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Keywords:
generalized fuzzy 2-metric space; generalized fuzzy $2$-normed space; tripled fuzzy metric space; Hausdorff topology
generalized fuzzy 2-metric space; generalized fuzzy $2$-normed space; tripled fuzzy metric space; Hausdorff topology
Summary:
The aim of the paper is to present three-variable generalizations of fuzzy metric spaces in sense of George and Veeramani from functional and topological points of view, respectively. From the viewpoint of functional generalization, we introduce a notion of generalized fuzzy 2-metric spaces, study their topological properties, and point out that it is also a common generalization of both tripled fuzzy metric spaces proposed by Tian et al. and $\mathcal{M}$-fuzzy metric spaces proposed by Sedghi and Shobe. Since the ordinary tripled norm is the same as the ordinary norm up to the induced topology, we keep our spirit on fuzzy normed structures and introduce a concept of generalized fuzzy 2-normed spaces from the viewpoint of topological generalization. It is proved that generalized fuzzy 2-normed spaces always induces a Hausdorff topology.
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