Editing Filter (mathematics) (section)

=== Filters on a set; subbases ===
{{Main|Filter (set theory)}}
{{Families of sets}}
Given a set&nbsp;{{Mvar|S}}, the [[power set]]&nbsp;{{Math|{{mathcal|P}}(''S'')}} is [[Partially ordered set|partially ordered]] by [[set inclusion]]; filters on this poset are often just called "filters on {{Mvar|S}}," in an [[abuse of terminology]].  For such posets, downward direction and upward closure reduce to:{{sfn|Dugundji|1966|pp=211-213}}
; Closure under finite intersections: If {{Math|''A'', ''B'' &isin; ''F''}}, then so too is {{Math|''A'' &cap; ''B'' &isin; ''F''}}.  
; Isotony:{{sfn|Dolecki|Mynard| 2016|pp=27-29}} If {{Math|''A'' &isin; ''F''}} and {{Math|''A'' &subseteq; ''B'' &subseteq; ''S''}}, then {{Math|''B'' &isin; ''F''}}.

A '''proper<ref>{{cite book |last1=Goldblatt |first1=R |url=https://archive.org/stream/springer_10.1007-978-1-4612-0615-6/10.1007-978-1-4612-0615-6#page/n31/mode/2up/search/proper+filter |title=Lectures on the Hyperreals: an Introduction to Nonstandard Analysis |page=32}}</ref>/non-degenerate{{sfn|Narici|Beckenstein|2011|pp=2-7}}''' filter is one that does not contain {{Math|&emptyset;}}, and these three conditions (including non-degeneracy) are [[Henri Cartan]]'s original definition of a filter.{{sfn|Cartan|1937a|p=}}{{sfn|Cartan|1937b|p=}}   It is common&nbsp;— ''though not universal''&nbsp;— to require filters on sets to be proper (whatever one's stance on poset filters); we shall again eschew this convention.

Prefilters on a set are proper if and only if they do not contain {{Math|&emptyset;}} either.

For every subset&nbsp;{{Mvar|T}} of {{Math|{{mathcal|P}}(''S'')}}, there is a smallest filter&nbsp;{{Mvar|F}} containing {{Mvar|T}}.  As with prefilters, {{Mvar|T}} is said to generate or span {{Mvar|F}}; a base for {{Mvar|F}} is the set&nbsp;{{Mvar|U}} of all finite intersections of {{Mvar|T}}.  The set {{Mvar|T}} is said to be a '''filter subbase''' when {{Mvar|F}} (and thus {{Mvar|U}}) is proper.

Proper filters on sets have the [[finite intersection property]].

If {{Math|''S'' {{=}} &emptyset;}}, then {{Mvar|S}} admits only the improper filter {{Math|{{brace|&emptyset;}}}}.

==== Free filters ====
A filter is said to be a '''free filter''' if the intersection of its members is empty.  A proper principal filter is not free.

Since the intersection of any finite number of members of a filter is also a member, no proper filter on a finite set is free, and indeed is the principal filter generated by the common intersection of all of its members.  But a nonprincipal filter on an infinite set is not necessarily free: a filter is free if and only if it includes the [[Fréchet filter]] (see {{Slink||Examples}}).