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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)], {\it 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 $x\ne y$, (3) $G(x,x,y)\le G(x,y,z)$ whenever $z\ne y$, (4) $G$ is a symmetric function of its three variables, and (5) $G(x,y,z)\le G(x,a,a)+ G(a,y,z)$. 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.

54E35Metric spaces, metrizability
47H10Fixed-point theorems for nonlinear operators on topological linear spaces
46B20Geometry and structure of normed linear spaces
54E50Complete metric spaces