A short and elementary proof of Hanner's theorem

TL;DR

This paper presents a concise and elementary proof of Hanner's theorem, simplifying the understanding of when local absolute neighborhood extenders (ANE) are globally extenders in topological spaces.

Contribution

It offers a new, straightforward proof of Hanner's theorem that relies solely on classical point-set topology, making the result more accessible.

Findings

Proof is shorter and more elementary than previous versions

Applicable to general topological spaces, not just separable ones

Simplifies understanding of local and global ANE relationships

Abstract

Hanner's theorem is a classical theorem in the theory of retracts and extensors in topological spaces, which states that a local ANE is an ANE. While Hanner's original proof of the theorem is quite simple for separable spaces, it is rather involved for the general case. We provide a proof which is not only short, but also elementary, relying only on well-known classical point-set topology.