Editing Finite intersection property (section)

==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}}