Ratio Mathematica Volume 43, 2022 The Extension Of Generalized Intuitionistic Topological Spaces Mathan Kumar GK* G. Hari Siva Annam† Abstract In this paper, irresolute functions in generalized intuitionistic topo- logical spaces were introduced. Regarding these functions, we at- tempted to unveil the notions of some minimal and maximal irreso- lute functions. In addition, the generalized intuitionistic topological spaces were extended by using their open sets which are finer than of it and their basic characterizations were investigated. Some continu- ous functions in the extension of generalized intuitionistic topological spaces are also been discussed in this paper. Keywords: mn-µI -ops, mx-µI -ops, PµI -ops, SµI -ops, mn-µI -cts, mx-µI -cts, mn-µI -irresolute, mx-µI irresolute. 2020 AMS subject classifications: 54A05, 54C08, 54C10. 1 *Research Scholar [19212102091012], PG and Research Department of Mathematics, Kama- raj College, Thoothukudi-628003, Tamil Nadu, India. mathangk96@gmail.com. Affiliated to Manonmaniam Sundaranar University, Tirunelveli-627012, Tamil Nadu, India. †Assistant Professor, PG and Research Department of Mathematics, Kamaraj College, Thoothukudi-628003, Tamil Nadu, India. hsannam84@gmail.com. Affiliated to Manonmaniam Sundaranar University, Tirunelveli-627012, Tamil Nadu, India. 1Received on November 1st, 2022. Accepted on December 29th, 2022. Published on December 30th, 2022. doi: 10.23755/rm.v41i0.949. ISSN: 1592-7415. eISSN: 2282-8214. ©The Authors. This paper is published under the CC-BY licence agreement. Mathan Kumar GK, G. Hari Siva Annam 1 Introduction The concept of an intuitionistic set which is a generalization of an ordinary set and the specialization of an intuitionistic fuzzy set was given by Coker[2]. After that time, intuitionistic topological spaces were introduced [3]. A.Csaszar[1] introduced many closed sets in generalized topological spaces based on their ba- sics. In 2019 [9], some new generalized closed sets in ideal nano topological spaces were developed. In 2022 [6], we have introduced a new type of topology called generalized intuitionistic topological spaces with the help of intuitionistic closed sets. After that time we introduced and studied µI -maps in generalized intuitionistic topological spaces. In addition we have introduced and defined a new structure of minimal and maximal µI -open sets in generalized intuitionistic topological spaces. In 2011 [10], the subject like minimal and maximal continu- ous, minimal and maximal irresolute, T-min space etc. were investigated on basic topological spaces. In 2022 [7], the characterizations of nIαg-closed sets are proved. In that paper au- thors has been used Kuratowski’s closure operator. Taking it as an inspiration we introduce µI -irresolute functions in generalized intuitionistic topological spaces throughout this paper. Also, some minimal and maximal µI -irresolute functions were introduced and studied in detail. The aim of this paper is, to introduce the µI (A)-topology which is finer than µI - topology by using the formula U ∪ (V ∩ A), where U and V are µI -open. In addition, some important and interesting results were discussed by using µI - continuous maps on the extension of µI -topology. Also, some counterexamples are given to support this work. 2 Preliminaries Definition 2.1 (6). A µI topology on a non-empty set X is a family of intuitionistic subsets of X satisfying the following axioms: 1. ∅ ∈ µI 2. Arbitrary union of elements of µI belongs to µI . For a GITS (X,µI ), the elements of µI are called µI -open sets(briefly µI -ops) and the complement of µI -open sets are called µI -closed sets(briefly µI -cds). Note:[6] CµI (∅) ̸= ∅, CµI (X) = X, IµI (∅) = ∅ and IµI (X) ̸= X. The Extension Of Generalized Intuitionistic Topological Spaces Definition 2.2 (6). Let (X,µI ) be a GITS. 1. A proper non-null µI -ops G of (X, µI ) is said to be a mn-µI -ops if any µI - ops which is contained in G is ∅ or G. 2. A proper non-null µI -ops G( ̸= MµI ) of (X,µI ) is said to be a mx-µI -ops set if any µI -ops which contains G is MµI or G. Definition 2.3 (6). Let (X,µI ) and (Y,σI ) be the topological spaces. A map f: (X,µI ) → (Y,σI ) is called, 1. mn-µI -cts if f−1(G) is a µI -ops in (X, µI ) for every mn-µI -ops G in (Y,σI ). 2. mx-µI -cts if f−1(G) is a µI -ops in (X, µI ) for every mx-µI -ops set G in (Y,σI ). Results: [6] 1. Every µI -cts map is mn-µI -cts. 2. Every µI -cts map is mx-µI -cts. 3. Mn-µI -cts and mx-µI -cts maps are independent of each other. 4. If f: (X,µI ) → (Y,σI ) is µI -cts and g: (Y,σI ) → (Z,ρI ) is mn-µI -cts then g◦f: (X,µI ) → (Z,ρI ) is mn-µI -cts. 5. f: (X,µI ) → (Y,σI ) is µI -cts and g: (Y,σI ) → (Z,ρI ) is mx-µI -cts then g◦f: (X,µI ) → (Z,ρI ) is mx-µI -ops. Definition 2.4 (4). Let X be a µI -topological spaces. A subset A of X is said to be µI -dense if CµI (A) = X. Clearly, X is the only µI -closed set dense in (X,µI ). Theorem 2.1. Let (X,µI ) be a GITS with closed under intersection property. Then CµI (A ∪ B) = CµI (A) ∪ CµI (B). Proof: Since A ⊂ A ∪ B and B ⊂ A ∪ B, CµI (A) ⊂ CµI (A ∪ B) and CµI (B) ⊂ CµI (A ∪ B). Now we have to prove the second part, Since A ⊆ CµI (A) and B ⊆ CµI (B), A∪B ⊆ CµI (A)∪CµI (B) which is µI -closed. Then CµI (A∪B) ⊆ CµI (A) ∪ CµI (B). Hence the theorem. Mathan Kumar GK, G. Hari Siva Annam 3 µI-irresolute in GITS Definition 3.1. A mapping k: (X,µI ) → (Y,σI ) is said to be a 1. semi µI -irresolute function(briefly SµI -irresolute) if the inverse image of semi µI -open sets(briefly SµI -ops) in (Y,σI ) is SµI -op in (X,µI ). 2. pre µI -irresolute function(briefly PµI -irresolute) if the inverse image of pre µI -open sets(briefly PµI -ops) in (Y,σI ) is PµI -op in (X,µI ). 3. αµI -irresolute function if the inverse image of αµI -ops in (Y,σI ) is αµI - open in (X, µI ). 4. βµI -irresolute function if the inverse image of βµI -ops in (Y,σI ) is βµI -open in (X,µI ). Theorem 3.1. Let k: (X,µI ) → (Y,σI ) be a semi µI -irresolute function if and only if the inverse image of semi µI -cds in (Y,σI ) is semi µI -closed in (X,µI ). Proof: Necessary part: Let k: (X,µI ) → (Y,σI ) be a semi µI -irresolute function and A be a semi µI -cds in (Y,σI ). Since f is SµI -irresolute, k−1(Y − A) = X − k−1(A) is SµI -open in (X,µI ). Hence k−1(A) is SµI -closed in (X,µI ). Sufficient part: Assume that k−1(A) is SµI -closed in (X,µI ) for each SµI -closed set in (Y,σI ). Let V be a SµI -ops in (Y,σI ) which yields that Y − V is SµI -cds in (Y,σI ). Then we get k(−1)(Y − V ) = X − k(−1)(V ) is SµI -closed in (X,µI ) this implies k−1(V ) is SµI -open in (X,µI ). Hence k is SµI -irresolute. Theorem 3.2. If k is SµI -irresolute then k is SµI -cts. Proof: Suppose k is SµI -irresolute. Let A be any SµI -ops in (Y,σI ). Since every µI -ops is SµI -open and since A is SµI -open, k−1(A) is SµI -open in (X,µI ). Hence k is SµI -cts. Remark 3.1. Since every SµI -ops need not be µI -open, we cannot deduce the reversal concept of the above statement. Theorem 3.3. Let (X,µI ), (Y,σI ) and (Z,ρI ) be three µI -topological spaces. For any SµI -irresolute map k: (X,µI ) → (Y,σI ) and any SµI -cts ℏ: (Y,σI ) → (Z,ρI ) the composition ℏ ◦ k: (X,µI ) → (Z,ρI ) is SµI -cts. Proof: Let A be any µI -ops in (Z,ρI ). Since ℏ is SµI -cts, ℏ−1(A) is SµI -open in (Y,σI ). By using k is semi µI -irresolute, we get k−1[ℏ−1(A)] is SµI -open in (X,µI ). The Extension Of Generalized Intuitionistic Topological Spaces But k−1[ℏ−1(A)] = (ℏ ◦ k)−1(A). Therefore, inverse image of µI -ops in (Z,ρI ) is SµI -open in (X,µI ). Hence ℏ ◦ k: (X,µI ) → (Z,ρI ) is SµI -cts. Theorem 3.4. If k: (X,µI ) → (Y,σI ) and ℏ: (Y,σI ) → (Z,ρI ) are both SµI -irresolute then ℏ ◦ k: (X,µI ) → (Z,ρI ) is also SµI -irresolute. Proof: Let A be any SµI -ops in (Z,ρI ). Since k and ℏ are SµI -irresolute, ℏ−1(A) is SµI -open in (Y,σI ) and k−1[ℏ−1(A)] is SµI -open in (X,µI ). Hence (ℏ ◦ k)−1(A) = k−1[ℏ−1(A)] is SµI -open and so ℏ ◦ k: (X,µI ) → (Z,ρI ) is SµI -irresolute. Theorem 3.5. Let k: (X,µI ) → (Y,σI ) be a PµI -irresolute(resp. αµI -irresolute and βµI -irresolute) function if and only if the inverse image of PµI -closed(resp. αµI -closed and βµI -closed) sets in (Y,σI ) is PµI -closed(resp. αµI -closed and βµI -closed) in (X,µI ). Proof: We can prove this theorem as we have done in the theorem 3.2. Theorem 3.6. If f is PµI -irresolute(resp. αµI -irresolute and βµI -irresolute) then f is PµI -continuous(resp. αµI -cts and βµI -cts). Proof: We can prove this theorem as we have done in the theorem 3.3. Remark 3.2. Since every PµI -open(resp. αµI -open and βµI -open) set need not be µI -open, we cannot deduce the reversal concept of the above statement. Theorem 3.7. Let (X,µI ), (Y,σI ) and (Z,ρI ) be three µI -topological spaces. For any PµI -irresolute(resp. αµI -irresolute and βµI -irresolute) map k: (X,µI ) → (Y,σI ) and any PµI -cts(resp. αµI -cts and βµI - cts) ℏ: (Y,σI ) → (Z,ρI ) the compo- sition ℏ ◦ k: (X,µI ) → (Z,ρI ) is PµI -cts(resp. αµI -cts and βµI -cts). Proof: We can prove this theorem as we have done in the theorem 3.5. Theorem 3.8. If k: (X,µI ) → (Y,σI ) and ℏ: (Y,σI ) → (Z,ρI ) are both PµI - irresolute(resp. αµI -irresolute and βµI -irresolute) then ℏ ◦ k: (X,µI ) → (Z,ρI ) is also PµI -irresolute(resp. αµI -irresolute and βµI -irresolute). Proof: We can prove this theorem as we have done in the theorem 3.6 4 Minimal and Maximal µI-irresolute Definition 4.1. Let (X,µI ) and (Y,σI ) be the topological spaces. A map k: (X,µI ) → (Y,σI ) is called, 1. mn-µI -irresolute if the inverse image of every mn-µI -ops in (Y,σI ) is mn-µI - open in (X,µI ). Mathan Kumar GK, G. Hari Siva Annam 2. mx-µI -irresolute if the inverse image of every mx-µI -ops in (Y,σI ) is mx-µI - open in (X,µI ). Example 4.1. Let X = {a, b, c, d} and Y = {t, u, v, w} with µI = {∅ , < X, ∅, {b} >, < X, ∅, {d} >, < X, {a, d}, ∅ >, < X, {a}, ∅ >, < X, ∅, ∅ >, < X, ∅, {c, d} >, < X, ∅, {c} >, < X, {d}, ∅ >, < X, {d}, {b} >} and σI = {∅, < X, ∅, {v} >, < X, ∅, {w} >, < X, ∅, {u, v} >, < X, ∅, ∅ >, < X, {v}, ∅ >, < X, {v}, {w} >}. Define k: (X,µI ) → (Y,σI ) by k(a) = t, k(b) = w, k(c) = u and k(d) = v. Hence k is a mn-µI -irresolute map. Theorem 4.1. Every mn-µI -irresolute map is mn-µI -cts. Proof: Let k: (X,µI ) → (Y,σI ) be a mn-µI -irresolute map. Let G be any mn-µI - ops in (Y,σI ). Since k is mn-µI -irresolute, k−1(A) is a mn-µI -ops in (X,µI ). That is k−1(A) is a µI -ops in (X,µI ) Hence k is mn-µI -cts. Remark 4.1. The reversal statement of the above theorem is not necessarily true. In example 4.3, k is mn-µI -cts but not mn-µI -irresolute. Since k−1(¡X,w,∅¿) = ¡X,b,∅¿ which is not minimal µI -open in (X,µI ). Theorem 4.2. Every mx-µI -irresolute map is mx-µI -cts. Proof: We can prove this theorem as we have done in the theorem 4.4. Remark 4.2. The reversal statement of the above theorem is not necessarily true. In example 4.2, k is mx-µI -cts but not mx-µI -irresolute. Since k−1(¡X,v,w¿ = ¡X,d,b¿ which is not mx-µI -open in (X,µI ). Remark 4.3. In example 4.2, k is a mn-µI -irresolute map but not mx-µI -irresolute. In example 4.3, k is a mx-µI -irresolute map but not mn-µI -irresolute. That is mn- µI -irresolute maps and mx-µI -irresolute maps are independent of each other. Remark 4.4. Since mn-µI -ops and mx-µI -ops are independent of each other, 1. mn-µI -irresolute and mx-µI -cts are independent of each other. 2. mx-µI -irresolute and mn-µI -cts are independent of each other. Theorem 4.3. Let k: (X,µI ) → (Y,σI ) be a mn-µI -irresolute map if and only if the inverse image of each mx-µI -closed in (Y,σI ) is a mx-µI -closed in (X,µI ). Proof: We can prove this theorem by using the result, if G is a mn-µI -ops if and only if Gc is a mx-µI -closed set. Theorem 4.4. If k: (X,µI ) → (Y,σI ) and ℏ: (Y,σI ) → (Z,ρI ) are mn-µI -irresolute then ℏ ◦ k: (X,µI ) → (Z,ρI ) is a mn-µI -irresolute map. Proof: Let G be any mn-µI -ops in (Z,ρI ). Since ℏ is mn-µI -irresolute, ℏ−1(G) is a mn-µI -ops in (Y,σI ). Also since k is mn-µI -irresolute, k−1[ℏ−1(G)] = (ℏ◦k)−1(G) is a mn-µI -ops in (X,µI ). Hence ℏ ◦ k is mn-µI -irresolute. The Extension Of Generalized Intuitionistic Topological Spaces Theorem 4.5. Let k: (X,µI ) → (Y,σI ) be a mx-µI -irresolute map if and only if the inverse image of each mn-µI -closed in (Y,σI ) is a mn-µI -closed in (X,µI ). Proof: We can prove this theorem by using the result, if G is a mx-µI -ops if and only if Gc is a mn-µI -cds. Theorem 4.6. If k: (X,µI ) → (Y,σI ) and ℏ: (Y,σI ) → (Z,ρI ) are mx-µI -irresolute then ℏ ◦ k: (X,µI ) → (Z,ρI ) is a mx-µI -irresolute map. Proof: Similar to that of theorem 4.11. 5 The Simple Extension of µI-topology over a µI-set In (X,µI ) a subset A of X, we denote by µI(A) the simple extension of µI over A, that is the collection of sets U∪(V∩A), where U,V ∈ µI . Note that µI(A) is finer than µI . Theorem 5.1. If A is µI -dense subset of the space (X,µI ), then A is also µI -dense in (X,µI(A)). Proof: Since µI(A) is finer than µI , µI ⊂ µI(A). This gives CµI(A)(A) ⊂ CµI (A). To prove CµI (A) ⊂ CµI(A)(A). Let x ∈ CµI (A) and let G be a µI -ops of x in µI(A). Then x∈G = H∪(J∩A) where H,J ∈ µI . If x∈H then H∩A ̸= ∅ and G∩A ̸= ∅. If x∈J∩A then J∩A ̸= ∅ and G∩A ̸= ∅. Hence x ∈ CµI(A)(A). Therefore CµI(A)(A) = CµI (A). Theorem 5.2. Let (X,µI ) be a µI -topological space with closed under intersection property. Let A be a µI -dense subset of the space (X,µI ). Then for every µI -open subset G of the space (X,µI(A)) we have CµI (G) = CµI(A)(G) and for every µI - closed subset F of the space (X,µI(A)) we have IµI (F) = IµI(A)(F). Proof: Let V ∈ µI . Since µI(A) is finer than µI , CµI(A)(V ) ⊂ CµI (V ). Now to prove, CµI (V ) ⊂ CµI(A)(V ). Let x ∈ CµI (V ) and let G be a µI -open neighborhood of x in (X,µI(A)). Then x∈G = H∪(J∩A) where H,J ∈ µI . If x∈H then H∩V ̸= ∅. Again if x ∈ J∩A⊂J then J∩V ̸= ∅ and hence J∩V∩A ̸= ∅, since J∩V ∈ µI and since A is µI -dense. Thus also in this case G∩V ̸= ∅ and hence x ∈ CµI(A)(V ). This implies CµI (V ) ⊂ CµI(A)(V ). Henceforth CµI (V ) = CµI(A)(V ) for each V ∈ µI . Let G ∈ µI(A) then G = H∪(J∩A) where H,J ∈ µI . Clearly CµI (H) = CµI(A)(H). Since J ∈ µI(A) and since A is a µI -dense subset of (X,µI(A)), CµI(A)(J∩A) = CµI(A)(J) = CµI (J) = CµI (J∩A). Thus CµI(A)(G) = CµI (H) ∪ CµI (J∩A) =CµI (H∪(J∩A)) = CµI (G). Proceeding like this we can prove IµI (F) = IµI(A)(F). Corolary 5.1. Let (X,µI ) be a GITS with closed under intersection property. If A is a µI -dense subset of the space (X,µI ). Then for every V ∈ µI(A) we have IµI (CµI (V )) = IµI(A)(CµI(A)(V )). Hence the set V is a regular µI -open subset of Mathan Kumar GK, G. Hari Siva Annam (X,µI ) if and only if it is regular µI -open in (X,µI(A)). Proof: From the previous theorem we have IµI (CµI (V )) = IµI (CµI(A)(V )) = IµI(A)(CµI(A)(V )). 6 The characterization of extension on µI-topology Remark 6.1. If k: (X,µI(A)) → (Y,σI ) is µI -cts. Then the restriction of k on (X,µI ) [Shortly, k|(X,µI )] need not be µI -cts. Example 6.1. Let X = {a, b, c} and Y = {u, v, w} with µI = {∅, < X, ∅, {a} >, < X, ∅, {b} >, < X, ∅, ∅ >, < X, ∅, {a, b} >, < X, {a, b}, ∅ >}, µI(A) = { ∅, < X, ∅, {a} >, < X, ∅, {b} >, < X, ∅, ∅ >, < X, ∅, {a, b} >, < X, {a, b}, ∅ >, < X, {b}, ∅ >} and σI = {∅, < X, ∅, {u} >, < X, ∅, {v} >, < X, ∅, ∅ >, < X, {v}, ∅ >}. Define k: (X,µI(A)) → (Y,σI ) by k(a) = u, k(b) = v and k(c) = w. Hence k is µI(A)-cts. But k|(X,µI(A)) is not µI -cts, since k−1(< X, {v}, ∅ >) = < X, {b}, ∅ > /∈ µI . Remark 6.2. Since µI(A) is finer than µI , some elements of µI(A) does not be- longs to µI and the elements of µI(A) which is not in µI need not be mn-µI -open in (X,µI ). For, U ⊂ U∪(V∩A) /∈ µI and U ∈ µI(A), U∪(V∩A) should not be mn-µI -open in (X,µI(A)). By the previous example, we may conclude that every mx-µI -ops in (X,µI(A)) need not be µI -open in (X,µI ). Remark 6.3. A function k is mn-µI(A)-cts in (X,µI(A)) then k|(X,µI ) is mn-µI - cts. In example 6.2, A function f is mx-µI(A)-cts in (X,µI(A)) then f|(X,µI ) need not be mx-µI -cts. 7 Conclusions In example 4.2, k is a mn-µI -irresolute map but not mx-µI -irresolute and in example 4.3, k is a mx-µI -irresolute map but not mn-µI -irresolute. This exam- ples evinces mn-µI -irresolute maps and mx-µI -irresolute maps are independent of each other. Remark 6.1 propounded the restriction of the function K on (X,µI ) need not be a µI -continuous function. In remark 6.3, we discussed the connec- tions between minimal µI -open sets in (X,µI ) and in (X,µI(A)). We hope that we improved some results concerning µI(A)-topological spaces. We will extend our research in kernel and contra continuous of µI -topological spaces. The Extension Of Generalized Intuitionistic Topological Spaces Acknowledgements My completion of this paper could not have been accomplished without the support of my guide and I cannot express enough thanks to my guide for the continued support and encouragement References [1] A.Csaszar, Generalized topology, generalized continuity, Acta Mathematics, Hungar, 96(2002). [2] Dogan Coker, A note on intuitionistic sets and intuitionistic points, Tr.J. of Mathematics, 20(1996), 343-351. [3] J.H.Kim, P.K.Lim, J.G.Lee, K.Hur, Intuitionistic topological spaces, Annals of Fuzzy Mathematics and Informations, 14 December 2017. 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