The Campbell-Hausdorff near-ring over N-A topological view

The famous Baker-Campbell-Hausdorff series defines a group composition on the set of the so called grouplike elements of a completed free Lie algebra as we can find in P. Cartier's paper on the Campbell-Hausdorff formula in 1956. According to A. Baider and R.C. Churchill, we call this group the Campbell-Hausdorff group over the alphabet X of the free Lie algebra. By a certain coproduct and certain bialgebra endomorphisms of the free associative algebra over a set X which should be at most countably infinite, we get an alternative realisation of this Campbell-Hausdorff group. This realisation is much more comfortable to deal with than the classical one. The usual composition of endomorphisms gives a near-ring structure to this group. In this paper we consider a metric on the Campbell-Hausdorff near-ring over the alphabet N and the compatibility of this metric with the near-ring compositions and we make some topological remarks.