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 $\{{A}_{1},{A}_{2},\mathrm{\dots},{A}_{n}\}$ of $\mathcal{A}$ 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 $\mathcal{C}$ is a countable^{} collection of non-empty nested sets, i.e., when we have
$${C}_{1}\supset {C}_{2}\supset {C}_{3}\supset \mathrm{\cdots}\text{.}$$ |
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^{}, ${\bigcap}_{i=1}^{\mathrm{\infty}}{C}_{i}\ne \mathrm{\varnothing}$.
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.
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 |