In this paper we introduce intuitionistic fuzzy lightly $\hat{\hat{g}}$-closed sets and intuitionistic fuzzy lightly $\hat{\hat{g}}$-open sets and study some of their properties with suitable examples are given

M. Rameshpandi

P. Deepika

R. Premkumar

English

ZENODO

Asia MathematikaVolume: 7 Issue: 2 , (2023) Pages: 35 – 42Available online at www.asiamath.orgLightly ˆ̂g -closed sets in intuitionistic fuzzy topological spacesP. Deepika1∗, M. Rameshpandi2 and R. Premkumar31Department of Mathematics, Sri Adichunchanagairi Women’s College,Kumuli Main Road, Cumbum-625 516, Tamil Nadu, India. ORCID iD: 0000-0003-3813-07002Department of Mathematics, P.M.T. College, Usilampatti,Madurai District, Tamil Nadu, India. ORCID iD: 0000-0002-5882-19903Department of Mathematics, Arul Anandar College, Karumathur,Madurai-625 514, Tamil Nadu, India. ORCID iD: 0000-0002-8088-5114Received: 28 Jun 2023 • Accepted: 30 Jul 2023 • Published Online: 31 Aug 2023Abstract: In this paper we introduce intuitionistic fuzzy lightly ˆ̂g -closed sets and intuitionistic fuzzy lightly ˆ̂g -opensets and study some of their properties with suitable examples are given.Key words: Intuitionistic fuzzy topology, Intuitionistic fuzzy ˆ̂g -closed set and Intuitionistic fuzzy Lˆ̂g -closed set1. IntroductionThe concept of fuzzy sets was introduced by Zadeh [17] and later Atanassov [1] generalized this idea tointuitionistic fuzzy sets using the notion of fuzzy sets. On the other hand Coker [4] introduced intuitionistic fuzzytopological spaces using the notion of intuitionistic fuzzy sets. Recently many fuzzy topological concepts havebeen extended to intuitionistic fuzzy topological spaces. Further, several researchers find real-life applications infuzzy topological spaces, soft fuzzy topological spaces and intuitionistic fuzzy topological space for example [2],[8], [9] and [13] and so on. In this paper we introduce intuitionistic fuzzy lightly ˆ̂g -closed sets and intuitionisticfuzzy lightly ˆ̂g -open sets and study some of their properties.2. PreliminariesThroughout this paper (X, τ) (briefly, X ) will denote an intuitionistic fuzzy topological space or IFTS (X, τ) .If H < X , cl(H) = C(H) and int(H) = I(H) will, respectively, denote the closure and interior of H inIFTS(X, τ) . We given some definitions and note some fundamental results necessary for our present study.Definition 2.1. An IFS A in an IFTS (X, τ ) is said to be an1. intuitionistic fuzzy regular closed set(IFRCS in short) if A = cl(int(A)), [6]2. intuitionistic fuzzy semi closed set (IFSCS in short) if int(cl(A)) ⊆ A, [3]3. intuitionistic fuzzy α -closed set (IFαCS in short) if cl(int(cl(A))) ⊆ A, [7]4. intuitionistic fuzzy semi pre closed set (IFSPCS in short) if int(cl(int(A))) ⊆ A. [16]Definition 2.2. An IFS A in (X, τ ) is said to be an©Asia Mathematika, DOI: 10.5281/zenodo.8369010∗Correspondence: deepiprabha11@gmail.com35P. Deepika, M. Rameshpandi and R. Premkumar1. intuitionistic fuzzy generalized closed set (IFGCS in short) if cl(A) ⊆ U whenever A ⊆ U and U is an IFOS in(X, τ ), [14]2. intuitionistic fuzzy generalized semi closed set (IFGSCS in short) if scl(A) ⊆ U whenever A ⊆ U and U is anIFOS in (X, τ ), [11]3. intuitionistic fuzzy semi generalized closed set (IFSGCS in short) if scl(A) ⊆ U whenever A ⊆ U and U is anIFSOS in (X, τ ), [15]4. intuitionistic fuzzy generalized semi pre closed set (IFGSPCS in short) if spcl(A) ⊆ U whenever A ⊆ U and Uis an IFOS in (X, τ ), [10]5. intuitionistic fuzzy α generalized closed set (IFαGCS in short) if αcl(A) ⊆ U whenever A ⊆ U and U is anIFOS in (X, τ ). [12]The complements of the above mentioned intuitionistic fuzzy closed sets are called their respectiveintuitionistic fuzzy open sets.Denote IFSGO(X), the set of all intuitionistic semi-generalized open sets of X.3. IF lightly ˆ̂g -closed setsDefinition 3.1. Let H be an IFS in an IFTS (X, τ) is called an intuitionistic fuzzy lightly ˆ̂g -closed set (briefly,IFL ˆ̂gCS) if H ⊆ G , G is an IFSGOS ⇒ C(I(H)) ⊆ G .The collection of all intuitionistic fuzzy lightly ˆ̂g -closed sets in X is denoted by IFL ˆ̂gC(X).Example 3.1. Consider X = {m,n} with τ = {0∼, A, 1∼} is an IFT on X , where A = ⟨α, (0.6, 0.7), (0.4, 0.3)⟩ .Then θA(m) = 0.6, θA(n) = 0.7, βA(m) = 0.4 and βA(n) = 0.3 . Let be an IFS H = ⟨α, (0.4, 0.3), (0.6, 0.7)⟩ =Ac . This verifies that H is an IFL ˆ̂gCS.2. Consider X = {m,n} with τ = {0∼, A, 1∼} is an IFT on X , where A = ⟨α, (0.6, 0.7), (0.4, 0.3)⟩ . ThenθA(m) = 0.6, θA(n) = 0.7, βA(m) = 0.4 and βA(n) = 0.3 . Let be an IFS H = ⟨α, (0.7, 0.8), (0.3, 0.2)⟩ . Thisverifies that H is not an IFL ˆ̂gCS.Proposition 3.1. Let H be in an IFTS (X, τ) , the following statements are true.1. If H is an IFCS then H is an IFL ˆ̂gCS.2. If H is an IF ˆ̂gCS then H is an IFL ˆ̂gCS.3. If H is an IFRCS then H is an IFL ˆ̂gCS.Proof.1 Let H be an IFCS in (X, τ) . Let G be an IFSGOS such that H ⊆ G . Since C(H) = H,C(I(H)) ⊆C(H) = H . Then C(I(H)) ⊆ H ⊆ G whenever H ⊆ G and G is IFSGOS. It is follows that H is an IFL ˆ̂gCSin X .2. Let H be an IF ˆ̂gCS in (X, τ) . Let G be an IFSGOS such that H ⊆ G . Since H is an IF ˆ̂gCS, C(H) ⊆ G .Since C(I(H)) ⊆ C(H) , then H ⊆ G , G is an IFSGOS ⇒ C(I(H)) ⊆ G . This verifies that H is an IFL ˆ̂gCSin X .36P. Deepika, M. Rameshpandi and R. Premkumar3. Let H be an IFRCS in (X, τ) . Let G be an IFSGOS in (X, τ) such that H ⊆ G . Since H is an IFRCS,C(I(H)) = H ⊆ G . Thus we have H ⊆ G , G is an IFSGOS ⇒ C(I(H)) ⊆ G . Which verifies that H is anIFL ˆ̂gCS.Remark 3.1. The following example shows that converse of Proposition 3.1 is not true in general.Example 3.2.1 Consider X = {m,n} and τ = {0∼, A, 1∼} is an IFT on X , where A = ⟨α, (0.3, 0.4), (0.7, 0.6)⟩ .Then θA(m) = 0.3, θA(n) = 0.4, βA(m) = 0.7 and βA(n) = 0.6 . Let be an IFS H = ⟨α, (0.2, 0.3), (0.8, 0.7)⟩ isan IFL ˆ̂gCS but not an IFCS.2. Consider X = {m,n} and τ = {0∼, A, 1∼} is an IFT on X , where A = ⟨α, (0.4, 0.3), (0.6, 0.7)⟩ . ThenθA(m) = 0.4, θA(n) = 0.3, βA(m) = 0.6 and βA(n) = 0.7 . Let be an IFS H = ⟨α, (0.3, 0.2), (0.7, 0.8)⟩ . Thisverifies that H is an IFL ˆ̂gCS but not an IF ˆ̂gCS.3. Consider X = {m,n} and τ = {0∼, A, 1∼} is an IFT on X , where A = ⟨α, (0.6, 0.7), (0.4, 0.3)⟩ . ThenθA(m) = 0.6, θA(n) = 0.7, βA(m) = 0.4 and βA(n) = 0.3 . Let be an IFS H = ⟨α, (0.4, 0.3), (0.6, 0.7)⟩ . Thisverifies that H is an IFL ˆ̂gCS but not an IFRCS.Proposition 3.2. If H is an IFL ˆ̂gCS in an IFTS (X, τ) , then I(H) is an IFGSPCS.Proof. Let H be an IFL ˆ̂gCS in (X, τ) . Let G be IFOS in X such that H ⊆ G . Then I(H) ⊆ I(G) = G . It isknown that every IFOS is an IFSGOS. Since H is IFL ˆ̂gCS in X , C(I(H)) ⊆ G . We have I(C(I(H))) ⊆ I(G) =G and I(H) ⊆ G . It implies that spcl(I(H)) = I(H) ∪ I(C(I(I(H)))) = I(H) ∪ I(C(I(H))) ⊆ G ∪ G = G .Thus I(H) is an IFGSPCS in X .Remark 3.2. The following example shows that converse of Proposition 3.2 is not true in general.Example 3.3. Consider X = {m,n} and τ = {0∼, A, 1∼} is an IFT on X , where A = ⟨α, (0.7, 0.6), (0.3, 0.4)⟩ .Then θA(m) = 0.7, θA(n) = 0.6, βA(m) = 0.3 and βA(n) = 0.4 . Let be an IFS H = ⟨α, (0.8, 0.7), (0.2, 0.3)⟩ .This verifies that I(H) = ⟨α, (0.7, 0.6), (0.3, 0.4)⟩ = A is an IFGSPCS but H is not an IFL ˆ̂gCS in X .Remark 3.3. The following example shows that the family of IFGCS and the family of IFL ˆ̂gCS are independentof each other.Example 3.4. Consider X = {m,n} and τ = {0∼, A, 1∼} is an IFT on X , where A = ⟨α, (0.2, 0.3), (0.7, 0.6)⟩ .Then θA(m) = 0.2, θA(n) = 0.3, βA(m) = 0.7 and βA(n) = 0.6 . Let be an IFS H = ⟨α, (0.8, 0.7), (0.1, 0.2)⟩ .This verifies that H is an IFGCS but H is not an IFL ˆ̂gCS in X .Example 3.5. Consider X = {m,n} and τ = {0∼, A, 1∼} is an IFT on X , where A = ⟨α, (0.6, 0.7), (0.3, 0.2)⟩ .Then θA(m) = 0.6, θA(n) = 0.7, βA(m) = 0.3 and βA(n) = 0.2 . Let be an IFS H = ⟨α, (0.4, 0.3), (0.5, 0.6)⟩ .This verifies that H is an IFL ˆ̂gCS but H is not an IFGCS in X .Remark 3.4. The following example shows that the family of IFSCS and the family of IFL ˆ̂gCS are independentof each other.37P. Deepika, M. Rameshpandi and R. PremkumarExample 3.6.1 Consider X = {m,n} and τ = {0∼, A, 1∼} is an IFT on X , where A = ⟨α, (0.3, 0.4), (0.6, 0.5)⟩ .Then θA(m) = 0.3, θA(n) = 0.4, βA(m) = 0.6 and βA(n) = 0.5 . Let be an IFS H = ⟨α, (0.4, 0.5), (0.5, 0.4)⟩ .This verifies that H is an IFSCS but H is not an IFL ˆ̂gCS in X .2. Consider X = {m,n} and τ = {0∼, A, 1∼} is an IFT on X , where A = ⟨α, (0.4, 0.3), (0.5, 0.6)⟩ . ThenθA(m) = 0.4, θA(n) = 0.3, βA(m) = 0.5 and βA(n) = 0.6 . Let be an IFS H = ⟨α, (0.3, 0.2), (0.6, 0.7)⟩ . Thisverifies that H is an IFL ˆ̂gCS but H is not an IFSCS in X .Theorem 3.1. An IFS H of an IFTS (X, τ) . If H is both IFCS and IFαGCS and IFO(X) = IFSGO(X),then H is an IFL ˆ̂gCS in X .Proof. Let H be an IFS in an IFTS (X, τ) . Given H is both IFCS and IFαGCS. Since H is an IFαGCS,we have αC(H) ⊆ G whenever H ⊆ G and G is an IFOS in X . Since H is an IFCS, we have C(I(H)) ⊆C(I(C(H))) ⊆ H ∪C(I(C(H))) = αC(H) . We have C(I(H)) ⊆ G whenever H ⊆ G and G is an IFOS in X .Also given IFO(X) = IFSGO(X), then C(I(H)) ⊆ G whenever H ⊆ G and G is an IFSGOS in X . Hence His an IFL ˆ̂gCS in X .Theorem 3.2. An IFS H of an IFTS (X, τ) . If H is both IFOS and IFL ˆ̂gCS, then H is an IFGCS in X .Proof. Since H is an IFOS in (X, τ) . Then H = I(H) . Let G be an IFOS in X such that H ⊆ G . ThenH ⊆ G and G is an IFSGOS in X . Since H is IFL ˆ̂gCS in (X, τ) . We have C(I(H)) = C(H) ⊆ G . Thisproves that H is an IFGCS in X .Theorem 3.3. An IFS H of an IFTS (X, τ) . If H is both IFOS and IFL ˆ̂gCS, then H is an IFCS in X .Proof. Since H is an IFOS and an IFL ˆ̂gCS in (X, τ) . Then H ⊆ H and H is an IFSGOS in X . Given H isan IFL ˆ̂gCS in X , C(I(H)) ⊆ H . Also given H is an IFOS, C(H) ⊆ H . Thus C(H) = H and H is an IFCSin X .Corollary 3.1. An IFS H of an IFTS (X, τ) . If H is both IFOS and IFL ˆ̂gCS, then H is both IFROS andIFRCS in X .Proof. Since H is both IFOS and IFL ˆ̂gCS in (X, τ) . By Theorem 3.3, H is an IFCS. Since H is both IFOSand IFCS, C(I(H)) = C(H) = H and I(C(H)) = I(H) = H . Thus H is both IFROS and IFRCS in X .4. IF lightly ˆ̂g -open setsDefinition 4.1. An IFS H of an IFTS (X, τ) is called an intuitionistic fuzzy lightly ˆ̂g -open set (briefly,IFL ˆ̂gOS ) if its complement Hc is an IFL ˆ̂gCS in (X, τ) .The family of all intuitionistic fuzzy lightly ˆ̂g -open sets in X is denoted by IFL ˆ̂gO(X).Example 4.1. Consider X = {m,n} and τ = {0∼, A, 1∼} is an IFT on X , where A = ⟨α, (0.6, 0.7), (0.4, 0.3)⟩ .Then θA(m) = 0.6, θA(n) = 0.7, βA(m) = 0.4 and βA(n) = 0.3 . Let be an IFS H = ⟨α, (0.4, 0.3), (0.6, 0.7)⟩ =Ac . This verifies that H is an IFL ˆ̂gOS.38P. Deepika, M. Rameshpandi and R. Premkumar2. Consider X = {m,n} and τ = {0∼, A, 1∼} is an IFT on X , where A = ⟨α, (0.6, 0.7), (0.4, 0.3)⟩ . ThenθA(m) = 0.6, θA(n) = 0.7, βA(m) = 0.4 and βA(n) = 0.3 . Let be an IFS H = ⟨α, (0.3, 0.2), (0.7, 0.8)⟩ . Thisverifies that H is not an IFL ˆ̂gOS.Theorem 4.1. An IFS H of an IFTS (X, τ) is IFL ˆ̂gOS ⇐⇒ G ⊆ I(C(H)) whenever G ⊆ H and G is anIFSGCS.Proof. Necessity: Given H is an IFL ˆ̂gOS in X . Then Hc is an IFL ˆ̂gCS. Let G be an IFSGCS such thatG ⊆ H . Then Gc is an IFSGOS such that Hc ⊆ Gc . Since Hc is an IFLˆ̂gCS, then C(I(Hc)) ⊆ Gc . ThusG ⊆ I(C(H)) .Sufficiency: Assuming that G ⊆ I(C(H)) whenever G ⊆ H and G is IFSGCS. Then Gc is an IFSGOSsuch that Hc ⊆ Gc and (I(C(H)))c ⊆ Gc . This implies C(I(Hc)) ⊆ Gc . Hence Hc is an IFLˆ̂gCS. Thisproves that H is an IFL ˆ̂gOS.Proposition 4.1. In an IFTS (X, τ) , every an IFOS is an IFL ˆ̂gOS.Proof. Since H is an IFOS in (X, τ) . Then Hc is an IFCS in X . By Proposition 3.1(1), since every IFCS isan IFL ˆ̂gCS in X . Therefore Hc is an IFL ˆ̂gCS in X . Which proves that H is an IFL ˆ̂gOS in X .Remark 4.1. The following example shows that converse of Proposition 4.1 is not true in general.Example 4.2. Consider X = {m,n} and τ = {0∼, A, 1∼} is an IFT on X , where A = ⟨α, (0.3, 0.4), (0.6, 0.7)⟩ .Then θA(m) = 0.3, θA(n) = 0.4, βA(m) = 0.6 and βA(n) = 0.7 . Let be an IFS H = ⟨α, (0.8, 0.7), (0.2, 0.3)⟩ .Which is an IFL ˆ̂gOS but not an IFOS.Proposition 4.2. In an IFTS (X, τ) , every an IF ˆ̂gOS is an IFL ˆ̂gOS.Proof. Let A be an IF ˆ̂gOS in (X, τ ). Then Ac is an IF ˆ̂gCS in X. By Proposition 3.1(2), we have every IF ˆ̂gCSis an IFL ˆ̂gCS in X. Therefore Ac is an IFL ˆ̂gCS in X. Hence A is an IFL ˆ̂gOS in X.Remark 4.2. The following example shows that converse of Proposition 4.2 is not true in general.Example 4.3. Consider X = {m,n} and τ = {0∼, A, 1∼} is an IFT on X , where A = ⟨α, (0.4, 0.3), (0.6, 0.7)⟩ .Then θA(m) = 0.4, θA(n) = 0.3, βA(m) = 0.6 and βA(n) = 0.7 . Let be an IFS H = ⟨α, (0.7, 0.8), (0.3, 0.2)⟩ .Which follows that H is an IFL ˆ̂gCS but not an IF ˆ̂gCS.Proposition 4.3. In an IFTS (X, τ) , every IFROS is an IFL ˆ̂gOS.Proof. Let A be an IFROS in (X, τ ). Then Ac is an IFRCS in X. By Proposition 3.1(3), we have every IFRCSis an IFL ˆ̂gCS in X. Therefore Ac is an IFL ˆ̂gCS in X. Hence A is an IFL ˆ̂gOS in X.Remark 4.3. The following example shows that converse of Proposition 4.3 is not true in general.Example 4.4. Consider X = {m,n} and τ = {0∼, A, 1∼} is an IFT on X , where A = ⟨α, (0.6, 0.7), (0.4, 0.3)⟩ .Then θA(m) = 0.6, θA(n) = 0.7, βA(m) = 0.4 and βA(n) = 0.3 . Let be an IFS H = ⟨α, (0.6, 0.7), (0.4, 0.3)⟩ .Which follows that H is an IFL ˆ̂gOS but not an IFROS.39P. Deepika, M. Rameshpandi and R. PremkumarProposition 4.4. An IFL ˆ̂gOS H of an IFTS (X, τ) . Then C(H) is an IFGSPOS.Proof. Let H be an IFL ˆ̂gOS in (X, τ) . Then Hc is an IFL ˆ̂gCS in X . By Theorem 3.2, then I(Hc ) is anIFGSPCS in X . Hence (I(Hc))c = C((Hc)c) = C(H) is an IFGSPOS in X .Remark 4.4. The following example shows that converse of Proposition 4.4 is not true in general.Example 4.5. Consider X = {m,n} and τ = {0∼, A, 1∼} is an IFT on X , where A = ⟨α, (0.7, 0.6), (0.3, 0.4)⟩ .Then θA(m) = 0.7, θA(n) = 0.6, βA(m) = 0.3 and βA(n) = 0.4 . Let be an IFS H = ⟨α, (0.2, 0.3), (0.8, 0.7)⟩ .Then C(H) = ⟨α, (0.3, 0.4), (0.7, 0.6)⟩ = Ac is an IFGSPOS but H is not an IFL ˆ̂gOS in X .Remark 4.5. The following example shows that an IFGSPOS is not an IFL ˆ̂gOS in general.Example 4.6. Consider X = {m,n} and τ = {0∼, A, 1∼} is an IFT on X , where A = ⟨α, (0.3, 0.4), (0.7, 0.6)⟩ .Then θA(m) = 0.3, θA(n) = 0.4, βA(m) = 0.7 and βA(n) = 0.6 . Let be an IFS H = ⟨α, (0.2, 0.3), (0.8, 0.7)⟩ .This verifies that H is an IFGSPOS but H is not an IFL ˆ̂gOS in X.Remark 4.6. The following example shows that the family of IFGOS and the family of IFL ˆ̂gOS are independentof each other.Example 4.7. Consider X = {m,n} and τ = {0∼, A, 1∼} is an IFT on X , where A = ⟨α, (0.2, 0.3), (0.7, 0.6)⟩ .Then θA(m) = 0.2, θA(n) = 0.3, βA(m) = 0.7 and βA(n) = 0.6 . Let be an IFS H = ⟨α, (0.1, 0.2), (0.8, 0.7)⟩ .This verifies that H is an an IFGOS but H is not an IFL ˆ̂gOS in X .Example 4.8. Consider X = {m,n} and τ = {0∼, A, 1∼} is an IFT on X , where A = ⟨α, (0.6, 0.7), (0.3, 0.2)⟩ .Then θA(m) = 0.6, θA(n) = 0.7, βA(m) = 0.3 and βA(n) = 0.2 . Let be an IFS H = ⟨α, (0.5, 0.6), (0.4, 0.3)⟩ .This verifies that H is an IFL ˆ̂gOS but H is not an IFGOS in X .Remark 4.7. The following example shows that the family of IFSOS and the family of IFL ˆ̂gOS are independentof each other.Example 4.9. Consider X = {m,n} and τ = {0∼, A, 1∼} is an IFT on X , where A = ⟨α, (0.3, 0.4), (0.6, 0.5)⟩ .Then θA(m) = 0.3, θA(n) = 0.4, βA(m) = 0.6 and βA(n) = 0.5 . Let be an IFS H = ⟨α, (0.5, 0.4), (0.4, 0.5)⟩ .This verifies that H is an IFSOS but H is not an IFL ˆ̂gOS in X .Example 4.10. Consider X = {m,n} and τ = {0∼, A, 1∼} is an IFT on X , where A = ⟨α, (0.4, 0.3), (0.5, 0.6)⟩ .Then θA(m) = 0.4, θA(n) = 0.3, βA(m) = 0.5 and βA(n) = 0.6 . Let be an IFS H = ⟨α, (0.6, 0.7), (0.3, 0.2)⟩ .This verifies that H is an IFL ˆ̂gOS but H is not an IFSOS in X .Theorem 4.2. An IFS H of an IFTS (X, τ) . If H is both an IFOS and an IFαGOS and IFO(X) =IFSGO(X), then H is an IFL ˆ̂gOS in X .Proof. Since H be an IFS in an IFTS (X, τ) . Since H is both an IFOS and an IFαGOS. Then Hc is bothIFCS and IFαGCS. Also given IFO(X) = IFSGO(X). By Theorem 3.1, then Hc is an IFL ˆ̂gCS in X . Thisproves that H is an IFL ˆ̂gOS in X .Theorem 4.3. An IFS H of an IFTS (X, τ) . If H is both IFCS and IFL ˆ̂gOS, then H is an IFGOS in X .40P. Deepika, M. Rameshpandi and R. PremkumarProof. Since H is an IFS in an IFTS (X, τ) . Since H is both IFCS and IFL ˆ̂gOS. Hence Hc is both IFOS andIFL ˆ̂gCS. By Theorem 3.2, then H is an IFGCS in X . This proves that H is an IFGOS in X .Theorem 4.4. An IFS H of an IFTS (X, τ) . If H is both IFCS and IFL ˆ̂gOS, then H is an IFOS in X .Proof. Since H is an IFS in an IFTS (X, τ) . Since H is both IFCS and IFL ˆ̂gOS. Therefore Hc is both IFOSand IFL ˆ̂gCS. By Theorem 3.3, then H is an IFCS in X . This proves that H is an IFOS in X .Corollary 4.1. An IFS H of an IFTS (X, τ) . If H is both IFCS and IFL ˆ̂gOS, then H is both IFROS andIFRCS in X .Proof. Since H is both IFCS and IFL ˆ̂gOS in (X, τ) . Then by Corollary 3.1, H is an IFOS. Since H is bothIFOS and IFCS, H is both IFROS and IFRCS in X .References[1] K. Atanassov, Intuitionistic fuzzy sets, Fuzzy S ets and Systems, 20(1986), 87-96.[2] Atiqe Ur Rahman, Muhammad Saeed, Saba Zahid, Application in decision making based on fuzzy parameterizedhypersoft set theory, Asia Mathematika 5(1)(2021), 19-27.[3] K. K. Azad, On fuzzy semi continuity, fuzzy almost continuity and fuzzy weakly continuity, Jour. Math. Anal.Appl., 82(1981), 14-32.[4] D. Coker, An introduction to intuitionistic fuzzy topological spaces, Fuzzy Sets and Systems, 88(1997), 81-89, doi:10.1016/S0165-0114(96)00076-0.[5] C. L. Chang, Fuzzy topological spaces, J. Math. Anal. Appl., 24(1968), 182-190.[6] H. Gurcay, D. Coker and Es. A. Haydar, On fuzzy continuity in intuitionistic fuzzy topological spaces, The J.Fuzzy Math., 5(1997), 365-378.[7] K. Hur and Y. B. Jun, On intuitionistic fuzzy alpha continuous mappings, Honam Math. J, 25(2003), 131-139.[8] Mohammed M. Khalaf, Modeling treatments prediction of coronary artery disease based on fuzzy soft expertsystem, Asia Mathematika, 4(2)(2020), 1-108.[9] Mohammed M. Khalaf, Strongly semi-(T , T ∗R∗) -L-fuzzy nano continuity, strongly semi-(T , T ∗R∗) -L-fuzzy nanoretracts and decision making, Asia Mathematika, 4(3)(2020), 24-40.[10] R. Santhi and D. Jayanthi, Intuitionistic fuzzy generalized semipreclosed mappings, NIFS, 16(2010), 28-39.[11] R. Santhi and K. Sakthivel, Intuitionistic fuzzy generalized semi continuous mappings, Advances in Theoreticaland Applied Mathematics, 5(2009), 73-82.[12] R. Santhi and K. Arun Prakash, On intuitionistic fuzzy semi generalized closed sets and its applications, Int. J.Contemp. Math. Sci., 5(2010), 1677-1688.[13] I. Silambarasa, Some new operations and their properties on q-rung orthopair fuzzy matrices, Asia Mathematika,5(1)(2021), 119-133.[14] S. S. Thakur and Rekha Chaturvedi, Generalized closed sets in intuitionistic fuzzy topology, The Journal of FuzzyMathematics, 16(2008), 559-572.[15] S. S. Thakur and Jyoti Pandey Bajpai, Intuitionistic fuzzy ω -closed sets and intuitionistic fuzzy ω -continuity,International Journal of Contemporary Advanced Mathematics, 1(2010), 1-15.41P. Deepika, M. Rameshpandi and R. Premkumar[16] Young Bae Jun and Seok-Zun Song, Intuitionistic fuzzy semi preopen sets and intuitionistic semi pre continuousmappings, J. Appl. Math. & Computing, 19(2005), 467-474.[17] L. A. Zadeh, Fuzzy sets, Information and Control, 8(1965), 338-353.42

Lightly g-closed sets in intuitionistic fuzzy topological spaces

https://zenodo.org/record/8369010/files/AM-2308-7204.pdf

Lightly g-closed sets in intuitionistic fuzzy topological spaces

Abstract

Similar works

Full text

Available Versions

ZENODO