Mathematical Statement
Definition. Let X and I be any sets. Then by family of elements in X indexed by I, we mean a function . An indexed family is denoted by, where it is understood that there is a function x that maps i to .
An indexed family can be turned into a set by considering the set, that is, the range of x, but this will collapse the elements with into one element in .
Definition. Let S be a set. An indexed family of sets is an indexed family that maps I to elements of the power set of S.
Hence, an indexed family of sets is conceptually different from a family of sets (which is just a synonym for "set of sets"), but in practice the distinction is sometimes fuzzy and the indexed family is identified with its range and treated like an ordinary family.
Read more about this topic: Indexed Family
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