Choice Function

A choice function is a function C: 2^{X} \to 2^{X} such that:
* C(A) \subseteq A, \forall A \in 2^{X}
* the choice function is non-empty: C(A) \neq \emptyset, \forall A \neq \emptyset.
where X is a finite set of alternatives, and 2^{X} = \{A | A \subseteq X \} is the set of all possible subsets A in X. Note, that the second line does not imply that the consumer can not choose to do nothing, but instead doing nothing is included as an option in the choice set. 
Intuitively, as the choice function is intended to describe observed behavior, an empty choice set would mean that there was no choice moment?