[Retracted] More on <span class="nowrap"><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.3240995pt" id="M1" height="12.223pt" version="1.1" viewBox="-0.0657574 -11.8989 25.1585 12.223" width="25.1585pt"><g transform="matrix(.017,0,0,-0.017,0,0)"><path id="g198-5" d="M810 587C770 574 761 568 725 543C654 630 540 687 408 687C217 687 69 577 69 430C69 319 160 247 276 247C369 247 511 319 511 495C511 510 512 523 509 538H490C490 354 393 277 274 277C174 277 110 344 110 422C110 559 219 655 415 655C524 655 636 602 695 519C628 463 572 392 516 303C433 170 377 97 313 57C252 89 199 109 143 109C79 109 36 87 36 54C36 15 82 -9 165 -9C222 -9 276 0 326 18C373 -1 423 -15 482 -15C645 -15 804 133 804 324C804 393 783 458 747 514C776 546 781 551 818 567L810 587ZM756 336C756 135 623 16 478 16C439 16 402 24 367 36C386 45 401 56 420 68C505 127 569 206 616 302C647 362 679 427 719 480C744 435 756 387 756 336ZM276 39C243 26 207 21 164 21C106 21 77 32 77 54C77 75 125 79 144 79C191 79 227 66 276 39Z"/></g><g transform="matrix(.017,0,0,-0.017,15.187,0)"><path id="g113-223" d="M545 106L524 126C493 85 467 65 455 65C438 65 427 113 405 238C448 295 498 362 543 439L533 448L478 435C453 386 423 331 398 295H395C370 404 347 448 282 448C169 448 23 309 23 153C23 54 65 -12 128 -12C203 -12 283 70 339 155H341C360 29 380 -12 411 -12C444 -12 491 11 545 106ZM333 204C265 95 210 54 169 54C137 54 113 96 113 171C113 302 191 405 252 405C301 405 318 306 333 204Z"/></g></svg>-</span>Closed Sets in Topological Spaces

Gao, Xiao-Yan; Khalil, Ahmed Mostafa

doi:https://doi.org/10.1155/2021/5525739

Journal of Mathematics

On this page

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

Research Article Retraction

!

This article has been Retracted. To view the article details, please click the ‘Retraction’ tab above.

Special Issue

Soft Computing Algorithms Based on Fuzzy Extensions

View this Special Issue

Research Article | Open Access

Volume 2021 | Article ID 5525739 | https://doi.org/10.1155/2021/5525739

[Retracted] More on -Closed Sets in Topological Spaces

Xiao-Yan Gao¹and Ahmed Mostafa Khalil²

Academic Editor: Sami Ullah Khan

Received23 Feb 2021

Revised01 Apr 2021

Accepted08 Apr 2021

Published26 Apr 2021

Abstract

The aim of this paper is to present and study topological properties of -derived, -border, -frontier, and -exterior of a set based on the concept of -open sets. Then, we introduce new separation axioms (i.e., and ) by using the notions of -open set and -closure. The space of (resp., ) is strictly between the spaces of (resp., ) and (resp., ). Further, we present the notions of -kernel and -convergent to a point and discuss the characterizations of interesting properties between -closure and -kernel. Finally, several properties of weakly space are investigated.

1. Introduction and Preliminaries

Many researchers (see [1–9]) were interested in general topology-like family (e.g., the family of all -open sets) and also the notion of generalized closed (briefly, g-closed) subset of a topological space [10–14]. In 1982, Dunham [14] used the generalized closed sets to define a novel closure operator and consequently a novel topology , on the space, and discussed several of the properties of this novel topology. Sayed and Khalil [15] introduced and studied a novel type of sets called -open sets in topological spaces and studied the notions of -continuous, -open, and -closed functions between topological spaces. Further, they investigated several properties of -closed and strongly -closed graphs. In fact, research on spaces analogous to topological spaces and generalized closed sets among topological spaces may have certain driving effect on research on theory of rough set, soft set, spatial reasoning, implicational spaces and knowledge spaces, and logic (see [16–18]). For this reason, we will define the notions of -derived, -border, -frontier, and -exterior of a set based on the notion of -open sets. We will also discuss new separation axioms ( and ) by using the notions of -open set and -closure operator.

The rest of this article is arranged as follows. In this section, we briefly recall several notions: -open set, an -closed set, generalized open set, generalized closed set, space, space, space, space, -derived, -border, -frontier, and -exterior of a set, which are used in the sequel. In Section 2, we define the notions of -derived, -border, -frontier, and -exterior of a set based on -open sets. In Section 3, we present the notions , , -kernel, and -convergent to a point and introduce the characterizations of interesting properties between -closure and -kernel. In Section 4, we define the weakly space and investigate some properties of weakly space.

Throughout the present paper, two subsets of a space , and , denote the closure and the interior of , respectively. Since we require the following known definitions, notations, and some properties, we recall in this section.

Definition 1. Let be a topological space and . Then,(1) is -open [1] if and -closed [1] if (2) is generalized closed (briefly, -closed) [10] if whenever and is open in (3) is generalized open (briefly, -open) [10] if is -closed(4) is -open [15] if and -closed [15] if The -closure of a subset of [2] is the intersection of all -closed sets containing and is denoted by . The -interior of a subset of [2] is the union of all -open sets contained in and is denoted by . The intersection of all -closed sets containing [14] is called the -closure of and is denoted by and the -interior of [19] is the union of all -open sets contained in and is denoted by . The intersection of all -closed sets containing [15] is called the -closure of and is denoted by and the -interior of [15] is the union of all -open sets contained in and is denoted by .
We need the following notations:(i) (resp., ) denotes the family of all -open sets (resp., -closed sets) in (ii) (resp., ) denotes the family of all generalized open sets (resp., generalized closed sets) in (iii) (resp., ) denotes the family of all -open sets (resp., -closed sets) in (iv), , and (v) and

Definition 2. A topological space is said to be(1) space [20] (resp., space [21]) if every -open (resp., -open) set contains the -closure (resp., -closure) of each of its singletons(2) space [20] (resp., space [21]) if, for in with (resp., ), there exist disjoint -open (resp., -open) sets and such that (resp., ) is a subset of and (resp., ) is a subset of

Definition 3 (see [22]). A point is said to be -limit point of in topological space if, for each -open set containing . The set of all -limit points of is called an -derived set of .

Definition 4 (see [22]). Let be a subset of a space :(1)An -border of is defined by (2)An -frontier of is defined by (3)An -exterior of is defined by

2. A -Derived, -Border, -Frontier, and -Exterior of a Set

Definition 5. Let be a subset of a space . A point is said to be -limit point of if it satisfies the following assertion:The set of all -limit points of is called a -derived set of and is denoted by .
Note that, for a subset of , a point is not a -limit point of if and only if there exists a -open set in such thator, equivalently,or equivalently,

Theorem 1. Let and be subsets of a topological space . Then, the following results hold:(1), where is the -derived set ([22], Definition 2.1) of (2)If , then (3) and (4)(5)

Proof. (1)It follows from ([15], Theorem 3.6 (i)).(2)Let and with . Then . Since , it follows that . Therefore .(3)It follows from (2) above.(4)Let and with . Then . Let . Then and , and so . If we take , then for and . Hence, . Therefore .(5)Let . If , the result is obvious. Suppose that . Then for all with . Hence, or . The first case implies that . If , then . Since , it follows similarly from (4) that . Therefore, holds.

Theorem 2. Let be a subset of a topological space . Then .

Proof. Let . If , then the proof is complete. If and with , then , and so . Hence, . The converse follows from ([15], Theorem 2.14 (i)) and . Thus, . Therefore .

Corollary 1. A subset is a -closed set if and only if it contains the set of the -limit points

Theorem 3. Let and be subsets of . If is -closed, then .

Proof. It follows from Theorems 2.13 and 2.14 (vi) in [15].

Lemma 1. Let be a subset of a topological space . If is -closed set, then .

Proof. Suppose that is a -closed set. Let ; that is, . Since it is a -open, is not a -limit point of , that is, , because . Hence, .

Theorem 4. Let be a subset of a topological space . If is a -closed set of , then .

Proof. By Theorem 1 (2) and Lemma 1, implies that .

Theorem 5. Let be a subset of a topological space . If a point is a -limit point of , then is also a -limit point of .

Proof. The proof is obvious.

Definition 6. Let be a subset of a topological space . The -border of , denoted by , is defined as .

Theorem 6. Let be a subset of a topological space . Then, the following results hold:(1), where is the -border ([22], Definition 2.8) of (2)(3)(4) is a -open set if and only if (5)(6)(7)

Proof. (1)Since ([1], Theorem 3.15 (i)), we have(2)and (3) are obvious.(4)Itfollows from Theorems 3.14 and 3.15 (i) in [15].(5)Since is a -open, it follows from (4) that .(6)Using ([15], Lemma 3.13 (i)), we have(7)Applying (6) and Theorem 3, we have

The converse of (1) of Theorem 6 is not true in general as shown in the following example.

Example 1. Let be a topological space, where and . Then, ,,,. Let . Then .

Definition 7. Let be a subset of a topological space . The -frontier of , denoted by , is defined as .

Lemma 2. Let be a subset of . If is a -closed subset of , then .

Proof. It follows from ([15], Theorem 2.13).

Theorem 7. Let be a subset of a topological space . Then the following results hold:(1), where is the -frontier ([22], Definition 2.11) of .(2).(3).(4).(5).(6)If is a -open set, then .(7).(8).(9) is a -closed set.(10).(11).(12).(13).(14).(15).(16).

Proof. (1)Since ([15], Theorem 2.14 (i)) and ([15], Theorem 3.15 (i)), we have .(2)It is obvious.(3).(4)Since ([15], Theorem 2.14 (i)), we have .(5)Using Theorem 2, we have(6)It follows from (5) above, Theorem 6 (4), (7), and ([15], Theorem 3.14).(7)It follows from ([15], Lemma 3.13 (ii)).(8)It follows from (7) above.(9) Obviously, ([15], Theorem 2.14 (i)), and so . Hence, is a -closed set.(10)It follows from (9) above and Lemma 2.(11)It follows from Definition 7 and ([15], Theorem 3.15 (vi)).(12)It follows from Definition 7 and ([15], Theorem 2.14 (vi)).(13)(14)It follows from (7) above and ([15], Lemma 3.13 (ii)).(15)(16)

The converse of (1) and (4) of Theorem 7 is not true as shown in the following examples.

Example 2. Consider the topological space which is given in Example 1. Let . Then .

Example 3. Let be a topological space, where and . Then . Let . Then .

Theorem 8. Let be a subset of a topological space . Then if and only if is a -closed set and a -open set.

Proof. Suppose that . First, we prove that is a -closed set. We have or . Hence, . Therefore, and so is a -closed set. Now, we prove that is a -open set. Indeed, we have or . Hence, and so . Therefore, is a -open set. Conversely, suppose that is a -closed set and a -open set. Then .

Theorem 9. Let be a subset of a topological space . Then,(1) is a -open set if and only if ;(2) is a -closed set if and only if .

Proof. (1)Let be a -open set. Then implies that (by Theorem 7 (3)). Conversely, suppose that . Then or , which implies that . Moreover, . Therefore, and thus is a -open set.(2)Let be a -closed set. Then . Now, . That is, . Conversely, suppose that . Then . Since (by Theorem 7 (8)), we have . By (1), is a -open set. Hence, is a -closed set.

Lemma 3. Let be a subset of a topological space . If is a -closed set, then .

Proof. It follows from ([15], Theorem 2.13) and Theorem 7 (14).

Theorem 10. Let and be subsets of . Then, the following results hold:(1).(2).(3).

Proof. Now considerTherefore,where . From (i) and (ii), we have

Definition 8. Let be a subset of a topological space . The -exterior of , denoted by , is defined as .

Theorem 11. Let and be subsets of . Then the following results hold:(1), where is the -exterior ([22], Definition 2.16) of .(2).(3).(4)If , then .(5).(6).(7) and .(8).(9).

Proof. (1)It follows from ([15], Theorem 3.15 (i)).(2)It follows from ([15], Lemma 3.13 (i)).(3)It follows from ([15], Lemma 3.13 (ii)).(4)It follows from ([15], Theorem 3.15 (iii)).(5)It follows from ([15], Theorem 3.15 (vi)).(6)It follows from ([15], Theorem 3.15 (v)).(7)It is obvious.(8)It follows from ([15], Theorem 3.15 (iv)).(9)It is obvious.

The opposite of (1) and (4) of Theorem 11 is not true as shown in the following examples.

Example 4. Consider the topological space which is given in Example 1. Let . Then .

Example 5. Consider the topological space which is given in Example 3. Let and . Then .

Remark 1. The equality in statements (5) of Theorem 11 need not be true as seen from Example 3. Let , and . Then . Furthermore, the equality in statement (6) of the above theorem need not be true as seen from Example 3. Let , and . Then .

3. and Spaces

Definition 9. Let be a subset of a topological space . The -kernel of , denoted by , is defined as .

Definition 10. Let be a point of a topological space . The -kernel of , denoted by , is defined as .

Lemma 4. Let be a topological space and .Then,(1) if and only if ;(2).

Proof. (1)Suppose that . Then there exists a -open set containing such that . Therefore, we have . The proof of the opposite case can be done similarly.(2)Let and . Hence, which is a -open set containing . This is impossible, since . Consequently, . Let and . Then, there exists a -open set containing and . Let . Hence, is a -neighborhood of where . By this contradiction, and the proof is completed.

Lemma 5. The following statements are equivalent for any points and in a topological space :(1).(2).

Proof. (i) Suppose that . Then there exists a point in such that and . It follows from that . This implies that . By , we have . Since and , . Now, implies that .(ii) Suppose that . Then there exists a point in such that and . Then, there exists a -open set containing and therefore but not , that is, . Hence, .

Definition 11. A topological space is said to be a space if every -open set contains the -closure of each of its singletons.

Theorem 12. Let be a topological space. Then,(1)every space is (2)every space is

Proof. It is obvious from ([15], Theorem 3.6).

From the above discussions, we have the following diagram in which the opposites of implications need not be true.

Theorem 13. A topological space is a space if and only if, for any and in implies that .

Proof. Necessity. Suppose that is a and such that . Then, there exists such that (or such that ). There exists such that and ; hence, . Therefore, we have . Thus, , which implies that and . The proof for otherwise is similar.(i)Sufficiency. Let and . We will show that . Let ; that is, . Then and . This shows that . By assumption, . Hence, and therefore .

Theorem 14. A topological space is a space if and only if, for any and in implies that .

Proof. Suppose that is a space. Then, by Lemma 5, for any points and in if , then . Now, we prove that . Assume that . By and Lemma 4 (1), it follows that . Since , by Theorem 13. Similarly, we have . This is a contradiction. Therefore, we have . Conversely, let be a topological space such that, for any points and in implies that . If , then, by Lemma 5, . Hence, , which implies that . Because implies that , . By hypothesis, we have . Then implies that . This is a contradiction. Hence, . By Theorem 13, we have that is a space.

Theorem 15. For a topological space , the following properties are equivalent:(1) is a space.(2)For any and such that , there exists such that and .(3)Any .(4)Any .(5)For any .

Proof. Let be a nonempty set of and such that . There exists . Since . Set ; then , and . Let . Then . Let be any point of . There exists such that and . Therefore, we have and hence . This is obvious. Let be any point of and . There exists such that and . Hence, . By (4) . There exists such that and . Therefore, and . Consequently, we obtain . Let and . Suppose that . Then and . This implies that . Therefore, is a space.

Corollary 2. For a topological space , the following properties are equivalent:(1) is a space.(2) for all .

Proof. Suppose that is a space. By Theorem 15, for each . Let . Then and so . Therefore, and hence . This shows that . This is obvious by Theorem 15.

Theorem 16. For a topological space , the following properties are equivalent:(1) is a space.(2) if and only if , for any points and in .

Proof. Assume that is a space. Let and let be any -open set such that . Now, by hypothesis, . Therefore, every -open set containing contains . Hence, . Let be a -open set and . If , then and hence . This implies that . Hence, is a space.

Theorem 17. For a topological space , the following properties are equivalent:(1) is a space.(2)If is -closed, then .(3)If is -closed and , then .(4)If , then .

Proof. Let be -closed and . Thus, is -open and contains . Since is ,. Thus and by Lemma 4 (2) . Therefore, . In general, implies that . Therefore, it follows from (2) that . Since and is -closed, by (3), . We show the implication by using Theorem 16. Let . Then, by Lemma 4 (1), . Since and is -closed, by (4), we obtain . Therefore, implies that . The opposite is obvious and is a space.

Definition 12. A filter base is called -convergent to a point in , if, for any -open set of containing , there exists in such that is a subset of .

Lemma 6. Let be a topological space and and are any two points in such that every net in -converging to -converges to . Then .

Proof. Suppose that for each . Then is a net in . Since -converges to , -converges to and this implies that .

Theorem 18. For a topological space , the following statements are equivalent:(1) is a space.(2)If , then if and only if every net in -converging to -converges to .

Proof. Let such that . Suppose that is a net in such that -converges to . Since and by Theorem 15, we have . Therefore, . This means that -converges to . Conversely, let such that every net in -converging to -converges to . Then by Lemma 4 (2). By Theorem 15, we have . Therefore, . Assume that and are any two points of such that . Let . So there exists a net in such that -converges to . Since , we have -converges to . It follows that . Similarly, we obtain . Therefore, and, by Theorem 15, is a space.

Definition 13. A topological space is said to be space if, for in with , there exist disjoint -open sets and such that is a subset of and is a subset of .

Theorem 19. Let be a topological space. Then,(1)every space is (2)every space is

Proof. It is obvious.

From the above discussions, we have the following diagram in which the opposite of implications need not be true.

Theorem 20. If is a space, then is a space.

Proof. Let be -open such that . If , then since . Hence, there exists -open such that and , which implies that . Thus, . Therefore, is a space.

Theorem 21. A topological space is said to be a space if and only if , and there exist disjoint -open sets and such that and .

Proof. It follows from Lemma 4 (1).

4. Weakly Space

Definition 14. A topological space is said to be weakly space if .

Theorem 22. A topological space is weakly space if and only if for every .

Proof. Assume that the space is weakly space. Suppose that there is a point in such that . Then , where is some proper -open subset of . This implies that . But this is a contradiction. Now suppose that for every . If there exists a point such that , then every -open set containing must contain every point of . This implies that the space is the unique -open set containing . Thus, , which is a contradiction. Hence, is a weakly space.

Theorem 23. A topological space is a weakly space if and only if for every .

Definition 15. A function is said to be always -closed if the image of every -closed subset of is -closed in .

Theorem 24. If is an always injective -closed function and is a weakly space, then is a weakly space.

Proof. The proof is clear.

Theorem 25. If the topological space is weakly and is any topological space, then the product is weakly .

Proof. If we show that , then we are done. Observe that and hence the proof is completed.

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 Yulin University Industry Collaboration Project (2019-75-3).

References

O. Njåstad, “On some classes of nearly open sets,” Pacific Journal of Mathematics, vol. 15, pp. 961–970, 1965.
View at: Google Scholar
D. Andrijevic, “Some properties of the topology of-sets,” Matematički Vesnik, vol. 36, pp. 1–10, 1984.
View at: Google Scholar
M. Caldas and J. Dontchev, “On spaces with hereditarily compact-topologies,” Acta Mathematica Hungarica, vol. 82, pp. 121–129, 1999.
View at: Google Scholar
M. Caldas and S. Jafari, “On some low separation axioms in topological spaces,” Houston Journal of Mathematics, vol. 29, pp. 93–104, 2003.
View at: Google Scholar
M. Caldas and G. Navalagi, “On weakly-open functions between topological spaces,” International Journal of Mathematics and Mathematical Sciences, vol. 39, pp. 39–51, 2004.
View at: Google Scholar
S. Jafari and T. Noiri, “Some remarks on weakly-continuity,” Far East Journal of Mathematical Sciences, vol. 6, pp. 619–625, 1998.
View at: Google Scholar
S. N. Maheshwari and S. S. Thakur, “On-irresolute mappings,” Tamkang Journal of Mathematics, vol. 11, pp. 209–214, 1980.
View at: Google Scholar
H. Maki, R. Devi, and K. Balachandran, “Generalized-closed sets in topology,” Fukuoka University of Education, vol. 42, pp. 13–21, 1993.
View at: Google Scholar
T. Noiri, “Weaklyα-continuous functions,” International Journal of Mathematics and Mathematical Sciences, vol. 10, no. 3, pp. 483–490, 1987.
View at: Publisher Site | Google Scholar
N. Levine, “Generalized closed sets in topology,” Rendiconti del Circolo Matematico di Palermo, vol. 19, no. 1, pp. 89–96, 1970.
View at: Publisher Site | Google Scholar
P. Agashe and N. Levine, “Adjacent topologies,” Journal of Mathematics, Tokushima University, vol. 7, pp. 21–35, 1973.
View at: Google Scholar
W. Dunham, “-spaces,” Kyungpook Mathematical Journal, vol. 17, pp. 161–169, 1977.
View at: Google Scholar
W. Dunham and N. Levine, “Further results on generalized closed sets in topology,” Kyungpook Mathematical Journal, vol. 20, pp. 169–175, 1980.
View at: Google Scholar
W. Dunham, “A new closure operator for non-topologies,” Kyungpook Mathematical Journal, vol. 22, pp. 55–60, 1982.
View at: Google Scholar
O. R. Sayed and A. M. Khalil, “Some applications of Dα-closed sets in topological spaces,” Egyptian Journal of Basic and Applied Sciences, vol. 3, no. 1, pp. 26–34, 2016.
View at: Publisher Site | Google Scholar
L. Li and Q. Li, “On enriched L-topologies: base and subbase,” Journal of Intelligent & Fuzzy Systems, vol. 28, no. 6, pp. 2423–2432, 2015.
View at: Publisher Site | Google Scholar
L. Zhou and J. Zhang, “Lukasiewicz semantic MV-topology for MV-algebra and its application to Lukasiewicz propositional logic,” Journal of Intelligent & Fuzzy Systems, vol. 33, no. 1, pp. 377–387, 2017.
View at: Publisher Site | Google Scholar
M. El-Dardery, A. A. Ramadan, and Y. C. Kim, “L-fuzzy topogenous orders and L-fuzzy topologies,” Journal of Intelligent & Fuzzy Systems, vol. 24, no. 4, pp. 685–691, 2013.
View at: Publisher Site | Google Scholar
A. Robert and S. P. Missier, “On semi-closed sets,” The Asian Journal of Mathematics, vol. 4, pp. 173–176, 2012.
View at: Google Scholar
M. Caldas, D. N. Georgiou, and S. Jafari, “Characterizations of low separation axioms via-open sets and -closure operator,” Bulletin of Parana’s Mathematical Society, , vol. 21, no. 1-2, pp. 1–14, 2003.
View at: Publisher Site | Google Scholar
B. Roy, “On generalized R 0 and R 1 spaces,” Acta Mathematica Hungarica, vol. 127, no. 3, pp. 291–300, 2010.
View at: Publisher Site | Google Scholar
M. Caldas, “A note on some applications of α-open sets,” International Journal of Mathematics and Mathematical Sciences, vol. 2003, pp. 125–130, 2003.
View at: Publisher Site | Google Scholar

Copyright

Copyright © 2021 Xiao-Yan Gao 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.

PDF Download Citation

Download other formats

Order printed copies

Views

1354

Downloads

696

Citations