Download finite intersection property

finite intersection property∗
azdbacks4234†
2013-03-21 15:53:25
A collection A = {Aα }α∈I of subsets of a set X is said to have the finite intersection property,
Tn abbreviated f.i.p., if every finite subcollection {A1 , A2 , . . . , An }
of A satisifes i=1 Ai 6= ∅.
The finite intersection property is most often used to give the following
equivalent condition for the compactness of a topological space (a proof of which
may be found here):
Proposition. A topological space X is compact if and only if for every collection
C
T = {Cα }α∈J of closed subsets of X having the finite intersection property,
α∈J Cα 6= ∅.
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
C
Ti∞is a closed subset of a compact topological space, then, by the proposition,
i=1 Ci 6= ∅.
The f.i.p. characterization of compactness 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.
∗ hFiniteIntersectionPropertyi
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