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.
Contents
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.
Refinement of the notion of choice function
A 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.
Bourbaki tau function
Nicolas Bourbaki used epsilon calculus for their foundations that had a
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.