Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 6s (2025) 295 https://internationalpubls.com New Kinds of Open Sets in Intuitionistic Interval-Valued Fuzzy Topological Space S.Sivaraja1, B.Sudha2, K. Srinivasan3, K.Bhuvaneswari4 1Assistant Professor , Department of Mathematics, K.Ramakrishnan college of Engineering(Autonomous),Trichy. 2Assistant Professor , Department of Mathematics, College of Engineering and Technology SRM Institute of Science and Technology, Kattankulathur. 3Associate Professor , Department of Mathematics, R.M.K Engineering college, Kavaraipettai. 4Assistant Professor , Department of Mathematics, Sathyabama Institute of Science and Technology, Chennai Article History: Received: 19-10-2024 Revised: 03-12-2024 Accepted: 11-12-2024 Abstract: In an intuitionistic interval valued fuzzy topological space (inshort IIVFTS) a new sort of open sets termed intuitionistic interval valued fuzzy minimum open(resp.maximal open) have been researched. Keywords: Intuitionistic interval valued fuzzy minimal open (resp.minimal open), intuitionistic interval valued fuzzy maximal closed (resp.minimal closed) in short IIVFMIO, IIVFMAO, IIVFMAC, IIVFMIC. 1. Introduction C.L. Chang[2] used the fuzzy set, which Zadeh[1] had established in 1965, to topology in 1968. Once more, Mondal and Samanta [4] used the interval-valued fuzzy set notion used by Zadeh [3] in 1975 to topology in 1999. Additionally, in 1997, Coker[6] applied the intuitionistic fuzzy sets, which Attanassov[5] had introduced in 1986, to topology.In 2012, Pyung Ki Lim et al. [8] used intuitionistic interval-valued fuzzy sets, which Cheong and Hur [7] had introduced in 2010, to topology. In fuzzy topological space and hesitant fuzzy topological space, A. Swaminathan [9,10] and S. Sivaraja established and expanded on new types of open and closed sets. 2. Preliminaries: Definition 2.1 A function ϑ:X → Λ(I ⊕ I) is an IIVFS for any nonempty X,denoted by ϑ = [ϑL, ϑU] = [(αϑ L, β ϑ L), (αϑ U, β ϑ U)]. The IIVFS whole (resp.empty) fuzzy set in X, is given by 1̃ = [(1,0), (1,0)] (resp 0̃ = [(0,1), (0,1)] for x ∈ X. Definition 2.2 Let X ≠ ϕ, ζ ∈ Λ(I ⊕ I)X then ζ is an intuitionistic interval valued fuzzy topology (inshort IIVFT) on X if (i) 0̃, 1̃ ∈ ζ. (ii) ϑ1 ∩ ϑ2 ∈ ζ, ∀{ϑ1, ϑ2} ∈ ζ. (iii) ⋃ p∈K ϑp ∈ ζ, {ϑp} p∈K ⊂ ζ. The pair (X, ζ) is IIVFTS and every ϑ ∈ ζ is open and ϑ is said to be closed in X if ϑ c ∈ ζ. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 6s (2025) 296 https://internationalpubls.com 3 INTUITIONISTIC INTERVAL VALUED FUZZY MINIMAL OPEN SETS Definition 3.1 A proper IIVFO set ϑ of IIVFTS (X, ζ) is said to be a IIVFMIO iff any IIVFO which is contained in ϑ is either ϑ or 0̃. Lemma 3.1 In a IIVFTS (X, ζ), (i) If ϑ1 is a IIVFMIO and ϑ2 is a IIVFO set in X then, ϑ1 ∩ ϑ2 = 0̃ or ϑ1 ⊂ ϑ2. (ii) If ϑ1, ϑ2 are IIVFMIO sets then ϑ1 ∩ ϑ2 = 0̃ or ϑ1 = ϑ2. Proof. (i) Suppose that ϑ1 ∩ ϑ2 ≠ 0̃ for any IIVFO set ϑ2 then (ϑ1 ∩ ϑ2) ⊂ ϑ1, a contradiction to minimality of ϑ1, then (ϑ1 ∩ ϑ2) = ϑ1 implying that ϑ1 ⊂ ϑ2. (ii) As ϑ1, ϑ2 are IIVFMIO sets,if ϑ1 ∩ ϑ2 ≠ 0̃ then ϑ1 ⊂ ϑ2 and ϑ2 ⊂ ϑ1, implying that ϑ1 = ϑ2. Theorem 3.2 If ϑ and ϑp are IIVFMIO sets for any p ∈ K. If ϑ ⊆ ⋃ p∈K ϑp then ∃ an element p ∈ K such that ϑ = ϑp. Proof. If ϑ ⊆ ⋃ p∈K ϑp then ϑ = ϑ ∩ (⋃ p∈K ϑp) = ⋃ p∈K (ϑ ∩ ϑp). Clearly (ϑ ∩ ϑp) = 0̃ or ϑ = ϑp as ϑ, ϑp are IIVFMIO sets by Lemma-3.1(ii). If (ϑ ∩ ϑp) = 0̃ then ϑ = 0̃, a contradiction to minimality of ϑ. Theorem 3.3 Suppose ϑ, ϑp are IIVFMIO sets for any p ∈ K, ϑ ≠ ϑp for any p ∈ K then ϑ ∩ (⋃ p∈K ϑp) = 0̃. Proof. Suppose that ϑ ∩ (⋃ p∈K ϑp) ≠ 0̃, then ∃ an element p ∈ K with (ϑ ∩ ϑp) ≠ 0̃. By Lemma- 3.1(ii) ϑ = ϑp, a contradiction. Theorem 3.4 A IIVFMIO set ϑp for any p ∈ K; |K| ≥ 2 and ϑl ≠ ϑp for any distinct l, p ∈ K, then for any l ∈ K, ϑl ∩ (⋃ p∈K ϑp) = 0̃. Proof. Let ϑl ∩ (⋃ p∈K/l ϑp) ≠ 0̃, then ⋃ p∈K/l (ϑl ∩ ϑp) ≠ 0̃. By Lemma-3.1(ii), ϑl = ϑp , a contradiction. Theorem 3.5 If ϑp is a IIVFMIO set for any k ∈ K; |K| ≥ 2 and ϑl ≠ ϑp for any distinct l, p ∈ K. If α is a proper IIVF subset of K,then (⋃ k∈K/α ϑl) ∩ (⋃ m∈α ϑm) = 0̃. Proof. By assuming the contrary, we have ∪ (ϑl ∩ ϑm) ≠ 0̃ for k ∈ K/α and m ∈ α implying (ϑl ∩ ϑm) ≠ 0̃ for some p ∈ K,m ∈ α.By Lemma-0.1(ii) we have, ϑl = ϑm , a contradiction. Theorem 3.6 If ϑp is a IIVFMIOs for any p ∈ K with ϑp ≠ ϑl for any distinct l, p ∈ K, then [⋃p∈K/l ϑp] ∩ [⋃l∈T ϑl] = 0̃ for any proper IIVF subset T of K. Proof. By assuming the contrary, we have ∪ [ϑp ∩ ϑl] ≠ 0̃, ∀p ∈ K/l, l ∈ T implying ∪ [ϑp ∩ ϑl] ≠ 0̃ for some p ∈ K; l ∈ T. By Lemma-3.1(ii) a contradiction to minimality of ϑp. Theorem 3.7 If ϑp, ϑl are IIVFMIO sets for any p ∈ K; l ∈ T respectively and if ∃n ∈ T such that ϑp ≠ ϑn, for any p ∈ K, then [⋃ n∈M ϑn][⋃ k∈K ϑp]. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 6s (2025) 297 https://internationalpubls.com Proof. By assuming the contrary, then ∃n ∈ T with ϑp ≠ ϑn for any p ∈ K, then [⋃ n∈M ϑn] ⊂ [⋃ p∈K ϑp]. ϑn ⊂ [⋃ p∈K ϑp], for some n ∈ M. Hence, ϑp ≠ ϑn , for any p ∈ K, a contradiction. Theorem 3.8 If ϑp is a IIVFMIO for any p ∈ K with ϑp ≠ ϑm for any distinct p,m ∈ K then [⋃ p∈T ϑp] ⫋ [⋃ m∈K ϑm] for any proper subset T of K. Proof. For any p ∈ K/T, ϑl is a IIVFMIO of family {ϑl|l ∈ K/T} of IIVFMIO sets. Clearly, ϑl ∩ [⋃ p∈T ϑp] = ⋃ p∈T [ϑl ∩ ϑp] = 0̃. Also, ϑl ∩ [⋃ m∈K ϑm] = ⋃ m∈K [ϑl ∩ ϑm] = ϑl. In case [⋃l∈T ϑp] = [⋃m∈k ϑm] then, ϑl = 0̃, a contradiction as ϑl is a IIVFMIO. Hence proved. Theorem 3.9 If ϑp is a IIVFMIO for any p ∈ T such that ϑp ≠ ϑm for any p,m ∈ T, then (i) ϑm ⊂ [⋃l∈T/m ϑl] c, for some m ∈ K. (ii) ⋃l∈K/m ϑm ≠ 1̃, ∀m ∈ K. Proof. (i) Given that ϑp ≠ ϑm for any p,m ∈ T, => ⋃p∈T [ϑp] ∩ ϑm = 0̃. => ⋃p∈T [ϑp ∩ ϑm] = 0̃. => [ϑp ∩ ϑm] = 0̃. => ϑp ⊂ ϑm c . => ϑm ⊂ [⋃l∈T/m ϑl] c. (ii) Suppose that, ⋃l∈T/m ϑl = 1̃. => ϑl = 0̃, a contradiction for minimality of ϑl. This completes the proof. Corollary 3.10 If ϑp is a IIVFMIO for any p ∈ T such that ϑp ≠ ϑm for any p,m ∈ T, then ϑp ∪ ϑm ≠ 1̃. Proof. Similar to the Previous theorem. Theorem 3.11 If ϑp is a IIVFMIO for any p ∈ T such that ϑp ≠ ϑm for any p,m ∈ T, then ϑm = [⋃ p∈T ϑp] ∩ [⋃ p∈T/m ϑp]c, for any m ∈ T. Proof. => [⋃ p∈T ϑp] ∩ [⋃ p∈T/m ϑp]c = [⋃ p∈T/m ϑp ∪ ϑm] ∩ [⋃ p∈T/m ϑp]c. = (⋃ p∈T/m ϑp ∩ [⋃ p∈T/m ϑp]c). Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 6s (2025) 298 https://internationalpubls.com =∪ (ϑm ∩ [⋃ p∈T/m ϑp]c). = 0̃ ∪ ϑm=ϑm. 4. INTUITIONISTIC INTERVAL VALUED FUZZY MAXIMAL OPEN SETS Definition 4.1 A proper IIVFO sets ϑ of a IIVFTS (X, ζ) is said to be IIVFMAO set if any IIVFO set containing ϑ is either 1̃ or itself. Example 4.2 Let X = {l,m, n} and (X, ζ) is IIVFTS with IIVFs ϑ given by: ϑ(l) = [(0.55,0.45), (0.65,0.25)], ϑ(m) = [(0.45,0.35), (0.55,0.35)], ϑ(n) = [(0.25,0.55), (0.45,0.35)] and ζ = {0̃, ϑ, 1̃} then ϑ is both IIVFMIO and IIVFMAO in (X, ζ). Lemma 4.2 Let (X, ζ) be a IIVFTS then, (i) If ϑ1 is IIVFMAO and ϑ2 is a IIVFO set in X, then either ϑ2 ⊂ ϑ1 or (ϑ1 ∪ ϑ2) = 1̃. (ii) If ϑ1, ϑ2 are IIVFMAOs in X, then eithet ϑ1 = ϑ2 or (ϑ1 ∪ ϑ2) = 1̃. Proof. (i) Suppose that (ϑ1 ∪ ϑ2) ≠ 1̃ for any IIVFMAO set ϑ1 and IIVFO ϑ2, if ϑ2ϑ1 then ϑ1 ⊂ (ϑ1 ∪ ϑ2), a contradiction. Hence, ϑ2 ⊂ ϑ1. (ii) Suppose that (ϑ1 ∪ ϑ2) ≠ 1̃ for any IIVFMAOs ϑ1, ϑ2, then ϑ1 ⊂ ϑ2 and ϑ2 ⊂ ϑ1 then ϑ1 = ϑ2. Theorem 4.3 If ϑl, ϑm, ϑp are IIVFMAO sets such that ϑl ≠ ϑm and (ϑl ∩ ϑm) ⊂ ϑp, then either ϑl = ϑp or ϑm = ϑp. Proof. Suppose that (ϑl ∩ ϑm) ⊂ ϑp and ϑl ≠ ϑp then, (ϑm ∩ ϑp) = (ϑm) ∩ (ϑp ∩ 1̃) =(ϑm) ∩ [(ϑp) ∩ (ϑl ∪ ϑm)] =(ϑm) ∩ [(ϑp) ∩ ϑl) ∪ (ϑp) ∩ ϑm)] =(ϑm ∩ ϑp ∩ ϑl) ∪ (ϑm ∩ ϑp ∩ ϑm)] =(ϑm ∩ ϑl) ∪ (ϑm ∩ ϑp) =ϑm ∩ (ϑl ∪ ϑp) =ϑm ∩ 1̃ =ϑm. (ϑm ∩ ϑ3) = ϑm => ϑm ⊂ ϑp. As ϑm is a IIVFMAO set ϑp ⊂ ϑm This implies ϑm = ϑp. Theorem 4.4 [ϑ1 ∩ ϑ2][ϑ1 ∩ ϑ3] for any distinct IIVFMAO sets ϑ1, ϑ2, ϑ3. Proof. Let us assume the contrary, [ϑ1 ∩ ϑ2] ⊂ [ϑ1 ∩ ϑ3] for any distinct IIVFMAO sets then, [ϑ1 ∩ ϑ2] ∪ [ϑ2 ∩ ϑ3] ⊂ [ϑ1 ∩ ϑ3] ∪ [ϑ2 ∩ ϑ3]. => [ϑ1 ∪ ϑ3] ∩ ϑ2 ⊂ [ϑ1 ∪ ϑ2] ∩ ϑ3 => 1̃ ∩ ϑ2 ⊂ 1̃ ∩ ϑ3 => ϑ2 ⊂ ϑ3, a contradiction. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 6s (2025) 299 https://internationalpubls.com Remark 4.5 These proofs are left out because Theorem 4.5, Corollary 4.6, Theorem 4.7, and Theorem 4.8 are comparable to Theorem 3.9, Corollary 3.10, Theorem 3.11, and Theorem 3.8. the 4.6 Theorem. Theorem 4.5 If ϑp is a IIVFMAO for any p ∈ K, a finite set and ϑp ≠ ϑm for any distinct m, p ∈ K then (i) [⋂K/m ϑp]c ⊂ ϑm for any m ∈ K. (ii) [⋂K/m ϑp] ≠ 0̃ for any m ∈ K. Corollary 4.6 If ϑp is a IIVFMAO for any p ∈ K, a finite set and ϑp ≠ ϑm for any distinct m, p ∈ K then [ϑp ∩ ϑm] ≠ 0̃. Theorem 4.7 If ϑp is a IIVFMAO for any p ∈ K,a finite set and ϑp ≠ ϑm for any distinct m, p ∈ K, then ϑm = [⋂p∈K ϑp] ∪ [⋂p∈K/m ϑp]c for any m ∈ K. Theorem 4.8 If ϑp is a IIVFMAO for any p ∈ K,a finite set and ϑp ≠ ϑm for any distinct m, p ∈ K, and if T is a proper nonempty subset of K, then ⋂p∈K ϑp ⊂ ⋂t∈T ϑt. Theorem 4.9 If ϑp is a IIVFMAO for any p ∈ K,a finite set and ϑp ≠ ϑm for any distinct m, p ∈ K and if ⋂p∈K ϑp is a IIVFC set, then ϑm is a IIVFC set for any j ∈ K. Proof. By Theorem-4.7, we have ϑm = [⋂p∈K ϑp] ∪ [⋂p∈K/m ϑp]c for any m ∈ K => ϑm = [⋂p∈K ϑp] ∪ [⋂p∈K/m (ϑp)c]. As K is finite , [⋂p∈K/m (ϑp)c] is IIVFC. Hence, ϑm is IIVFC. Theorem 4.10 If ϑp is a IIVFMAO for any p ∈ K,a finite set and ϑp ≠ ϑm for any distinct m, p ∈ K. If ⋂p∈K ϑp = 0̃, then {ϑp|p ∈ K} is a set of all IIVFMAO sets of X. Proof. Suppose ∃ϑn another IIVFMAO set of X such that ϑn ≠ ϑp∀p ∈ K. Clearly, 0̃ = ⋂p∈K ϑp = ⋂p∈(K∪n)/n ϑp ≠ 0̃, a contradiction. Hence, proved. References [1] L.A.Zadeh, Fuzzy sets, Inf.Control.8 (1965), 338-353. [2] C.L.Chang, Fuzzy topological spaces. J.Math. Anal. Appl. 24 (1968), 182-190. [3] L.A.Zadeh, The Concept of a linguistic variable and its application to approximate reasoning I,Inform.Sci. 8 (1975),199-249. [4] T.K.Mondal and S.K.Samanta, Topology of interval valued fuzzy sets,Indian J.Pure Appl.Math. (January 1999). [5] K.Attanassov, Intuitionistic fuzzy sets, Fuzzy Sets and Systems 20 (1986), 87-96. [6] D.Coker,An introduction to intuitionistic fuzzy topological spaces, Fuzzy Sets and Systems 88 (1987), 81-89. [7] M.S.Cheong and K.Hur, Instutionistic interval valued fuzzy sets, J.Korean Institute of Intelligent systems 20 (6) (2010), 864-874. [8] Pyung Ki Lim et.all,Instutionistic interval valued fuzzy topological spaces,JKIIS (22) (2012), 126-134. [9] A.Swaminathan and S.Sivaraja, Fuzzy maximal,minimal open and closed sets, Journal of advances in mathematics 9(10),2020 7741-7747. [10] A.Swaminathan and S.Sivaraja,Hesitant Fuzzy minimal and maximal open sets, Journal of appl.and Pure mathematics 5(2),2023 121-128.