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In mathematics, a non-empty collection of sets is called a **δ-ring** (pronounced "*delta-ring*") if it is closed under union, relative complementation, and countable intersection. The name "delta-ring" originates from the German word for intersection, "Durschnitt", which is meant to highlight the ring's closure under countable intersection, in contrast to a 𝜎-ring which is closed under countable unions.

A family of sets is called a **δ-ring** if it has all of the following properties:

- Closed under finite unions: for all
- Closed under relative complementation: for all and
- Closed under countable intersections: if for all

If only the first two properties are satisfied, then is a ring of sets but not a δ-ring. Every 𝜎-ring is a δ-ring, but not every δ-ring is a 𝜎-ring.

δ-rings can be used instead of σ-algebras in the development of measure theory if one does not wish to allow sets of infinite measure.

The family is a δ-ring but not a 𝜎-ring because is not bounded.

- Field of sets – Algebraic concept in measure theory, also referred to as an algebra of sets
- 𝜆-system (Dynkin system) – Family closed under complements and countable disjoint unions
- Monotone class – theorem
- π-system – Family of sets closed under intersection
- Ring of sets – Family closed under unions and relative complements
- σ-algebra – Algebraic structure of set algebra
- 𝜎-ideal – Family closed under subsets and countable unions
- 𝜎-ring – Family of sets closed under countable unions

- Cortzen, Allan. "Delta-Ring." From MathWorld—A Wolfram Web Resource, created by Eric W. Weisstein. http://mathworld.wolfram.com/Delta-Ring.html

Families of sets over | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|

Is necessarily true of or, is closed under: |
Directed by |
F.I.P. | ||||||||

π-system | ||||||||||

Semiring | Never | |||||||||

Semialgebra (Semifield) | Never | |||||||||

Monotone class | only if | only if | ||||||||

𝜆-system (Dynkin System) | only if |
only if or they are disjoint |
Never | |||||||

Ring (Order theory) | ||||||||||

Ring (Measure theory) | Never | |||||||||

δ-Ring | Never | |||||||||

𝜎-Ring | Never | |||||||||

Algebra (Field) | Never | |||||||||

𝜎-Algebra (𝜎-Field) | Never | |||||||||

Dual ideal | ||||||||||

Filter | Never | Never | ||||||||

Prefilter (Filter base) | Never | Never | ||||||||

Filter subbase | Never | Never | ||||||||

Open Topology | (even arbitrary ) |
Never | ||||||||

Closed Topology | (even arbitrary ) |
Never | ||||||||

Is necessarily true of or, is closed under: |
directed downward |
finite intersections |
finite unions |
relative complements |
complements in |
countable intersections |
countable unions |
contains | contains | Finite Intersection Property |

Additionally, a A is a semiring where every complement is equal to a finite disjoint union of sets in semialgebraare arbitrary elements of and it is assumed that |