Editing Finite intersection property (section)

=== 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).