The Doitchinov Completion of a Regular Paratopological Group

In memory of Professor D. Doitchinov ∗ This paper was written while the first author was supported by the Swiss National Science Foundation under grants 21–30585.91 and 2000-041745.94/1 and by the Spanish Ministry of Education and Sciences under DGES grant SAB94-0120. The second author was supported under DGES grant PB95-0737. During her stay at the University of Berne the third author was supported by the first author’s grant 2000-041745.94/1 from the Swiss National Science Foundation.

We show that the two-sided quasi-uniformity UB of a regular paratopological group (G, ·) is quiet. The Doitchinov completion (G, UB ) of (G, UB ) can be considered a paratopological group containing G as a doubly dense subgroup whenever G is Abelian. Furthermore UB is the two-sided quasi-uniformity of (G, ·). These results generalize in an appropriate way important results about topological groups to regular paratopological groups. A counterexample dealing with the non-Abelian case is presented. Furthermore we give conditions, depending on quasi-uniform completeness properties, under which a paratopological group is a topological group.