Abstract
A strong regularity condition is given which generalizes the notion of a completely regular mapping defined by Dyer and Hamstrom. These mappings are shown to be fiber spaces of various types with some additional hypotheses. Strongly regular mappings on manifolds whose point inverses are ANR or AR are considered. For example, it is shown that with AR fibers the map is either a homeomorphism or the manifold has boundary which meets all non- degenerate point inverses.
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