"Some generalization of Godement's theorem on division"

Proceedings of the Winter School on Geometry and Physics, SRNI, 10-17 January, 1987, Supplemento ai Rendiconti del Circolo Matematico di Palermo, Serie II - numero 16 - 1987, 119-124.
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Some generalization of Godement's theorem on division is found. This generalization characterizes all equivalence relation R (on a C^ˇ-manifold) such that every abstract class of R has a countable number of arcwise connected components and the family of all such components is a regular foliation. Using it, another proof of that classical Godement's theorem is obtained. As a corollary we also have obtained a groupoid's characterization of regular foliations.