Metric spaces

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Warsaw, Poland: Polish Academy of Sciences; Diening L. Theoretical and numerical results for electrorheological fluids [Ph.

A new generalization of metric spaces: rectangular M-metric spaces | SpringerLink

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External link. In this sense, the real numbers form the completion of the rational numbers. The proof of this fact, given in by the German mathematician Felix Hausdorff , can be generalized to demonstrate that every metric space has such a completion.

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Metric space. Info Print Cite. Submit Feedback. Thank you for your feedback. Written By: Stephan C. See Article History. Start your free trial today for unlimited access to Britannica. Learn More in these related Britannica articles:. Other examples of topologies on sets occur purely in terms of set theory. In this paper, we introduce the concept of the rectangular M -metric spaces, along with its topology and we prove some fixed-point theorems under different contraction principles with various techniques.


Metric Spaces, Convexity and Nonpositive Curvature (Second edition)

Also we give an application to the fixed-circle problem. The well-known Banach contraction principle has been studied and generalized in many different directions such as generalizing the used metric spaces. Recently, new generalized metric spaces have been presented for this purpose. For example, M -metric spaces, rectangular metric spaces, partial rectangular metric spaces have been introduced and studied see [ 2 , 3 , 7 ].

Branciari in [ 3 ] defined rectangular metric spaces as follows:. Inspired by the work of Branciari, Shukla in [ 7 ] defined rectangular partial metric spaces which are generalizations of rectangular metric spaces. Asadi et al. On these new spaces, some generalized fixed-point results have been obtained see [ 1 , 2 , 3 , 6 , 7 ].

In this paper, we introduce the concept of a rectangular M -metric space, along with proving some fixed-point theorems for self-mappings in rectangular M -metric spaces. In Sect. Using these concepts, we present an application to fixed-circle problem.

Notice that every M -metric is also a rectangular M -metric. Also it can be easily verified the following inequality under some cases:. Note that we can obtain a rectangular metric space from a rectangular M -metric space as seen in the following examples. In the following proposition, we see the relationship between a rectangular partial metric and a rectangular M -metric. Every partial rectangular metric is a rectangular M - metric.

It is known that every metric space is a rectangular metric space see [ 4 ] and that every rectangular metric space is a partial rectangular metric space with zero self-distance see [ 7 ]. Also every metric space is a partial metric space and every partial metric space is an M -metric space see [ 2 , 5 ]. Consequently, we can give the following diagram. Here, arrows stand for inclusions.

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In this section, we investigate some topological properties of rectangular M -metric spaces. Without loss of generality, let us consider the following cases:. We reach subcase 1, so u is a fixed point of T. The notions of a circle and of a fixed circle on a rectangular M -metric space are defined as follows:. Skip to main content Skip to sections.