Editing Finite intersection property (section)

==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).