[Retracted] On Strongly <span class="nowrap"><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.4618998pt" id="M1" height="12.7225pt" version="1.1" viewBox="-0.0657574 -12.2606 34.5686 12.7225" width="34.5686pt"><g transform="matrix(.017,0,0,-0.017,0,0)"><path id="g113-99" d="M452 333C452 398 425 448 375 448C329 448 235 404 140 292H138L229 683C233 701 234 712 225 712C208 712 149 686 66 677L64 652H97C135 652 140 651 129 598L38 167C23 96 29 57 44 33C60 7 98 -12 132 -12C169 -12 232 3 290 41C383 102 452 223 452 333ZM365 316C365 199 308 87 239 49C221 39 200 35 183 35C137 35 99 70 112 163C116 191 123 215 129 233C190 316 290 392 330 392C348 392 365 372 365 316Z"/></g><g transform="matrix(.017,0,0,-0.017,11.889,0)"><path id="g117-33" d="M535 230V280H52V230H535Z"/></g><g transform="matrix(.017,0,0,-0.017,25.797,0)"><path id="g113-230" d="M475 507C475 612 440 712 326 712C139 712 23 420 23 215C23 96 58 -12 180 -12C369 -12 475 293 475 507ZM391 522C391 486 387 448 379 394H126C155 538 222 677 310 677C386 677 391 571 391 522ZM373 346C344 193 283 22 189 22C126 22 106 114 106 196C106 243 111 293 118 346H373Z"/></g></svg>-</span>Continuous Mappings in Fuzzifying Topology

Yang, Ting; Khalil, Ahmed Mostafa

doi:https://doi.org/10.1155/2021/3244618

Mathematical Problems in Engineering

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Abstract Introduction and Preliminaries Conclusions Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

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Volume 2021 | Article ID 3244618 | https://doi.org/10.1155/2021/3244618

[Retracted] On Strongly -Continuous Mappings in Fuzzifying Topology

Ting Yang¹and Ahmed Mostafa Khalil²

Academic Editor: Lazim Abdullah

Received18 Apr 2021

Accepted11 May 2021

Published04 Jun 2021

Abstract

In this article, we will define the new notions (e.g., -neighborhood system of point, -closure (interior) of a set, and -closed (open) set) based on fuzzy logic (i.e., fuzzifying topology). Then, we will explain the interesting properties of the above five notions in detail. Several basic results (for instance, Definition 7, Theorem 3 (iii), (v), and (vi), Theorem 5, Theorem 9, and Theorem 4.6) in classical topology are generalized in fuzzy logic. In addition to, we will show that every fuzzifying -closed set is fuzzifying -closed set (by Theorem 3 (vi)). Further, we will study the notion of fuzzifying -derived set and fuzzifying -boundary set and discuss several of their fundamental basic relations and properties. Also, we will present a new type of fuzzifying strongly -continuous mapping between two fuzzifying topological spaces. Finally, several characterizations of fuzzifying strongly -continuous mapping, fuzzifying strongly -irresolute mapping, and fuzzifying weakly -irresolute mapping along with different conditions for their existence are obtained.

1. Introduction and Preliminaries

In classical topology, the notions of -open set, -closed set, and strongly -continuous mapping are presented in [1, 2]. After that, Hanafy [3] used the term -open sets instead of -open sets and studied the notions of -open sets and -continuous mapping in fuzzy topology [4]. Benchalli and Karnel [5] presented a novel form of fuzzy subset named fuzzy b-open (closed) set, and some basic properties are proved and also their relations with different fuzzy sets in fuzzy topological spaces are investigated. In 2017, Dutta and Tripathy [6] introduced a new kind of open set named fuzzy open set (i.e., which is a generalization of open set). Ying [7] extended the basic notions in classical topology to fuzzifying topology based on fuzzy logic (i.e., as considered a novel approach of fuzzy topology, which depends on the various basic relations of topological spaces and the logical analysis of topological axioms). Many researchers are interested in fuzzifying topology (such as fuzzifying semiopen sets [8], fuzzifying preopen sets [9], fuzzifying -open sets [10], fuzzifying -open sets [11], and fuzzifying -open sets [12]). Therefore, in this article, we will extend the notions of -neighborhood system of a point, -closure (interior) of a set, -open (closed) set, -derived sets, and -boundary sets in fuzzifying topology. Also, we introduce the notion of fuzzifying strongly -continuous mapping, fuzzifying strongly -irresolute mapping, and fuzzifying weakly -irresolute mapping between two fuzzifying topological spaces.

The rest of this article is arranged as follows. In this section, we briefly recall several notions: closed (open) set, closure (interior) of a set, neighborhood system of point, -closed (open) set, -closure (interior) of a set, -neighborhood system of point, continuous mapping, and -continuous mapping in fuzzifying topology which are used in the sequel. In Section 2, we define the notions of -neighborhood system of a point, -closure (interior) of a set, and -open (closed) set in fuzzifying topology. The interesting relation properties of the above notions are explained in detail. In Section 3, we present the notions -derived set and -boundary set in fuzzifying topology and introduce the characterizations of interesting properties between fuzzifying -derived set and fuzzifying -closure of a set. In Section 4, we define the fuzzifying strongly -continuous mapping, fuzzifying strongly -irresolute mapping, and fuzzifying weakly -irresolute mapping between two fuzzifying topological spaces and investigate some properties of them.

Firstly, we give the notions of a fuzzy logical [7, 13] as follows:(1)We give the symbol which is the truth value of (i.e., the set of truth values means the unit interval ).(2)We can write (i.e., is valid).(3)For (i.e., mean the whole of fuzzy subsets of a set ) have the following for :

Secondly, we present the basic notions related to fuzzifying topological space as follows.

Definition 1. (see [7]). (i.e., , the set of all subsets of a set ) is called a fuzzifying topological space (for short, ) if we have the following three conditions:(i)(ii)(iii)

Definition 2. (see [7, 13]). The several notions of are given as follows :(i) (i.e., ) is called the set of all fuzzifying closed sets if , where is the complement of (ii) (i.e., ) is called a fuzzifying neighborhood system of if (iii) is called a fuzzifying closure of if (iv) is called a fuzzifying interior of if

Noiri and Sayed [12] presented and studied the following notions in as indicated below.

Definition 3. (see [12]). (i) (i.e.) is called the set of all fuzzifying -open sets if i.e.,(ii) (i.e., ) is called the set of all fuzzifying -closed sets if , where is the complement of .(iii) (i.e., ) is called a fuzzifying -neighborhood system of if .(iv) is called a fuzzifying -closure of if .(v) is called a fuzzifying -interior of if .

Definition 4. (see [14]). (i.e., ) (a unary fuzzy predicate) is called fuzzifying continuous mappings between and ifi.e.,

Definition 5. (see [12]). (i.e., ) (a unary fuzzy predicate) is called fuzzifying -continuous mappings between and ifi.e.,

2. On Fuzzifying -Neighborhood System

Definition 6. (i.e., ) is called a fuzzifying -neighborhood system of if

Theorem 1. Let be a mapping from to s.t. and is a family of N-fuzzy (Normal fuzzy) subsets of ) having the following three properties :(i)(ii)(iii)

Proof. (i)Case 1: if , then . Case 2: if , then . There exists such that . From Theorems 4.2 (1) and 5.2 (3) in [12], we obtain Consequently, , and so . If , then we have ; it is contradiction. Thus, , and hence, .(ii)Case 1: if , then . Case 2: if and , then and so . Thus, . Hence, .(iii). SinceThus,

Next, we will generalize the notion of -closure [2] in as follows.

Definition 7. is called a fuzzifying -closure of if

Theorem 2. The following relation is holding in :

Proof.

Theorem 3. The following relations are holding in :(i)(ii)(iii)(iv)(v)(vi)(vii)

Proof. (i)(ii)From (i) above and since is normal, we have .(iii)Case 1: if , then . Case 2: if , then . Thus, .(iv).(v)Case 1: if , then . Case 2: if , then .(vi)From Theorems 5.2 (3) and 5.3 (2) in [12], we have .(vii)It is easy to get . Conversely, for every ,

Lemma 1. .

Proof. It follows from Theorem 3 (vi).

Clearly, if we consider and in .

Definition 8. (i) (i.e., ) is called the set of all fuzzifying -closed sets if i.e.,(ii) (i.e., is called the set of all fuzzifying -open sets if , where is the complement of .

By Definition 8, we can conclude .

Definition 9. is called a fuzzifying -interior of ifi.e.,

Theorem 4. The following relations are holding in :(i)(ii)(iii)(iv)(v)(vi)(vii)(viii)

Proof. (i)By Theorem 3 (i), we obtain . Hence, .(ii)By (i) above, we obtain .(iii)By Theorem 3 (iii), we obtain .(iv)(v)Case 1: if , then . Case 2: if , then from Theorem 1 (ii) and by (v) above.(vi)Similar to Theorem 3 (vi).(vii)By Lemma 1, we obtain .(viii)By Theorem 3 (vii), we obtain

Theorem 5. The following two relations are holding in :(i)(ii)

Proof. (i)(ii)Similar to (i).

Theorem 6. (i.e., is a set) is a if we have the following three conditions:(i)(ii)(iii)

Proof. (i)Clear.(ii)By Theorem 4 (v) and Theorem 1 (iii), we obtain(iii)By Theorem 4 (v), we obtainThis completes the proof.

Theorem 7. The following two relations are holding in :(i)(ii)

Proof. (i). Firstly, we obtain . Further, assume that . Hence, , we obtain and .(ii)By (i) we obtain

Theorem 8. The following two relations are holding in :(i)(ii)

Proof. (i)By Corollary 4.1 in [12] and Theorem 3 (vi), we obtain .(ii)Follows from (i) above.

Next, we will generalize Theorem 3.8 (a) in [2] in by the following theorem.

Theorem 9. The following relation is holding in :

Proof. Firstly, we prove that . By Corollary 4.1 and Theorem 5.3 (1) in [12], we obtain Case 1: if , then Case 2: if , thenConsequently, .
Thus,This completes the proof.

Corollary 1. The following relation is holding in :

Theorem 10. The following relation is holding in :

Proof.

3. Fuzzifying -Derived Sets and Fuzzifying -Boundary Sets

Definition 10. is called a fuzzifying -derived set of ifi.e.,

Theorem 11. The following relations are holding in :(i)(ii)(iii)

Proof. (i).(ii)By (i) above, we obtain (i.e., is normal).(iii)Case 1: , then . Case 2: if and by Theorem 1 (ii), we obtain .

Theorem 12. The following relation is holding in :

Proof. By Theorem 7, we obtain

Theorem 13. The following two relations are holding in :(i)(ii)

Proof. (i)By Theorem 11 (i) and Theorem 1 (iii), we obtain(ii)Case 1: if , then by Theorem 1 (i), . Hence, . Case 2: if , then .

Theorem 14. The following two relations are holding in :(i)(ii)

Proof. (i)Case 1: if , . Case 2: if , then(ii)If , then . Now, we suppose that and have , . So . If , then . For any , we get , i.e., there exists such that and . Next, we will show that . If not, then there is with . Thus, we get ; it is contradiction. Therefore, . Since is arbitrary, it holds that .

Definition 11. is called a fuzzifying -boundary set of ifi.e.,

Theorem 15. The following relation is holding in :

Proof. Similar to Lemma 2.1 in [13].

Theorem 16. The following two relations are holding in :(i) and so (ii) and so

Proof. (i)Case 1: if , then (by Theorem 3 (iii)). Case 2: if , then . Therefore, by Theorem 12 and Theorem 14 (i), we get(ii)By Theorem 4 (i) and (i) above, we get . Also, from Theorem 5 (i), we obtain .

4. Fuzzifying Strongly -Continuous Mapping, Fuzzifying Strongly -Irresolute Mapping, and Fuzzifying Weakly -Irresolute Mapping

In the following section, we define a fuzzifying strongly -continuous mapping, fuzzifying strongly -irresolute mapping, and fuzzifying weakly -irresolute mapping between and .

Definition 12. (i.e., (a unary fuzzy predicate) is called fuzzifying strongly -continuous mappings between and ifi.e.,

Definition 13. Assuming that and and , we put(i), where is a family of fuzzifying closed subset of and also is a family of fuzzifying -closed subset of (ii), where is a fuzzifying neighborhood system of of and is a fuzzifying -neighborhood system of of (iii)(iv)(v)(vi)

Theorem 17. .

Proof. (i)We show that .(ii)We prove that . Firstly, we show that . If , then . Assume that . Thus, if , then . Hence, Consequently, . Thus, Therefore, . Secondly, we show that . From Corollary 4.1 in [12] and Theorem 4 (iv), we have(iii)We show that . By Theorem 1 (ii), we obtain Thus,(iv)We show that . Firstly, we prove that . For every , we have that and . By Lemma 1.2 (2) in [14], we obtain Thus, Secondly, , there exists s.t. and . By Lemma 1.2 (1) in [14], we obtain(v)We show that . Thus,(vi)We prove that . Thus, This completes the proof.

Remark 1. The following relation is holding in crisp setting, while in does not hold by the next example.

Example 1. If we consider and , and in and defined the identity mapping between and byThus, .

Definition 14. (i.e., (a unary fuzzy predicate) is called fuzzifying strongly -irresolute mapping between and ifi.e.,

Definition 15. Assuming that and and , we put(i), where is a family of fuzzifying -closed subset of and is a family of fuzzifying -closed subset of (ii), where is a fuzzifying -neighborhood system of of and is a fuzzifying -neighborhood system of of (iii)(iv)(v)(vi)

Theorem 18. .

Proof. Similar to Theorem 17.

Definition 16. (i.e., ) (a unary fuzzy predicate) is called fuzzifying weakly -irresolute mapping between and ifi.e.,

Definition 17. Assuming that and and , we put(i), where is a family of fuzzifying -closed subset of and is a family of fuzzifying -closed subset of (ii), where is a fuzzifying -neighborhood system of of and is a fuzzifying -neighborhood system of of (iii)(iv)(v)(vi).

Theorem 19. .

Proof. Similar to Theorem 17.

Next, we will generalize Theorem 5.8 (a) in [2] in by the following theorem.

Theorem 20. Assume that , , , , and . The following two relations are holding:(i)(ii)

Proof. (i)We will show that . Case 1: if , the results holds. Case 2: if , then Therefore, . (ii)

5. Conclusions

The present paper investigates topological notions when these are planted into the framework of Ying’s fuzzifying topological spaces (in the semantic method of continuous-valued logic). It continues various investigations into fuzzy topology in a legitimate way and extends some fundamental results in general topology to fuzzifying topology. An important virtue of our approach (in which we follow Ying) is that we define topological notions as fuzzy predicates (by formulae of Łukasiewicz fuzzy logic) and prove the validity of fuzzy implications (or equivalences). Unlike the (more widespread) style of defining notions in fuzzy mathematics as crisp predicates of fuzzy sets, fuzzy predicates of fuzzy sets provide a more genuine fuzzification; furthermore, the theorems in the form of valid fuzzy implications are more general than the corresponding theorems on crisp predicates of fuzzy sets. The main contributions of the present paper are to define fuzzifying -neighborhood system of a point, fuzzifying -closure of a set, fuzzifying -interior of a set, fuzzifying -open sets, fuzzifying -closed sets, fuzzifying -derived sets, and fuzzifying -boundary sets in the setting fuzzifying topological space. Also, we define the concepts of fuzzifying strongly -continuous mapping, fuzzifying strongly -irresolute mapping, and fuzzifying weakly -irresolute mapping of fuzzifying topological spaces and obtain some basic properties of such spaces. There are some problems for further study:(1)One obvious problem is our results are derived in the Łukasiewicz continuous logic. It is possible to generalize them to a more general logic setting, like resituated lattice-valued logic considered in [13, 14].(2)What is the justification for fuzzifying strongly -continuous functions in the setting of topologies?(3)Obviously, fuzzifying topological spaces in [15] form a fuzzy category. Perhaps, this will become a motivation for further study of the fuzzy category.(4)What is the justification for fuzzifying strongly -continuous functions in -topologies, etc.?

Furthermore, the future possible research of the authors will be to give several new results (e.g., fuzzifying semi--open sets and fuzzifying -open sets), which are similar to the results of fuzzifying -open sets, fuzzifying semipre--open sets [16] and fuzzifying pre--open sets [17] based on fuzzifying topology.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (Grant no. 31660203).

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Copyright © 2021 Ting Yang and Ahmed Mostafa Khalil. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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