Addendum: New Proof of Finite-Dimensional Selection Theorem

Abstract

The main goal of this chapter is to present a new approach to the proof of the Finite-dimensional selection theorem (see §5) which was recently proposed by Scepin and Brodskiĭ [373]. First, note that there is only one proof of the Finite-dimensional selection theorem [259]. (Observe that the proof [131] is a reformulation of Michael’s proof in terms of coverings and provides a way to avoid uniform metric considerations.) Second, [373] gives in fact a generalization of Michael’s theorem. Third, and most important, this approach is based on the technique, which is widely exploited in other branches of the theory of multivalued mappings. Namely, in the fixed-point theory, where proofs are often based on UV n -mappings and on (graph) approximations of such mappings (see [16]).