Abstract
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.
Highlights
An important class of spaces was introduced by I.A
We prove the theorem in the case of ⊕-sb-metric spaces
We have used the concept of an extended t-conorm ⊕ in order to define ⊕-metric
Summary
An important class of spaces was introduced by I.A. Bakhtin (under the name almost metric spaces) and rediscovered by S. As shown by Kirk and Shahzad [1], in sb-metric spaces open balls are really open in the induced topology Thanks to this fact, sb-metric spaces have many useful properties common with ordinary metric spaces. Mathematics 2020, 8, 1097 of strong b-metric spaces that fail to be metric Having this primary aim in mind, we decide that it could be interesting, and probably useful, to develop our work on the basis of the operation ⊕, which is a kind of extended t-conorm.
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