Axiom of choice

The axiom of choice states that every family of nonempty sets admits a function selecting one element from each set, even when no selection rule is specified.

Choice function: A function on a family of sets that assigns each set one of its elements. It is the function whose existence the axiom asserts.
Ernst Zermelo: A German mathematician who formulated the axiom of choice and used it to prove the well-ordering theorem. His 1904 proof made the principle's role in set theory explicit.
Axiom of countable choice: The principle that every countable family of nonempty sets has a choice function. It restricts the axiom to countable families and is strictly weaker in general.
Vector space basis theorem: The theorem that every vector space has a basis, including vector spaces of arbitrary dimension. Its general proof follows from Zorn's lemma, an equivalent form of choice.
Cartesian product: The set of functions that choose one element from each set in a given family. A choice function is exactly an element of this product when every factor is nonempty.
Zermelo–Fraenkel set theory: An axiomatic foundation for mathematics built from sets and governing their construction and membership. The axiom of choice is studied as an addition to this foundational theory.
Constructive mathematics: Mathematics that requires proofs of existence to provide, or justify, a construction of the object asserted to exist. An unrestricted existence claim need not provide the explicit selection rule constructive methods demand.
Hahn–Banach theorem: A theorem extending bounded linear functionals from a subspace to a larger normed space. Standard proofs use choice principles to extend functionals across arbitrary spaces.
Finite choice principle: The principle that a finite family of nonempty sets has a choice function, provable in ordinary set theory. Unlike the unrestricted axiom, finite choice requires no additional set-theoretic assumption.
Zermelo–Fraenkel set theory without choice: Set theory using the Zermelo–Fraenkel axioms while omitting the axiom of choice. It provides a setting for separating choice from the other standard set-theoretic axioms.

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