The author establishes a topological version of the correspondence between regular semigroups and inductive groupoids given by \textit{K. S. S. Nambooripad} [Mem. Am. Math. Soc. 224, 117 p. (1979; Zbl 0457.20051)]. He defines concepts of topological regular semigroup and topological inductive groupoid, and proves that the regular semigroup corresponding to a topological inductive groupoid is a topological regular semigroup. Given a topological regular semigroup, he proves that the corresponding inductive groupoid \(G\) is a topological inductive groupoid and, further, has a basis of open sets \(W\) such that if \(\gamma\in G\) is such that there exist \(\alpha, \beta\in W\) with \(\alpha p \gamma\) and \(\beta^{-1} p \gamma^{-1}\), then \(\gamma\in W\); here \(p\) is the standard equivalence relation on \(G\) (op. cit.). These constructions are mutually inverse for the classes described.