In general topology, a branch of mathematics, a non-empty family A of subsets of a set
Sets with the finite intersection property are also called centered systems and filter subbases.
[1] The finite intersection property can be used to reformulate topological compactness in terms of closed sets; this is its most prominent application.
Other applications include proving that certain perfect sets are uncountable, and the construction of ultrafilters.
has the FIP if, for any choice of a finite nonempty subset
[1] In the study of filters, the common intersection of a family of sets is called a kernel, from much the same etymology as the sunflower.
[2] The empty set cannot belong to any collection with the finite intersection property.
A sufficient condition for the FIP intersection property is a nonempty kernel.
The converse is generally false, but holds for finite families; that is, if
has the finite intersection property if and only if it is fixed.
is a decreasing sequence of non-empty sets, then the family
admits the strong finite intersection property as well.
is an infinite set, then the Fréchet filter (the family
All of these are free filters; they are upwards-closed and have empty infinitary intersection.
th decimal place, then any finite intersection of
The (strong) finite intersection property is a characteristic of the family
has the FIP; this family is called the principal filter on
has the FIP for much the same reason: the kernels contain the non-empty set
A proper filter on a set has the finite intersection property.
A π–system is a non-empty family of sets that is closed under finite intersections.
, the finite intersection property is equivalent to any of the following: The finite intersection property is useful in formulating an alternative definition of compactness: Theorem — A space is compact if and only if every family of closed subsets having the finite intersection property has non-empty intersection.
[6][7] This formulation of compactness is used in some proofs of Tychonoff's theorem.
Another common application is to prove that the real numbers are uncountable.
be a non-empty compact Hausdorff space that satisfies the property that no one-point set is open.
Then by the Hausdorff condition, choose disjoint neighbourhoods
Corollary — Every perfect, locally compact Hausdorff space is uncountable.
be a perfect, compact, Hausdorff space, then the theorem immediately implies that
[8] Additionally, a semiring is a π-system where every complement
is equal to a finite disjoint union of sets in
is equal to a finite disjoint union of sets in