linguistic:voice_change
Examples

See the image at the top of this article for a simple example of filters on the finite poset P ({1, 2, 3, 4}).

Partially order ℝ → ℝ, the space of real-valued functions on ℝ, by pointwise comparison. Then the set of functions "large at infinity,"

is a filter on ℝ → ℝ. One can generalize this construction quite far by compactifying the domain and completing the codomain: if X is a set with distinguished subset S and Y is a poset with distinguished element m, then { f : f | ≥ m } is a filter in X → Y.The set { { k : k ≥ N } : N ∈ ℕ } is a filter in P (ℕ). More generally, if D is any directed set, then

is a filter in P (D), called the tail filter. Likewise any net { x } generates the eventuality filter { { x : α ≤ β } : α ∈ Α }. A tail filter is the eventuality filter for x = α.The Fréchet filter on an infinite set X is

If (X, μ) is a measure space, then the collection { A : μ (X ∖ A) = 0 } is a filter. If μ (X) = ∞, then { A : μ (X ∖ A) < ∞ } is also a filter; the Fréchet filter is the case where μ is counting measure.

Given an ordinal a, a subset of a is called a club if it is closed in the order topology of a but has net-theoretic limit a. The clubs of a form a filter: the club filter, ♣(a).

The previous construction generalizes as follows: any club C is also a collection of dense subsets (in the ordinal topology) of a, and ♣(a) meets each element of C. Replacing C with an arbitrary collection C ̃ of dense sets, there "typically" exists a filter meeting each element of C ̃, called a generic filter. For countable C ̃, the Rasiowa–Sikorski lemma implies that such a filter must exist; for "small" uncountable C ̃, the existence of such a filter can be forced through Martin's axiom.

Let P denote the set of partial orders of limited cardinality, modulo isomorphism. Partially order P by:

Then the subset of non-atomic partial orders forms a filter. Likewise, if I is the set of injective modules over some given commutative ring, of limited cardinality, modulo isomorphism, then a partial order on I is:

Given any infinite cardinal κ, the modules in I that cannot be generated by fewer than κ elements form a filter.

Every uniform structure on a set X is a filter on X × X.