  We introduce the general notions of an index and a core of a relation. We
postulate a limited form of the axiom of choice -- specifically that all
partial equivalence relations have an index -- and explore the consequences of
adding the axiom to standard axiom systems for point-free reasoning. Examples
of the theorems we prove are that a core/index of a difunction is a bijection,
and that the so-called ``all or nothing'' axiom used to facilitate pointwise
reasoning is derivable from our axiom of choice.
