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In mathematics, a family is a collection of objects that is usually indexed by another set. An indexed family is denoted by (A_i)_{i in I}, where I is the index set and each A_i is an object in a given universe. The phrase “a family consisting of” describes a collection where every element satisfies certain specified properties or belongs to a particular type of object.

Notation and concepts: A family can be finite or infinite. It is common to write F = {A_i : i in I} or (A_i)_{i in I}. Subfamilies are formed by selecting a subset J of the index set, yielding a new family (A_j)_{j in J}. A subfamily may preserve or alter properties of interest, such as disjointness or coverage.

Examples: A family of subsets of a set X is a collection {A_i ⊆ X : i in I}. A covering of X is a family whose union equals X. A partition is a family of pairwise disjoint sets whose union is the entire X. In analysis and topology, one often works with families of open sets or measurable sets. In algebra and combinatorics, families of functions, subsets, or sequences are common.

Applications: Families organize and structure collections of objects for operations like union, intersection, or composition. They are used to define topological covers, measure-theoretic constructions, probabilistic events, and combinatorial configurations. The term highlights the idea that the collection is built by indexing a common type of object and may be manipulated through subfamilies, refinements, or selections.

See also: indexed family, family of sets, subfamily, covering, partition.