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.
Keywords: groupoid, Urysohn-type lemma, metrization theorem
2010 MSC: 22A22; 54E15, 54E35
Theory and Applications of Categories, Vol. 32, 2017, No. 28, pp 970-994.