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We introduce the concept of cyclic morphisms with respect to a morphism in the category of pairs as a generalization of the concept of cyclic maps and we use the concept to obtain certain sets of homotopy classes in the category of pairs. For these sets, we get complete or partial answers to the following questions: (1) Is the concept the most general concept in the class of all concepts of generalized Gottlieb subsets intro- duced by many authors until now? (2) Are they homotopy invariants in the category of pairs? (3) When do they have a group structure?

GOTTLIEB SUBSETS WITH RESPECT TO A MORPHISM IN THE CATEGORY OF PAIRS