Abstract
In this paper, we study those fuzzy metrics M on X, in the George and Veeramani’s sense, such that ⋀ t > 0 M ( x , y , t ) > 0 . The continuous extension M 0 of M to X 2 × 0 , + ∞ is called extended fuzzy metric. We prove that M 0 generates a metrizable topology on X, which can be described in a similar way to a classical metric. M 0 can be used for simplifying or improving questions concerning M; in particular, we expose the interest of this kind of fuzzy metrics to obtain generalizations of fixed point theorems given in fuzzy metric spaces.
Highlights
In 1975, Kramosil and Michalek introduced in [1] a notion of fuzzy metric space
In 1975, Kramosil and Michalek introduced a notion of fuzzy metric space in [1]
It was slightly modified by George and Veeramani in [2]. Both notions share many properties. They are topologically equivalent to classical metric spaces
Summary
In 1975, Kramosil and Michalek introduced in [1] a notion of fuzzy metric space. Later, George and Veeramani in [2] strengthened some conditions on this concept. George and Veeramani studied some aspects of the above concept in [2] They proved that every fuzzy metric M on X generates a topology τM on X. From the topological point of view (see Remark 3), the class of extended fuzzy metrics ( X, M0 , ∗) are so close to metrics that topological results related to M0 can be established as a simple extension of classical concepts to the fuzzy setting, only by modifying the notation (which is left to the reader). Mimicking arguments in the literature, one can give fixed point theorems for extendable fuzzy metrics in a more general version The reader can find in this example a method for obtaining more general results in fixed point theory, but for extendable fuzzy metrics.
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