finite. Then, since X has the finite intersection property,


This contradicts the assumptionPlanetmathPlanetmath that {Fi}iI is a cover for X.


Title a space is compact iff any family of closed sets having fip has non-empty intersection
Canonical name ASpaceIsCompactIffAnyFamilyOfClosedSetsHavingFipHasNonemptyIntersection
Date of creation 2013-03-22 13:34:10
Last modified on 2013-03-22 13:34:10
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 20
Author CWoo (3771)
Entry type Theorem
Classification msc 54D30