Editing Finite intersection property (section)

=== "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.