Choice function

A choice function (selector, selection) is a mathematical function "f" that is defined on some collection "X" of nonempty sets and assigns to each set "S" in that collection some element "f"("S") of "S". In other words, "f" is a choice function for "X" if and only if it belongs to the direct product of "X".
An example.
Let "X" = { {1,4,7}, {9}, {2,7} }. Then the function that assigns 7 to the set {1,4,7}, 9 to {9}, and 2 to {2,7} 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.
Choice function of a multivalued map.
Given two sets "X" and "Y", let "F" be a multivalued map from "X" and "Y" (equivalently, formula_8is a function from "X" to the power set of "Y").
A function formula_9 is said to be a selection of "F", 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 for their foundations that had a formula_11 symbol that could be interpreted as choosing an object (if one existed) that satisfies a given proposition. So if formula_12 is a predicate, then formula_13 is one particular object that satisfies formula_14 (if one exists, otherwise it returns an arbitrary object). Hence we may obtain quantifiers from the choice function, for example formula_15 was equivalent to formula_16.
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.