An almost uniquely homogeneous subgroup of R

AbstractAn example is constructed of a connected locally connected (n–1)-dimensional subgroup X of R with the property that for every homeomorphism g in H(X) there is an h ϵ X such that g(x)=x+h or g(x)=-x+h. The dimension of the space of homeomorphisms H(X) will also be n-1. A new proof of a theorem of Barit and Renaud is also given. In particular, let X be a separable metric space which is either compact or locally connected. If X is uniquely homogeneous, then X is trivial or the group of order two