Choice function

A choice function on a family of sets assigns each set one of its elements. It formalizes simultaneous selection from a collection, including collections with infinitely many sets.

Set: A collection of distinct objects, treated as a single mathematical object. Each input to a choice function is a set.
Choice axiom: The principle that every family of nonempty sets has a choice function. It guarantees choice functions even when no explicit selection rule is given.
Well-ordering theorem: The theorem that every set can be given a well-order. It is equivalent to the axiom guaranteeing choice functions for arbitrary nonempty families.
Constructive mathematics: Mathematics that requires existence proofs to provide constructions or explicit witnesses. A choice function may be asserted to exist without a construction, unlike constructive selection.
Family of sets: A collection whose members are themselves sets. A choice function takes a family of sets as its domain.
Partial choice function: A function selecting an element from only some members of a family of sets. It contrasts with a choice function defined on the entire family.
Zorn's lemma: A maximality principle asserting that certain partially ordered sets contain maximal elements. It is equivalent to the axiom of choice and often proves existence without constructing selections.
Countable choice: The principle that every countable family of nonempty sets has a choice function. It is a restricted choice principle weaker than unrestricted choice.
Cartesian product: The set of all tuples formed by choosing one element from each of several sets. Choice functions correspond to elements of products of nonempty sets.
Choice sequence: A sequence selecting one element from each set in a countable family. For countable families, a choice function can be represented by such a sequence.

Linked from 11 pages