# Field of sets

[https://en.wikipedia.org/wiki/Field\_of\_sets](https://en.wikipedia.org/wiki/Field_of_sets)

A **field of sets** is a mathematical structure consisting of a pair ⟨X, 𝓕⟩ where X is a set and 𝓕 is a family of subsets of X called an algebra over X that contains _the empty set_ as an element (identity), and is closed under the operations of taking _complements_ in X, finite _unions_, and finite _intersections_.

Equivalently, an algebra over X is a subset 𝓕 of the power set of X such that

* ∅ ∈ 𝓕              (𝓕 has the identity element)
* ∀A ∈ 𝓕. A' ∈ 𝓕     (𝓕 is closed under compliment)
* ∀AB ∈ 𝓕. A  B ∈ 𝓕 (𝓕 is closed under intersection)
* ∀AB ∈ 𝓕. A ∪ B ∈ 𝓕 (𝓕 is closed under union)
