An example of a non-Borel locally-connected finite-dimensional topological group - Banakh, I.Ya.; Banakh, T.O.; Vovk, M.I. (Carpathian Mathematical Publications, 2017)
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 n2 we construct a locally connected subgroup GR^n+1 of dimension (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.