Let X be a set of sets none of which are empty. Then a choice function (selector, selection) on X is a mathematical function f that is defined on X such that f is a mapping that assigns each element of X to one of its elements.

An example

Let X={{1,4,7},{9},{2,7}}. Then the function f defined by f({1, 4, 7}) = 7, f({9}) = 9 and f({2, 7}) = 2 is a choice function on X.

History and importance

Ernst Zermelo (1904) introduced choice functions as well as the axiom of choice (AC) and proved the well-ordering theorem, which states that every set can be well-ordered. AC states that every set of nonempty sets has a choice function. A weaker form of AC, the axiom of countable choice (ACω) states that every countable set of nonempty sets has a choice function. However, in the absence of either AC or ACω, some sets can still be shown to have a choice function.

• is a finite This requires only finitely many choices, so neither AC or ACω is needed.

• is a nonempty set, and the union by making just one choice of a well-order of the union, so neither AC nor ACω was needed. (This example shows that the well-ordering theorem implies AC. The converse is also true, but less trivial.)

Choice function of a multivalued map

be a multivalued map to the power set).

is said to be a selection, if:

The existence of more regular choice functions, namely continuous or measurable selections is important in the theory of differential inclusions, optimal control, and mathematical economics. See Selection theorem.

Bourbaki tau function

Nicolas Bourbaki used epsilon calculus.

However, Bourbaki's choice operator is stronger than usual: it's a global choice operator. That is, it implies the axiom of global choice. Hilbert realized this when introducing epsilon calculus.

See also

• Axiom of countable choice

• Axiom of dependent choice

• Hausdorff paradox

• Hemicontinuity

Notes

References