A URYSOHN TYPE LEMMA FOR GROUPOIDS.

Starting from the observation that through groupoids we can express in a unified way the notions of fundamental system of entourages of a uniform structure on a space X, respectively the system of neighborhoods of the unity of a topological group that determines its topology, we introduce in this paper a notion of G-uniformity for a groupoid G. The topology induced by a G-uniformity turns G into a topological locally transitive groupoid. We also prove a Urysohn type lemma for groupoids and obtain metrization theorems for groupoids unifying in two ways the Alexandroff--Urysohn Theorem and Birkhoff- Kakutani Theorem. [ABSTRACT FROM AUTHOR]