Editing Finite intersection property (section)

== Relationship to {{pi}}-systems and filters ==
{{Main|Pi-system|Upward closure}}
A [[Pi-system|{{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|{{pi}}–system]] generated by {{Nowrap|<math display=inline>\mathcal{A}</math>,}} because it is the [[Minimal element|smallest]] {{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 {{Nowrap|<math display=inline>\mathcal{A}</math>,}} the finite intersection property is equivalent to any of the following:

* The [[Pi-system|{{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 {{em|{{visible anchor|proper}}}} or {{em|{{visible anchor|non-degenerate}}}} 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 (set theory)|filter]]".</ref> [[prefilter]].
* The family <math>\mathcal{A}</math> is a subset of some (proper) [[prefilter]].{{sfn|Joshi|1983|pp=242−248}}
* The upward closure <math>\pi(\mathcal{A})^{\uparrow X}</math> is a [[Proper filter (set theory)|(proper) filter]] on {{Nowrap|<math>X</math>.}} In this case, <math>\pi(\mathcal{A})^{\uparrow X}</math> is called the filter on <math>X</math> generated by {{Nowrap|<math>\mathcal{A}</math>,}} 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.{{sfn|Joshi|1983|pp=242−248}}