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.
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. Ernst Zermelo (1904) introduced choice functions as well as the axiom of choice (AC) and proved the well-ordering theorem,[1] 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.
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.
Nicolas Bourbaki used epsilon calculus for their foundations that had a
symbol that could be interpreted as choosing an object (if one existed) that satisfies a given proposition.
(if one exists, otherwise it returns an arbitrary object).
Hence we may obtain quantifiers from the choice function, for example
[3] However, Bourbaki's choice operator is stronger than usual: it's a global choice operator.
That is, it implies the axiom of global choice.
[4] Hilbert realized this when introducing epsilon calculus.
[5] This article incorporates material from Choice function on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.