Compact-covering-properties of finite-to-one mappings
An inductively perfect map is one that is perfect when restricted to a suitable closed subset of its domain (it should remain surjective). Such maps are compact-covering (because preimages of compact sets are compact under perfect maps). In [Compact covering and game determinacy, Topology Appl. 68, 153-185 (1996; Zbl 0848.54024)] the authors showed that, given enough determinacy, this implication may be reversed in the class of separable metric spaces. In the present paper the authors investigate under what conditions countable-compact-covering maps are inductively perfect. The main results are: a negative answer for finite-to-one maps and a positive answer for at-most-\(n\)-to-one maps. The proof of the positive result involves a detailed study of certain kinds of multi-valued maps into the hyperspace of compact sets.
scientific article; zbMATH DE number 1088821
scientific article