On Projections and Limit Mappings of Inverse Systems of Compact Spaces

For some classes K of mappings we discuss two problems connected with limits of inverse systems: (1) Does the condition that all bonding mappings are in K imply that all projections are in K? (2) Does the condition that all mappings between factor spaces of two given inverse systems are in K imply that the limit mapping between the inverse limit spaces is in K? We answer both these questions in the affirmative for the classes of monotone, of confluent and of weakly confluent mappings of compact spaces, and for some generalizations of these mappings.