Constructing Selections Stepwise Over Skeletons of Nerves of Covers

TL;DR

This paper presents a simplified, self-contained proof of Michael's finite-dimensional selection theorem using stepwise approximate selections over skeletons of nerves, also extending to Schepin--Brodsky's generalization.

Contribution

It introduces a new method for constructing selections via skeletons of nerves, simplifying existing proofs of key theorems in selection theory.

Findings

Simplified proof of Michael's finite-dimensional selection theorem

Extension of the method to Schepin--Brodsky's generalization

Enhanced understanding of selection construction techniques

Abstract

It is given a simplified and self-contained proof of the classical Michael's finite-dimensional selection theorem. The proof is based on approximate selections constructed stepwise over skeletons of nerves of covers. The method is also applied to simplify the proof of the Schepin--Brodsky's generalisation of this theorem.