Abstract
The theory of topological semigroups has emerged as a separate discipline during the fifties (cf. Hofmann and Mostert [5] and Carruth, Koch and Hildebrant [3]), but the idea of topologically consistent regularity in such semigroups does not seem to have gained much attention. In this paper topological regular semigroups are introduced where the regularity is topologically consistent. The topological version of the structure theorem for regular semigroups given by Nambooripad [7] is provided here. Some special cases are also discussed. The necessary terminology on topological groupoids are available in Brown and Hardy [1] and Mackenzie [6].
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