finite intersection property
A collection 𝒜={Aα}α∈I 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 ⋂ni=1Ai≠∅.
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 C={Cα}α∈J of closed subsets of X having the finite intersection property, ⋂α∈JCα≠∅.
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
C1⊃C2⊃C3⊃⋯. |
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 proposition, ⋂∞i=1Ci≠∅.
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 |