Near-selection theorems for C-spaces

TL;DR

This paper extends Haver's near-selection theorem from $\sigma$-compact metrizable $C$-spaces to all paracompact $C$-spaces using oriented simplicial complexes, simplifying proofs and broadening applicability.

Contribution

The paper introduces a new approach based on oriented simplicial complexes to generalize near-selection theorems to all paracompact $C$-spaces, simplifying proofs and extending previous results.

Findings

Extended near-selection theorem to all paracompact $C$-spaces.

Simplified proof using oriented simplicial complexes.

Broadened applicability from compact metric to all paracompact finite $C$-spaces.

Abstract

Haver's near-selection theorem deals with approximate selections of Hausdorff continuous CE-valued mappings defined on $\sigma$-compact metrizable $C$-spaces. In the present paper, we extend this theorem to all paracompact $C$-spaces. The technique developed to achieve this generalisation is based on oriented simplicial complexes. This approach makes not only a considerable simplification in the proof but is also successful in generalising the special case of compact metric $C$-spaces to all paracompact finite $C$-spaces.