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'finite intersection property'
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| Title of object: |
finite intersection property |
| Canonical Name: |
FiniteIntersectionProperty |
| Type: |
Definition |
| Created on: |
2003-04-12 09:03:45 |
| Modified on: |
2007-06-18 14:25:09 |
| Classification: |
msc:54D30 |
| Defines: |
f.i.p. |
| Synonyms: |
finite intersection property=finite intersection condition
finite intersection property=f.i.c. |
Preamble:
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\usepackage{amssymb}
\usepackage{amsmath}
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\usepackage{amsthm}
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%\usepackage{psfrag}
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%\usepackage{graphicx}
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\theoremstyle{plain}
\newtheorem*{thm}{Theorem}
\newtheorem*{lem}{Lemma}
\newtheorem*{cor}{Corollary}
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Content:
A collection $\mathcal{A}=\{A_\alpha\}_{\alpha\in I}$ of subsets of a set $X$ is said to have the \emph{finite intersection property}, abbreviated f.i.p, if every finite subcollection $\{A_1,A_2,\ldots,A_n\}$ of $\mathcal{A}$ satisifes $\bigcap_{i=1}^nA_i\neq\emptyset$.
Notice that an implicit requirement imposed by the definition of the f.i.p. is that each set in the collection $\mathcal{A}$ be nonempty. The finite intersection property is most often used to give an alternative characterization of compactness of a topological space.
\begin{thm}
A topological space $X$ is compact if and only if for every collection $\mathcal{C}=\{C_\alpha\}_{\alpha\in J}$ of closed subsets of $X$ having the finite intersection property, $\bigcap_{\alpha\in J}C_\alpha\neq\emptyset$.
\end{thm}
An important special case of the preceding theorem is that in which $\mathcal{C}$ is a countable collection of nested sets, i.e., when we have
\begin{equation*}
C_1\subset C_2\subset\cdots\subset C_n\subset C_{n+1}\subset\cdots
\end{equation*}
As long as each $C_i$ is nonempty, the collection $\mathcal{C}$ will have the finite intersection property; moreover, if each $C_i$ is a closed subset of a compact topological space $X$, then by the theorem, $\bigcap_{i=1}^\infty C_i\neq\emptyset$.
The f.i.p. characterization of compactness may also be used to show that the interval $[0,1]$ is uncountable, as well as in a proof of Tychonoff's Theorem.
Please note this object is currently under construction.
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