Ratio Mathematica Volume 46, 2023 IFG#α-CS in intuitionistic fuzzy topological spaces Christy Jenifer J* Kokilavani V † Abstract The primary aim of this prospectus is to introduce and study the basic properties of Intuitionistic fuzzy generalized #α-closed sets, Intu- itionistic fuzzy generalized #α-open sets. Here we, compare the g #α- closed sets with the existing closed sets with proper examples given. Keywords: IFS, IFT, IFG#αCS, IFG #αOS. 2020 AMS subject classifications: 54A40. 1 *Kongunadu Arts and Science College, Coimbatore, Tamil Nadu, India; christi- jeni94@gmail.com. †Kongunadu Arts and Science College, Coimbatore, Tamil Nadu,India; vanikasc@yahoo.co.in. 1Received on September 15, 2022. Accepted on December 15, 2022. Published on March 20, 2023. DOI: 10.23755/rm.v46i0.1058. ISSN: 1592-7415. eISSN: 2282-8214. ©Christy Jenifer J et al. This paper is published under the CC-BY licence agreement. 63 Christy Jenifer J and Kokilavani V 1 Introduction Earlier g alpha closed sets has been introduced in general topology, fuzzy topology, supra topology, nano topology, we wish to introduce g #α in intuition- istic fuzzy topological spaces. Following authors motivated me to further continue my research in intuitionistic fuzzy topology. A. Zadeh A.Zadeh [1965] initiated the concept of fuzzy sets in which the values are taken between 0 and 1. Fur- ther Atanassov [1986] established the idea of IFS by generalizing fuzzy sets, here similar to fuzzy topology values are taken between 0 and 1 but it is defined as membership value and non-membership value. Later on IFTS was initiated using the notion of IFSs which was proposed by Coker Coker [1997], using the mem- bership and non-membership values it was applied in general topological axioms. In continuation of above we initiate IFG #α -closed sets and IFG alpha-open sets and establish its characterization and find the weaker and stronger forms of topol- ogy by comparing it to other existing sets and also find whether it satisfies the topological axioms as union and finite intersection properties 2 Preliminaries In this segment, few basic definitions and results are reviewed. Definition 2.1. Thakur and Chaturvedi [2008]Let A be an IFS in (X,τ), is pro- posed to be IFGCS if cl(A) ⊆ U whenever A ⊆ U and U is an IFOS in X. Similarly, IFαGCS and IFGαCS Kalamani et al. [2012], , IFGSPCS Santhi and Jayanthi [2010] , IFGSCS Santhi and Sakthivel [2009], IFGSRCS Anitha and Mohana [2018], were introduced. With the help of above closed sets we, initiate new set IFG#α- closed set. 3 IFG#α- closed sets Definition 3.1. An IFS C in (X, τ) is proposed to be an IFG#α-closed set if αcl(C) ⊆ C, whenever C ⊆ U and U is an IFGOS in (E, τ). The family of all IFG#αCS of an IFTS (E, τ) is defined by IFG#αC(X). Example 3.1. Consider E = {p, q}, τ = {0∼, J, 1∼} is IFT on E, in that J =< e, (0.3, 0.2), (0.5, 0.6) >. In this the only α-open sets are 0∼, 1∼, J. At that point IFS, C =< e, (0.1, 0), (0.6, 0.8) > an IFG#αCS in (E, τ). Theorem 3.1. Every IFCS is IFG#αCS, but reverse implication is not possible. 64 IFG#α-CS in intuitionistic fuzzy topological spaces Proof. ConsiderC is IFCS in (E, τ). Suppose an IFS, C ⊆ U where U is IFGOS. Considering αcl(C) ⊆ cl(C) and C is an IFCS in E, αcl(C) ⊆ cl(C) = C ⊆ U and U is IFGOS. That is αcl(C) ⊆ U. Consequently C is IFG#αCS in E. Example 3.2. Let E = {p, q} and let τ = {0∼, J, 1∼} is an IFT on E, where J =< e, (0.4, 0.1), (0.5, 0.6) >. Let C =< e, (0.3, 0.1), (0.7, 0.9) > be an IFS in E. Here C is an IFG#αCS but not IFCS in (E, τ). Theorem 3.2. Every IFαCS is IFG#αCS but, reverse implication is not possible. Proof. Consider C is a IFαCS in E. Suppose an IFS, C ⊆ U, where U is IFGOS. Considering C is IFαCS, α cl(C) = C. Hence αcl(C) ⊆ U once C ⊆ U, U is IFGOS. Consequently IFG#αCS in E. Example 3.3. Let E = {p, q} and let τ = {0∼, J, 1∼} be IFT, we have J =< e, (0.2, 0.4), (0.6, 0.5) >. Consider C =< e, (0.2, 0.4), (0.6, 0.5) > be an IFS on E. Then C is IFG#αCS but not IFα CS in (E, τ). Theorem 3.3. Every IFRCS is IFG#αCS but, reverse implication is not possible. Proof. Consider C is an IFRCS. We know that C=cl(int(C)) using definition. This signifies cl(C)=cl(int(C)). Consequently cl(C)=C. By which C is IFCS in E. C is an IFG#αCS in E. Example 3.4. Consider E = {p, q}, τ = {0∼, J, 1∼} be IFT, we have J =< e, (0.4, 0.4), (0.5, 0.5) >. Here an IFS,C =< e, (0.2, 0.2), (0.7, 0.8) > is an IFG#αCS but not IFRCS in (E, τ). Theorem 3.4. Every IFG#αCS is IFSGCS but, reverse implication is not possible. Proof. Consider C an IFG#αCS. Suppose an IFS, C ⊆ U where U is IFSO set. Since, every IFSO set is IFGO set and C be an IFG#αCS. We have scl(C) ⊆ αcl(C) ⊆ U. Therefore C is IFSGCS. Example 3.5. Let E = {p, q} τ = {0∼, J, 1∼} be IFT we have J =< e, (0.5, 0.7), (0.6, 0.7) >. Let C =< e, (0.5, 0.6), (0.8, 0.9) >. Then C is an IFSGCS but not IFG#αCS in (E, τ). Theorem 3.5. Every IFG#αCS is IFGSCS but, reverse implication is not possible. Proof. Consider C an IFG#αCS in (E, τ). Suppose an IFS, C ⊆ U where U is IFOS. Since, every IFOS set is an IFGOS and A be an IFG#αCS. We have scl(C) ⊆ αcl(C) ⊆ U. Therefore, C is IFGSCS set. 65 Christy Jenifer J and Kokilavani V Example 3.6. Let E = {p, q}, τ = {0∼, J, 1∼} be IFT, we have J =< e, (0.1, 0.2), (0.6, 0.6) >. Let, C =< e, (0, 0.2), (0.9, 0.6) >. Here C is an IFGSCS but not an IFG#αCS. Theorem 3.6. Every IFG#αCS is IFGSRCS in (E, τ) but, reverse implication is not possible. Proof. Consider C an IFG#αCS in (E, τ). Suppose an IFS, C ⊆ U where U is an IFROS. Since, every IFROS set is an IFGSROS and C be an IFG#αCS. We have scl(C) ⊆ αcl(C) ⊆ U. Therefore, C is IFGSRCS set. Example 3.7. Let E = {p, q}, τ = {0∼, J, 1∼}, here J =< e, (0.3, 0.6), (0.7, 0.4) >. Consider, C =< e, (0.3, 0.4), (0.7, 0.6) >. Here C is an IFGSRCS but not an IFG#αCS. Remark 3.1. For any two IFG#αCS intersection is also IFG#αCS. Proof. Consider C and D any two IFG#αCS. That is Iαcl(C) ⊆ G. Once C ⊆ G and G is IFGOS and is Iαcl(D) ⊆ G whenever D ⊆ G and G is IFGOS. Now, Iαcl(C ⊆ D) = Iαcl(C) ∩ Iαcl(D) ⊆ G, where (C ∩ D) ⊆ G and G is IFGOS. Thus, intersection of any two IFG#α- closed set is IFG#αCS. Theorem 3.7. Let (E, τ) be IFTS. Then for every C ⊆ IFG#αCS(E) and for every D ∈ IFS(E), C ⊆ D ⊆ αcl(C) implies D ∈ IFG#αCS(E). Proof. Consider IFS D ⊆ U and U be IFGOS, considering C ⊆ D, C ⊆ U and C is IFG#αCS, αcl(C) ⊆ U. By assumption, D ⊆ αcl(C), α cl(D) ⊆ α cl(C) ⊆ U. Consequently αcl(D) ⊆ U. Thus C is IFG#αCS of E. Theorem 3.8. Consider E an IFTS. Then IFGO(E) = IFGC(E) if and only if every IFS in E an IFG#αCS in E. Proof. Necessity : Assume IFGO(E) = IFGC(E). Consider C ⊆ G, G an IFGOS. This signifies αcl(C) ⊆ αcl(G). Considering G an IFGOS in E, by assumption G is IFGCS in E, αcl(C) ⊆ G. This signifies αcl(C) ⊆ G. Consequently C is IFG#αCS in E. Sufficiency : Assume, every IFS is IFG#αCS. Consider G∈ IFO(E), we have G∈ IFGO(E) and so C ⊆ G also G is IFOS in E, by assumption αcl(C) ⊆ G. That is G∈ IFGC(E). Accordingly IFGO(E)⊆ IFGCS(E). Consider C∈ IFGC(E), we have Cc is IFGOS in E. But IFGO(E)⊆ IFGC(E). Consequently Cc ∈ IFGC(E). Here C∈ IFGO(E). Consequently IFGC(E)⊆ IFGO(E). We know that IFGO(E) ⊆ IFGC(E). Theorem 3.9. Let C be IFG#αCS of E, then αcl(C)-C contains no non-empty IFGCS. 66 IFG#α-CS in intuitionistic fuzzy topological spaces Proof. Suppose C is IFG#αCS of E and consider F to be non-empty IFGCS of E, and so F ⊆ αcl(C)-C. We have A ⊆ E - F. Considering C is IFG#αCS and E-F is IFGOS, and so αcl(C) ⊆ E − F . This signifies F ⊆ E − αcl(C). Also F ⊆ (E −αcl(C))∩(αcl(C)−C) ⊆ (E −αcl(C))∩αcl(C) = ϕ. Consequently F is empty. Theorem 3.10. Let C ⊆ D ⊆ E and assume that C is IFG#αCS in E then C is an IFG#αCS relative to D. Proof. Here we have, C ⊆ D ⊆ E also C an IFG#αCS. Considering C ⊆ D ∩ F where F is IFGOS in E. Since C is an IFG#αCS in E, C ⊆ F implies, αcl(C) ⊆ F . It follows that D ∩ αcl(C) ⊆ D ∩ F = F . Thus C is an IFG#αCS relative to D. 4 IFG#α-open sets In this segment, we define and establish the idea of IFG#α –open sets (briefly IFG#αOS) in IFTS and establish its characterizations. Definition 4.1. A subset B of IFTS E is proposed to be an IFG#α-open if Ac is IFG#α-open set. Theorem 4.1. Consider E an IFTS we have, (i) . Every IF-open set is IFG#αOS. (ii) . Every IFα-open set is IFG#αOS. (iii) . Every IFR-open set is IFG#αOS. Proof. Proof is obvious Theorem 4.2. Let (E, τ)be the IFTS then, (i) Every IFG#α-open set is IFSGOS. (ii) Every IFG#α-open set is IFGSPOS. (iii) Every IFG#α-open set is IFGSOS. (iv) Every IFG#α-open set is IFGSROS. Proof. Proof is obvious Theorem 4.3. An IFS C of IFTS E is IFG#αOS on the condition that D ⊆ αint(C) at any moment D is IFGCS in E also D ⊆ F . 67 Christy Jenifer J and Kokilavani V Proof. Necessity: Let C is IFG#αOS in E. Consider D be IFGCS in E also D ⊆ C. We have Dc is IFGOS in E in this extent Cc ⊆ Dc. Considering Cc is IFG#αCS, then αcl(Cc) ⊆ Dc. So (αcl(C))c ⊆ Dc. Consequently D ⊆ αcl(C). Sufficiency: Consider D ⊆ αint(C) at any moment D is IFGCS also D ⊆ C. We have Cc ⊆ Dc also Dc an IFGOS. By assumption, (αcl(C))c ⊆ Dc.Therefore Cc is IFG#αCS of E. Consequently C is IFG#αOS. Theorem 4.4. Consider E an IFTS. We have for all C ⊆ IFG#αOS also probably D ∈ IFS(E), αint(C) ⊆ D ⊆ C signifies D∈ IFG#αOS. Proof. Here αint(C) ⊆ D ⊆ C implies Cc ⊆ Dc ⊆ (αint(C))c. Consider Dc ⊆ F also F is IFGOS in E. Considering Cc ⊆ Dc, Cc ⊆ F . Since Cc is IFG#αCS, αcl(Cc) ⊆ F and so Dc ⊆ (αint(C))c = αcl(Cc). Consequently αcl(Dc) ⊆ αcl(Cc) ⊆ F . Accordingly Dc is IFG#αCS in E. This signifies D is IFG#αOS in E. Therefore D ∈ IFG#αCS. 5 Conclusions Here we have derived a new concept of closed set called IFG #α CS. We have proved that IFG#αCS is stronger than IFCS, IFCS, IFRCS. Also, IFGSCS, IFSGCS and IFGSRCS is stronger than IFG#αCS. Further, it satisfies the inter- section axiom but it does not satisfy union property. Thus we can conclude that IFG#αCS does not form a topology. This concept can further be extended to continuous functions, irresolute functions, various forms of continuous functions such as completely continuous, perfectly continuous, contra continuous, almost continuous, slightly continuous and spaces can be introduced which are normal space and regular space, also connectedness can be described. Application can be done based on membership and non-membership values and find the MCDM problems. References S. Anitha and K. Mohana. Ifgsr – closed sets in intuitionistic fuzzy topological spaces. International Journal of Innovative Research in Technology, 5(2):365– 369, 2018. L. A.Zadeh. Fuzzy sets. Information and control, 8:338–353, 1965. D. Coker. An introduction to fuzzy topological space. Fuzzy sets and systems, 88: 81–89, 1997. 68 IFG#α-CS in intuitionistic fuzzy topological spaces D. Kalamani, K. Sakthivel, and C. S. Gowri. Generalized alpha closed sets in intuitionistic fuzzy topological spaces. Applied mathematical Sciences, 6(94): 4691–4700, 2012. R. Santhi and D. Jayanthi. Intuitionistic fuzzy generalized semipreclosed map- pings. Notes on Intuitionistic Fuzzy Sets, 16:28–39, 2010. R. Santhi and K. Sakthivel. Intuitionistic fuzzy generalized semicontinuous map- pings. Advances in Theoretical and Applied Mathematics, 5:73–82, 2009. S. S. Thakur and R. Chaturvedi. Generalized closed sets in intuitionistic fuzzy topology. The Jornal of Fuzzy Mathematics, 16:559–572, 2008. 69