Closed locally path-connected subspaces of finite-dimensional groups are locally compact

TL;DR

The paper proves that closed locally continuum-connected subspaces of finite-dimensional topological groups are locally compact, leading to new characterizations of Lie groups and counterexamples to previous assumptions.

Contribution

It establishes that such subspaces are locally compact and constructs examples of 1-dimensional spaces not embeddable in finite-dimensional groups, answering a longstanding question.

Findings

Closed locally continuum-connected subspaces are locally compact

Constructed 1-dimensional spaces not homeomorphic to closed subsets of finite-dimensional groups

Characterized Lie groups as finite-dimensional locally continuum-connected topological groups

Abstract

We prove that each closed locally continuum- connected subspace of a finite dimensional topological group is locally compact. This allows us to construct many 1-dimensional metrizable separable spaces that are not homeomorphic to closed subsets of finite-dimensional topological groups, which answers in negative a question of D.Shakhmatov. Another corollary is a characterization of Lie groups as finite-dimensional locally continuum-connected topological groups. For locally path connected topological groups this characterization was proved by Gleason and Palais in 1957.