Finite intersection property

Template:Short description In general topology, a branch of mathematics, a non-empty family <math>A</math> of subsets of a set <math>X</math> is said to have the finite intersection property (FIP) if the intersection over any finite subcollection of <math>A</math> is 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.Template:Sfn

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.

DefinitionEdit

Let <math display="inline">X</math> be a set and <math display="inline">\mathcal{A}</math> a nonempty family of subsets of Template:Nowrap that is, <math display="inline">\mathcal{A}</math> is a nonempty subset of the power set of Template:Nowrap 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.Template:Sfn

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 Template:Nowrap 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 Template:Nowrap there are infinitely many such Template:Nowrap Template:Sfn

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.Template:Sfn

Families of examples and non-examplesEdit

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 intersectionEdit

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, Template:Nowrap and Template:Nowrap 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 constructionsEdit

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|Template:Pi–system]]). If the inclusions <math>A_1 \supseteq A_2 \supseteq A_3 \cdots</math> are 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 totally ordered by inclusion has the FIP.

At the same time, the kernel of <math display="inline">\mathcal{A}</math> may be empty: if Template:Nowrap then the 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 propertiesEdit

The family of all 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 Template:Nowrap has the FIP. All of these are free filters; they are upwards-closed and have empty infinitary intersection.Template:Sfn Template:Sfn

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>0</math> in the <math>i</math>^th decimal place, then any finite intersection of <math>X_i</math> is non-empty — 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 setEdit

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

Generated filters and topologiesEdit

Template:See also 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 Template:Nowrap 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 Template:Nowrap 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 Template:Nowrap 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 Template:Pi-systems and filtersEdit

{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} A [[Pi-system|Template: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|Template:Pi–system]] generated by Template:Nowrap because it is the smallest Template: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 Template:Nowrap the finite intersection property is equivalent to any of the following:

The [[Pi-system|Template: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 Template:Em or Template:Em 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".</ref> prefilter.
The family <math>\mathcal{A}</math> is a subset of some (proper) prefilter.Template:Sfn
The upward closure <math>\pi(\mathcal{A})^{\uparrow X}</math> is a (proper) filter on Template:Nowrap In this case, <math>\pi(\mathcal{A})^{\uparrow X}</math> is called the filter on <math>X</math> generated by Template:Nowrap 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.Template:Sfn

ApplicationsEdit

CompactnessEdit

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

Template:Math theorem

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

Uncountability of perfect spacesEdit

Another common application is to prove that the real numbers are uncountable. Template:Math theoremAll 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 numbers shows.
We cannot eliminate the condition that one point sets cannot be open, as any finite space with the discrete topology shows.

Template:Math proof Template:Math theorem

Template:Math theorem

Template:Math proof

UltrafiltersEdit

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 (in <math>2^X</math>) such that <math>F \subseteq U.</math> This result is known as the ultrafilter lemma.<ref>Template:Citation.</ref>