zbMATH — the first resource for mathematics

Some remarks concerning \(D\)-metric spaces. (English) Zbl 1079.54017
García-Falset, Jesús (ed.) et al., Proceedings of the international conference on fixed-point theory and its applications, Valencia, Spain, July 13–19, 2003. Yokohama: Yokohama Publishers (ISBN 4-946552-13-8/hbk). 189-198 (2004).
Motivated by Gähler’s concept of 2-metric [see S. Gähler, Math. Nachr. 26, 115–148 (1963; Zbl 0117.16003)], B. C. Dhage [Bull. Calcutta Math. Soc. 84, No. 4, 329–336 (1992; Zbl 0782.54037)] proposed the notion of a \(D\)-metric on a nonempty set. This and subsequent works have been the basis for a substantial number of results in fixed point theory. In this paper, the authors show that most of the claims concerning the fundamental topological properties of \(D\)-metric spaces are incorrect. So the validity of fixed point theorems and other results in \(D\)-metric spaces becomes doubtful.
For the entire collection see [Zbl 1063.47054].

54E99 Topological spaces with richer structures
54H25 Fixed-point and coincidence theorems (topological aspects)