In mathematics, a filter on a set X {\displaystyle X} is a family B {\displaystyle {\mathcal {B}}} of subsets such that: X ∈ B {\displaystyle X\in {\mathcal {B}}} and ∅ ∉ B {\displaystyle \emptyset \notin {\mathcal {B}}} if A ∈ B {\displaystyle A\in {\mathcal {B}}} and B ∈ B {\displaystyle B\in {\mathcal {B}}}, then A ∩ B ∈ B {\displaystyle A\cap B\in {\mathcal {B}}} If A, B ⊂ X, A ∈ B {\displaystyle A,B\subset X,A\in {\mathcal {B}}}, and A ⊂ B {\displaystyle A\subset B}, then B ∈ B {\displaystyle B\in {\mathcal {B}}} A filter on a set may be thought of as representing a "collection of large subsets", one intuitive example being the neighborhood filter.
