Two Selection Theorems for Extremally Disconnected Spaces

TL;DR

This paper presents simplified proofs of classical and recent theorems related to continuous selections in extremally disconnected spaces, extending existing results to broader classes of mappings with minimal complexity.

Contribution

It provides straightforward proofs of Hasumi's theorem and a recent extension result, broadening understanding of continuous selections in extremally disconnected spaces.

Findings

Simplified proof of Hasumi's theorem for usco mappings

Extension of continuous selections to all usco mappings with regular range

Generalization of previous selection extension results

Abstract

The paper contains a very simple proof of the classical Hasumi's theorem that each usco mapping defined on an extremally disconnected space has a continuous selection. The paper also contains a very simple proof of a recent result about extension of densely defined continuous selections for compact-valued continuous mappings, in fact a generalisation of this result to all usco mappings with a regular range.