In this paper, two interesting classes of sets with respect to a multi
valued mapping are considered: the class of stable sets and the class
of pure sets. In addition to the results existing in the literature, s
everal new properties of stable and pure sets are established. A gener
alization of the concepts of stable and pure sets to fuzzy multivalued
mappings is developed. The properties of stable and pure fuzzy sets a
re studied extensively. It is obtained that the stable (resp. pure) fu
zzy sets with respect to a normalized fuzzy multivalued mapping consti
tute a complete lattice and also a stratified fuzzy topology. The noti
ons of level stable and level pure fuzzy sets are introduced. Interact
ions between stability (resp. purity) and level stability (resp. level
purity) are established and expressed in terms of relationships betwe
en particular stratified fuzzy topologies. Some additional relationshi
ps between the stability (resp. purity) of a fuzzy set with respect to
a multivalued mapping and the stability (resp. purity) of its cuts wi
th respect to this mapping are established. By imposing certain condit
ions on the triangular norm and the implication operator involved a on
e-to-one correspondence between the class of stable fuzzy sets and the
class of pure fuzzy sets with respect to a normalized fuzzy multivalu
ed mapping that has a normalized inverse is obtained. In the last sect
ion, it is shown that a normalized fuzzy multivalued mapping is contin
uous with respect to the stratified fuzzy topologies constituted by th
e stable fuzzy sets and the pure fuzzy sets with respect to this fuzzy
multivalued mapping. (C) 1998 Elsevier Science B.V. All rights reserv
ed.