Family (mathematics)

A family is a collection of objects indexed by a set. Its indices distinguish objects even when some of those objects are equal.

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Set: A set is a collection of distinct objects treated as a single mathematical object. The index set supplies the labels that organize the family.
Indexed notation: Indexed notation names objects using subscripts or other labels drawn from an index set. Symbols such as Aᵢ identify each set in a family.
Cover (topology): A cover of a space is a family of subsets whose union contains the space. Open covers are a central topological use of families of sets.
Sequence: A sequence is a function from the natural numbers, or another ordered index set, into a set. Sequences are families with a particular, usually countable, indexing structure.
Index set: An index set is the set whose elements label members of a family. Each member of a family is associated with an element of this set.
Image (mathematics): The image of a function is the set of outputs it attains on its domain. The image records the distinct objects appearing in a family while discarding repeated values.
Open cover: An open cover is a collection of open sets whose union contains a given space. Compactness is defined by extracting finite subfamilies from open covers.
Multiset: A multiset is a collection in which elements can occur with specified multiplicities. A multiset records repetition counts, while a family records the indices causing repetition.
Indexed collection: An indexed collection assigns an object to each element of a specified set. This is the general structure captured by a mathematical family.
Union: The union of sets contains every element belonging to at least one of them. The union of a family collects elements that occur in any indexed member.