In this study, we define the fuzzy cone metric space, the topology induced by this space and some related results of them. Also we state and prove the fuzzy cone Banach contraction theorem. c ⃝2015 All rights reserved.

We prove Baire's theorem for fuzzy cone metric spaces in the sense ofÖner et al. [T.Öner, M. B. Kandemir, B. Tanay, J. Nonlinear Sci. Appl., 8 (2015), 610-616]. A necessary and sufficient condition for a fuzzy cone metric space to be precompact is given. We also show that every separable fuzzy cone metric space is second countable and that a subspace of a separable fuzzy cone metric space is separable.

Kirk and Shahzad introduced the class of strong b-metric spaces lying between the class of b-metric spaces and the class of metric spaces. As compared with b-metric spaces, strong b-metric spaces have the advantage that open balls are open in the induced topology and, hence, they have many properties that are similar to the properties of classic metric spaces. Having noticed the advantages of strong b-metric spaces Kirk and Shahzad complained about the absence of non-trivial examples of such spaces. It is the main aim of this paper to construct a series of strong b-metric spaces that fail to be metric. Realizing this programme, we found it reasonable to consider these metric-type spaces in the context when the ordinary sum operation is replaced by operation ⊕, where ⊕ is an extended t-conorm satisfying certain conditions.

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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.