An example of a non-Borel locally-connected finite-dimensional topological group

TL;DR

This paper constructs specific locally connected subgroups within Euclidean spaces that are finite-dimensional but not locally compact, addressing a question in topological group theory.

Contribution

It provides explicit examples of non-locally compact, locally connected subgroups of Euclidean spaces of arbitrary finite dimension, answering a previously open question.

Findings

Constructed such subgroups for all natural numbers n

Demonstrated these groups are not locally compact

Addressed a question posed by S. Maillot

Abstract

Answering a question posed by S.Maillot in MathOverFlow, for every $n\in\mathbb N$ we construct a locally connected subgroup $G\subset\mathbb R^{n+1}$ of dimension $dim(G)=n$, which is not locally compact.