Finite intersection property

Ingeneral topology,a branch ofmathematics,a non-empty familyAofsubsetsof aset$X$is said to have thefinite intersection property(FIP) if theintersectionover any finite subcollection of$A$isnon-empty.It has thestrong finite intersection property(SFIP) if the intersection over any finite subcollection of$A$is infinite. Sets with the finite intersection property are also calledcentered systemsandfilter subbases.

The finite intersection property can be used to reformulate topologicalcompactnessin terms ofclosed sets;this is its most prominent application. Other applications include proving that certainperfect setsare uncountable, and the construction ofultrafilters.

Definition

Let${\textstyle X}$be a set and${\textstyle {\mathcal {A}}}$anonemptyfamily of subsetsof${\textstyle X}$;that is,${\textstyle {\mathcal {A}}}$is asubsetof thepower setof${\textstyle X}$.Then${\textstyle {\mathcal {A}}}$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.

In symbols,${\textstyle {\mathcal {A}}}$has the FIP if, for any choice of a finite nonempty subset${\textstyle {\mathcal {B}}}$of${\textstyle {\mathcal {A}}}$,there must exist a point$$x\in \bigcap _{B\in {\mathcal {B}}}{B}{\text{.}}$$Likewise,${\textstyle {\mathcal {A}}}$has the SFIP if, for every choice of such${\textstyle {\mathcal {B}}}$,there are infinitely many such${\textstyle x}$.

In the study offilters,the common intersection of a family of sets is called akernel,from much the same etymology as thesunflower.Families with empty kernel are calledfree;those with nonempty kernel,fixed.

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${\mathcal {A}}$is finite, then${\mathcal {A}}$has the finite intersection property if and only if it is fixed.

Pairwise intersection

The finite intersection property isstrictly strongerthan pairwise intersection; the family$\{\{1,2\},\{2,3\},\{1,3\}\}$has pairwise intersections, but not the FIP.

More generally, let${\textstyle n\in \mathbb {N} \setminus \{1\}}$be a positive integer greater than unity,${\textstyle [n]=\{1,\dots,n\}}$,and${\textstyle {\mathcal {A}}=\{[n]\setminus \{j\}:j\in [n]\}}$.Then any subset of${\mathcal {A}}$with fewer than${\textstyle n}$elements has nonempty intersection, but${\textstyle {\mathcal {A}}}$lacks the FIP.

End-type constructions

If$A_{1}\supseteq A_{2}\supseteq A_{3}\cdots$is a decreasing sequence of non-empty sets, then the family${\textstyle {\mathcal {A}}=\left\{A_{1},A_{2},A_{3},\ldots \right\}}$has the finite intersection property (and is even aπ–system). If the inclusions$A_{1}\supseteq A_{2}\supseteq A_{3}\cdots$arestrict,then${\textstyle {\mathcal {A}}}$admits the strong finite intersection property as well.

More generally, any${\textstyle {\mathcal {A}}}$that istotally orderedby inclusion has the FIP.

At the same time, the kernel of${\textstyle {\mathcal {A}}}$may be empty: if${\textstyle A_{j}=\{j,j+1,j+2,\dots \}}$,then thekernelof${\mathcal {A}}$is theempty set.Similarly, the family of intervals$\left\{[r,\infty ):r\in \mathbb {R} \right\}$also has the (S)FIP, but empty kernel.

"Generic" sets and properties

The family of allBorel subsetsof$[0,1]$withLebesgue measure${\textstyle 1}$has the FIP, as does the family ofcomeagresets. If${\textstyle X}$is an infinite set, then theFréchet filter(the family${\textstyle \{X\setminus C:C{\text{ finite}}\}}$)has the FIP. All of these arefree filters;they are upwards-closed and have empty infinitary intersection.

If$X=(0,1)$and, for each positive integer$i,$the subset$X_{i}$is precisely all elements of$X$havingdigit$0$in the$i$^thdecimal place,then any finite intersection of$X_{i}$is non-empty — just take$0$in those finitely many places and$1$in the rest. But the intersection of$X_{i}$for all$i\geq 1$is empty, since no element of$(0,1)$has all zero digits.

Extension of the ground set

The (strong) finite intersection property is a characteristic of the family${\textstyle {\mathcal {A}}}$,not the ground set${\textstyle X}$.If a family${\textstyle {\mathcal {A}}}$on the set${\textstyle X}$admits the (S)FIP and${\textstyle X\subseteq Y}$,then${\textstyle {\mathcal {A}}}$is also a family on the set${\textstyle Y}$with the FIP (resp. SFIP).

Generated filters and topologies

See also:§ Relationship to π-systems and filters

If$K\subseteq X$are sets with$K\neq \varnothing$then the family${\mathcal {A}}=\{S\subseteq X:K\subseteq S\}$has the FIP; this family is called the principal filter on${\textstyle X}$generated by${\textstyle K}$.The subset${\mathcal {B}}=\{I\subseteq \mathbb {R}:K\subseteq I{\text{ and }}I{\text{ an open interval}}\}$has the FIP for much the same reason: the kernels contain the non-empty set${\textstyle K}$.If${\textstyle K}$is an open interval, then the set${\textstyle K}$is in fact equal to the kernels of${\textstyle {\mathcal {A}}}$or${\textstyle {\mathcal {B}}}$,and so is an element of each filter. But in general a filter's kernel need not be an element of the filter.

Aproper filter on a sethas the finite intersection property. Everyneighbourhood subbasisat a point in atopological spacehas the FIP, and the same is true of everyneighbourhood basisand everyneighbourhood filterat a point (because each is, in particular, also a neighbourhood subbasis).

Relationship toπ-systems and filters

Main articles:Pi-systemandUpward closure

Aπ–systemis a non-empty family of sets that is closed under finite intersections. The set$$\pi ({\mathcal {A}})=\left\{A_{1}\cap \cdots \cap A_{n}:1\leq n<\infty {\text{ and }}A_{1},\ldots,A_{n}\in {\mathcal {A}}\right\}$$of all finite intersections of one or more sets from${\mathcal {A}}$is called theπ–systemgenerated by${\textstyle {\mathcal {A}}}$,because it is thesmallestπ–system having${\textstyle {\mathcal {A}}}$as a subset.

The upward closure of$\pi ({\mathcal {A}})$in${\textstyle X}$is the set$$\pi ({\mathcal {A}})^{\uparrow X}=\left\{S\subseteq X:P\subseteq S{\text{ for some }}P\in \pi ({\mathcal {A}})\right\}{\text{.}}$$

For any family${\textstyle {\mathcal {A}}}$,the finite intersection property is equivalent to any of the following:

• Theπ–systemgenerated by${\mathcal {A}}$does not have theempty setas an element; that is,$\varnothing \notin \pi ({\mathcal {A}}).$
• The set$\pi ({\mathcal {A}})$has the finite intersection property.
• The set$\pi ({\mathcal {A}})$is a (proper)prefilter.
• The family${\mathcal {A}}$is a subset of some (proper)prefilter.
• The upward closure${\textstyle \pi ({\mathcal {A}})^{\uparrow X}}$is a(proper) filteron${\textstyle X}$.In this case,$\pi ({\mathcal {A}})^{\uparrow X}$is called the filter on${\textstyle X}$generated by${\textstyle {\mathcal {A}}}$,because it is the minimal (with respect to$\,\subseteq \,$) filter on$X$that contains${\mathcal {A}}$as a subset.
• ${\mathcal {A}}$is a subset of some (proper)filter.

Applications

Compactness

The finite intersection property is useful in formulating an alternative definition ofcompactness:

Theorem—Aspaceis compact if and only if every family ofclosed subsetshaving the finite intersection property hasnon-empty intersection.

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

Uncountability of perfect spaces

Another common application is to prove that thereal numbersareuncountable.

Theorem—Let$X$be a non-emptycompactHausdorff spacethat satisfies the property that no one-point set isopen.Then$X$isuncountable.

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 theindiscrete topologyis 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 ofrational numbersshows.
3. We cannot eliminate the condition that one point sets cannot be open, as any finite space with thediscrete topologyshows.

Proof

We will show that if$U\subseteq X$is non-empty and open, and if$x$is a point of$X,$then there is aneighbourhood$V\subset U$whoseclosuredoes not contain$x$($x$' may or may not be in$U$). Choose$y\in U$different from$x$(if$x\in U$then there must exist such a$y$for otherwise$U$would be an open one point set; if$x\notin U,$this is possible since$U$is non-empty). Then by the Hausdorff condition, choose disjoint neighbourhoods$W$and$K$of$x$and$y$respectively. Then$K\cap U$will be a neighbourhood of$y$contained in$U$whose closure doesn't contain$x$as desired.

Now suppose$f:\mathbb {N} \to X$is abijection,and let$\left\{x_{i}:i\in \mathbb {N} \right\}$denote theimageof$f.$Let$X$be the first open set and choose a neighbourhood$U_{1}\subset X$whose closure does not contain$x_{1}.$Secondly, choose a neighbourhood$U_{2}\subset U_{1}$whose closure does not contain$x_{2}.$Continue this process whereby choosing a neighbourhood$U_{n+1}\subset U_{n}$whose closure does not contain$x_{n+1}.$Then the collection$\left\{U_{i}:i\in \mathbb {N} \right\}$satisfies the finite intersection property and hence the intersection of their closures is non-empty by the compactness of$X.$Therefore, there is a point$x$in this intersection. No$x_{i}$can belong to this intersection because$x_{i}$does not belong to the closure of$U_{i}.$This means that$x$is not equal to$x_{i}$for all$i$and$f$is notsurjective;a contradiction. Therefore,$X$is uncountable.

Corollary—Everyclosed interval$[a,b]$with$a<b$is uncountable. Therefore,$\mathbb {R}$is uncountable.

Corollary—Everyperfect,locally compactHausdorff spaceis uncountable.

Proof

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

Ultrafilters

Let$X$be non-empty,$F\subseteq 2^{X}.$$F$having the finite intersection property. Then there exists an$U$ultrafilter(in$2^{X}$) such that$F\subseteq U.$This result is known as theultrafilter lemma.

See also

• 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 xPages displaying wikidata descriptions as a fallback
• Ultrafilter (set theory)– Maximal proper filterPages displaying short descriptions of redirect targets

References

General sources

• Bourbaki, Nicolas(1989) [1966].General Topology: Chapters 1–4[Topologie Générale].Éléments de mathématique.Berlin New York: Springer Science & Business Media.ISBN978-3-540-64241-1.OCLC18588129.
• Bourbaki, Nicolas(1989) [1967].General Topology 2: Chapters 5–10[Topologie Générale].Éléments de mathématique.Vol. 4. Berlin New York: Springer Science & Business Media.ISBN978-3-540-64563-4.OCLC246032063.
• Bourbaki, Nicolas(1987) [1981].Topological Vector Spaces: Chapters 1–5.Éléments de mathématique.Translated by Eggleston, H.G.; Madan, S. Berlin New York: Springer-Verlag.ISBN3-540-13627-4.OCLC17499190.
• Comfort, William Wistar; Negrepontis, Stylianos (1974).The Theory of Ultrafilters.Vol. 211. Berlin Heidelberg New York:Springer-Verlag.ISBN978-0-387-06604-2.OCLC1205452.
• Császár, Ákos(1978).General topology.Translated by Császár, Klára. Bristol England: Adam Hilger Ltd.ISBN0-85274-275-4.OCLC4146011.
• Dolecki, Szymon;Mynard, Frédéric (2016).Convergence Foundations Of Topology.New Jersey: World Scientific Publishing Company.ISBN978-981-4571-52-4.OCLC945169917.
• Dugundji, James(1966).Topology.Boston: Allyn and Bacon.ISBN978-0-697-06889-7.OCLC395340485.
• Joshi, K. D.(1983).Introduction to General Topology.New York: John Wiley and Sons Ltd.ISBN978-0-85226-444-7.OCLC9218750.
• Koutras, Costas D.; Moyzes, Christos; Nomikos, Christos; Tsaprounis, Konstantinos; Zikos, Yorgos (20 October 2021). "On Weak Filters and Ultrafilters: Set Theory From (and for) Knowledge Representation".Logic Journal of the IGPL.31:68–95.doi:10.1093/jigpal/jzab030.
• MacIver R., David (1 July 2004)."Filters in Analysis and Topology"(PDF).Archived fromthe original(PDF)on 2007-10-09.(Provides an introductory review of filters in topology and in metric spaces.)
• Munkres, James R.(2000).Topology(Second ed.).Upper Saddle River, NJ:Prentice Hall, Inc.ISBN978-0-13-181629-9.OCLC42683260.
• Narici, Lawrence; Beckenstein, Edward (2011).Topological Vector Spaces.Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press.ISBN978-1584888666.OCLC144216834.
• Wilansky, Albert(2013).Modern Methods in Topological Vector Spaces.Mineola, New York: Dover Publications, Inc.ISBN978-0-486-49353-4.OCLC849801114.
• Wilansky, Albert(17 October 2008) [1970].Topology for Analysis.Mineola, New York: Dover Publications, Inc.ISBN978-0-486-46903-4.OCLC227923899.

External links

• "Finite intersection property".PlanetMath.

Families${\mathcal {F}}$of setsover$\Omega$
Is necessarily true of${\mathcal {F}}\colon$ or, is${\mathcal {F}}$closed under:	Directed by$\,\supseteq$	$A\cap B$	$A\cup B$	$B\setminus A$	$\Omega \setminus A$	$A_{1}\cap A_{2}\cap \cdots$	$A_{1}\cup A_{2}\cup \cdots$	$\Omega \in {\mathcal {F}}$	$\varnothing \in {\mathcal {F}}$	F.I.P.
π-system
Semiring										Never
Semialgebra(Semifield)										Never
Monotone class						only if$A_{i}\searrow$	only if$A_{i}\nearrow$
𝜆-system(Dynkin System)				only if $A\subseteq B$			only if$A_{i}\nearrow$or they aredisjoint			Never
Ring(Order theory)
Ring(Measure theory)										Never
δ-Ring										Never
𝜎-Ring										Never
Algebra(Field)										Never
𝜎-Algebra(𝜎-Field)										Never
Dual ideal
Filter				Never	Never				$\varnothing \not \in {\mathcal {F}}$
Prefilter(Filter base)				Never	Never				$\varnothing \not \in {\mathcal {F}}$
Filter subbase				Never	Never				$\varnothing \not \in {\mathcal {F}}$
Open Topology							(even arbitrary$\cup$)			Never
Closed Topology						(even arbitrary$\cap$)				Never
Is necessarily true of${\mathcal {F}}\colon$ or, is${\mathcal {F}}$closed under:	directed downward	finite intersections	finite unions	relative complements	complements in$\Omega$	countable intersections	countable unions	contains$\Omega$	contains$\varnothing$	Finite Intersection Property
Additionally, asemiringis aπ-systemwhere every complement$B\setminus A$is equal to a finitedisjoint unionof sets in${\mathcal {F}}.$ Asemialgebrais a semiring where every complement$\Omega \setminus A$is equal to a finitedisjoint unionof sets in${\mathcal {F}}.$ $A,B,A_{1},A_{2},\ldots$are arbitrary elements of${\mathcal {F}}$and it is assumed that${\mathcal {F}}\neq \varnothing.$