An example of a non-Borel locally-connected finite-dimensional   topological group
  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. 
I.Banakh, T.Banakh, M.Vovk
math.GN math.GR