Preopenness Degree in <span class="nowrap"><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.2723999pt" id="M1" height="11.6718pt" version="1.1" viewBox="-0.0657574 -11.3994 20.4562 11.6718" width="20.4562pt"><g transform="matrix(.017,0,0,-0.017,0,0)"><path id="g113-83" d="M610 18C585 26 567 34 540 68C517 97 499 128 476 171C452 215 425 276 413 304C496 332 570 394 570 494C570 555 545 595 509 619S419 650 364 650H139L133 622C216 615 219 612 203 527L129 132C112 40 105 36 23 28L17 0H279L285 28C199 34 194 40 211 132L239 284H284C320 284 334 275 351 236C374 182 394 140 420 93C459 23 495 -1 592 -8H600L610 18ZM480 485C480 424 449 372 403 342C374 323 338 316 293 316H245L291 562C296 589 301 601 311 608S337 618 358 618C432 618 480 575 480 485Z"/></g><g transform="matrix(.017,0,0,-0.017,10.759,0)"><path id="g113-77" d="M541 160L512 170C485 116 462 88 439 67C411 41 376 35 318 35C272 35 238 37 224 48C207 61 205 85 212 121L290 533C305 611 308 615 391 622L397 650H139L133 622C217 615 221 612 206 533L126 118C111 41 103 34 23 26L17 0H474C489 31 528 124 541 160Z"/></g></svg>-</span>Fuzzy Bitopological Spaces

Khalil, O. H.; El-Helow, K. E.; Ghareeb, A.

doi:https://doi.org/10.1155/2022/9210694

Abstract and Applied Analysis

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Research Article | Open Access

Volume 2022 | Article ID 9210694 | https://doi.org/10.1155/2022/9210694

Preopenness Degree in -Fuzzy Bitopological Spaces

O. H. Khalil,¹K. E. El-Helow,²and A. Ghareeb³

Academic Editor: Victor Kovtunenko

Received08 Nov 2021

Revised20 Dec 2021

Accepted31 Jan 2022

Published11 Feb 2022

Abstract

Based on the concept of pseudocomplement, we introduce a new representation of preopenness of -fuzzy sets in -fuzzy bitopological spaces. The concepts of pairwise -fuzzy precontinuous and pairwise -fuzzy preirresolute functions are extended and discussed based on the --preopen gradation. Further, we follow up with a study of pairwise -fuzzy precompactness in -fuzzy bitopological spaces of an -fuzzy set. We find that our paper offers more general results since -fuzzy bitopology is a generalization of -bitopology, -bitopology, and -fuzzy topology.

1. Introduction

The fuzzy set theory was introduced in the year 1965 by Zadeh [1]. The development of fuzzy set theory, since its introduction, has been dramatic and breathtaking! Thousands of research papers have appeared in various journals devoted entirely to theoretical and application aspects of fuzzy sets. Artificial intelligence, automata theory, computer science, control theory, decision-making, expert systems, medical diagnosis, neural networks, pattern recognition, robotics, and social sciences are a few fields where fuzzy sets find application. Within a short time since its introduction, the fuzzy sets have permeated almost every academic discipline and have made their way into consumer products! Apart from this, it is also used for the construction of machines by way of intelligent robotics (engines, cars, ships, turbines, etc.) and controls (Sendai subway train in Japan, etc.) as well as for military purposes.

In 1991, Bin Shahna [2] introduced the concept of -open and preopen sets in the context of fuzzy sets, and he introduced a preliminary study of fuzzy strong semicontinuity and fuzzy precontinuity as well. Later, the concepts of fuzzy -open sets, fuzzy preopen, fuzzy -continuous mappings, and fuzzy precontinuous mappings have been generalized to the setting of fuzzy bitopological spaces in [3], where some of their fundamental properties have been studied.

Shi [4] presented the concept of an -fuzzy preopen degree of -fuzzy set in -fuzzy topological spaces. In addition, he discussed the fundamental properties of -fuzzy precontinuous and -fuzzy preirresolute mappings. It has been found that Shi’s operator is incredibly useful in introducing other gradations as well as in analyzing many topological characteristics [5].

The concept of -topology has recently been introduced and studied by H. Li and Q. Li [6] as an extension of -topology. A detailed discussion is also presented concerning -continuous mapping and -compactness by means of an inequality. As a generalization of -topology and -fuzzy topology, H. Li and Q. Li [7] defined -fuzzy topology. Further investigations are conducted regarding the -fuzzy compactness in -fuzzy topological spaces. As a consequence of their work, Zhang et al. [8] presented the Lindelöf property degree as well as the countable -fuzzy compactness degree of an -subset. It is clear that the gradation of fuzzy compactness and Lindelöf property in the sense of Kubiak and Šostak are special cases of the corresponding degrees in -fuzzy topology.

In this paper, the pseudocomplement of -fuzzy sets is put forward as a basis for the definition of --preopen degree in -fuzzy bitopology. Additionally, we introduce and discuss pairwise -fuzzy precontinuous, pairwise -fuzzy preirresolute mappings, and pairwise -fuzzy precompactness.

2. Preliminaries

Throughout this paper, denotes a complete DeMorgan algebra [9, 10] and is a nonempty crisp set. By , we refer to the family of all -fuzzy sets presented on . The greatest and the smallest elements in and are and , respectively. For any , , means that the element is wedge below in [11]. (resp. ) denotes the collection of nonzero coprime (resp. nonunit prime) members in . The greatest minimal collection and the greatest maximal collection of are denoted by and , respectively. Moreover, and . The valuable -fuzzy set is an -fuzzy set that achieves the condition . refers to the family of valuable -fuzzy sets on , i.e, . Moreover, for every . Let be a mapping, , and , with , is called an -fuzzy function restriction (briefly, -fuzzy function), defined by with . Moreover, for each . Clearly, . Let and , then if and if . The operation is called the pseudocomplement of with respect to [6, 7]. The following proposition lists some of its properties:

Proposition 1 (see [6, 7]). If , , and, then: (1)(2)(3)(4), provided that

Lemma 2 (see [6]). For any , , , , and , we have Equivalently [8],

A subcollection is called an -topology [9, 10, 12] (briefly, -t) if includes the smallest and the greatest -fuzzy sets in . Moreover, the subcollection is closed for every suprema and finite infima. If is an -topology on , then the paire is said to be an -topological space on . Furthermore, elements of are said to be open, and their complements are said to be closed. For any mapping , is said to be -continuous iff for every .

The mapping is called an -fuzzy topology on the set [13–15] if it achieves the following statements: (1)(2), for every (3), for every

If is an -topology on , then the pair is said to be an -fuzzy topological space (briefly, -fts). The degree of openness and the degree of closeness of are represented by and , respectively. The mapping is said to be an -fuzzy continuous if and only if for every .

Definition 3 (see [6]). Let and ; then is called a relative -topology (briefly, -t) if it achieves the next conditions: (1) and , for each (2), for any , (3), for any

If is an -topology on , then, the pair is called an -topological space on (briefly, -ts). The relative open -fuzzy sets (briefly, -open fuzzy set) are the members of while their pseudocomplements are said to be -closed fuzzy sets, i.e., . The family of all -closed fuzzy sets with respect to is denoted by . Let , and , be two -ts. The -fuzzy mapping is called an -continuous if and only if for each . Equivalently, is called an -continuous if and only if for each . If and are -topologies on , then is called an -bitopological space (briefly, -bts). --open (resp. closed) refers to the open (resp. closed) -fuzzy set with respect to , where . In case of , we will get back to -topology and -bitopology.

The concepts of -cover, strong -cover, -cover, -shading, strong -shading, -remote collection, and strong -remote family [16] are extended to -topological spaces in [17] as follows:

Definition 4 (see [17]). For each , -topology on , , and , a set is said to be (1)-cover of if for all and is said to be strong -cover of if (2)-cover of if for all

Definition 5 (see [17]). For each , -topology on , , and , a family is said to be (1)-shading of if for all (2)strong -shading of if (3)-remote family of if for all (4)strong -remote family of if

Theorem 6 (see [6]). For each -ts , the next statements are valid: (1) and for every (2) for every , (3) for every

Definition 7 (see [7]). The mapping such that is called an -fuzzy topology on if achieves the next statements: (1), for every , (2), for every , (3), for every

If is an -fuzzy topology on , then the pair is called an -fuzzy topological space (briefly, -fts). For every , (resp. ) refers to the degree of openness (resp. closeness) of relative to , respectively. Moreover, if (resp. ), then the -openness (resp. -closeness) of an -fuzzy set is confirmed. Clearly, if , then -fuzzy topology on turn into Kubiak-Šostak’s -fuzzy topology. Further, if is an -topology on and is a mapping defined by if , and if , then introduces a special -ft on .

Theorem 8 (see [7]). For every and -ft on . The mapping defined by for every achieves the next statements: (1), for every , (2), for every , (3), for every is called an -fuzzy cotopology (briefly, -fct) on .

Definition 9 (see [7]). Let and be two -fts on and , respectively. The -fuzzy mapping is called an -fuzzy continuous if and only if that is, for every . Further, if and are -fcts with respect to and respectively, then is called an -fuzzy continuous if and only if for every .

Definition 10 (see [17, 18]). Let , and be an -fts on . The mapping given by for every and is said to be the induced -fuzzy closure operator by .

Definition 11 (see [7]). For every and -ft on , an -fuzzy set is called an -fuzzy compact (briefly, -fc) if for every ; the next inequality is true:

Theorem 12 (see [7]). For , the next statements are true: (1), iff (2)-fc is turned into -fc(3) is -fc iff is -fc

Theorem 13 (see [7]). For every and -fts , the next conclusions are true: (1)If and are -fc, then is -fc(2)If such that is an -fc and is an -closed, then is an -fc

3. The --Preopenness Degree of -Fuzzy Set

If and be two -fuzzy topologies on , the the triple is called an -fuzzy bitopological space (briefly, -fbts). Moreover, if any topological property, then we refer to with respect the -ft by -. An -fuzzy set of an -bts is said to be an --preopen if and only if . In the remainder of this paper , such that .

Definition 14. Let and be an -fbts on . The --preopenness gradation of with respect to and is the mapping given by

The value introduces the gradation of --preopenness of is and introduces the gradation of --precloseness of .

The next corollary is a direct consequence of the above definition and Definition 10:

Corollary 15. Let be an -fbts on . Then, for every , we have

Theorem 16. Let be -ts on , and be the degree of --preopenness with respect to and with . Then if and only if is an --preopen.

Proof. The next inequality provides the proof:

Theorem 17. Let be an -fbts on and be the degree of --preopenness with respect to and with . Then, for every , we have .

Proof. The next inequality provides the proof:

Corollary 18. Let be an -fbts on and be the degree of --preopenness with respect to and with . Then, for every , we have .

Theorem 19. If be an -fbts on , and be the degree of --preopenness with respect to and with , then for every .

Proof. Let and . Then for any and for any , there exists such that Hence, By we have Then, .

Corollary 20. Let be an -fbts on , and be the degree of --preopenness with respect to and with . Then, for any .

4. A New Representation of Pairwise Fuzzy Precontinuous (Preirresolute) Functions

Let be -fbts on and , respectively. The -fuzzy mapping is called pairwise -fuzzy continuous (briefly, -fco) if and only if and are -fuzzy continuous. In a similar way, we define the concept pairwise -fuzzy open mapping (briefly, -fo).

Definition 21. Let and be -fbts on and , respectively, and are the corresponding --preopenness degrees. An -fuzzy mapping is said to be (1)pairwise -fuzzy precontinuous (briefly, -fpco) if and only if is true for every (2)pairwise -fuzzy preirresolute (briefly, -fpirr) if and only if is true for every

Corollary 22. Let and be -fbts on and , respectively, and be the corresponding --preopenness degrees. Then, (1) is -fpco if and only if for every (2) is -fpirr if and only if for every

Theorem 23. Let and be -fbts on and , respectively, and be the corresponding --preopenness degrees. Then, (1) is -fpco if and only if is -pco for every (2) is -fpirr if and only if is -pirr for every

Proof. (1)If for every and , then . Since is -fpco, then , i.e., . Subsequently, is --preopen -fuzzy set in . Thus, is -pco mappingNow, if for every and , then . Since is -pco, we have is --preopen in . Then, for every , where . Hence . (2)Let be an --preopen -fuzzy set with respect to , then . Since is -fpirr, then , so , then is --preopen -fuzzy set with respect to . Hence, is -pirrNow, if for every , then is an ---preopen with respect to . Since is -pirr, is ---preopen with respect to . Therefore, for all , where . Hence, .

Theorem 24. Let be -fbts on and , respectively. If is -fco, then, is also -fpco.

Proof. Let be -fco; then for every and . Based on Theorem 17, we have for every . Hence, is -fpco.

Theorem 25. Let , be two -fbts on and , respectively. If is -fpirr, then is -fprco.

Proof. If be -fpirr, then for every . Based on Theorem 17, we have . Hence, is -fpco.

Theorem 26. Let , , and be -fbts on , , and , respectively. If is -fpco and is -fco, then is -fpco.

Proof. Straightforward.

5. A New Representation of Pairwise Fuzzy Precompactness

Definition 27. For any -fbt on , an -fuzzy set is called a pairwise -fuzzy precompact (briefly, -fpcom) with respect to if for every , the next inequality is true: where denotes the family of all finite subfamilies of .

Theorem 28. Let be -fbt on . An -fuzzy set is called a -fpcom with respect to if for every , we have

Proof. Straightforward.

Theorem 29. If is an -fbt on , and , then the following conditions are equivalent: (1) is a -fpcom(2)For every , each strong -remote family of with has a finite subfamily which is a (strong) -remote family of (3)For every , in each strong -remote family of with , there exists a finite subfamily of and with is a (strong) -remote family of (4)For every , each strong -shading of with has a finite subfamily which is a (strong) -shading of (5)For every , in every strong -shading of with , there exists a finite family of and with is a (strong) -shading of (6)For every and , every -cover of with (for every ) has a finite subfamily which is a -cover of (7)For every and every , -cover of with (for every ) has a finite subfamily which is a (strong) -cover of

Proof. Straightforward.

Theorem 30. Let be an -fbt on , and for all ; then, the following conditions are equivalent: (1) is -fpcom(2)For every , every strong -cover of with has a finite subfamily which is a (strong) -cover of (3)For every , in every strong -cover of with , there exists a finite subfamily of and such that with is a (strongly) -cover of

Proof. Straightforward.

Definition 31. Let , , , and be an -bts. An -fuzzy set is said to be an -pairwise -fuzzy precompact (briefly, -fcom) if and only if for each , ---preopen cover of has a finite subfamily which is a ---preopen cover of .

Theorem 32. Let , and be an -bts. An -fuzzy set is -fpcom if and only if is -fpcom for every .

Proof. If is a-fpcom, then for every , , and which can be any ---preopen cover of , we obtain and , so that By , we have Then, there is such that . It follows that is ---preopen cover of .
Now, suppose that every ---preopen cover of has a finite subfamily which is a ---preopen cover of for each . Then, implies that . Hence, implies that . So implies that that is, yields Therefore,

Theorem 33. Let and be an -fbts. An -fuzzy set is a -fpcom in iff is an -fpcom in for each .

Proof. If be a -fpcom in , then for every family , we have Hence, for all and , we have that Hence, for each , there is such that , i.e., for each and , each ---preopen cover of in has a finite subfamily which is a ---preopen cover. Then, for each , is -fpcom in .
Now, suppose that for each , is -fpcom in and let for each , then and , i.e, and . Then, for every , there is suth that Then, So that is a -fpcom in .

Lemma 34. Let and be an -bitopological space, , and . If is -fpcom and is --preclosed, then is -fpcom.

As an immediate consequence from the above lemma, we have the following theorem:

Theorem 35. Let and be an -fbts, and . If is a -fpcom and , then is a -fpcom.

Lemma 36. Let and be an -bts, , and . If are -fpcom, then is -fpcom.

Theorem 37. Let and be an -fbts, and . If are -fpcom, then is -fpcom.

Proof. Straightforward.

Lemma 38. Let be -bts’s on and , respectively, , , and be a -irresolute function. If is -fpcom in , then is fpcom in .

Theorem 39. Let and be two -fbts’s on and , respectively, , and be a -fuzzy irresolute function. If is a -fpcom in , then is a -fpcom in .

Proof. Let be a -fpcom in . By Theorem 33, we have which is -fpcom in for every . By Theorem 33, is -irresolute. Therefore, based on Lemma 38, is -fpcom in . Thus, is -fpcom in .

6. Conclusion

In this paper, we presented the gradation of preopenness of -fuzzy sets in -fuzzy bitopological spaces relied on pseudo-complement. The new gradation is used to extend and characterize pairwise -fuzzy precontinuous and pairwise -fuzzy preirresolute functions. Moreover, we discussed pairwise -fuzzy precompactness of an -fuzzy set in -fuzzy bitopological spaces. We think that our findings present more general results and it will open the way for many other studies.

Data Availability

No data were used to support this study.

Disclosure

O. H. Khalil’s current address is Department of Mathematics, Faculty of Science in Al-Zulfi, Majmaah University, 11952, Saudi Arabia. A. Ghareeb’s current address is Department of Mathematics, College of Science, Al-Baha University, Al-Baha 65799, Saudi Arabia

Conflicts of Interest

The authors declare that they have no conflict of interest.

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Copyright

Copyright © 2022 O. H. Khalil et al. 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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