Metric spaces with additional structure and mapping selections (Q1909850)

The concept of a space with \(M\)-structure is introduced. It is shown that spaces with \(M\)-structure are linear normal spaces, superextensions, hyperspaces, spaces of order arcs and others. The main theorem of the paper is a theorem on the selection of multivalued maps to \(M\)-spaces. This theorem extends E. Michael's well-known classical theorem on selections of multivalued maps to Banach spaces. Applications of the selection theorem in concrete cases of spaces with a mixer and of spaces of order arcs are considered, too.