Vol. 45, No. 2, 1973

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ISSN: 0030-8730
Induced topologies for quasigroups and loops

Kenneth Paul Baclawski and Kenneth Kapp

Vol. 45 (1973), No. 2, 393–402
Abstract

The concept of a [semi] topological quasigroup is defined and the notions of induced groupoids and isotopes are extended to the topological case. Necessary and sufficient conditions are found in order for a continuously induced isotope of a semitopological quasigroup to be a semitopological quasigroup. Given an injection i of a topological space (A,𝒜) into a set S acted on by a group, G, a topology 𝒯A on S is introduced in a natural fashion under which i is continuous. When S = Q is itself a semitopological quasigroup and G is generated by the left or right translations of Q the continuity or openness of i can be checked by comparing the topology 𝒯A with that of Q. In particular this method is applied in §3 to the study of topologically invariant subloops.

Mathematical Subject Classification 2000
Primary: 22A99
Milestones
Received: 30 August 1971
Revised: 14 September 1972
Published: 1 April 1973
Authors
Kenneth Paul Baclawski
Kenneth Kapp