[Retracted] On Fuzzy <span class="nowrap"><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-3.3211pt" id="M1" height="16.6565pt" version="1.1" viewBox="-0.0657574 -13.3354 18.6799 16.6565" width="18.6799pt"><g transform="matrix(.017,0,0,-0.017,0,0)"><path d="M368 293C352 315 336 339 324 364C317 380 309 397 309 415C309 440 323 462 341 479C363 501 391 514 421 522C440 510 462 500 485 500C506 500 523 516 538 530C567 556 590 586 613 617V644C588 618 552 580 528 580C505 580 483 590 464 601C424 625 384 647 341 665C314 676 286 686 256 686C206 686 158 668 123 633C81 591 65 532 59 474L75 463C84 507 96 553 128 585C147 604 171 617 197 617C224 617 250 608 275 598C313 583 349 564 384 543C343 518 305 489 274 452C254 428 235 400 235 367C235 346 245 326 255 307C272 274 293 243 315 213C339 178 363 143 382 106C393 84 405 61 405 37C405 -4 392 -45 362 -75C338 -99 307 -117 272 -117C247 -117 225 -103 208 -85C192 -69 184 -46 181 -23C177 8 180 39 187 69C193 95 196 122 192 149C190 170 184 191 171 207C156 226 130 219 110 214C74 204 42 186 11 165L16 144C31 152 47 159 64 161C75 163 88 163 96 154C106 144 108 130 109 116C111 93 110 70 107 48C103 14 101 -20 107 -54C110 -77 117 -101 134 -117C156 -140 186 -155 219 -155C297 -155 371 -119 426 -64C463 -27 485 22 485 74C485 103 471 128 458 153C435 195 407 234 378 271C400 289 431 314 448 314C473 314 489 292 501 271L544 330C533 361 513 386 490 410L368 293Z"/></g><g transform="matrix(.012,0,0,-0.012,10.485,-7.578)"><path id="g50-43" d="M486 158C486 177 478 202 466 220C413 228 386 236 336 262C386 288 413 297 466 304C478 323 486 347 485 366C470 376 444 381 422 380C389 338 368 319 321 288C323 345 329 372 349 422C339 442 322 461 305 470C289 461 271 442 262 422C281 372 287 345 290 288C243 319 222 338 189 380C167 381 142 376 125 366C125 347 133 322 145 304C198 296 225 288 275 262C225 236 198 227 145 220C133 201 125 177 126 158C141 148 167 143 189 144C222 186 243 205 290 236C288 179 282 152 262 102C272 82 289 63 306 54C322 63 340 82 350 102C330 152 324 179 321 236C368 205 390 186 422 144C444 143 470 148 486 158Z"/></g></svg>-</span>Simply Connected Spaces in Fuzzy <span class="nowrap"><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-3.3211pt" id="M2" height="16.6565pt" version="1.1" viewBox="-0.0657574 -13.3354 18.6799 16.6565" width="18.6799pt"><g transform="matrix(.017,0,0,-0.017,0,0)"><path d="M368 293C352 315 336 339 324 364C317 380 309 397 309 415C309 440 323 462 341 479C363 501 391 514 421 522C440 510 462 500 485 500C506 500 523 516 538 530C567 556 590 586 613 617V644C588 618 552 580 528 580C505 580 483 590 464 601C424 625 384 647 341 665C314 676 286 686 256 686C206 686 158 668 123 633C81 591 65 532 59 474L75 463C84 507 96 553 128 585C147 604 171 617 197 617C224 617 250 608 275 598C313 583 349 564 384 543C343 518 305 489 274 452C254 428 235 400 235 367C235 346 245 326 255 307C272 274 293 243 315 213C339 178 363 143 382 106C393 84 405 61 405 37C405 -4 392 -45 362 -75C338 -99 307 -117 272 -117C247 -117 225 -103 208 -85C192 -69 184 -46 181 -23C177 8 180 39 187 69C193 95 196 122 192 149C190 170 184 191 171 207C156 226 130 219 110 214C74 204 42 186 11 165L16 144C31 152 47 159 64 161C75 163 88 163 96 154C106 144 108 130 109 116C111 93 110 70 107 48C103 14 101 -20 107 -54C110 -77 117 -101 134 -117C156 -140 186 -155 219 -155C297 -155 371 -119 426 -64C463 -27 485 22 485 74C485 103 471 128 458 153C435 195 407 234 378 271C400 289 431 314 448 314C473 314 489 292 501 271L544 330C533 361 513 386 490 410L368 293Z"/></g><g transform="matrix(.012,0,0,-0.012,10.485,-7.578)"><use xlink:href="#g50-43"/></g></svg>-</span>Homotopy

Madhuri, V.; Bazighifan, Omar; Ali, Ali Hasan; El-Mesady, A.

doi:https://doi.org/10.1155/2022/9926963

Journal of Function Spaces

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Abstract Introduction Preliminaries Conclusion Data Availability Conflicts of Interest References Copyright Related Articles

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Volume 2022 | Article ID 9926963 | https://doi.org/10.1155/2022/9926963

[Retracted] On Fuzzy -Simply Connected Spaces in Fuzzy -Homotopy

V. Madhuri,¹Omar Bazighifan,^2,3Ali Hasan Ali,^4,5and A. El-Mesady⁶

Academic Editor: Muhammad Gulzar

Received18 Feb 2022

Revised24 Mar 2022

Accepted05 Apr 2022

Published28 Apr 2022

Abstract

In this paper, the notions of fuzzy -simply connected spaces and fuzzy -structure homeomorphisms are introduced, and further fuzzy -structure homeomorphism between fuzzy -path-connected spaces are studied. Also, it is shown that every fuzzy -structure subspace of fuzzy -simply connected space is fuzzy -simply connected subspace. Further, the concepts of fuzzy -contractible spaces and fuzzy -retracts are introduced, and it is proved that every fuzzy -contractible space is fuzzy -simply connected.

1. Introduction

Homotopy theory is the main part of algebraic topology which studies topological objects up to homotopy equivalence. Homotopy equivalence is a weaker relation than topological equivalence. The homotopy theory is one among the foremost branches of algebraic topology. The thoughts and techniques of homotopy theory have pervaded many components of topology. Several topologists like Massey [1], Munkres [2], and Hatcher [3], introduced and studied concepts of homotopy theory and fundamental groups. The most appropriate theory for dealing with uncertainties is the theory of fuzzy sets developed by Zadeh [4]. The fuzzy homotopy theory was introduced by Culvacioglu and Citil [5]. Salleh and Tap [6, 7] introduced the concept of the fundamental group to fuzzy topological spaces based on the definition of fuzzy topology introduced by Chang [8]. Based on Chang’s [8] description of fuzzy topology, Salleh and Tap [6, 7, 9] introduced the concept of basic fundamental group of fuzzy topological spaces. Several recent papers contain extensive investigations and applications about fuzzy topological space and fuzzy homotopy theory; see [10–12].

Later on, Guner [13] intensively investigated the concept of fuzzy contractible spaces. Rodabaugh [14, 15] introduced the concept of L-fuzzy retract in 1981. Further fuzzy homotopy theory was intensively developed by Aras [16], Citil and Cuvalcioglu [17], Palmeira and Bedregal [18], etc. and gave many interesting results on fuzzy homotopy theory. In this connection, Madhuri and Amudhambigai [19, 20] introduced the concepts of fuzzy -homotopy and -fundamental group. Also the concept of fuzzy -structure isomorphisms between -fundamental groups are studied in [19, 20]. In this recent work, the concept of fuzzy -structure homeomorphism between fuzzy -path-connected spaces are studied. The notion of fuzzy -simply connected space is introduced, and it is shown that every fuzzy -structure subspace of fuzzy -simply connected space is fuzzy -simply connected. Further, the concepts of fuzzy -contractible spaces and fuzzy -retracts are introduced and it is proved that every fuzzy -contractible space is fuzzy -simply connected. Also in fuzzy -contractible space, each fuzzy -loop based at any fuzzy point is equivalent to the constant fuzzy -loop.

2. Preliminaries

In this section, some basic concepts of fuzzy homotopy have been recalled from the previous literature. Also related results and propositions are studied from various research articles. Some definitions and preliminary results are presented in this section.

Definition 1. Let be a fuzzy topological space. A fuzzy set is called fuzzy irreducible if , and for all fuzzy closed sets with , it follows that either or .

Remark 2. Let be a fuzzy topological space. Any is said to be fuzzy irreducible closed if it is both fuzzy irreducible and fuzzy closed.

Definition 3. Suppose that be a fuzzy topological space and assume that be a fuzzy open set in . Then the collection and is a fuzzy irreducible closed set in }. Then the collection which is finer than the fuzzy topology on is said to be a -structure on . A nonempty set with a -structure denoted by is said to be fuzzy -structure space. Each member of is said to be fuzzy -structure open set, and the complement of each fuzzy -structure open set is said to be fuzzy -structure closed. A -structure on a nonempty set together with is said to be fuzzy -structure. Then is called a fuzzy -structure space generated by .

Definition 4. Suppose that be any topological space and assume that be fuzzy -structure space. Assume also that and denotes the so-called fuzzy characteristic function of the subset . Then the fuzzy -structure given by is , and the pair is called a fuzzy -structure space given by .

Note 5. Let be the unit interval. Let be an Euclidean topology on , and then is a fuzzy -structure space introduced by the (usual) topological space .

Definition 6. Let and be any two fuzzy -fundamental groups of at and , respectively. A function is called a fuzzy -structure homomorphism if for every . Moreover the fuzzy -structure homomorphism is called a fuzzy -structure isomorphism if it is bijective.

Proposition 7. Let be any fuzzy -structure space and let . Let be any fuzzy -loops in . If and , then .

Proposition 8. Let be any fuzzy -structure space. Let , where is a fuzzy point in . Then .

Proposition 9. Let be any fuzzy -structure space and let be any fuzzy -structure space introduced by . Also, let be the fuzzy -path defined by for each in , and is fuzzy point in . Then , for each .

Proposition 10. Let be any fuzzy -structure space and be fuzzy point in . Let . Then there exists a such that .

Remark 11. From Proposition 8, Proposition 9, and Proposition 10, it is seen that forms a group under an operation (namely, multiplication). It is called -fundamental group of based at .

Definition 12. (see [21]). Let be fuzzy continuous maps and . If is a constant, then is called a fuzzy nulhomotopic map.

Definition 13. Let . is an L-fuzzy retract of in (abbreviated F-retract) if there is an F-continuous such that for each .

Definition 14. Let be an identity mapping. If is fuzzy homotopic to a constant, then is called a fuzzy contractible space.

Proposition 15. Let , , and be any three fuzzy -structure spaces. If and are fuzzy -structure continuous functions, then is a fuzzy -structure continuous function.

Remark 16. Let be any fuzzy -structure space and let . Let be any fuzzy -path joining with . Also, let be any two fuzzy -loops defined such that Then (i) on replacing by in the proof of Proposition 9(ii) on replacing by in the proof of Proposition 9(iii) where on replacing by in the proof of Proposition 10(iv) on replacing by in the proof of Proposition 10(v)If , then Therefore, serves as the left identity and serves as the right identity for any

Proposition 17. Let and be any two fuzzy -structure spaces. Let be any two fuzzy -structure continuous functions, where is fuzzy -structure homotopy between and . Let be any fuzzy -structure space introduced by . Also if is a fuzzy -path joining with defined by , where and , then the following Figure 1 of induced fuzzy -structure homomorphisms is commutative.

3. Properties of Fuzzy -Simply Connected Spaces

Throughout this paper, the collection of all fuzzy points , where over , is denoted by , and the set of all fuzzy points over is denoted by . In this section, the concepts of fuzzy -simply connected spaces, fuzzy -null-homotopic functions, fuzzy -contractible spaces, and fuzzy -retractions are introduced, and some interesting properties are studied.

Definition 18. Let and be any two fuzzy -structure spaces. Let be any two fuzzy -structure continuous functions and . If is a constant function, then is said to be a fuzzy -null-homotopic function.

Definition 19. Any fuzzy -structure space is said to be a fuzzy -simply connected space if it is fuzzy -path-connected and every fuzzy -loop in is a fuzzy -null-homotopic function.
A fuzzy -structure space which is not fuzzy -simply connected is said to be fuzzy -multiply connected.

Definition 20. Let and be any two fuzzy -structure spaces. If the bijective function and its inverse function are fuzzy -structure continuous functions, then the function is said to be a fuzzy -structure homeomorphism.

Proposition 21. Let and be any two fuzzy -structure spaces and let be a fuzzy -structure homeomorphism. If is fuzzy -path-connected, then is also a fuzzy -path-connected space.

Proof. Let be fuzzy -path-connected and be a fuzzy -structure homeomorphism. Let be a fuzzy -structure space introduced by . For , let be any two fuzzy points. Since is fuzzy -path connected, there exists a fuzzy -path such that and . Thus, is such that Also, , as is onto.
Therefore, is a fuzzy -path in joining from to . Hence, is a fuzzy -path-connected space.

Proposition 22. Let and be any two fuzzy -structure spaces and be a fuzzy -structure space introduced by . Let be any two fuzzy -paths joining from to such that , where . If is a fuzzy -structure continuous function, then are fuzzy -structure continuous functions and .

Proof. Let be any fuzzy -structure space introduced by . Since are fuzzy -structure continuous functions and by Proposition 15, , are also fuzzy -structure continuous functions. Furthermore, implies that there exists a fuzzy -structure continuous function such that where . Now, is such that , for each fuzzy point in and in . Since and are fuzzy -structure continuous functions, by Proposition 15, is also a fuzzy -structure continuous function. Moreover, satisfies the following conditions. for each fuzzy point in and in . Hence, .

Proposition 23. Let and be any two fuzzy -structure spaces. Let be any fuzzy -structure homeomorphism. Then is fuzzy -simply connected if and only if is fuzzy -simply connected.

Proof. Let be any fuzzy -structure space introduced by . Since is fuzzy -simply connected, by Definition 19, is fuzzy -path-connected. Let be a fuzzy -structure homeomorphism. Then by Proposition 21, is fuzzy -path-connected. Thus, it is enough to prove that every fuzzy -loop in is fuzzy -null-homotopic. Let be a fuzzy -loop in . Since is fuzzy -simply connected, is a fuzzy -loop in which is fuzzy -path-homotopic to some constant fuzzy -loop , i.e., .
Since is a fuzzy -structure continuous function and by Proposition 22which implies that , by using associative property, and so , as is an identity function.
Thus, .
Also, is fuzzy -path-homotopic to some constant fuzzy -loop . Thus, every fuzzy -loop in is fuzzy -null-homotopic. Thus, is fuzzy -simply connected.
The converse part is also proved in the reverse direction of and .

Proposition 24. Let be a fuzzy -structure space. Then, any fuzzy -path-connected space is fuzzy -simply connected if and only if any two fuzzy -paths in having same endpoints are fuzzy -path-homotopic.

Proof. Let be any fuzzy -structure space introduced by . Let be any two fuzzy -paths in such that where . Thus, is a fuzzy -loop based at . Since is fuzzy -simply connected, every fuzzy -loop based at in is fuzzy -null-homotopic, that is which implies that , then .
Thus, , as in (i) of Remark 16, , as in (iv) of Remark 16, ,as in (ii) of Remark 3.
Hence, .
Therefore, any two fuzzy -paths in having same endpoints are fuzzy -path-homotopic.
Conversely, let be any two fuzzy -paths in such that . Thus, , , and , as in (ii) of Remark 16.
Thus,, as in (iv) of Remark 16, and then, as in (i) of Remark 16, which implies that
Thus,
Therefore, for every fuzzy -loop based at in is fuzzy -null-homotopic. Thus, is fuzzy -simply connected.

Definition 25. Let be a fuzzy -structure space and . Let be a fuzzy -structure subspace and be any fuzzy -structure continuous function. Then is said to be fuzzy -retract of if there exists a fuzzy -structure continuous function such that , for all . Also, the function is called as fuzzy -retraction.

Proposition 26. Let be a fuzzy -structure space and . If is fuzzy -simply connected and is the fuzzy -retract of , then is also a fuzzy -simply connected space.

Proof. Let be fuzzy -simply connected and be the fuzzy -retract of . Then is a fuzzy -retraction function. If are any two fuzzy points, then there is a fuzzy -path such that and . Then is a fuzzy -path in such that where . Thus, is a fuzzy -path in . Therefore, is fuzzy -path-connected.
Let be the inclusion function. Thus, , that is, where is an identity function on . Also, let is a fuzzy -loop in at . Thus, is a fuzzy -loop in at . Since is fuzzy -simply connected, is fuzzy -path homotopic in to the constant fuzzy -loop such that , for all and and hence . Also, by Proposition 22implies that , and so
Thus, and is a constant fuzzy -loop (i.e.,) , for all . Therefore is fuzzy -null-homotopic in . Hence, is a fuzzy -simply connected space.

Definition 27. Let and be any two fuzzy -structure spaces and . Any fuzzy -structure continuous function is said to be a fuzzy -homotopy equivalence if there exists a fuzzy -structure continuous function such that is fuzzy -homotopic to where is an identity function on and is fuzzy -homotopic to where is an identity function on .
Also, any two fuzzy -structure spaces and are said to be of the same fuzzy -homotopy type if there exists a fuzzy -structure continuous function which is fuzzy -homotopy equivalence.

Definition 28. Let be a fuzzy -structure space and be an identity function. If is a fuzzy -null-homotopic function, then is said to be a fuzzy -contractible space.

Proposition 29. Let be a fuzzy -structure space. If is fuzzy -contractible, then each fuzzy -loop based at any fuzzy point is equivalent to the constant fuzzy -loop at .

Proof. Let be a fuzzy -contractible space. Then is an identity function and it is fuzzy -homotopic to the constant fuzzy -loop by a fuzzy -homotopy . Thus there is a fuzzy -loop at which is defined as , where . Let and be any two induced fuzzy -structure homomorphisms. Also, let be defined as , where . Then by Proposition 17, the following Figure 2 shows that .
For each , and, as in (i) of Remark16; and, as in (iv) of Remark16; and, as in (ii) of Remark16,
Therefore, each fuzzy -loop is equivalent to the constant fuzzy -loop .

Proposition 30. Every fuzzy -contractible space is fuzzy -simply connected.

Proof. Let be a fuzzy -contractible space and let be the constant fuzzy -loop at . Since is fuzzy -contractible, is a fuzzy -null-homotopic function by a fuzzy -homotopy , so . Thus, the fuzzy -homotopy is defined as: where and . Let be a fuzzy -path such that . Thus, and . Thus, is a fuzzy -path from to . Similarly, is also fuzzy -path from to where . Therefore, is also a fuzzy -path from to . Thus, is fuzzy -path connected.
Let and also let us define a fuzzy -homotopy . Thus, and , where is a constant fuzzy -loop at . Thus, . Thus, . Hence, every fuzzy -loop in is fuzzy -null-homotopic. Therefore, is fuzzy -simply connected. Hence, every fuzzy -contractible space is fuzzy -simply connected.

4. Conclusion

In this paper, the notions of fuzzy -simply connected spaces and fuzzy -contractible spaces are introduced, and some important characterizations related to fuzzy -homotopy are discussed. Also it is proved that in fuzzy -contractible space, each fuzzy -loop based at any fuzzy point is equivalent to the constant fuzzy -loop. This is just a beginning of studying fuzzy -simply connected spaces. There is a huge scope of further study in extending the results of fuzzy -homotopy to fuzzy -covering spaces. Further using fuzzy compact and fuzzy Lindelof spaces, types of fuzzy -homotopy can be developed.

Data Availability

No data were used to support this study.

Conflicts of Interest

All authors have declared they do not have any competing interests.

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Copyright © 2022 V. Madhuri 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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