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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 created: h2013-03-21i by: hazdbacks4234i version: h34178i Privacy setting: h1i hDefinitioni h54D30i † This text is available under the Creative Commons Attribution/Share-Alike License 3.0. You can reuse this document or portions thereof only if you do so under terms that are compatible with the CC-BY-SA license. 1