A new approach to generalized metric spaces Zead Mustafa and Brailey Sims Abstract. To overcome fundamental flaws in B. C. Dhage’s theory of generalized metric spaces, flaws that invalidate most of the results claimed for these spaces, we in- troduce an alternative more robust generalization of metric spaces. Namely, that of a G-metric space, where the G-metric satisfies the axioms: (1) G(x, y, z)=0 if x = y = z, (2) 0 <G(x, x, y); whenever x = y, (3) G(x, x, y) ≤ G(x, y, z) whenever z = y, (4) G is a symmetric function of its three variables, and (5) G(x, y, z) ≤ G(x, a, a)+ G(a, y, z) 1. Introduction During the sixties, 2-metric spaces were introduced by Gahler [6], [7]. Definition 1. Let X be a nonempty set, and let R denote the real numbers. A function d : X × X × X → R + satisfying the following properties: (A1) For distinct points x, y ∈ X, there is z ∈ X, such that d(x, y, z) =0. (A2) d(x, y, z) = 0 if two of the triple x, y, z ∈ X are equal. (A3) d(x, y, z)= d(x,z,y)= ··· (symmetry in all three variables), (A4) d(x, y, z) ≤ d(x, y, a)+ d(x, a, z)+ d(a, y, z), for all x,y,z,a ∈ X, is called a 2-metric, on X. The set X equipped with such a 2-metric is called a 2-metric space. It is clear that taking d(x, y, z) to be the area of the triangle with vertices at x, y and z in R 2 provides an example of a 2 -metric. 1991 Mathematics Subject Classification. Primary 47H10, Secondary 46B20. Key words and phrases. Metric space, generalized metric space, D-metric space, 2-metric space. 1