finite intersection propertyFiniteIntersectionProperty2003-04-12 09:03:452007-06-22 01:05:47Definitionfinite intersection propertycompactintersectionfinite%%packages
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A collection $\mathcal{A}=\set{A_\alpha}_{\alpha\in I}$ of subsets of a set $X$ is said to have the \emph{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 \PMlinkid{equivalent}{3769} condition for the \PMlinkid{compactness}{503} of a topological space (a proof of which may be found \PMlinkid{here}{4181}):
\begin{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, $\bigcap_{\alpha\in J}C_\alpha\neq\emptyset$.
\end{proposition*}
An important special case of the preceding \PMlinkescapeword{proposition} 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 \PMlinkescapetext{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.
\begin{thebibliography}{1}
\bibitem{Munkres}
J. Munkres, \emph{Topology}, 2nd ed. Prentice Hall, 1975.
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