Neutrosophic Crisp Supra Bi- and Tri-Topological Spaces PDF
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Advances and Applications in Mathematical Sciences Volume 20, Issue 5, March 2021, Pages 917-929 © 2021 Mili Publications NEUTROSOPHIC CRISP SUPRA BI AND TRI-TOPOLOGICAL SPACES V. AMARENDRA BABU and P. RAJASEKHAR Department of Mathematics Acharya Nagarjuna University Nagrjuna Nagar, A.P, India E-mail: [email protected] [email protected] Abstract The aim of this paper is to introduced neutrosophic crisp supra topological spaces (NCSTS) and neutrosophic crisp supra bi and tri-topological spaces, new types of open and closed sets in neutrosophic crisp supra bi and tri-topological spaces, the closure and interior of neutrosophic crisp supra bi and tri-topological space, new concepts of open and closed sets, their properties are investigated. 1. Introduction The idea of degree of membership and the concept of fuzzy set [7] was introduced by Zadeh in 1965. In 1983 generalization of fuzzy set intuitionistic fuzzy set was introduced by K. Atanassov [1] as a beyond the degree of membership and the degree of non membership of each element. Neutrosophic set is a generalization of intuitionistic fuzzy set. The idea of “neutrosophic set” was first proposed by Smarandache [6, 5]. Neutrosophic operations have been developed by Salama et al. [4, 2]. Salama and Alblowi [2] define neutrosophic topological space, established some of its properties. Salama and Smarandache [3, 6, 4] introduced the concept of neutrosophic crisp set and neutrosophic crisp operators have been investigated. In this paper we introduced neutrosophic crisp supra topological spaces and neutrosophic crisp supra bi and tri-topological spaces, new types of open and closed sets in neutrosophic crisp supra bi and tri-topological spaces, the 2010 Mathematics Subject Classification: 03E72, 18B05. Keywords: NCS-bi-OS, NCS-bi-CS, NCS-tri-OS & NCS-tri-CS. Received January 23, 2020; Accepted May 20, 2020 918 V. AMARENDRA BABU and P. RAJASEKHAR closure and interior of neutrosophic crisp supra bi and tri-topological spaces and new concept of open and closed sets, their properties are investigated. 2. Preliminaries Definition 2.1 [2]. Let X be a non-empty fixed set. A neutrosophic crisp set (NCS for short) E is an object having the form E E , E , E , where 1 2 3 E , E and E are sub sets of X. Then the object having the form 1 2 3 E E , E , E is called a NCS if satisfying E E , E E 1 2 3 1 2 1 3 andE E . 2 3 Definition 2.2 [2]. Types of NCSs and X in X may be defined as N N following 1. (a) , , X or (b) , X, or (c) , , . N N N 2. (a) X X, , n Definition 2.3 [2]. If F and E are two NCSs, then E F can be defined as (a) E F E , F , E F and F E . 1 1 2 2 3 3 (b) E F E F , E F and F E . 1 1 2 2 3 3 Definition 2.4 [2]. Let X be a non-empty set, and the NCSs E and F in the form E E , E , E , F F , F , F then 1 2 3 1 2 3 (1) E F may be defined as following (a) E F E F , E F , E F or 1 1 2 2 3 3 (b) E F E F , E F , E F 1 1 2 2 3 3 (2) E F may be defined as following (a) E F E F , E F , E F 1 1 2 2 3 3 or (b) E F E F , E F , E F 1 1 2 2 3 3 Advances and Applications in Mathematical Sciences, Volume 20, Issue 5, March 2021 NEUTROSOPHIC CRISP SUPRA BI and TRI-TOPOLOGICAL … 919 Definition 2.5 [2]. Let X be a non-empty set, and the NCS E E , E , E . Then EC may be defined as following 1 2 3 (a) EC E , E , E . 3 2 1 3. Neutrosophic Crisp Supra Topological Spaces Definition 3.1. A neutrosophic crisp supra topology (NCST for short) on a non empty set X is a family of neutrosophic crisp subset in X if satisfying the following (a) and X N N (b) E E : j J . j j Then the pair X, is called neutrosophic crisp supra topological space (NCSTS) in X. The elements in are said to be neutrosophic crisp supra open sets (NCSOS). The complement of are called neutrosophic crisp supra closed sets (NCSCS). Example 3.2. Let X , , , , , X , L, M, N, 1 2 3 4 N N where L , ,, ,M , ,, , ,N ,, , . 1 2 3 1 2 4 3 2 1 3 Then X, is a NCSTS. 4. Neutrosophic Crisp Supra Bi-Topological Spaces Definition 4.1. Let , be any two neutrosophic crisp supra topology 1 2 (NCST) on a non empty set X then X, , is a neutrosophic crisp supra 1 2 bi-topological space (NCS-bi-TS). Example 4.2. Let X , , , , , X , L, M, N, 1 2 3 4 1 N N ,X ,O,P,Q where L , ,, ,M , ,, , , 2 N N 1 2 3 1 2 4 3 N , , , , O , , , P , , , , 2 1 3 1 3 1 4 3 Q , , , . Then X, , is NCS-bi-TS. 1 2 3 1 2 Advances and Applications in Mathematical Sciences, Volume 20, Issue 5, March 2021 920 V. AMARENDRA BABU and P. RAJASEKHAR Definition 4.3. Let X, , be a NCS-bi-TS. Then is said to 1 2 1 2 be neutrosophic crisp supra bi-open sets (NCS-bi-OSs for short) and the complement of NCS-bi-OSs are called neutrosophic crisp supra bi-closed sets (NCS-bi-CSs for short). Note 4.4. (a) The family of all NCS-bi-OSs can be denoted as NCS-bi- OS(X). (b) The family of all NCS-bi-CSs can denoted as NCS-bi-CS(X). Example 4.5. (a) From example 4.2 NCS-bi-OSs are NCS-bi-OS) X , X , L, M, N, O, P, Q. 1 2 N N (b) From example 4.2 NCS-bi-CSs are NCS-bi-CS X 1 2 , X , LC, MC, NC, OC, PC, QC. where LC , , , , N N 3 1 2 MC , , , , , NC , , , , OC , , , 4 3 1 2 1 3 2 3 1 PC 4, 3, , 1 , QC 2, 3, , 1 . Remark 4.6. (a) Every NCSOS (NCSCS) in X, or X, is a NCS- 1 2 bi-OS (NCS-bi-CS). Remark 4.7. Every NCS-bi-TS X, , induces two NCSTSs as 1 2 X, , X, . 1 2 Theorem 4.8. Let X, , be a NCS-bi-TS. Then the union of two 1 2 NCS-bi-OS (NCS-bi-CS) need not a NCS-bi-OS (NCS-bi-CS). Proof. The proof of this theorem follows from the example 4.9. Example 4.9. Let X , , , , , X , L, M, N, 1 2 3 4 1 N N ,X ,O,P,Q where L , ,, ,M , ,, , , 2 N N 1 2 3 1 2 4 3 N ,, , ,O ,, ,P ,, , ,Q ,, , . 2 1 3 3 1 3 4 1 3 2 1 Hence X, , X, are NCSTS. Therefore X, , is NCS-bi-TS. 1 2 1 2 N, O are NCS-bi-OSs but N O , , , is not a NCS-bi-OS. 2 3 1 Advances and Applications in Mathematical Sciences, Volume 20, Issue 5, March 2021 NEUTROSOPHIC CRISP SUPRA BI and TRI-TOPOLOGICAL … 921 NC , , , and OC , , are two NCS-bi-CSs but 1 3 2 1 3 NC OC , , , is not a NCS-bi-CS. 1 3 Theorem 4.10. Let X, , be a NCS-bi-TS. Then the intersection of 1 2 two NCS-bi-OS (NCS-bi-CS) need not a NCS-bi-OS (NCS-bi-CS). Proof. The proof of this theorem follows from the example 4.11. Example 4.11. Let X , , , , , X , L, M, N, 1 2 3 4 1 N N ,X ,O,P,Q where L , ,, ,M , ,, , , 2 N N 1 2 3 1 2 4 3 N , , , , O , , , P , , , , 2 1 3 3 1 3 4 1 Q , , , . Hence X, , X, are NCSTS. Therefore 3 2 1 1 2 X, , is NCS-bi-TS. 1 2 L, O are two NCS-bi-OSs but L O , , , is not a NCS-bi- 3 1 OS. LC , , , and OC , , are two NCS-bi-CSs 3 1 2 1 3 but LC OC , , , , is not a NCS-bi-CS. 1 2 3 5. The Closure and Interior via NCS-bi-OS and NCS-bi-CS Definition 5.1. Let X, , be a NCS-bi-TS and A is NCS. Then 1 2 neutrosophic crisp supra bi-interior of A can be defined as NCS-bi-Int A : A, is a NCS-bi-OS}. Definition 5.2. Let X, , be a NCS-bi-TS and A is NCS. Then 1 2 neutrosophic crisp supra bi-closure of A can be defined as NCS-bi-Cl A : A ; is a NCS-bi-CS}. Theorem 5.3. Let X, , be NCS-bi-TS, A is a NCS then 1 2 (a) NCS-bi-int A A. (b) NCS-bi-int (A) is not a NCS-bi-OS. Proof. (a) it is clear from definition 5.1. Advances and Applications in Mathematical Sciences, Volume 20, Issue 5, March 2021 922 V. AMARENDRA BABU and P. RAJASEKHAR (b) follows from Theorem 4.8. Theorem 5.4. Let X, , be NCS-bi-TS, A is a NCS then 1 2 (a) A NCS-bi-clA. (b) NCS-bi-cl (A) is not a NCS-bi-CS. Proof. (a) it is clear from definition 5.2. (b) follows from theorem 4.10. 6. Neutrosophic Crisp Supra S-Open Sets (NCS-SOS) and Neutrosophic Crisp Supra S-closed Set (NCS-SCS) The concepts of open and closed sets in NCS-bi-TS were introduced in this section. Definition 6.1. Let X, , be a NCS-bi-TS. Then a subset A of space 1 2 X is said to be neutrosophic crisp supra S-open set (NCS-SOS) if A and 1 A or A and A and the complement of NCS-SOS is said to be 2 2 1 neutrosophic crisp supra S-closed set (NCS-SCS). Example 6.2. From example 4.9 L and O are any two NCS-SOSS. Theorem 6.3. Let X, , be a NCS-bi-TS then 1 2 (a) Every NCS-SOS is NCS-bi-OS. (b) Every NCS-SCS is NCS-bi-CS. Proof. (a) Let A be a NCS-SOS, then A and A or A and 1 2 2 A , therefore A is NCS-bi-OS. 1 (b) Let A be a NCS-SCS, then AC is NCS-SOS therefore AC and 1 AC or AC and AC hence AC is NCS-bi-OS therefore A is 2 2 1 NCS-bi-CS. Remark 6.4. The converse of 6.2 is not true as seen from the following 6.5. Advances and Applications in Mathematical Sciences, Volume 20, Issue 5, March 2021 NEUTROSOPHIC CRISP SUPRA BI and TRI-TOPOLOGICAL … 923 Example 6.5. From any NCS-bi-TS, , X are two NCS-bi-OS but not N N NCS-SOS and also , X are two NCS-bi-CS but not NCS-SCS N N Theorem 6.6. Let X, , be a NCS-bi-TS, then the union of two 1 2 NCS-SOS (NCS-SCS) is not a NCS-SOS (NCS-SCS). Proof. Proof follows from the following example. Example 6.7. Let X , , , , , X , L, M, N, 1 2 3 4 1 N N ,X ,O,P,Q where L , ,, ,M , ,, , , 2 N N 1 2 3 1 2 4 3 N ,, , ,O ,, , P ,, , ,Q , 2 1 3 3 1 3 4 1 3 , , . Hence X, , X, are NCSTS. Therefore X, , is 2 1 1 2 1 2 NCS-bi-TS. N, O are two NCS-SOSs but N O , , , is not a NCS- 2 3 1 SOS. NC , , , and OC , , are two NCS-bi-CSs 1 3 2 1 3 but NC OC , , , is not a NCS-SCS. 1 3 Theorem 6.8. Let X, , be a NCS-bi-TS, then the intersection of 1 2 two NCS-SOS (NCS-SCS) is not a NCS-SOS (NCS-SCS). Proof. Proof follows from the following example. Example 6.9. Let X , , , , , X , L, M, N, 1 2 3 4 1 N N ,X ,O,P,Q where L , ,, ,M , ,, , , 2 N N 1 2 3 1 2 4 3 N , , , , O , , , P , , , , 2 1 3 3 1 3 4 1 Q , , , . Hence X, , X, are NCSTS. Therefore 3 2 1 1 2 X, , is NCS-bi-TS. 1 2 L, O are two NCS-SOS but L O , , , is not a NCS-SOS. 3 1 LC , , , and OC , , are two NCS-SCSs but 3 1 2 1 3 LC OC , , , , is not a NCS-SCS. 1 2 3 Advances and Applications in Mathematical Sciences, Volume 20, Issue 5, March 2021 924 V. AMARENDRA BABU and P. RAJASEKHAR 7. Neutrosophic Crisp Supra Tri-Topological Spaces Definition 7.1. Let , and are three NCSTs on a non-empty set X 1 2 3 then X, , , is called neutrosophic crisp supra tri-topological spaces 1 2 3 (NCS-tri-TS for short). Example 7.2. Let X , , , , , X , L, M, N, 1 2 3 4 1 N N ,X ,O,P,Q, ,X ,R,S,T, where L , ,, , 2 N N 3 N N 1 2 3 M , , , , , N , , , , O , , , 1 2 4 3 2 1 3 1 3 P ,, , ,Q ,, , ,R ,, ,S ,, , , 1 4 3 1 2 3 3 1 3 4 1 T , , , . Hence X, , X, and X, are NCSTS. 3 2 1 1 2 2 Therefore X, , , is NCS-tri-TS. 1 2 3 Definition 7.3. Let X, , , be a (NCS-tri-TS) then 1 2 3 1 2 3 are called neutrosophic crisp supra tri-open sets (NCS-tri-OSs for short) and the complement of NCStri-OSs are called neutrosophic crisp supra bi-closed sets (NCS-tri-CSs for short). Note 7.4. (a) The family of all NCS-tri-OSs can be written as NCS-tri- OS(X). (b) The family of all NCS-tri-CSs can be written as NCS-tri-CS(X). Example 7.5. From Example 7.2 NCS-tri-OSs are NCS-tri-OS X , X , L, M, N, O, P, Q, R, S, T and NCS-tri- 1 2 3 N N CSs are NCS-tri-CS X , X , LC, MC, NC, OC, PC, 1 2 3 N N QC,RC,SC,TC, where LC ,, , ,MC , ,, , , 3 1 2 4 3 1 2 NC , , , , OC , , , PC , , , , 1 3 2 3 1 4 3 1 QC , , , , RC , , , SC , , , , 2 3 1 1 3 4 1 3 TC , , , . 2 1 3 Remark 7.6. (a) Every NCSOS in X, or X, or X, is NCS- 1 2 3 tri-OS. Advances and Applications in Mathematical Sciences, Volume 20, Issue 5, March 2021 NEUTROSOPHIC CRISP SUPRA BI and TRI-TOPOLOGICAL … 925 (b) Every NCSCS in X, or X, or X, is NCS-tri-CS. 1 2 3 Theorem 7.7. Let X, , , be a NCS-tri-TS then the union of two 1 2 3 NCS-tri-OS (NCS-tri-CS) need not a NCS-tri-OS (NCS-tri-CS). Proof. Proof follows 7.8. Example 7.8. Let X , , , , , X , L, M, N, 1 2 3 4 1 N N ,X ,O,P,Q, ,X ,R,S,T, where L , ,, , 2 N N 3 N N 1 2 3 M , , , , , N , , , , O , , , 1 2 4 3 2 1 3 1 3 P , , , , Q , , , , R , , , 1 4 3 1 2 3 3 1 S , , , , T , , , . Hence X, , X, and 3 4 1 3 2 1 1 2 X, are NCSTS. Therefore X, , , is NCS-tri-TS. 2 1 2 3 M, R are NCS-tri-OSs but M R , , , , is not a NCS-tri- 1 2 3 OS. MC , , , , and RC , , are two NCS-tri- 4 3 1 2 1 3 CSs but MC RC , , , , is not a NCS-tri-CS. 1 4 3 Theorem 7.9. Let X, , , be a NCS-tri-TS then the intersection of 1 2 3 two NCS-tri-OS (NCS-tri-CS) need not a NCS-tri-OS (NCS-tri-CS). Proof. Follows from the following example. Example 7.10. Let X , , , , , X , L, M, N, 1 2 3 4 1 N N ,X ,O,P,Q, ,X ,R,S,T, where L , ,, , 2 N N 3 N N 1 2 3 M , , , , , N , , , , O , , , 1 2 4 3 2 1 3 1 3 P , , , , Q , , , ,R , , , 1 4 3 1 2 3 3 1 S , , , , T , , , . Hence X, , X, and 3 4 1 3 2 1 1 2 X, are NCSTS. Therefore X, , , is NCS-tri-TS. 2 1 2 3 M, R are two NCS-tri-OSs but M R , , , , is not a NCS- 1 3 4 tri-OS. MC , , , , and RC , , are two NCS- 4 3 1 2 1 3 tri-CSs but MC RC , , , , is not a NCS-tri-CS. 1 2 3 Advances and Applications in Mathematical Sciences, Volume 20, Issue 5, March 2021 926 V. AMARENDRA BABU and P. RAJASEKHAR 8. The Closure and Interior via NCS-tri-OS and NCS-tri-CS Definition 8.1. Let X, , , be a NCS-tri-TS and A is NCS. Then 1 2 3 neutrosophic crisp supra tri-interior of A can be defined as NCS-tri-Int A : A; is a NCS-tri-OS}. Definition 8.2. Let X, , , be a NCS-tri-TS and A is a NCS. 1 2 3 Then neutrosophic crisp supra tri-closure of A can be defined as NCS-tri-Cl A : A ; is a NCS-tri-CS}. Theorem 8.3. Let X, , be NCS-bi-TS, A is NCS then 1 2 (a) NCS-tri-int A A. (b) NCS-tri-int (A) is not a NCS-tri-OS. Proof. (a) It is clear from Definition 8.1. (b) follows from Theorem 7.7. Theorem 8.4. Let X, , be NCS-tri-TS, A is a NCS then 1 2 (a) A NCS-tri-cl (A). (b) NCS-tri-cl (A) is not a NCS-tri-CS. Proof. (a) It is clear from Definition 8.2. (b) follows from Theorem 7.9. 9. The Neutrosophic Crisp Supra Tri S-Open Sets (NCS-tri-SOS) and Neutrosophic Crisp Supra Tri S-Closed Set (NCS-tri-SCS) In this section we introduced new concept of open and closed sets in NCS- tri-TS. Also we introduced the basic properties of this new concept. Definition 9.1. Let X, , , be a NCS-tri-TS. Then the 1 2 3 neutrosophic crisp supra open set only in one of the three neutrosophic crisp supra topological spaces X, , X, and X, are called the 1 2 3 neutrosophic crisp tri-S-open set (NCS-tri-SOS). Advances and Applications in Mathematical Sciences, Volume 20, Issue 5, March 2021
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