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finite intersection property (Definition)

A collection $\mathcal{A}=\set{A_\alpha}_{\alpha\in I}$ of subsets of a set $X$ is said to have the finite intersection property, abbreviated f.i.p., if every finite subcollection $\set{A_1,A_2,\ldots,A_n}$ of $\mathcal{A}$ satisifes $\bigcap_{i=1}^nA_i\neq\emptyset$

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 $\mathcal{C}=\set{C_\alpha}_{\alpha\in J}$ of closed subsets of $X$ having the finite intersection property, $\bigcap_{\alpha\in J}C_\alpha\neq\emptyset$

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 \begin{equation*} C_1\supset C_2\supset C_3\supset\cdots\text{.} \end{equation*}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, $\bigcap_{i=1}^\infty C_i\neq\emptyset$

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.

Bibliography

1
J. Munkres, Topology, 2nd ed. Prentice Hall, 1975.




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See Also: compact, intersection, finite

Other names:  finite intersection condition, f.i.c., f.i.p.
Also defines:  finite intersection property
Keywords:  compact, intersection, finite
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Cross-references: proof of Tychonoff's theorem, Hausdorff spaces, characterization, countable, closed subsets, compact, proof, topological space, finite, subsets, collection
There are 16 references to this entry.

This is version 14 of finite intersection property, born on 2003-04-12, modified 2007-06-22.
Object id is 4178, canonical name is FiniteIntersectionProperty.
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Classification:
AMS MSC54D30 (General topology :: Fairly general properties :: Compactness)

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