Completeness of hyperspaces on topological groups

We prove that if the bilateral uniformity of a topological group of pointwise countable type is complete, then the Hausdorff–Bourbaki uniformity of is complete. It follows that for every ech complete topological group, the Hausdorff–Bourbaki uniformity of is complete. Finally, we prove that if X is a compact topological space, then the Hausdorff–Bourbaki uniformity of the bilateral uniformity of the free Abelian topological group over X, is complete.