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Maejo Int. J. Sci. Technol. 2016, 10(02), 187-196 Maejo International Journal of Science and Technology ISSN 1905-7873 Available online at www.mijst.mju.ac.th Full Paper On -open set and topological spaces -open set in intuitionistic Basker Palaniswamy 1, * and Kokilavani Varadharajan 2 1 Department of Mathematics, Jansons Institute of Technology, Coimbatore-641 659, Tamil Nadu, India 2 Department of Mathematics, Kongunadu Arts and Science College, Coimbatore - 641 029, Tamil Nadu, India * Correspondingauthor, e-mail: [email protected]; [email protected] Received: 11 November 2014 / Accepted: 26 March 2016 / Published: 15 July 2016 Abstract: We introduce some new types of sets called intuitionistic semi-open set, open set, intuitionistic pre-open set and Also, we discuss some of their properties. Keywords: intuitionistic semi-open set, open set - -open set in intuitionistic topological spaces. -open set, intuitionistic pre-open set, - _______________________________________________________________________________________ INTRODUCTION After Atanassov [1] introduced the concept of ‘intuitionistic fuzzy sets’ as a generalisation of fuzzy sets, it became a popular topic of investigation in the fuzzy set community. Many mathematical advantages of intuitionistic fuzzy sets have been discussed. Coker [2] generalised topological structures in fuzzy topological spaces to intuitionistic fuzzy topological spaces using intuitionistic fuzzy sets. Later many researchers have studied topics related to intuitionistic fuzzy topological spaces. On the other hand, Coker [3] introduced the concept of ‘intuitionistic sets’ in 1996. This is a discrete form of intuitionistic fuzzy set, where all the sets are entirely crisp sets. Still, it has membership and non-membership degrees, so this concept gives us more flexible approaches to representing vagueness in mathematical objects including those in engineering fields with classical set logic. In 2000 Coker [4] also introduced the concept of intuitionistic topological spaces with intuitionistic set and investigated the basic properties of continuous functions and compactness. He and his colleague [5, 6] also examined separation axioms in intuitionistic topological spaces. 188 Maejo Int. J. Sci. Technol. 2016, 10(02), 187-196 PRELIMINARIES Definition 1 [3]. Let be a non-empty fixed set. An intuitionistic set ( ) is an object having the form , where and are subsets of satisfying . The set is called the set of members of , while is called the set of non-members of . Every crisp set on a non-empty set is obviously an having the form , and one can define several relations and operations between as follows: Definition 2 [3]. Let let be a non-empty set, be an arbitrary family of if and only if and if and only if ; ; ; and in , where ; ; be . Then if and only if ; ; ; and on , and and ; . The following are the basic properties of inclusion and complementation: Corollary 1 [3]. Let for each , and be in . Then for each ; ; ; ; . ; Now we generalise the concept of ‘intuitionistic fuzzy topological space’ to intuitionistic sets: Definition 3 [2]. An intuitionistic topology ( satisfying the following axioms: ) on a non-empty set is a family of in , for any , for any arbitrary family . In this case the pair is called an intuitionistic topological space ( known as an intuitionistic open set ( ) in . Example 1 [4]. Any topological space is obviously an whenever we identify a subset in with its counterpart Example 2 [4]. Let where ) and any in the form as before. and consider the family . Then is an Example 3 [4]. Let , on . , . Then is , and and consider the family where in , , is an on . Definition 4 [4]. The complement of an intuitionistic open set called an intuitionistic closed set ( ) in . Now we define closure and interior operations in and : in an is 189 Maejo Int. J. Sci. Technol. 2016, 10(02), 187-196 Definition 5 [7]. Let closure of are defined by be an and be an in . Then the interior and , . In this paper we use shown that is an and is an instead of and is an in if and only if Example 4 [4]. Let where and in Proposition 2 [4]. Let (a) ; (c) (e) (g) (i) ; , then we can write . in we have be an ; in an (a) intuitionistic semi-open ( (b) intuitionistic -open ( in . Then the following properties hold: ; ; ; is said to be -open) if ; -open) if (c) intuitionistic pre-open ( ; -open) if -open ( , . -open, -open) sets of an is denoted by ). An IS in an ( , Example 5. Let and is an . -OPEN SET Definition 6. An IS The family of all and and , be (b) ; (d) (f) ; (h) (j) . ; -OPEN SET AND if , and Proposition 1 [4]. For any IS ( in is said to be -closed ( -closed, -closed) ). and consider the IT . Let , where . Here -open set since Example 6. Let and . and consider the IT Let , where is an - . Here open set since . Example 7. Let and since . It can also be if and only if and consider the family . If and ON instead of , and is an . and consider the IT Let . . Here is an where -open set 190 Maejo Int. J. Sci. Technol. 2016, 10(02), 187-196 Theorem 1. (a) Every -open set is be an IOS in Proof. (a) Let . Since Hence (b) -open set; (b) every and Let be -open set. , then is an -open -open set is . -open set. set in . . Therefore, Then is an it follows that -open set. Remark 1. The separate converses need not be true in general, which is shown by the following examples. and consider the IT Let Example 8. Let and open set but not an IOS since and consider the IT Let and -open set since and . Here is not contained in Theorem 2. The finite union of Proof. Let is an , where -open set . Example 9. Let but not an . Here , where is an - . -open set is always an be two and -open set. -open sets. Then imply that is an Proposition . Therefore, -open set. Let 3. be ITS an and let . Then . Proof. If , then . Conversely, let and so . Take be such that . Then and . Then . Theorem 3. Let be an ITS. A sub-set open set and -open set. Proof. Necessity: Let A be that of be -open set if and only if it is -open set. Then we have and Sufficiency: Let is an . Hence -open set and . This implies is Let is said to be an be an ITS and . -open set and -open set. -open set. Then we have . This shows that Definition 7. A sub-set of open set, which is equivalent. - be a sub-set is an -open set. -closed set if and only if . Then is an is an - -closed set if and only if 191 Maejo Int. J. Sci. Technol. 2016, 10(02), 187-196 Definition 8. Let (a) -interior of . be an ITS and be a sub-set . Then is the union of all -open sets contained in and it is denoted by . (b) -closure of . is the intersection of all -closed sets containing and it is denoted by . (c) -interior of . is the union of all -preopen sets contained in and it is denoted by . (d) -closure of . is the intersection of all -preclosed sets containing and it is denoted by . (e) -interior of . is the union of all -semiopen sets contained in and it is denoted by . (f) by -closure of . is the intersection of all -semiclosed sets containing and it is denoted . Remark 2. It is clear that is an Theorem 4. An IS (a) (b) -open set and in an . . is an -closed set. . Then Proof. (a) and (b) are clear. Observation 1. The following statements are true for every (a) . (b) . and : Proof. Obvious. Observation 2. Let be a sub-set of a space (a) . Then . (b) . (c) . (d) . (e) . (f) . Proof. Obvious. Example 10. Let and and consider the IT Let , where . Then 192 Maejo Int. J. Sci. Technol. 2016, 10(02), 187-196 (a) . (b) . (c) . (d) . (e) . (f) . Observation 3. Let A be a sub-set of a space . Then (a) . (b) . (c) . Example 11. Let and and consider the IT Let (a) , where . Then . , , , , (1) (2) From (1) and (2), . (b) , , (3) (4) From (3) and (4), . 193 Maejo Int. J. Sci. Technol. 2016, 10(02), 187-196 (c) , (5) , (6) , (7) , (8) , (9) From (5), (6), (7), (8) and (9), . Definition 9. An IS in an is said to be (a) intuitionistic semi-preopen ( -open or -open) if there exists such that . The family of all -open sets of an will be denoted by . (b) intuitionistic semi-preclosed ( -closed or -closed) if there exists an intuitionistic preclosed set such that . The family of all -closed sets of an will be denoted by . Let . Then Example 12. Let and , , is an IT on and in . Then be an . , and hence . Theorem 5. For every Proof. Straightforward. Theorem 6. Every Proof. Let be in -open set is an , we have . -open set. -open set in . Then it follows that . Hence is an -open set. Remark 3. The converse of the above theorem need not be true in general. It is shown by the following example. Example 13. Let but not an and -open set since Theorem 7. For every Proof. Straightforward. and consider the IT Let is not contained in in , we have Theorem 8. Let be an . Then (a) Any union of -open sets is an -open set. (b) Any intersection of -closed sets is an -closed set. . Here is an , where -open set . . 194 Maejo Int. J. Sci. Technol. 2016, 10(02), 187-196 Proof. (a) Let be a collection of -open sets of such that and (b) This is from (a) by taking compliments. for . Hence Theorem 9. For any IS . Then there exists each . , , then we can take so that and assume that there exists , and so Theorem 10. Let (a) be an follows that . in an Proof. If Let be an in Then set by Theorem 8(a). It if and only if for every such that , which is an . . -open . Then . (b) . Proof. (a) Assume that be such that that so that (b) This follows from (a). Definition 10. Let (a) -interior of . for every . Obviously, and . From . Hence be an ITS and be a sub-set . Then is the union of all -open sets contained in . Let , it follows . and it is denoted by . (b) -closure of . is the intersection of all -closed sets containing and it is denoted by . Observation 4. Let be a sub-set of a space (a) . Then . (b) . Example 14. Let and and consider the IT Let (a) (b) Observation 5. Let (a) , where . Then . . be a sub-set of a space . Then . (b) . 195 Maejo Int. J. Sci. Technol. 2016, 10(02), 187-196 Example 15. Let and (a) and consider the IT Let . , where . Then . (10) , (11) From (10) and (11), . (b) . , (12) , (13) (14) , (15) (16) From (12), (13), (14), (15) and (16), . CONCLUSIONS We have introduced the sets in intuitionistic topological spaces called intuitionistic semiopen set, intuitionistic -open set, intuitionistic pre-open set, -interior of , -closure of , -interior of , -closure of , -interior of , -closure of , intuitionistic semipreopen set, intuitionistic semi-preclosed set, -interior of and -closure of , and studied some of their properties. ACKNOWLEDGEMENTS We would like to express our sincere gratitude to the referees and editor for their valuable suggestions and comments which improved the paper. REFERENCES 1. K. T. Atanassov, “Intuitionistic fuzzy sets”, Fuzzy Sets Syst., 1986, 20, 87-96. 2. D. Coker, “An introduction to intuitionistic fuzzy topological spaces”, Fuzzy Sets Syst., 1997, 88, 81-89. 3. D. Coker, “A note on intuitionistic sets and intuitionistic points”, Tr. J. Math., 1996, 20, 343351. 4. D. Coker, “An introduction to intuitionistic topological spaces”, Bull. Stud. Exchanges Fuzz. Appl. 2000, 81, 51-56. 196 Maejo Int. J. Sci. Technol. 2016, 10(02), 187-196 5. S. Bayhan and D. Coker, “On separation axioms in intuitionistic topological spaces”, Int. J. Math. Math. Sci., 2001, 27, 621-630. 6. S. Bayhan and D. Coker, “Pairwise separation axioms in intuitionistic topological spaces”, Hacet. J. Math. Stat., 2005, 34S, 101-114. 7. S. J. Lee and J. M. Chu, “Categorical property of intuitionistic topological spaces”, Commun. Korean Math. Soc., 2009, 24, 595-603. © 2016 by Maejo University, San Sai, Chiang Mai, 50290 Thailand. Reproduction is permitted for noncommercial purposes.