Idealizing Rough Topological Structures Generated by Several Types of Maximal Neighborhoods and Exploring Their Applications
Abstract
1. Introduction
2. Preliminaries
- (i)
- (ii)
- (iii)
- (iv)
- (v)
- (vi)
- (vii)
- (viii)
- (i)
- (ii)
- and when ⨿ is symmetric;
- (iii)
- when ⨿ is symmetric and transitive;
- (iv)
- All types of are equal when ⨿ is equivalence.
3. Idealization of Topologies Derived from Various Kinds of Maximal Neighborhoods
- (i)
- Clearly,
- (ii)
- Let and . Then,and so
- (iii)
- Let andand
- 1.
- and
- 2.
- and
- 3.
- and
- 4.
- and
- 5.
- and ;
- 6.
- and
- 7.
- 8.
- and
- 1.
- and
- 2.
- and
- 3.
- and
- 4.
- and
- (1)
- Let . Then, Thus, Hence, and . Therefore, and Hence, and Statement 3 can be evidenced in a similar manner.
- (2)
- Let . Then, Thus, Consequently, Hence, . Therefore, Statement 4 can be evidenced in a similar manner.
- 1.
- 2.
- 3.
- 4.
- 1.
- 2.
- and
- 3.
- 4.
- and
- 5.
- 6.
- 7.
- 8.
- 9.
- 10.
- 11.
- 12.
- 13.
- (it is obvious if we set );
- 14.
- is not a dual of (it is obvious if we set ).
- (i)
- and when ⨿ is symmetric;
- (ii)
- when ⨿ is symmetric and transitive;
- (iii)
- when ⨿ is equivalence.
- (i)
- Let Then,(by Theorem 2).It follows that , and consequently,
- (ii)
- Let Then,(by Theorem 2).It follows that the remaining cases are comparable.
- (iii)
- The proof is straightforward.
- 1.
- Similarity in (i) is crucial because and
- 2.
- Similarity and transitivity in (ii) are crucial because
- 3.
- Equivalence in (iii) is crucial because not all topologies are always equal in the absence of this restriction.
4. Idealization of Approximate Models
- (i)
- ;
- (ii)
- ;
- (iii)
- ;
- (iv)
- If , then ;
- (v)
- (vi)
- (vii)
- .
- (iv)
- Let . Then, , and so .
- (v)
- by (iv). Since and it follows that Consequently, Then, Thus,
- (vi)
- Let . Then, such that and so . Therefore, . Thus, . Let . Then, , and so such that and . Therefore, . Hence, .
- (vii)
- From (i) we obtain . Conversely, let . Then, such that . (by (iv)). By Definition 10, we observe that , so . Thus, .
- (i)
- ;
- (ii)
- but ;
- (iii)
- .
- (i)
- ;
- (ii)
- ;
- (iii)
- ;
- (iv)
- If , then ;
- (v)
- ;
- (vi)
- ;
- (vii)
- .
- (i)
- ;
- (ii)
- but ;
- (iii)
- .
- 1.
- 2.
- 3.
- 4.
- (by Proposition 3). Hence,
- Let (by Proposition 3). Hence,
- The proof is straightforward using statement 3.
- (i)
- (ii)
- .
- (i)
- Let Then, Therefore, and Thus, and . Hence, Therefore,
- (ii)
- This is obtained from Proposition 6.
5. Comparisons of the Proposed Methods and Their Advantages over Earlier Ones
5.1. Relationships Among Different Sorts of the Current Approximations
- (i)
- ;
- (ii)
- ;
- (iii)
- ;
- (iv)
- ;
- (v)
- ;
- (vi)
- ;
- (vii)
- ;
- (viii)
- .
- (i)
- ;
- (ii)
- ;
- (iii)
- ;
- (iv)
- ;
- (v)
- ;
- (vi)
- ;
- (vii)
- ;
- (viii)
- .
- (i)
- (ii)
- (iii)
- (iv)
- (v)
- (vi)
- (vii)
- (viii)
- (i)
- (ii)
- (iii)
- (iv)
- ;
- (v)
- (vi)
- ;
- (vii)
- (viii)
- (i)
- (ii)
- (iii)
- (iv)
- (i)
- If then
- (ii)
- If then
- (iii)
- If then
- (i)
- and ;
- (ii)
- and .
- (i)
- ;
- (ii)
- (iii)
- ;
- (iv)
- (i)
- (ii)
- .
- (i)
- (ii)
- (i)
- (ii)
- (iii)
5.2. Evaluating Our Approach Against Previous Methods
- 1.
- 2.
- (1)
- (by Theorem 5). The proof of (2) can be expounded similarly.
- 1.
- 2.
- 3.
- Every -exact subset in is £--exact;
- 4.
- Every £--rough subset in is -rough.
- 1.
- If then
- 2.
- If then
- 1.
- 2.
- Let Then, such that Thus, Consequently, Hence,
- This proof can be elaborated in the same manner as the proof of (1).
6. An Application of the Suggested Method to Diagnose Dengue Sickness
- 1.
- Hosny’s method [36] in Definition 7:
- 2.
- The proposed Definitions 10–12:
- 1.
- Hosny’s method [36] in Definition 7:
- 2.
- The proposed Definitions 10–12:
7. Discussions: Merits and Demerits
- The merits of the new models are as follows:
- (i)
- The prior topologies [35,36] are weaker than the present one. That is, the current topologies are stronger and include a lot of information that we need when studying rough set. These topologies are more effective for handling large samples and are crucial for making more accurate decisions in cases where the framework is appropriate, such as in infectious diseases, where the spread of infection depends on the sample size.
- (ii)
- (iii)
- even though as in [17].
- (iv)
- (v)
- (vi)
- (vii)
- The current technique satisfies all of Pawlak’s criteria without conditions, whereas the methods in some earlier studies either failed to meet all of Pawlak’s characteristics or achieved them only under certain constraints [18].
- The demerits of the present models are as follows:
8. Conclusions
- Introducing two ideals in lieu of one to generalize the present method;
- Generating supra-topology to extend the current work;
- Generalizing the current paper to soft and fuzzy sets.
Funding
Data Availability Statement
Conflicts of Interest
References
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V | Definitions 10 and 11 for | Definitions 7 [36] for | ||||
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V | H | P | J | B | Dengue Fever |
---|---|---|---|---|---|
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Hosny, M. Idealizing Rough Topological Structures Generated by Several Types of Maximal Neighborhoods and Exploring Their Applications. Axioms 2025, 14, 333. https://doi.org/10.3390/axioms14050333
Hosny M. Idealizing Rough Topological Structures Generated by Several Types of Maximal Neighborhoods and Exploring Their Applications. Axioms. 2025; 14(5):333. https://doi.org/10.3390/axioms14050333
Chicago/Turabian StyleHosny, Mona. 2025. "Idealizing Rough Topological Structures Generated by Several Types of Maximal Neighborhoods and Exploring Their Applications" Axioms 14, no. 5: 333. https://doi.org/10.3390/axioms14050333
APA StyleHosny, M. (2025). Idealizing Rough Topological Structures Generated by Several Types of Maximal Neighborhoods and Exploring Their Applications. Axioms, 14(5), 333. https://doi.org/10.3390/axioms14050333