Towards Strong Convergence and Cauchy Sequences in Binary Metric Spaces
A Kuratowski topology is a topology specified in terms of closed sets rather than open sets. Recently, the binary metric was introduced as a symmetric, distributive-lattice-ordered magma-valued function of two variables satisfying a “triangle inequality” and subsequently proved that every Kuratowski...
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MDPI AG
2022-08-01
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| Series: | Axioms |
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| Online Access: | https://www.mdpi.com/2075-1680/11/8/383 |
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| author | Shubham Yadav Dhananjay Gopal Parin Chaipunya Juan Martínez-Moreno |
| author_facet | Shubham Yadav Dhananjay Gopal Parin Chaipunya Juan Martínez-Moreno |
| author_sort | Shubham Yadav |
| collection | DOAJ |
| description | A Kuratowski topology is a topology specified in terms of closed sets rather than open sets. Recently, the binary metric was introduced as a symmetric, distributive-lattice-ordered magma-valued function of two variables satisfying a “triangle inequality” and subsequently proved that every Kuratowski topology can be induced by such a binary metric. In this paper, we define the strong convergence of a sequence in a binary metric space and prove that strong convergence implies convergence. We state the conditions under which strong convergence is equivalent to convergence. We define a strongly Cauchy sequence and a strong complete binary metric space. Finally, we give the strong completion of all binary metric spaces with a countable indexing set. |
| first_indexed | 2024-03-09T11:54:32Z |
| format | Article |
| id | doaj.art-6c77134037be4ec481983771906b0aea |
| institution | Directory Open Access Journal |
| issn | 2075-1680 |
| language | English |
| last_indexed | 2024-03-09T11:54:32Z |
| publishDate | 2022-08-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Axioms |
| spelling | doaj.art-6c77134037be4ec481983771906b0aea2023-11-30T23:11:14ZengMDPI AGAxioms2075-16802022-08-0111838310.3390/axioms11080383Towards Strong Convergence and Cauchy Sequences in Binary Metric SpacesShubham Yadav0Dhananjay Gopal1Parin Chaipunya2Juan Martínez-Moreno3Department of Applied Mathematics and Humanities, S. V. National Institute of Technology Surat, Surat 395007, IndiaDepartment of Mathematics, Guru Ghasidas Vishwavidyalaya (A Central University), Bilaspur 495009, IndiaDepartment of Mathematics, King Mongkut’s University of Technology Thonburi, Bangkok 10140, ThailandDepartment of Mathematics, University of Jaén Campus Las Lagunillas s/n, 23071 Jaén, SpainA Kuratowski topology is a topology specified in terms of closed sets rather than open sets. Recently, the binary metric was introduced as a symmetric, distributive-lattice-ordered magma-valued function of two variables satisfying a “triangle inequality” and subsequently proved that every Kuratowski topology can be induced by such a binary metric. In this paper, we define the strong convergence of a sequence in a binary metric space and prove that strong convergence implies convergence. We state the conditions under which strong convergence is equivalent to convergence. We define a strongly Cauchy sequence and a strong complete binary metric space. Finally, we give the strong completion of all binary metric spaces with a countable indexing set.https://www.mdpi.com/2075-1680/11/8/383binary metricgeneralized metricconvergence in binary metric |
| spellingShingle | Shubham Yadav Dhananjay Gopal Parin Chaipunya Juan Martínez-Moreno Towards Strong Convergence and Cauchy Sequences in Binary Metric Spaces Axioms binary metric generalized metric convergence in binary metric |
| title | Towards Strong Convergence and Cauchy Sequences in Binary Metric Spaces |
| title_full | Towards Strong Convergence and Cauchy Sequences in Binary Metric Spaces |
| title_fullStr | Towards Strong Convergence and Cauchy Sequences in Binary Metric Spaces |
| title_full_unstemmed | Towards Strong Convergence and Cauchy Sequences in Binary Metric Spaces |
| title_short | Towards Strong Convergence and Cauchy Sequences in Binary Metric Spaces |
| title_sort | towards strong convergence and cauchy sequences in binary metric spaces |
| topic | binary metric generalized metric convergence in binary metric |
| url | https://www.mdpi.com/2075-1680/11/8/383 |
| work_keys_str_mv | AT shubhamyadav towardsstrongconvergenceandcauchysequencesinbinarymetricspaces AT dhananjaygopal towardsstrongconvergenceandcauchysequencesinbinarymetricspaces AT parinchaipunya towardsstrongconvergenceandcauchysequencesinbinarymetricspaces AT juanmartinezmoreno towardsstrongconvergenceandcauchysequencesinbinarymetricspaces |