On punctured locally compact spaces

TL;DR

This paper presents a new proof demonstrating that certain locally compact spaces, including Euclidean spaces, are not homeomorphic to their punctured versions, using only local compactness and connectedness near infinity, without relying on algebraic topology.

Contribution

It extends previous results by removing the need for metrizability and allows deleting entire compact subsets, not just points, to show non-homeomorphism.

Findings

Locally compact and connected spaces are not homeomorphic to their punctured versions.

The method applies to Euclidean spaces and many-holed Euclidean balls.

No algebraic topology tools are needed in the proof.

Abstract

In a recent paper \cite{T} the fact that a class of locally compact metric spaces $X$, among which are Euclidean spaces, are not homemorphic to their punctured version $X\men\{p\}$, was given an interesting new proof which does not use algebraic topology; essential tools of this proof are a boundedly compact metric structure, and path--connectedness near infinity. Here we show that local compactness and ordinary connectedness near infinity suffice; no metrizability is needed, and moreover we can also delete whole compact subsets, not only single points. Some non--homeomorphism results on many--holed Euclidean balls are also obtained. This note ought to distil the essence of the method developed in \cite{T}.