Finite intersection property

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.

Definition

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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

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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

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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

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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

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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

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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

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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

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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 π–system generated by   does not have the empty set as an element; that is,  
  • The set   has the finite intersection property.
  • The set   is a (proper)[note 1] prefilter.
  • The family   is a subset of some (proper) prefilter.[1]
  • The upward closure   is a (proper) filter on  . In this case,   is called the filter on   generated by  , because it is the minimal (with respect to  ) filter on   that contains   as a subset.
  •   is a subset of some (proper)[note 1] filter.[1]

Applications

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Compactness

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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

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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.
Proof

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.

Proof

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.

Ultrafilters

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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

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  • Filter (set theory) – Family of sets representing "large" sets
  • Filters in topology – Use of filters to describe and characterize all basic topological notions and results.
  • Neighbourhood system – (for a point x) collection of all neighborhoods for the point x
  • Ultrafilter (set theory) – Maximal proper filter

References

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Notes

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  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".

Citations

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  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

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