Editing Filter (mathematics) (section)

==Motivation==

Fix a [[Partially ordered set|partially ordered set (poset)]]&nbsp;{{Mvar|P}}.  Intuitively, a filter&nbsp;{{Mvar|F}}  is a subset of {{Mvar|P}} whose members are elements large enough to satisfy some criterion.{{sfn|Koutras|Moyzes|Nomikos|Tsaprounis|2021|p=}} For instance, if {{Math|''x'' &isin; ''P''}}, then the set of elements above {{Mvar|x}} is a filter, called the principal filter at {{Mvar|x}}. (If {{Mvar|x}} and {{Mvar|y}} are [[Comparability|incomparable]] elements of {{Mvar|P}}, then neither the principal filter at {{Mvar|x}} nor {{Mvar|y}} is contained in the other.)

Similarly, a filter on a set&nbsp;{{Mvar|S}} contains those subsets that are sufficiently large to contain some given {{em|thing}}. For example, if {{Mvar|S}} is the [[real line]] and {{Math|''x'' &isin; ''S''}}, then the family of sets including {{Mvar|x}} in their [[Interior (topology)|interior]] is a filter, called the neighborhood filter at {{Mvar|x}}. The {{em|thing}} in this case is slightly larger than {{Mvar|x}}, but it still does not contain any other specific point of the line.

The above considerations motivate the upward closure requirement in the [[Filter (mathematics)#Definition|definition below]]: "large enough" objects can always be made larger.

To understand the other two conditions, reverse the roles and instead consider {{Mvar|F}} as a "locating scheme" to find {{Mvar|x}}.  In this interpretation, one searches in some space&nbsp;{{Mvar|X}}, and expects {{Mvar|F}} to describe those subsets of {{Mvar|X}} that contain the goal. The goal must be located somewhere; thus the [[empty set]]&nbsp;{{Math|&emptyset;}} can never be in {{Mvar|F}}.  And if two subsets both contain the goal, then should "zoom in" to their common region.

An ultrafilter describes a "perfect locating scheme" where each scheme component gives new information (either "look here" or "look elsewhere").  [[Compactness#Ordered Spaces|Compactness]] is the property that "every search is fruitful," or, to put it another way, "every locating scheme ends in a search result."

A common use for a filter is to define properties that are satisfied by "generic" elements of some topological space.<ref>{{cite arXiv|last1=Igarashi|first1=Ayumi|last2=Zwicker|first2=William S.|date=16 February 2021|title=Fair division of graphs and of tangled cakes|class=math.CO|eprint=2102.08560}}</ref>  This application generalizes the "locating scheme" to find points that might be hard to write down explicitly.