On the concepts of balls in a D-metric space

Dhage [1, 2, 3] introduced the concept of open balls in a D-metric space in two different ways and discussed at length the properties of the topologies generated by the family of all open balls of each kind. Here we observe that many of his results are either false or of doubtful validity. In some cases we give examples to show that either the results are false or that the proofs given by him are not valid. With regard to one type of open balls we observe that some of them may be empty and that the ball with a given center may not increase as the radius increases. The latter is contrary to a remark made by Dhage based on which he proves that the family of all open balls forms a base for a topology. Definition 1.1 [1]. Let X be a nonempty set. A function ρ : X ×X ×X → [0,∞) is called a D-metric on X if (i) ρ(x, y,z)= 0 if and only if x = y = z (coincidence), (ii) ρ(x, y,z) = ρ(p(x, y,z)) for all x, y,z ∈ X and for any permutation p(x, y,z) of x, y, z (symmetry), (iii) ρ(x, y,z) ≤ ρ(x, y,a) + ρ(x,a,z) + ρ(a, y,z) for all x, y,z,a ∈ X (tetrahedral inequality). If X is a nonempty set and ρ is a D-metric on X , then the ordered pair (X ,ρ) is called a D-metric space. When the D-metric ρ is understood, X itself is called a D-metric space.