an:00977861
Zbl 0861.22002
Szambien, H.
Completion of topological loops
EN
Abh. Math. Semin. Univ. Hamb. 66, 135-142 (1996).
00037817
1996
j
22A30
Hausdorff topological group; group completion; Cauchy filters; topological loops; locally uniform spaces
Summary: A Hausdorff topological group equipped with the right uniformity admits a group completion iff the inversion mapping preserves Cauchy filters [cf. \textit{N. Bourbaki}, Topologie g??n??ral (Paris 1971; Zbl 0249.54001)], III. \S 3, No. 5, Th??or??me 1]. Up until today a general theorem on the completion of topological loops is not available, for partial results see \textit{H. Wefelscheid} [Math. Z. 99, 279-298 (1967; Zbl 0153.35901)]. This is among others due to the fact that topological loops will not necessarily have a compatible right uniformity. The main results of this paper are the following: All topological loops are locally uniform in the sense of \textit{J. Williams} [Trans. Am. Math. Soc. 168, 435-469 (1972; Zbl 0235.54026)], 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 developed in sections 1-5 of this paper.
Zbl 0249.54001; Zbl 0153.35901; Zbl 0235.54026