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.

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