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Generalized quotient maps that are inductively index-σ-discrete

£. Michael recently showed that a continuous quotient 5-map between metrizable spaces can be contracted onto a G 8-set so that the resulting map is index-σ-discr ete; i.e., one that preserves σ-discretely decomposable families. Because of the potential utility of this result in descriptive set theory, we give a refinement that is less dependent upon the behavior of open sets under the map. Several types of generalized quotient maps are defined, not necessarily continuous, and we show that these are precisely the maps that are "inductively" index-σ-discrete under certain conditions similar to the above. The inter-relationships among these maps are also described. We further show that when the given map has a nice property (such as Borel measurability), then the restriction can be defined on a similarly nice subset of the domain. An application is made to maps that preserve analytic metric spaces; and additional applications to the existence of Borel measurable inverses will be given elsewhere.