Abstract
Fuzzy strong b-metrics called here by fuzzy sb-metrics, were introduced recently as a fuzzy version of strong b-metrics. It was shown that open balls in fuzzy sb-metric spaces are open in the induced topology (as different from the case of fuzzy b-metric spaces) and thanks to this fact fuzzy sb-metrics have many useful properties common with fuzzy metric spaces which generally may fail to be in the case of fuzzy b-metric spaces. In the present paper, we go further in the research of fuzzy sb-metric spaces. It is shown that the class of fuzzy sb-metric spaces lies strictly between the classes of fuzzy metric and fuzzy b-metric spaces. We prove that the topology induced by a fuzzy sb-metric is metrizable. A characterization of completeness in terms of diameter zero sets in these structures is given. We investigate products and coproducts in the naturally defined category of fuzzy sb-metric spaces.
Highlights
The concept of a fuzzy sb-metric as a strengthening of the concept of a fuzzy b-metric on one side and as a fuzzy version of the notion of a strong b-metric was introduced in Reference [11,12]
We further develop the study of fuzzy sb-metrics
We study continuity of a fuzzy sb-metric M( x, y, −) and prove metrizability of the topology induced by a fuzzy sb-metric, investigate diameter zero sets in fuzzy sb-metric spaces and use them for characterization of completeness of fuzzy sb-metric spaces
Summary
The class of b-metric spaces, one of the generalizations of metric spaces, has been introduced by different authors under different names (b-metric by Czerwik [1], quasimetric by Bakhtin [2] and by Heinhonen [3], NEMr (nonlinear elastic matching) by Fagin et al [4], metric type by Khamsi et al [5]. In the papers [11,12], a fuzzy version of an sb-metric was introduced and the basic properties of fuzzy strong b-metric spaces (renamed here as fuzzy sb-metric spaces) were studied This notion can be viewed as a generalization of fuzzy metric spaces in the sense of George and Veeramani [13] since its definition is obtained by replacing the fuzzy triangularity axiom in the definition of a fuzzy metric with M ( x, y, t) ∗ M (y, z, s) ≤ M ( x, z, t + k · s) for some k ≥ 1. Sedghi et al [23] introduced the concept of partial fuzzy metric as a fuzzy analogy of partial metric spaces and gave the topological structure and proved some fixed point results. In the last section we sketch some directions where the research in the field of fuzzy sb-metrics can be continued and its results could find some applications
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