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, -metric spaces. In the present paper, the authors introduce an alternative, more robust generalization of metric spaces, namely, -metric spaces, where the -metric satisfies the following axioms:
(1) if , (2) whenever , (3) whenever , (4) is a symmetric function of its three variables, and (5) .
In Section 2, some properties of -metric spaces are studied. Section 3, entitled “The -metric topology”, contains: Convergence and continuity in -metric spaces; Completeness of -metric spaces and compactness in -metric spaces. In the last section, products of -metric spaces are studied.