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# finite intersection property

A collection $\mathcal{A}=\{A_{\alpha}\}_{{\alpha\in I}}$ of subsets of a set $X$ is said to have the *finite intersection property*, abbreviated f.i.p., if every finite subcollection $\{A_{1},A_{2},\ldots,A_{n}\}$ of $\mathcal{A}$ satisifes $\bigcap_{{i=1}}^{n}A_{i}\neq\emptyset$.

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 $\mathcal{C}=\{C_{\alpha}\}_{{\alpha\in J}}$ 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

$C_{1}\supset C_{2}\supset C_{3}\supset\cdots\text{.}$ |

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.

# References

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

## Mathematics Subject Classification

54D30*no label found*

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