In general topology, a branch of mathematics, a non-empty family A of subsets of a set is said to have the finite intersection property (FIP) if the intersection over any finite subcollection of is non-empty. It has the strong finite intersection property (SFIP) if the intersection over any finite subcollection of is infinite. 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.


Let be a set and a nonempty family of subsets of ; that is, is a subset of the power set of . Then is said to have the finite intersection property if every nonempty finite subfamily has nonempty intersection; it is said to have the strong finite intersection property if that intersection is always infinite.[1]

In symbols, has the FIP if, for any choice of a finite nonempty subset of , there must exist a point

Likewise, has the SFIP if, for every choice of such , there are infinitely many such .[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. Families with empty kernel are called free; those with nonempty kernel, fixed.[2]

Families of examples and non-examples

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 is finite, then has the finite intersection property if and only if it is fixed.

Pairwise intersection

The finite intersection property is strictly stronger than pairwise intersection; the family has pairwise intersections, but not the FIP.

More generally, let be a positive integer greater than unity, , and . Then any subset of with fewer than elements has nonempty intersection, but lacks the FIP.

End-type constructions

If is a decreasing sequence of non-empty sets, then the family has the finite intersection property (and is even a π–system). If the inclusions are strict, then admits the strong finite intersection property as well.

More generally, any that is totally ordered by inclusion has the FIP.

At the same time, the kernel of may be empty: if , then the kernel of is the empty set. Similarly, the family of intervals also has the (S)FIP, but empty kernel.

"Generic" sets and properties

The family of all Borel subsets of with Lebesgue measure has the FIP, as does the family of comeagre sets. If is an infinite set, then the Fréchet filter (the family ) has the FIP. All of these are free filters; they are upwards-closed and have empty infinitary intersection.[3][4]

If and, for each positive integer the subset is precisely all elements of having digit in the th decimal place, then any finite intersection of is non-empty — just take in those finitely many places and in the rest. But the intersection of for all is empty, since no element of has all zero digits.

Extension of the ground set

The (strong) finite intersection property is a characteristic of the family , not the ground set . If a family on the set admits the (S)FIP and , then is also a family on the set with the FIP (resp. SFIP).

Generated filters and topologies

See also: § Relationship to π-systems and filters

If are sets with then the family has the FIP; this family is called the principal filter on generated by . The subset has the FIP for much the same reason: the kernels contain the non-empty set . If is an open interval, then the set is in fact equal to the kernels of or , and so is an element of each filter. But in general a filter's kernel need not be an element of the filter.

A proper filter on a set has the finite intersection property. Every neighbourhood subbasis at a point in a topological space has the FIP, and the same is true of every neighbourhood basis and every neighbourhood filter at a point (because each is, in particular, also a neighbourhood subbasis).

Relationship to π-systems and filters

Main articles: Pi-system and Upward closure

A π–system is a non-empty family of sets that is closed under finite intersections. The set

of all finite intersections of one or more sets from is called the π–system generated by , because it is the smallest π–system having as a subset.

The upward closure of in is the set

For any family , 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.[5][6]

This formulation of compactness is used in some proofs of Tychonoff's theorem.

Uncountability of perfect spaces

Another common application is to prove that the real numbers are uncountable.

Theorem — Let be a non-empty compact Hausdorff space that satisfies the property that no one-point set is open. Then is uncountable.

All the conditions in the statement of the theorem are necessary:

  1. We cannot eliminate the Hausdorff condition; a countable set (with at least two points) with the indiscrete topology is compact, has more than one point, and satisfies the property that no one point sets are open, but is not uncountable.
  2. We cannot eliminate the compactness condition, as the set of rational numbers shows.
  3. We cannot eliminate the condition that one point sets cannot be open, as any finite space with the discrete topology shows.

We will show that if is non-empty and open, and if is a point of then there is a neighbourhood whose closure does not contain (' may or may not be in ). Choose different from (if then there must exist such a for otherwise would be an open one point set; if this is possible since is non-empty). Then by the Hausdorff condition, choose disjoint neighbourhoods and of and respectively. Then will be a neighbourhood of contained in whose closure doesn't contain as desired.

Now suppose is a bijection, and let denote the image of Let be the first open set and choose a neighbourhood whose closure does not contain Secondly, choose a neighbourhood whose closure does not contain Continue this process whereby choosing a neighbourhood whose closure does not contain Then the collection satisfies the finite intersection property and hence the intersection of their closures is non-empty by the compactness of Therefore, there is a point in this intersection. No can belong to this intersection because does not belong to the closure of This means that is not equal to for all and is not surjective; a contradiction. Therefore, is uncountable.

Corollary — Every closed interval with is uncountable. Therefore, is uncountable.

Corollary — Every perfect, locally compact Hausdorff space is uncountable.


Let be a perfect, compact, Hausdorff space, then the theorem immediately implies that is uncountable. If is a perfect, locally compact Hausdorff space that is not compact, then the one-point compactification of is a perfect, compact Hausdorff space. Therefore, the one point compactification of is uncountable. Since removing a point from an uncountable set still leaves an uncountable set, is uncountable as well.


Let be non-empty, having the finite intersection property. Then there exists an ultrafilter (in ) such that This result is known as the ultrafilter lemma.[7]

See also



  1. ^ a b A filter or prefilter on a set is proper or non-degenerate if it does not contain the empty set as an element. Like many − but not all − authors, this article will require non-degeneracy as part of the definitions of "prefilter" and "filter".


  1. ^ a b c d e Joshi 1983, pp. 242−248.
  2. ^ Dolecki & Mynard 2016, pp. 27–29, 33–35.
  3. ^ Bourbaki 1987, pp. 57–68.
  4. ^ Wilansky 2013, pp. 44–46.
  5. ^ Munkres 2000, p. 169.
  6. ^ A space is compact iff any family of closed sets having fip has non-empty intersection at PlanetMath.
  7. ^ Csirmaz, László; Hajnal, András (1994), Matematikai logika (In Hungarian), Budapest: Eötvös Loránd University.

General sources