Finite intersection property

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In general topology, the finite intersection property is a property of a collection of subsets of a set X. A collection has this property if the intersection over any finite subcollection of the collection is nonempty.

1 Definition
2 Discussion
3 Examples
4 Theorems
5 Variants
6 References

[edit] Definition

Let $X$ be a set with $A=\{A_i\}_{i\in I}$ a family of subsets of $X$ . Then the collection $A$ has the finite intersection property (fip), if any finite subcollection $J\subset I$ has non-empty intersection $\bigcap_{i\in J} A_i$ .

[edit] Discussion

This condition is trivially satisfied if the intersection over the entire collection is nonempty (in particular, if the collection itself is empty), and it is also trivially satisfied if the collection is nested, meaning that for any finite subcollection, a particular element of the subcollection is contained in all the other elements of the subcollection, e.g. the nested sequence of intervals

(0, 1/n).

These are not the only possibilities however. For example, if X = (0, 1) and for each positive integer i, X_i is the set of elements of X having a decimal expansion with digit 0 in the i'th decimal place, then any finite intersection is nonempty (just take 0 in those finitely many places and 1 in the rest), but the intersection of all X_i for i≥1 is empty, since no element of (0, 1) has all zero digits.

The finite intersection property is useful in formulating an alternative definition of compactness: a space is compact if and only if every collection of closed sets satisfying the finite intersection property has nonempty intersection itself^[1]. This formulation of compactness is used in some proofs of Tychonoff's theorem and the uncountability of the real numbers.

[edit] Examples

A filter has the finite intersection property by definition.

[edit] Theorems

Let $X \neq \emptyset$ , $F \subseteq 2^X$ , F having the finite intersection property. Then there exists an $F^\prime$ ultrafilter (in $2 X$ ) such that $F \subseteq F^\prime$ . See details and proof in ^[2].