indexijoukko

An indexijoukko, also known in Finnish as indeksijoukko, is a set I used to label members of a family {A_i}_{i∈I} of objects. The elements A_i are indexed by i from I, so each i yields A_i. The index set need not have any particular algebraic structure; it is simply a bookkeeping device that enables concise formulations of families that may be finite or infinite, large or small.

Common examples include sequences {a_n}_{n∈N}, which is a family indexed by the natural numbers; in this case I = N and the i-th object is a_n. More generally, one talks about families of sets, vector spaces, topological spaces, or modules {X_i}_{i∈I}.

Two standard constructions use the index set: the Cartesian product ∏_{i∈I} X_i, which consists of all choices x_i ∈ X_i for each i; and the direct sum ⊕_{i∈I} X_i, consisting of those families with finite support (nonzero for finitely many i).

Properties: the index set may be finite or infinite; it can be empty, and conventions differ on whether the empty product is a singleton or not. In many theories, I is assumed nonempty, especially when talking about products; nets and filters use directed index sets to express convergence, and ordering or topology on I can be meaningful.

Usage: index sets allow uniform statements across a family, unify constructions across many objects, and stabilize notation in proofs and definitions. In Finnish mathematics, the term indeksijoukko is commonly used for index set.