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A new approach to generalized metric spaces. (English) Zbl 1111.54025
In [Bull. Calcutta Math. Soc. 84, No. 4, 329–336 (1992; Zbl 0782.54037)], B. C. Dhage initiated the study of a generalized metric spaces, namely, \(D\)-metric spaces. In the present paper, the authors introduce an alternative, more robust generalization of metric spaces, namely, \(G\)-metric spaces, where the \(G\)-metric satisfies the following axioms:
(1) \(G(x,y,z)= c\) if \(x= y= z\), (2) \(0< G(x,x,z)\) whenever \(x\neq y\), (3) \(G(x,x,y)\leq G(x,y,z)\) whenever \(z\neq y\), (4) \(G\) is a symmetric function of its three variables, and (5) \(G(x,y,z)\leq G(x,a,a)+ G(a,y,z)\).
In Section 2, some properties of \(G\)-metric spaces are studied. Section 3, entitled “The \(G\)-metric topology”, contains: Convergence and continuity in \(G\)-metric spaces; Completeness of \(G\)-metric spaces and compactness in \(G\)-metric spaces. In the last section, products of \(G\)-metric spaces are studied.

54E35 Metric spaces, metrizability
47H10 Fixed-point theorems
46B20 Geometry and structure of normed linear spaces
54E50 Complete metric spaces