stylistic:tone
Motivation

Fix a partially ordered set (poset) P. Intuitively, a filter F is a subset of P whose members are elements large enough to satisfy some criterion. For instance, if x ∈ P, then the set of elements above x is a filter, called the principal filter at x. (If x and y are incomparable elements of P, then neither the principal filter at x nor y is contained in the other.)

Similarly, a filter on a set S contains those subsets that are sufficiently large to contain some given thing. For example, if S is the real line and x ∈ S, then the family of sets including x in their interior is a filter, called the neighborhood filter at x. The thing in this case is slightly larger than x, but it still does not contain any other specific point of the line.

The above considerations motivate the upward closure requirement in the definition below: "large enough" objects can always be made larger.

To understand the other two conditions, reverse the roles and instead consider F as a "locating scheme" to find x. In this interpretation, one searches in some space X, and expects F to describe those subsets of X that contain the goal. The goal must be located somewhere; thus the empty set ∅ can never be in 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 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. This application generalizes the "locating scheme" to find points that might be hard to write down explicitly.