Editing Finite intersection property

{{Short description|Property in general topology}}
In [[general topology]], a branch of [[mathematics]], a non-empty family <math>A</math> of [[subset]]s of a [[Set (mathematics)|set]] <math>X</math> is said to have the '''finite intersection property''' (FIP) if the [[intersection (set theory)|intersection]] over any finite subcollection of <math>A</math> is [[Empty set|non-empty]]. It has the '''strong finite intersection property''' (SFIP) if the intersection over any finite subcollection of <math>A</math> is infinite.  Sets with the finite intersection property are also called '''centered systems''' and '''filter subbases'''.{{sfn|Joshi|1983|pp=242−248}}  

The finite intersection property can be used to reformulate topological [[compactness]] in terms of [[Closed set|closed sets]]; this is its most prominent application.  Other applications include proving that certain [[Perfect set|perfect sets]] are uncountable, and the construction of [[Ultrafilter (set theory)|ultrafilters]].  

==Definition==

Let <math display="inline">X</math> be a set and <math display="inline">\mathcal{A}</math> a [[NonEmpty|nonempty]] [[Family of sets|family of subsets]] of {{Nowrap|<math display=inline>X</math>;}} that is, <math display="inline">\mathcal{A}</math> is a nonempty [[subset]] of the [[power set]] of {{Nowrap|<math display=inline>X</math>.}}  Then <math display="inline">\mathcal{A}</math> 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.{{sfn|Joshi|1983|pp=242−248}}  

In symbols, <math display="inline">\mathcal{A}</math> has the FIP if, for any choice of a finite nonempty subset <math display="inline">\mathcal{B}</math> of {{Nowrap|<math display=inline>\mathcal{A}</math>,}} there must exist a point <math display="block">x\in\bigcap_{B\in \mathcal{B}}{B}\text{.}</math>  Likewise, <math display="inline">\mathcal{A}</math> has the SFIP if, for every choice of such {{Nowrap|<math display=inline>\mathcal{B}</math>,}} there are infinitely many such {{Nowrap|<math display=inline>x</math>.}}{{sfn|Joshi|1983|pp=242−248}}  

In the study of [[Filter (set theory)|filters]], the common intersection of a family of sets is called a [[Kernel (filters)|kernel]], from much the same etymology as the [[Sunflower (mathematics)|sunflower]].  Families with empty kernel are called [[Free filter|free]]; those with nonempty kernel, [[Fixed filter (math)|fixed]].{{sfn|Dolecki|Mynard|2016|pp=27–29, 33–35}}

==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 <math>\mathcal{A}</math> is finite, then <math>\mathcal{A}</math> 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 <math>\{\{1,2\}, \{2,3\}, \{1,3\}\}</math> has pairwise intersections, but not the FIP.  

More generally, let <math display="inline">n \in \N\setminus\{1\}</math> be a positive integer greater than unity, {{Nowrap|<math display=inline>[n]=\{1,\dots,n\}</math>,}} and {{Nowrap|<math display=inline>\mathcal{A}=\{[n]\setminus\{j\}:j\in[n]\}</math>.}}  Then any subset of <math>\mathcal{A}</math> with fewer than <math display="inline">n</math> elements has nonempty intersection, but <math display="inline">\mathcal{A}</math> lacks the FIP.

=== End-type constructions ===
If <math>A_1 \supseteq A_2 \supseteq A_3 \cdots</math> is a decreasing sequence of non-empty sets, then the family <math display="inline">\mathcal{A} = \left\{A_1, A_2, A_3, \ldots\right\}</math> has the finite intersection property (and is even a [[Pi-system|{{pi}}–system]]).  If the inclusions <math>A_1 \supseteq A_2 \supseteq A_3 \cdots</math> are [[Strict subset|strict]], then <math display="inline">\mathcal{A}</math> admits the strong finite intersection property as well.   

More generally, any <math display="inline">\mathcal{A}</math> that is [[Total order|totally ordered]] by inclusion has the FIP.   

At the same time, the kernel of <math display="inline">\mathcal{A}</math> may be empty: if {{Nowrap|<math display=inline>A_j=\{j,j+1,j+2,\dots\}</math>,}} then the [[#kernel|kernel]] of <math>\mathcal{A}</math> is the [[empty set]].  Similarly, the family of intervals <math>\left\{[r, \infty) : r \in \R\right\}</math> also has the (S)FIP, but empty kernel.  

=== "Generic" sets and properties ===
The family of all [[Borel set|Borel subsets]] of <math>[0, 1]</math> with [[Lebesgue measure]] <math display="inline">1</math> has the FIP, as does the family of [[comeagre]] sets.  If <math display="inline">X</math> is an infinite set, then the [[Fréchet filter]] (the family {{Nowrap|<math display=inline>\{X\setminus C:C\text{ finite}\}</math>)}} has the FIP.  All of these are [[Free filter|free filters]]; they are upwards-closed and have empty infinitary intersection.{{sfn|Bourbaki|1987|pp=57–68}}{{sfn|Wilansky|2013|pp=44–46}} 

If <math>X = (0, 1)</math> and, for each positive integer <math>i,</math> the subset <math>X_i</math> is precisely all elements of <math>X</math> having [[Digit (math)|digit]] <math>0</math> in the <math>i</math><sup>th</sup> [[decimal place]], then any finite intersection of <math>X_i</math> is non-empty&nbsp;— just take <math>0</math> in those finitely many places and <math>1</math> in the rest.  But the intersection of <math>X_i</math> for all <math>i \geq 1</math> is empty, since no element of <math>(0, 1)</math> has all zero digits.

=== Extension of the ground set ===
The (strong) finite intersection property is a characteristic of the family {{Nowrap|<math display=inline>\mathcal{A}</math>,}} not the ground set {{Nowrap|<math display=inline>X</math>.}}  If a family <math display="inline">\mathcal{A}</math> on the set <math display="inline">X</math> admits the (S)FIP and {{Nowrap|<math display=inline>X\subseteq Y</math>,}} then <math display="inline">\mathcal{A}</math> is also a family on the set <math display="inline">Y</math> with the FIP (resp.&nbsp;SFIP).  

=== Generated filters and topologies ===
{{See also|#Relationship to π-systems and filters}}
If <math>K \subseteq X</math> is a non-empty set, then the family <math>\mathcal{A}=\{S \subseteq X : K \subseteq S\}</math> has the FIP; this family is called the principal filter on <math display="inline">X</math> generated by {{Nowrap|<math display=inline>K</math>.}}  The subset <math>\mathcal{B} = \{I \subseteq \R : K \subseteq I \text{ and } I \text{ an open interval}\}</math> has the FIP for much the same reason: the kernels contain the non-empty set {{Nowrap|<math display=inline>K</math>.}}  If <math display="inline">K</math> is an open interval, then the set <math display="inline">K</math> is in fact equal to the kernels of <math display="inline">\mathcal{A}</math> or {{Nowrap|<math display=inline>\mathcal{B}</math>,}} 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 (set theory)|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 {{pi}}-systems and filters ==
{{Main|Pi-system|Upward closure}}
A [[Pi-system|{{pi}}–system]] is a non-empty family of sets that is closed under finite intersections. The set <math display="block">\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\}</math>of all finite intersections of one or more sets from <math>\mathcal{A}</math> is called the [[Pi-system|{{pi}}–system]] generated by {{Nowrap|<math display=inline>\mathcal{A}</math>,}} because it is the [[Minimal element|smallest]] {{pi}}–system having <math display="inline">\mathcal{A}</math> as a subset.  

The upward closure of <math>\pi(\mathcal{A})</math> in <math display="inline">X</math> is the set <math display="block">\pi(\mathcal{A})^{\uparrow X} = \left\{S \subseteq X : P \subseteq S \text{ for some } P \in \pi(\mathcal{A})\right\}\text{.}</math>For any family {{Nowrap|<math display=inline>\mathcal{A}</math>,}} the finite intersection property is equivalent to any of the following:

* The [[Pi-system|{{pi}}–system]] generated by <math>\mathcal{A}</math> does not have the [[empty set]] as an element; that is, <math>\varnothing \notin \pi(\mathcal{A}).</math>
* The set <math>\pi(\mathcal{A})</math> has the finite intersection property.
* The set <math>\pi(\mathcal{A})</math> is a (proper)<ref name="ProperDef">A filter or prefilter on a set is {{em|{{visible anchor|proper}}}} or {{em|{{visible anchor|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 (set theory)|filter]]".</ref> [[prefilter]].
* The family <math>\mathcal{A}</math> is a subset of some (proper) [[prefilter]].{{sfn|Joshi|1983|pp=242−248}}
* The upward closure <math>\pi(\mathcal{A})^{\uparrow X}</math> is a [[Proper filter (set theory)|(proper) filter]] on {{Nowrap|<math>X</math>.}} In this case, <math>\pi(\mathcal{A})^{\uparrow X}</math> is called the filter on <math>X</math> generated by {{Nowrap|<math>\mathcal{A}</math>,}} because it is the minimal (with respect to <math>\,\subseteq\,</math>) filter on <math>X</math> that contains <math>\mathcal{A}</math> as a subset.
* <math>\mathcal{A}</math> is a subset of some (proper)<ref name="ProperDef" /> filter.{{sfn|Joshi|1983|pp=242−248}}
==Applications==

=== Compactness ===
The finite intersection property is useful in formulating an alternative definition of [[Compact space|compactness]]: 

{{math theorem | math_statement =A [[Topological space|space]] is compact if and only if every family of [[Closed set|closed subset]]s having the finite intersection property has [[#non-empty intersection|non-empty intersection]].{{sfn|Munkres|2000|p=169}}<ref>{{planetmath| urlname=ASpaceIsCompactIffAnyFamilyOfClosedSetsHavingFipHasNonemptyIntersection| title=A space is compact iff any family of closed sets having fip has non-empty intersection}}</ref>}}

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 number|real numbers]] are [[Uncountable set|uncountable]].  {{math theorem | math_statement = Let <math>X</math> be a non-empty [[Compact space|compact]] [[Hausdorff space]] that  satisfies the property that no one-point set is [[Open set|open]]. Then <math>X</math> is [[uncountable]].}}All the conditions in the statement of the theorem are necessary:
# 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.
# We cannot eliminate the compactness condition, as the set of [[rational number]]s shows.
# We cannot eliminate the condition that one point sets cannot be open, as any finite space with the [[discrete topology]] shows.

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

Now suppose <math>f : \N \to X</math> is a [[bijection]], and let <math>\left\{ x_i : i \in \N \right\}</math> denote the [[Image (mathematics)#Image of a function|image]] of <math>f.</math> Let <math>X</math> be the first open set and choose a neighbourhood <math>U_1 \subset X</math> whose closure does not contain <math>x_1.</math> Secondly, choose a neighbourhood <math>U_2 \subset U_1</math> whose closure does not contain <math>x_2.</math> Continue this process whereby choosing a neighbourhood <math>U_{n+1} \subset U_n</math> whose closure does not contain <math>x_{n+1}.</math> Then the collection <math>\left\{ U_i : i \in \N \right\}</math> satisfies the finite intersection property and hence the intersection of their closures is non-empty by the compactness of <math>X.</math> Therefore, there is a point <math>x</math> in this intersection. No <math>x_i</math> can belong to this intersection because <math>x_i</math> does not belong to the closure of <math>U_i.</math> This means that <math>x</math> is not equal to <math>x_i</math> for all <math>i</math> and <math>f</math> is not [[Surjection|surjective]]; a contradiction. Therefore, <math>X</math> is uncountable.
}}{{math theorem | name=Corollary | math_statement=Every [[closed interval]] <math>[a, b]</math> with <math>a < b</math> is uncountable. Therefore, <math>\R</math> is uncountable.}}

{{math theorem | name=Corollary | math_statement=Every [[Perfect set|perfect]], [[Locally compact space|locally compact]] [[Hausdorff space]] is uncountable.}}

{{math proof | proof =
Let <math>X</math> be a perfect, compact, Hausdorff space, then the theorem immediately implies that <math>X</math> is uncountable. If <math>X</math> is a perfect, locally compact Hausdorff space that is not compact, then the [[one-point compactification]] of <math>X</math> is a perfect, compact Hausdorff space. Therefore, the one point compactification of <math>X</math> is uncountable. Since removing a point from an uncountable set still leaves an uncountable set, <math>X</math> is uncountable as well.
}}

=== Ultrafilters ===
Let <math>X</math> be non-empty, <math>F \subseteq 2^X.</math> <math>F</math> having the finite intersection property. Then there exists an <math>U</math> [[Ultrafilter (set theory)|ultrafilter]] (in <math>2^X</math>) such that <math>F \subseteq U.</math>  This result is known as the [[ultrafilter lemma]].<ref>{{citation |last1=Csirmaz |first1=László |title=Matematikai logika |url=http://www.renyi.hu/~csirmaz/ |year=1994 |location=Budapest |publisher=[[Eötvös Loránd University]] |format=In Hungarian |last2=Hajnal |first2=András |author2-link=András Hajnal}}.</ref>  

==See also==

* {{annotated link|Filter (set theory)}}
* {{annotated link|Filters in topology}}
* {{annotated link|Neighbourhood system}}
* {{annotated link|Ultrafilter (set theory)}}

==References==
===Notes===
{{reflist|group=note}}

===Citations===
{{reflist}}

===General sources===
* {{Bourbaki General Topology Part I Chapters 1-4}} <!--{{sfn|Bourbaki|1989|p=}}-->
* {{Bourbaki General Topology Part II Chapters 5-10}} <!--{{sfn|Bourbaki|1989|p=}}-->
* {{Bourbaki Topological Vector Spaces Part 1 Chapters 1–5}} <!--{{sfn|Bourbaki|1987|p=}}-->
* {{Comfort Negrepontis The Theory of Ultrafilters 1974}} <!--{{sfn|Comfort|Negrepontis|1974|p=}}-->
* {{Császár General Topology}} <!--{{sfn|Császár|1978|p=}}-->
* {{Dolecki Mynard Convergence Foundations Of Topology}} <!--{{sfn|Dolecki|Mynard|2016|p=}}-->
* {{Dugundji Topology}} <!--{{sfn|Dugundji|1966|p=}}-->
* {{Joshi Introduction to General Topology}} <!--{{sfn|Joshi|1983|p=}}-->
* {{cite journal|last1=Koutras|first1=Costas D.|last2=Moyzes|first2=Christos|last3=Nomikos|first3=Christos|last4=Tsaprounis|first4=Konstantinos|last5=Zikos|first5=Yorgos|title=On Weak Filters and Ultrafilters: Set Theory From (and for) Knowledge Representation|journal=[[Logic Journal of the IGPL]]|date=20 October 2021|volume=31 |pages=68–95 |doi=10.1093/jigpal/jzab030}} <!--{{sfn|Koutras|Moyzes|Nomikos|2021|p=}}-->
* {{cite web|last=MacIver R.|first=David|title=Filters in Analysis and Topology|date=1 July 2004|url=http://www.efnet-math.org/~david/mathematics/filters.pdf|archive-url=https://web.archive.org/web/20071009170540/http://www.efnet-math.org/~david/mathematics/filters.pdf |archive-date=2007-10-09 }} (Provides an introductory review of filters in topology and in metric spaces.)
* {{Munkres Topology|edition=2}} <!--{{sfn|Munkres|2000|p=}}-->
* {{Narici Beckenstein Topological Vector Spaces|edition=2}} <!--{{sfn|Narici|Beckenstein|2011|p=}}-->
* {{Wilansky Modern Methods in Topological Vector Spaces|edition=1}} <!--{{sfn|Wilansky|2013|p=}}-->
* {{Wilansky Topology for Analysis 2008}} <!--{{sfn|Wilansky|2008|p=}}-->

==External links==

* {{planetmathref|urlname=FiniteIntersectionProperty|title=Finite intersection property}}

{{Families of sets}}

{{Set theory}}

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