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An indexed family {S_i : i ∈ I} associates each index i in set I with a set S_i. This formalism enables rigorous definition of generalized union ⋃_{i∈I} S_i and intersection ⋂_{i∈I} S_i, extending binary operations to arbitrary collections.
You already know what sets are and how subset relationships work: A ⊆ B means every element of A is also an element of B. You've worked with binary union A ∪ B and binary intersection A ∩ B. But mathematics constantly requires operating on *infinitely many* sets at once — the intersection of all open sets containing a point, the union of all sets in a collection indexed by the integers, or the product of a family of groups. Indexed families provide the rigorous framework for making these operations precise.
An indexed family of sets is a function f: I → V where I is any set (the index set) and V is a collection of sets (or a universe). We write the family as {S_i : i ∈ I} or simply (S_i)_{i ∈ I}, where S_i = f(i) is the set assigned to index i. The index set I can be anything: {1, 2, 3} gives a finite family, ℕ gives a countably infinite family, ℝ gives an uncountably infinite family. The key point is that the index labels the sets without requiring the sets to be distinct — S_i and S_j can be equal for i ≠ j, and the indexing still makes sense.
Generalized union ⋃_{i∈I} S_i is the set of all elements that belong to *at least one* S_i: x ∈ ⋃_{i∈I} S_i if and only if there exists some i ∈ I with x ∈ S_i. Generalized intersection ⋂_{i∈I} S_i is the set of all elements belonging to *every* S_i: x ∈ ⋂_{i∈I} S_i if and only if for all i ∈ I, x ∈ S_i. Both definitions generalize the binary operations directly — set I = {1, 2} and you recover the binary case. For a concrete example: let S_n = {n, n+1, n+2, ...} for n ∈ ℕ. Then ⋃_{n∈ℕ} S_n = ℕ (every natural number is in some S_n) and ⋂_{n∈ℕ} S_n = ∅ (no natural number is in every S_n, since n is not in S_{n+1}).
The importance of indexed families extends beyond the operations themselves. When you build functions, sequences, and products in later topics, you will rely on this framework. A sequence (a_n)_{n∈ℕ} is just an indexed family with index set ℕ. A Cartesian product ∏_{i∈I} S_i is the set of all functions f: I → ⋃ S_i such that f(i) ∈ S_i for each i — this is the rigorous definition of an arbitrary product, which collapses to S × T when I = {1, 2}. The indexed family is the organizational primitive that lets set theory scale from finite combinatorics to transfinite constructions.