  The last sentence of this quotation—which asserts, in effect, that coverings always exist for the collection of nonempty subsets of any (nonempty) set—is Zermelo’s first formulation of the Axiom of   Choice[1].  This is now usually stated in terms of choice functions: here a choice function on a collection \(\sH\)   of nonempty sets is a map \(f\) with domain   \(\sH\)   such that \(f(X) \in X\) for every \(X \in \sH\). 