Int. J. Anal. Appl. (2022), 20:37 On Intuitionistic Fuzzy β Generalized α Normal Spaces Abdulgawad A. Q. Al-Qubati∗, Hadba F. Al-Qahtani Department of Mathematics, College of Science and Arts, Najran University, Saudi Arabia ∗Corresponding author: gawad196999@yahoo.com Abstract. In this paper a new concept of generalized intuitionistic fuzzy topological space called intuitionistic fuzzy β generalized α normal space is introduced. Several characterizations of intuitionistic fuzzy β generalized α normal space, intuitionistic fuzzy strongly β generalized α normal and intuitionistic fuzzy strongly β generalized α regular spaces are studied. Moreover, the related intuitionistic fuzzy functions with intuitionistic fuzzy β generalized α normal spaces are investigated. 1. Introduction The notion of intuitionistic fuzzy set was first defined by Atanassov [7, 8] as a generalization of Zadeh [21] fuzzy set. This notion of intuitionistic fuzzy set has been developed by the same author and appeared in the literature [7,8]. Using the notion of intuitionistic fuzzy sets, Coker [13] introduced the notion of intuitionistic fuzzy topological spaces as a generalization of Chang [11] fuzzy topological spaces. Recently many concepts of fuzzy topological space have been extended in intuitionistic fuzzy topological spaces. Separation axioms in intuitionistic fuzzy topological space have been studied by some authors [1–4, 6, 9, 10]. Jayanthi [17] introduced the generalized β closed set in intuitionistic fuzzy topological spaces and intuitionistic fuzzy generalized closed sets are introduced by Saranya and Jayanthi [18]. Then Gomathi and Jayanthi [15, 16] have studied intuitionistic fuzzy β generalized α closed sets and intuitionistic fuzzy β generalized α continuous functions respectively. Thanh and Quang [19] have studied πgp-normality in topological spaces by using πgp-closed and πgp-open sets. In this paper, we introduce a new class of spaces called an intuitionistic fuzzy β generalized α normal Received: Jun. 13, 2022. 2010 Mathematics Subject Classification. 54A05, 54A08, 54D10. Key words and phrases. intuitionistic fuzzy topology; intuitionistic fuzzy β generalized α closed sets; intuitionistic fuzzy β generalized α normal spaces. https://doi.org/10.28924/2291-8639-20-2022-37 ISSN: 2291-8639 © 2022 the author(s). https://doi.org/10.28924/2291-8639-20-2022-37 2 Int. J. Anal. Appl. (2022), 20:37 spaces, and investigate some of their properties. Some interesting characterizations such as an IF β generalized α-normality is hereditary property with respect to an open and IF β generalized α-closed subspace, and equivalently of intuitionistic fuzzyβ generalized α-normal space and other meanings are introduced. In addition, we introduce the concept of intuitionistic fuzzy β generalized α-regular spaces, some of their properties are investigated such as showing that an IF strongly-regular β∗- T1/2 space is an IF β generalized α-regular, and every IF β generalized α-normal - R0 space is an IF β generalized α-regular space. In the last section of this paper, we study some properties of related intuitionistic fuzzy functions with intuitionistic fuzzy β generalized α normal spaces. 2. Preliminaries Definition 2.1 [7] Let T be a non empty fixed set. An intuitionistic fuzzy set H (IFS for short )in T is an object having the form H = {〈t,µH(t),γH(t)〉 : t ∈ T} where the function µH : T → I and γH : T → I denote the degree of membership (namely µH(t)) and the degree of non-membership (namely γH(t)) of each element t ∈ T to the set H respectively, and 0 ≤ µH(t) + γH(t) ≤ 1, for each t ∈ T. Definition 2.2 [7] Let H and J be IF sets of the form H = {〈t,µH(t),γH(t)〉 : t ∈ T} and J = {〈x,µJ(t),γJ(t)〉 : t ∈ T}. Then (a) H ⊆ J if and only if µH(t) ≤ µJ(t) and γH(t) ≥ γJ(t) for all t ∈ T . (b) H = J if and only if H ⊆ J and J ⊆ H. (c) Hc = {〈t,γH(t),µH(t)〉 : t ∈ T}. (d) H ⋂ J = {〈t,µH(t) ∧µJ(t),γH(t) ∨γJ(t)〉 : t ∈ T}. (e) H ⋃ J = {〈t,µH(t) ∨µJ(t),γH(t) ∧γJ(t)〉 : t ∈ T}. (f) 0∼T = {〈t, , 0, 1〉 : t ∈ T} and 1 ∼ T = {〈t, 1, 0〉 : t ∈ T}. Definition 2.3 [5] Let {Hi : i ∈ I} be an arbitrary family of IFS in T. Then (a) ⋂ Hi = {〈t,∧µHi (t),∨γHi (t) : t ∈ T}. (b) ⋃ Hi = {t,∨µHi (t),∧γHi (t) : t ∈ T}. Definition 2.4 [12] Let α,β ∈ [0, 1],α + β ≤ 1 An intuitionistic fuzzy point (IFP for short) of nonempty set T is an IFS of T denoted by P = t(α,β) and defined by P = t(α,β)(y) =  (α,β)if t = y (0, 1)if t 6= y (2.1) Int. J. Anal. Appl. (2022), 20:37 3 In this case, t is called the support of t(α,β) and α,β are called the value and no value of t(α,β) respectively. Clearly an intuitionistic fuzzy point can be represented by an ordered pair of fuzzy point as follows: t(α,β) = (tα, 1 − t(1−β)). An IFP,t(α,β) is said to belong to an IFS H = {〈t,µH(t),γH(t)〉 : t ∈ T} denoted by P = t(α,β) ∈ H(orP ⊆ H), if α ≤ µH(t) and β ≥ γH(t). We identify a fuzzy point tr in T by the IF point t(r,(1−r)) in T . For more details about operations on IF-sets, IF points and IF functions, we can see [7,8,12–14]. Definition 2.5 [13] An intuitionistic fuzzy topology (briefly IFT) on a nonempty set T is a family ψ of intuitionistic fuzzy sets in T satisfy the following axioms: (T1) 0∼T , 1 ∼ T ∈ ψ. (T2) If H1,H2 ∈ ψ, then H1 ⋂ H2 ∈ ψ. (T3) If Hλ ∈ ψ for each λinΛ, then ⋃ λ∈Λ Hλ ∈ ψ. In this case the pair (T,ψ) is called an intuitionistic fuzzy topological space (briefly IFTS) denoted by T, and each intuitionistic fuzzy set in ψ is known as an intuitionistic fuzzy open set (briefly IFOS ) of T. The complement Hc of an IFOS H in IFTS (T,ψ) is an intuitionistic fuzzy closed set (briefly IFCS) in T. Definition 2.6. An IFS H of an IFTS (T,ψ) is an (i) IF α-open set (resp.α - closed ) if H ⊆ (int(cl(int(H))) ( resp.cl(int(cl(H))) ⊆ H. [14] (ii) IF β −open( resp.β - closed ) if H ⊆ (cl(int(cl(H))) ( resp.int(cl(int(H))) ⊆ H. [14] (iii) IF semi-open if H ⊆ cl(int(H)). [14] (iv) IF pre-open if H ⊆ (int(cl(H))). [14] (v) IF generalized closed (briefly IFgC) if cl(H) ⊆ J whenever H ⊆ J and J is an IFO. [18] 4 Int. J. Anal. Appl. (2022), 20:37 Definition 2.7 [5] Let H be any IFS in IFTS (T,ψ). Then the IF β closure and IF interior of H are defined as follows, IFβcl(H) = ⋂ {F : H ⊆ F,F is IFβCS in T} . IFβint(H) = ⋃ {J : J ⊆ H,J is IFβOS in T} . Definition 2.8 [15] An IFS H of an IFTS (T,ψ) is said to be an IF β generalized α -closed set (IFβGαCS for short) if βcl(H) ⊆ J whenever H ⊆ J and J is an IFαOS in (T,ψ). The complement Hc of an IFβGαCS H in an IFTS (T,ψ) is called an IF β generalized α - open set(IFβGαOS for short) in T. The family of all IFβGα CS of an IFTS (T,ψ) is denoted by IFβGαCS(T). Definition 2.9 Let H be any IFS in IFTS (T,ψ). Then the IF β generalized α closure and IF β generalized α interior of H are defined as follows, IFβgαcl(H) = ⋂ {F : H ⊆ F,F is IFβgαCS in T}. IFβgαint(H) = ⋃ {J : J ⊆ H,J is IFβgαOS in T}. 3. Intuitionistic Fuzzy β Generalized α Normal Spaces In this section, we have introduced IF β generalized α -normal space and studied some of its characterizations. Definition 3.1 An IF topological space T is said to be IF β generalized α-normal(in short IF β g α-normal) if for every pair of IF disjoint β generalized α- closed sets H1 and H2 of T, there exist IF disjoint β-open sets R1, R2 of T such that H1 ⊆ R1 and H2 ⊆ R2. Example 3.2 Let T = {a,b} and R1, R2 are IF sets on T defined as follows: R1 = 〈t, ( a1.0, b 0.0 ), ( a 0.0 , b 1.0 )〉. R2 = 〈t, ( a0.0, b 1.0 ), ( a 1.0 , b 0.0 )〉. Then the family ψ = {0∼T , 1 ∼ T ,R1,R2} is an IFT on T. The IF sets R1,R2 in T are IF disjoint β open sets and the IF sets H1 = 〈t, ( a0.7, b 0.0 ), ( a 0.3 , b 1.0 )〉 ,H2 = 〈t, ( a0.0, b 0.6 ), ( a 1.0 , b 0.4 )〉 are IF β g α -CSs such tha H1 ⋂ H2 = 0 ∼ T and H1 ⊆ R1 and H2 ⊆ R2. Int. J. Anal. Appl. (2022), 20:37 5 Then T is an IF β g α-normal space. Theorem 3.3 For an IFTS (T,ψ), the following are equivalent: (1) T is β g α-normal. (2) For any pair of IF disjoint β g α OSs R1 and R2 of T whose union is 1∼T , there exist an IF disjoint β-CSs H1 and H2 of T such that H1 ⊆ R1 and H2 ⊆ R2 and H1 ⋃ H2 = 1 ∼ T . (3) For each IFβ g α-CS H and an IFβ g α-OS K containing H, there exists an IFβ-OS R2 such that H ⊆ R2 ⊆ IFβ −cl(R2) ⊆ K. (4) For any pair of IF disjoint β g α-CSs H and K of T there exists an IFβ-OS R2 of T such that H ⊆ R2 and IFβ −cl(R2) ⋂ K = 0∼T . (5) For any pair of IF disjoint β gα-CSs H and K of T there exists an IFβ-OSs R1 and R2 of T such that H ⊆ R1, K ⊆ R2 and IFβ −cl(R1) ⋂ IFβ −cl(R2) = 0∼T . Proof.(1) ⇒ (2) Let R1 and R2 be two IF β g α- OSs in an IF β g α-normal space T such that R1 ⋃ R2 = 1 ∼ T . Then R c 1, R c 2 are IF disjoint β g α- CSs. Since T is an IF β g α-normal space there exist IF disjoint β-OSs R and Q such that Rc1 ⊆ R and R c 2 ⊆ Q. Let H1 = R c ,H2 = Qc. Then H1 and H2 are IF β-CSs H1 ⊆ R1, H2 ⊆ R2 and H1 ⋃ H2 = 1 ∼ T . (2) ⇒ (3) Let H be an IFβ g α- CS and K be an IFβ g α- OS containing H. Then Hc and K are IFβ g α -OSs such that Hc ⋃ K = 1∼T . Then by (2) there exist an IF β-CSs H1 and H2 such that H1 ⊆ Hc and H2 ⊆ K and H1 ⋃ H2 = 1 ∼ T . Thus, we obtain H ⊆ H c 1, K c ⊆ Hc2 and H c 1 ⋂ Hc2 = 0 ∼ T . Let R2 = Hc1 and R1 = H c 2. Then R1 and R2 are IF disjoint β -OSs such that H ⊆ H c 1 ⊆ R2 ⊆ K. As Rc2 an IF β-CS, we have H ⊆ R2 ⊆ IFβ −cl(R2) ⊆ K. (3) ⇒ (4) Let H and K be IF disjoint β g α -CSs of T. Then H ⊆ Kc where Kc is IFβ g α -open. By the part (3), there exists an IF β-OS R2 of T such that H ⊆ R2 ⊆ β − cl(R2) ⊆ Kc. Thus, IF β −c1(R2) ⋂ K = 0∼T . (4) ⇒ (5) Let H and K be any IF disjoint β g α -CSs of T. Then by the part (4), there exists an IF β-OS R1 containing H such that IFβ−c1(R1) ⋂ K = 0∼T . Since IFβ−c1(R1) is an IFβ g α - closed, then it is IFβ g α -closed. Thus IFβ − c1(R1) and K are IF disjoint β g α-CSs of T. Again by the part (4), there exists an IF β-OS R2 in T such that K ⊆ R2 and IFβ−c1(R1) ⋂ IFβ−c1(R2) = 0∼T . (5) ⇒ (1) Let H and K be any IF disjoint β g α-CSs of T. Then by the part (5), there exist IF β-OSs R1 and R2 such that H ⊆ R1, K ⊆ R2, and IFβ −c1(R1) ⋂ IFβ −c1(R2) = 0∼T . Therefore, 6 Int. J. Anal. Appl. (2022), 20:37 we obtain that R1 ⋂ R2 = 0 ∼ T . Hence T is IFβ g α-normal. � Definition 3.4 [16] A space (T,ψ) is called β∗ - T1/2 if every IF β g α-CS in T is β closed. Definition 3.5 [15] If every IF β g α-CS is an IFCS in (T,ψ), then the space can be called as an IF β g α T1/2 space. For the IF regularity we give the following definition. Definition 3.6 An IFTS T is said to be IF β g α-regular if for every β g α-CS F of T and an IF point P = t(α,β) not in F there exist IF disjoint β-OSs R1, R2 of T such that P ∈ R1 and F ⊆ R2. Example 3.7 Let T = {a,b,c} and M,N are IF sets on T defined as follows: R1 = 〈t, ( a1.0, b 0.0 , c 1.0 ), ( a 0.0 , b 1.0 , c 0.0 )〉. R2 = 〈t, ( a0.0, b 1.0 , c 0.0 ), ( a 1.0 , b 0.0 , c 1.0 )〉. Rc1 = 〈t, ( a 0.0 , b 1.0 , c 0.0 ), ( a 1.0 , b 0.0 , c 1.0 )〉. Rc2 = 〈t, ( a 1.0 , b 0.0 , c 1.0 ), ( a 0.0 , b 1.0 , c 0.0 )〉. Let P = t(α,β) = a(0.5, 0.3) with P * Rc1. Then there exist IF disjoint β-OSs R1, R2 of T such that P ⊆ R1, Rc1 ⊆ R2, and R1 ⋂ R2 = 0 ∼ T . Then the family ψ = {0∼T , 1 ∼ T ,R1,R2} is an IFT on T. Which is an β g α-regular space. Definition 3.8 [20] An IFTS T is called IF strongly-regular if for each IF β g α-CS H and an IF point P = t(α,β) not in H, there exist an IF β g α-OSs U, V of T such that P ∈ U and H ⊆ V . Definition 3.9 [20] An IF topological space T is called IF strongly-normal if for each IF β g α-CSs H1 and H2, there exist an IF β g α-OSs U, V such that H1 ⊆ U and H2 ⊆ V . Since every IF β -OS is an IF β g α-OS then we have. IF β g α normal (resp.regular) space ⇒ IF strongly-normal (resp. regular) space. Lemma 3.10 An IF strongly-regularβ∗-T1/2 space is an IF β g α-regular. Int. J. Anal. Appl. (2022), 20:37 7 Proof Let (T,ψ) be an IF strongly-regular space as well as β∗ —T1/2 space. Since, (T,ψ) is a β∗-T1/2 space, then every IF β g α-CS in T is β closed i.e. the class of IF β g α-CSs and β-closed sets coincide. Now, (T,ψ) is strongly regular space which provides that for each IF β g α-CS H of T and an IF point P = t(α,β) not in H there exist IF disjoint β-OSs U, V such that p ∈ U and H ⊆ V . Combining these facts, it is concluded that for each IF β g α-CS H and each IF point P = t(α,β) there exist an IF disjoint β-OSs U and V such that H ⊆ U and p ∈ V , which turns (T,ψ) to be an IF β g α-regular.� Definition 3.11 An IF β g α space is said to be IF R0 if for IF β - OS R and each IF point P = t(α,β) ∈ R, then IF βcl{p}⊆ R. Theorem 3.12 Every IF β g α-normal-R0 space is an IF β g α-regular space. Proof Let H be an IF β gα-CS in T and an IF point P = t(α,β) in T such that P is not in H. Then, p ∈ Hc, where Hc is an IF β g α OS in T. Since T is an IF β g α-normal R0 space, we have IFβcl{p} ⊆ Hc, then H ⋂ IFβcl{p} = 0∼T . Thus H and IFβcl{p} are IF disjoint β g α-CSs in T. By β g α-normality of T, there exist IF disjoint β-OSs R1, R2 of T such that H ⊆ R1 and IF βcl{p}⊆ R2. Therefore, there exist an IFβ-OSs R1, R2 of T such that H ⊆ R1 and p ∈ R2. Hence, T is an IF β g α-regular space.� Lemma (3.13) [17] Suppose H ⊆ Y ⊆ T and (T,ψ) is an IF β g α space. If Y is open and an IF β g α-closed in (T,ψ) and H is an IF β g α-closed in (Y,TY ), then H is also β g α-closed in (T,ψ). IF β generalized α-normality is hereditary property with respect to an open and IF β generalized α-closed subspace. Theorem 3.14 If (T,ψ) is an IF β generalized α-normal space and Y is an IF open and β g α-CS of (T,ψ), then (Y,TY ) is an IF β generalized α-normal subspace. Proof Let H1 and H2 be any two IF disjoint β g α-CSs of (Y,TY ). Since Y is an IF open and β g α-CS of (T,ψ), hence, in view of Lemma (3.13), H1 and H2 are IF β g α -closed in (T,ψ), and since (T,ψ) is an IF β g α-normal, then there exist an IF disjoint β-OSs R1 and R2 of (T,ψ) such that H1 ⊆ R1 and H2 ⊆ R2. As Y is also an IF open so Y is an IF α- open and then we get R1 ⋂ Y and R2 ⋂ Y as an IF disjoint β-OSs of the IF subspace (Y,TY ) such that H1 ⊆ R1 ⋂ Y and H2 ⊆ R2 ⋂ Y . Hence, (Y,TY ) is an IF β g α-normal space.� 8 Int. J. Anal. Appl. (2022), 20:37 4. The Related Intuitionistic Fuzzy Functions with Intuitionistic Fuzzy β Generalized α Normal Spaces We start by the following definition. Definition 4.1 A function f : (T,ψ) → (Y,δ) is called: (1) IF β g α-closed if f (H) is IF β -gα-closed in Y for each IF β g α-CS H of T. (2) IF M-β-open if f (H) is an IFβ -open in Y for each IF β-OS H of T. (3) IF β g α-irresolute if f−1(H) is IF β g α-closed in T for each IFβ g α - CS H in Y. Definition 4.2 [16] An IF function f : (T,ψ) → (Y,δ) is said to be an IF β g α-continuous function if f−1(F ) is an IF β g α-CS in T for every IF-CS H in Y. Theorem 4.3 Let f : (T,ψ) → (Y,δ) be an IF continuous β gα -closed injection and if (Y,δ) is an IF β g α-normal, then (T,ψ) is an IF β g α -normal. Proof Let H1 and H2 are IF disjoint β g α-CSs in (T,ψ), since f is injective, f (H1) and f (H2) are IF β g α -CSs in (Y,δ), there exist an IF disjoint β -OSs R1 and R2 such that f (Hi ) ⊆ Ri for i = 1, 2. Since f is an IF β g α-continuous,f−1(R1) andf−1(R2) are IF β g α -CSs in (T,ψ) and Hi ⊆ f−1(Ri ) for i = 1, 2. Put Qi = IFβ − int(f−1(Ri ) for i = 1, 2. Then Q1 and Q2 are IFβ OSs with H1 ⊆ Q1 and H2 ⊆ Q2, and Q1 ⋂ Q2 = 0 ∼ T . Then (T,ψ) is an IF β g α -normal.� The following important Lemma can be proved easily. Lemma 4.4 (a) The image of IF β g α-OS under an IF-open continuous function is β g α-open. (b) The inverse image of IF β g α-O(resp.β gα-C) set under an open continuous function is IF β g α-O (resp. β g α-C) set. Proposition 4.5 The image of IF β g α-OS under IF-open and IF-closed continuous function is IF β g α-open. Proof Clearly.� Int. J. Anal. Appl. (2022), 20:37 9 Theorem 4.6 If f : (T,ψ) → (Y,δ) be an IF-open and IF-closed continuous bijection function and H be a IF β g α -CS in (Y,δ), then f−1(H) is IF β g α-CS in (T,ψ). Proof Let H be an IF β g α-CS in (Y,δ) and R be any IF β g α -OS of (T,ψ) such that f−1(H) ⊆ R. Then by the Proposition (4.5), we have f (R) is IF β g α-OS of (Y,δ) such that H ⊆ f (R). Since H is an IF β g α -CS of (Y,δ) and f (R) is IFβ g α-OS in (Y,δ), thus IF β − cl(H) ⊆ R. By Lemma (4.4) we obtain that f−1(H) ⊆ f−1(IFβ − cl(H)) ⊆ R, where f−1(IFβ − cl(H)) is β- closed in (T,ψ). This implies that IFβ − cl(f−1(H)) ⊂ R. Therefore f−1(H) is IF β g α-CS in (T,ψ).� We show that an IFβ generalized α-normality is a topological property with respect to an IF open-and-closed bijection continuous function. Theorem 4.7 An IF β g α-normality is a topological property. Proof Let (T,ψ) be an IF β g α-normal space and be an open-and-closed bijection continuous function. We need to show that (Y,δ) is IF β generalized α-normal. Let H1 and H2 be any IF disjoint β generalized α-CSs in (Y,δ). Then by the Theorem (4.6) f−1(H1) and f−1(H1) are IF disjoint β generalized α-CSs of (T,ψ). By IF β g α-normality of (T,ψ), there exist β-OSs R1 and R2 of (T,ψ) such that f−1(H1) ⊆ R1, f−1(H2) ⊆ R2 and R1 ⋂ R2 = 0 ∼ T . Then, we have H1 ⊆ f (R1), H2 ⊆ f (R2) and f (R1) ⋂ f (R2) = 0 ∼ T . Thus, f (R1) and f (R2) are IF disjoint β-OSs of (Y,δ) such that H1 ⊆ f (R1) and H2 ⊆ f (R2). Hence, (Y,δ) is IF β g α-normal. �. Theorem 4.8 If f : (T,ψ) → (Y,δ) be an IFβ g α -irresolute, M-β-open bijection function from an IF β g α -normal (T,ψ) to an IF space (Y,δ), then (Y,δ) is an IF β g α -normal space. Proof Let H1 and H2 be any two IF disjoint β g α-CSs in (Y,δ). Since f is an IF β g α -irresolute, we have f−1(H1) and f−1(H2) are IF disjoint β g α-CSs in (T,ψ). By IF β gα-normality of (T,ψ), there exist β -OSs R1 and R2 in (T,ψ) such that f−1(H1) ⊆ R1, f−1(H2) ⊆ R2 and R1 ⋂ R2 = 0 ∼ T . Since f is an IF M-β-open and bijection function, we have f (R1) and f (R2) are IF β -OSs in (Y,δ) such that H1 ⊆ f (R1), H2 ⊆ f (R2) and f (R1) ⋂ f (R2) = 0 ∼ T . Therefore, (Y,δ) is an IF β g α -normal.� Theorem 4.9 If f : (T,ψ) → (Y,δ) is an IF β g α - closed continuous surjection and (T,ψ) is an IFβ normal, then (Y,δ) is an IF β g α-normal. 10 Int. J. Anal. Appl. (2022), 20:37 Proof Since every IF β normal is an IF β g α -normal, the proof is clear.� 5. Conclusion In this paper, we introduced the concept of IF β g α normal spaces with study some of its properties. We also investigated the related intuitionistic fuzzy functions with intuitionistic fuzzy β g α normal spaces. In the future, based on some recent intuitionistic fuzzy β g α spaces studies, we will expand the research content of this paper further. Also,the entire content will be a successful tool for the researchers for finding the path to obtain the results in the context of intuitionistic fuzzy strongly-regular and strongly-normal spaces. Authors’ Contributions: All authors read and approved the final manuscript. Acknowledgment: The authors would like to express their Gratitudes to the ministry of education and the deanship of scientific research-Najran University -Kingdom of Saudi Arabia for their financial and Technical support under code number [NU/SERC/10/564]. Conflicts of Interest: The authors declare that there are no conflicts of interest regarding the publi- cation of this paper. References [1] A. Al-Qubati, H.F. Al-Qahtani, On b-Separation Axioms in Intuistionistic Fuzzy Topological Spaces, Int. J. Math. Trends Technol. 21 (2015), 83-93. [2] A. Al-Qubati, On b-Regularity and Normality in Intuistionistic Fuzzy Topological Spaces, J. Inform. Math. Sci. 9 (2017), 89-100. [3] A. Al-Qubati, On Intuitionistic Fuzzy β and β∗-Normal Spaces, Int. J. Math. Anal. 12 (2018), 517 - 531. https: //doi.org/10.12988/ijma.2018.8859. [4] A. Al-Qubati, M.E. Sayed, H.F. Al-Qahtani, Small and Large Inductive Dimensions of Intuitionistic Fuzzy Topological Spaces, Nanosci. Nanotechnol. 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Control. 8 (1965), 338-353. https://doi.org/10.1016/S0019-9958(65) 90241-X. https://doi.org/10.1016/S0165-0114(96)00076-0 https://doi.org/10.1016/S0165-0114(96)00076-0 https://doi.org/10.20454/jast.2013.458 https://doi.org/10.20454/jast.2013.458 https://doi.org/10.1016/S0019-9958(65)90241-X https://doi.org/10.1016/S0019-9958(65)90241-X 1. Introduction 2. Preliminaries 3. Intuitionistic Fuzzy Generalized Normal Spaces 4. The Related Intuitionistic Fuzzy Functions with Intuitionistic Fuzzy Generalized Normal Spaces 5. Conclusion References