              To see the connection between the idea of a maximal element and AC, let us return to the latter’s formulation AC2 in terms of indexed sets. Accordingly suppose we are given an indexed family of nonempty sets \(\sA = \{A_{i}:i \in I\}.\) Let us define a potential choice function on \(\sA\) to be a function \(f\) whose domain is a subset of \(I\) such that \(f(i) \in A_{i}\) for all \(i\in J\).  (Here the use of the qualifier potential is suggested by the fact that the domain is a subset of \(I\); recall that a choice function \(f\) on \(\sA\) has the same properties as what we are now calling potential choice functions except that the domain of \(f\) is required to be all of \(I\), not just a subset.) The set \(P\) of potential choice functions on \(\sA\) can be partially ordered by inclusion: we agree that, for potential choice functions \(f, g \in P\), the relation \(f \le g\) holds provided that the domain of \(f\) is included in that of \(g\) and the value of \(f\) at an element of its domain coincides with the value of \(g\) there. It is now easy to see that the maximal elements of \(P\) with respect to the partial ordering \(\le\) are precisely the choice functions on \(\sA\). 