Expansion Theory of Deng’s Metric in [0,1]-Topology

Round 1

Reviewer 1 Report

Report - reviewer comment

Expansion Theory of Deng's Metric in [0, 1]-topology

    In the first parts, the authors presented introduction and related results from the literature. In the second part, they show that Deng's metric can be equivalently characterized by using M0 and M, and then its corresponding metric topology. In the third part, they proved the main results They prove that a Deng metric must be a Yang-Shi metric on IX, and then it also must be an Erceg metric on IX, Therefore, we further will acquire that a Deng's metric on IX must be Q − C1. In the section, they show some interesting properties about Deng's pseudo-metric.

1. Some notations and equations are messed up. Please use a proper equation editor format to make them clearly visible and understandable.

2. The authors are suggested to apply proofs of your paper in the actual problem (some examples).

3. All the notations and abbreviations should be checked carefully.

4. The references should have a clear and unified format.

5. Please, rewrite the abstract section again.

6. Are these new results sharp and more accurate compared with the others?

7. Conclusion section should be enriched with some future remarks.

8. please clearly describe the contributions of your paper in the Abstract.

9. The linguistic review of the entire paper must be done well because

10. The references should have a clear and unified format .

11. Please cite more papers in your paper to catch current significant developments in Metric space and [0, 1]-topology.

12. it contains many shortcomings in the language, especially taking into account the indefinite letters a, an, the ..etc. and unifying its use in all papers 2

13. I have reviewed the paper. Presentation, the related literature, and technical soundness are weak. It presents good method. After made some additions to the paper, I recommend for publication

14. Check the holy paper, and Please add (,) and (.) in all suitable places they are too missing characters .

15. In line 280: the sentence is not clear please fix that.

16. I suggest adding some new relevant references to this topic, especially in metric space and L-fuzzy topology.

(a) Degree of (L, M)-fuzzy semi-precontinuous and (L, M)-fuzzy semipreirresolute functions, Demonstr. Math. 51 (1), 182197. On Raoperator in ideal topological spaces, Creat. Math. Inform 25,2016, 1-10.

Recommendation. I recommend accepting and publishing the paper after taking all the previous comments and observations and making some minor modifications mentioned above.

This is a good survey article. Emphasis is placed on the theory of mathematics to define new [0, 1]-topology space and Metric space theory, introducing various independent concepts of Metric space conditions which are generalization of existing results in the literature The article will not be useful for graduate course, but useful for students study by research to build a good radicals.

Received: Month 6, day 26

Minor editing of English language required

Author Response

Dear reviewer, hello. This is my response and article modification instructions based on your insightful suggestions and very good feedback. Due to the almost extensive revision of the article and considering the comments and suggestions of the reviewers, our modifications are as follows:

Based on the comments provided by the reviewers, we have made significant revisions to our paper ({\bf Title:} Expansion Theory of Deng's Metric in $[0,1]$-topology,  MDPI-mathematics-2447960) during this period, with the following specific modifications as follows:

1. We have rewritten the abstract.

2. We have added {\bf 2010 MSC}: 54A40; 03E72; 54E35. verification and added it.

3. We have made adjustments to the order in which the reference literature appears in the article.

4. We have made many modifications to the introduction section of this article to make it more rigorous.

5. In Preliminaries, we have added three theorems: Theorem 6,Theorem 7 and Theorem 8, which will be clearer in the later proof of Lemma 3 and Lemma 4.

6. We have rewritten the proofs of Lemma 3 and Lemma 4.

7. We have made extensive and detailed modifications to the language of the paper.

8.  We modify the contributions of authors: Conceptualization, Peng Chen; formal analysis, Peng Chen, Bin Meng and Xiaohui Ba; funding acquisition, Peng Chen; investigation, Bin Meng; methodology, Peng Chen; project administration, Bin Meng; supervision, Bin Meng; validation, Xiaohui Ba; visualization, Xiaohui Ba, Bin Meng; writing-original draft, Bin Meng and Peng Chen; writing-review and editing, Bin Meng, Peng Chen and Xiaohui Ba. All authors have read and agreed to the published version of the manuscript.

9. We have revised and confirmed the author's address unit and email address.

 

10. We have added fund support projects: The project is funded by Development of Integrated Communication and Navigation Chips and Modules (2021000056).

In short, I am particularly grateful to the unknown reviewer for providing me with such good suggestions and revisions. This has almost made me rewrite this article, and according to the reviewer's feedback, I feel that after revising and proofreading, the article has become fuller and smoother. We once again thank the editor for their selfless dedication and the reviewers for their many appropriate and highly valuable suggestions and opinions. Thank you.

The corresponding author and his address and other information are as follows:

Address: Institute of Microelectronics, Chinese Academy of Sciences, Beijing 100029, P.R. China; University of Chinese Academy of Sciences, Beijing 100049, P.R. China

E-mail: [email protected]; [email protected].

Sincerely Yours.

Dr. Peng Chen.

Author Response File: Author Response.pdf

Reviewer 2 Report

The referred manuscript is devoted to some problems of the fuzzy set theory.  Firstly, it is demonstrated how the famous Deng’s metrics can be extended to a larger set. Secondly, some of its properties are discussed and compared with other known objects of the theory,

The proofs are complete, the results appear novel, and some new methods are introduced. The references and introduction are adequate to the current state of the problem. In general I believe that such a results could ne interesting to specialists in the area.

Meanwhile, there are some remarks to the text of the manuscript.

1) The Abstract contains many formulas and is difficult to read unless one reads the Introduction. I would suggest to rewrite the Abstract using less formulas. 

2) If the journal requirements allow to do so, I would recommend to add 'fuzzy sets' to the list of keywords. 

3) Page 2, line 77. The property $Q-C_1$ has never been introduced in the text. I suggest to define the concept or to provide a reference.

4) The last paragraph of page 2 contains some basic definitions e.g. those of fuzzy sets. I suggest to shift those definitions closer to the beginning of Introduction.

5) Page 3, Theorem 1. It should be better to explain what is 'members of open spheres'.

6) I suggest to clarify the difference between Definition 1 and Definition 8.

7) Page 12, line 12 from below. 'Thereby, it is true for ...'. I would suggest to write explicitly what does this 'it' mean.

There are some minor remarks.

1) Page 1, lines 27-28. The sentence 'fuzzy metrics introduced in the branch of learning' is not totally clear for me. Please consider reformulating it.

2) Page 4, Definition 10.  Please, change ’an open neighborhood' to 'open neighborhood'.

3) Page 3, line 199. 'By above Case 1 and (A4), we exchange $x_1$ and $y_\lambda$ to fulfill.' The sentence looks incomplete. I suggest rewriting it.

4) Page 9, line 215. Please replace 'Hence we only need prove' with  'Hence we only need to prove'.

5) Page 9, line 223. Please replace 'and then we have $e\le A$. Which shows' with  'and then we have $e\le A$ which shows'.

6) Page 17, line 326. The point is omitted after the word ‘Example'. Please add it.

 

 

Author Response

Dear reviewer, hello. This is my response and article modification instructions based on your insightful suggestions and very good feedback. Due to the almost extensive revision of the article and considering the comments and suggestions of the reviewers, our modifications are as follows:

Based on the comments provided by the reviewers, we have made significant revisions to our paper ({\bf Title:} Expansion Theory of Deng's Metric in $[0,1]$-topology,  MDPI-mathematics-2447960) during this period, with the following specific modifications as follows:

1. We have rewritten the abstract.

2. We have added {\bf 2010 MSC}: 54A40; 03E72; 54E35. verification and added it.

3. We have made adjustments to the order in which the reference literature appears in the article.

4. We have made many modifications to the introduction section of this article to make it more rigorous.

5. In Preliminaries, we have added three theorems: Theorem 6,Theorem 7 and Theorem 8, which will be clearer in the later proof of Lemma 3 and Lemma 4.

6. We have rewritten the proofs of Lemma 3 and Lemma 4.

7. We have made extensive and detailed modifications to the language of the paper.

8.  We modify the contributions of authors: Conceptualization, Peng Chen; formal analysis, Peng Chen, Bin Meng and Xiaohui Ba; funding acquisition, Peng Chen; investigation, Bin Meng; methodology, Peng Chen; project administration, Bin Meng; supervision, Bin Meng; validation, Xiaohui Ba; visualization, Xiaohui Ba, Bin Meng; writing-original draft, Bin Meng and Peng Chen; writing-review and editing, Bin Meng, Peng Chen and Xiaohui Ba. All authors have read and agreed to the published version of the manuscript.

9. We have revised and confirmed the author's address unit and email address.

 

10. We have added fund support projects: The project is funded by Development of Integrated Communication and Navigation Chips and Modules (2021000056).

In short, I am particularly grateful to the unknown reviewer for providing me with such good suggestions and revisions. This has almost made me rewrite this article, and according to the reviewer's feedback, I feel that after revising and proofreading, the article has become fuller and smoother. We once again thank the editor for their selfless dedication and the reviewers for their many appropriate and highly valuable suggestions and opinions. Thank you.

The corresponding author and his address and other information are as follows:

Address: Institute of Microelectronics, Chinese Academy of Sciences, Beijing 100029, P.R. China; University of Chinese Academy of Sciences, Beijing 100049, P.R. China

E-mail: [email protected]; [email protected].

Sincerely Yours.

Dr. Peng Chen.

Author Response File: Author Response.pdf