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

Keywords: topological group, Lie group
Published online: 2017-06-07

Abstract


According to a classical theorem of Gleason and Montgomery, every finite-dimensional locally path-connected topological group is a Lie group. In the paper for every $n\ge 2$ we construct a locally connected subgroup $G\subset{\mathbb R}^{n+1}$ of dimension $\dim(G)=n$, which is not locally compact. This answers a question posed by S. Maillot on MathOverflow and shows that the local path-connectedness in the result of Gleason and Montgomery can not be weakened to the local connectedness.

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