Completion of topological loops

Abstract

A Hausdorff topological group equipped with the right uniformity admits a group completion iff the inversion mapping preserves Cauchy filters, cf. [1], III. §3, No.5, Théorème 1. Up until today a general theorem on the completion of topological loops is not available, for partial results see [9], [10]. This is among others due to the fact that topological loops will not necessarily have a compatible right uniformity.

The main results (6–8) of this paper are the following: All topological loops are locally uniform in the sense of [11], and, provided the notion of “Cauchy filter” is suitably chosen, they can be completed. An analogue of the completion theorem for groups cited above holds for topological loops. According to these aims the theory of completion of locally uniform spaces is developped in 1–5 of this paper.