In set theory and related branches of mathematics, a family (or collection) can mean, depending upon the context, any of the following: set, indexed set, multiset, or class.
is called a family of subsets of
, in which case the sets of the family are indexed by members of
[1] In some contexts, a family of sets may be allowed to contain repeated copies of any given member,[2][3][4] and in other contexts it may form a proper class.
The subject of extremal set theory concerns the largest and smallest examples of families of sets satisfying certain restrictions.
is called the power set of
of all ordinal numbers is a large family of sets.
That is, it is not itself a set but instead a proper class.
is itself a subset of the power set
Any family of sets without repetitions is a subclass of the proper class of all sets (the universe).
Hall's marriage theorem, due to Philip Hall, gives necessary and sufficient conditions for a finite family of non-empty sets (repetitions allowed) to have a system of distinct representatives.
Certain types of objects from other areas of mathematics are equivalent to families of sets, in that they can be described purely as a collection of sets of objects of some type: A family of sets is said to cover a set
belongs to some member of the family.
A family is called a point-finite collection if every point of
lies in only finitely many members of the family.
If every point of a cover lies in exactly one member of
is a topological space, a cover whose members are all open sets is called an open cover.
A family is called locally finite if each point in the space has a neighborhood that intersects only finitely many members of the family.
A σ-locally finite or countably locally finite collection is a family that is the union of countably many locally finite families.
is said to refine another (coarser) cover
A Helly family is a set family such that any minimal subfamily with empty intersection has bounded size.
Helly's theorem states that convex sets in Euclidean spaces of bounded dimension form Helly families.
An abstract simplicial complex is a set family
A matroid is an abstract simplicial complex with an additional property called the augmentation property.
A convexity space is a set family closed under arbitrary intersections and unions of chains (with respect to the inclusion relation).
Other examples of set families are independence systems, greedoids, antimatroids, and bornological spaces.
Additionally, a semiring is a π-system where every complement
is equal to a finite disjoint union of sets in
A semialgebra is a semiring where every complement
is equal to a finite disjoint union of sets in