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On Intutionistic Fuzzy Gpr-closed Sets

On Intutionistic Fuzzy Gpr-closed Sets Fuzzy Inf. Eng. (2012) 4: 425-444 DOI 10.1007/s12543-012-0125-x ORIGINAL ARTICLE S. S. Thakur · Jyoti Pandey Bajpai Received: 13 January 2011/ Revised: 25 October 2012/ Accepted: 15 November 2012/ © Springer-Verlag Berlin Heidelberg and Fuzzy Information and Engineering Branch of the Operations Research Society of China Abstract In this paper, a new class of intuitionistic fuzzy closed sets called in- tuitionistic fuzzy generalized preregular closed sets (briefly intuitionistic fuzzy gpr- closed sets) and intuitionistic fuzzy generalized preregular open sets (briefly intuition- istic fuzzy gpr-open sets) are introduced and their properties are studied. Further the notion of intuitionistic fuzzy preregular T -spaces and intuitionistic fuzzy general- 1/2 ized preregular continuity (briefly intuitionistic fuzzy gpr-continuity) are introduced and studied. Keywords Intuitionistic fuzzy gpr-closed sets · Gpr-open sets· Gpr-connectedness · Gpr-compactness· Gpr-continuous and gpr-irresolute mappings 1. Introduction In 1970, Levine [12] introduced the concept of g-closed sets in general topology. Many researchers like Balchandran [3], Bhattacharya and Lahiri [4], Palaninappan and Rao [13] etc. have worked on g-closed sets, their generalization in general topol- ogy. After the introduction of fuzzy sets by Zadeh [27] in 1965 and fuzzy topol- ogy by Chang [6] in 1967, several researchers were conducted on generalization of the notion of fuzzy sets and fuzzy topology. The concept of intuitionistic fuzzy sets was introduced by Atanassov [1] as a generalization of fuzzy sets. In 1997, Coker [7] introduced the concept of intuitionistic fuzzy topological spaces. In 2008, Thakur and Chaturvedi extended the concepts of fuzzy g-closed sets [17] and fuzzy g-continuity[18] in intuitionistic fuzzy topological spaces. Recently, many general- izations of intuitionistic fuzzy g-closed sets like intuitionistic fuzzy rg-closed sets [19], intuitionistic fuzzy sg-closed sets [23], intuitionistic fuzzy w-closed sets [22], intuitionistic fuzzy αg-closed sets [14], intuitionistic fuzzy gsp-closed sets [15] have been appeared in the literature. S. S. Thakur () · Jyoti Pandey Bajpai () Department of Applied Mathematics, Jabalpur Engineering College, Jabalpur, India email: samajh singh@redifmail.com yk1305@gmail.com 426 S. S. Thakur · Jyoti Pandey Bajpai (2012) In the present paper, we extend the concepts of gpr-closed sets due to Gnanam- bal [11] in intuitionistic fuzzy topology which is the weaker form of intuitionistic fuzzy ag-closed sets [14] and intuitionistic fuzzy rg-closed sets [19]. In Section 3, we define the concepts of intuitionistic fuzzy gpr-closed sets and intuitionistic fuzzy gpr-open sets and obtain some of their properties and characterizations. In Section 4, we introduce the concepts of intuitionistic fuzzy gpr-connectedness and intuitionistic fuzzy gpr-compactness. Furthermore we discuss the concepts of intuitionistic fuzzy gpr-continuous mappings and intuitionistic fuzzy gpr-irresolute mappings in Section 2. Preliminaries Let X be a nonempty fixed set. An intuitionistic fuzzy set [1] A in X is an object having the form A = {< x,μ (x),γ (x) >: x ∈ X}, where the functions μ : X → A A A [0, 1] and γ : X → [0, 1] denotes the degree of membership μ (x) and the degree A A of nonmembership γ (x) of each element x ∈ X to the set A respectively and 0 ≤ μ (x)+γ (x) ≤ 1 for each x ∈ X. The intuitionistic fuzzy sets 0 = {< x, 0, 1 >: x ∈ X} A A and 1 = {< x, 1, 0 >: x ∈ X} are respectively called empty and whole intuitionistic fuzzy set on X. An intuitionistic fuzzy set A = {< x,μ (x),γ (x) >: x ∈ X} is called A A a subset of an intuitionistic fuzzy set B = {< x,μ (x),γ (x) >: x ∈ X} (for short B B A ⊆ B)if μ (x) ≤ μ (x) and γ (x) ≥ γ (x) for each x ∈ X. The complement of A B A B an intuitionistic fuzzy set A = {< x,μ (x),γ (x) >: x ∈ X} is the intuitionistic fuzzy A A set A = {< x,γ (x),μ (x) >: x ∈ X}. The intersection (respectively union) of any A A arbitrary family of intuitionistic fuzzy sets A = {< x,μ (x),γ (x) >: x ∈ X, i ∈∧} i A A i i of X are the intuitionistic fuzzy set ∩ A = {< x, ∧ μ (x), ∨ γ (x) >: x ∈ X } (resp. i Ai Ai ∪A ={<x,∨μ (x),∧γ (x)>: x∈ X }). Two intuitionistic fuzzy sets A = {<x,μ (x), i Ai Ai A γ (x)>: x ∈ X } and B = {<x,μ (x), γ (x)>: x ∈ X } are said to be q-coincident (A B A B B q for short) if and only if ∃ an element x∈ X such that μ (x)>γ (x) or γ (x)<μ (x). A B A B A family of intuitionistic fuzzy sets on a nonempty set X is called an intuitionistic fuzzy topology [7] on X if the intuitionistic fuzzy sets 0 and 1∈ , and is closed under arbitrary union and finite intersection. The ordered pair (X, ) is called an intuitionistic fuzzy topological space and each intuitionistic fuzzy set in is called an intuitionistic fuzzy open set. The complement of an intuitionistic fuzzy open set in X is known as intuitionistic fuzzy closed set. The intersection of all intuitionistic fuzzy closed sets containing A is called the closure of A, which is denoted by cl(A). The union of all intuitionistic fuzzy open subsets of A is called the interior of A.It is denoted by int(A)[7]. Lemma 2.1 [7] Let A and B be any two intuitionistic fuzzy sets of an intuitionistic fuzzy topological space (X, ). Then (1) (A B)⇔ A⊆ B . (2) A is an intuitionistic fuzzy closed set in X ⇔ cl (A) = A. (3) A is an intuitionistic fuzzy open set in X ⇔ int(A) = A. c c (4) cl (A ) = (int (A)) . Fuzzy Inf. Eng. (2012) 4: 425-444 427 c c (5) int (A ) = (cl (A)) . (6) A⊆ B⇒ int (A) ⊆ int (B). (7) A⊆ B⇒ cl (A) ⊆ cl (B). (8) cl (A∪ B)=cl(A) ∪ cl(B). (9) int(A∩ B) = int (A) ∩ int(B). Definition 2.1 [8] Let X be a nonempty set and c∈X a fixed element in X. If α∈(0, 1] andβ∈[0, 1) are two real numbers such that α+β≤1, then (1) c(α,β) = < x,α,1- β> is called an intuitionistic fuzzy point in X, where α denotes the degree of membership of c(α,β) and β denotes the degree of non- membership of c(α,β). (2) c(β) = <x, 0,β> is called a vanishing intuitionistic fuzzy point in X, where β denotes the degree of nonmembership of c( β). Definition 2.2 [9] A family {G :i∈∧} of intuitionistic fuzzy sets in X is called an intuitionistic fuzzy open cover of X if ∪{G :i∈∧} = 1 and a finite subfamily of an intuitionistic fuzzy open cover {G :i∈∧} of X which also an intuitionistic fuzzy open cover of X is called a finite subcover of {G :i∈∧}. Definition 2.3 [9] An intuitionistic fuzzy topological space (X, ) is called intuition- istic fuzzy compact if every intuitionistic fuzzy open cover of X has a finite subcover. Definition 2.4 [10] An intuitionistic fuzzy set A of an intuitionistic fuzzy topological space (X, ) is called (1) Intuitionistic fuzzy semi-open if A ⊆ cl(int(A)) and intuitionistic fuzzy semi- closed if int(cl(A)) ⊆ A. (2) Intuitionistic fuzzy α-open if A ⊆ int(cl(int(A))) and intuitionistic fuzzy α- closed if cl(int(cl(A))) ⊆ A. (3) Intuitionistic fuzzy preopen if A ⊆ int(cl(A)) and intuitionistic fuzzy preclosed if cl(int(A)) ⊆ A. (4) Intuitionistic fuzzy regular open if A = int(cl(A)) and intuitionistic fuzzy reg- ular closed if A = cl(int(A)). Definition 2.5 [26] An intuitionistic fuzzy set A in intuitionistic topological spaces (X, ) is said to be (1) Intuitionistic fuzzy semi-preopen if there exists an intuitionistic fuzzy preopen set B such that B ⊆ A ⊆ cl(B). (2) Intuitionistic fuzzy semi-preclosed if there exists an intuitionistic fuzzy pre- closed set B such that int(B) ⊆ A ⊆ B. Remark 2.1 [10] If (X, ) is an intuitionistic fuzzy topological space, then 428 S. S. Thakur · Jyoti Pandey Bajpai (2012) (1) Every intuitionistic fuzzy regular closed set in X is intuitionistic fuzzy closed in X. (2) Every intuitionistic fuzzy closed set in X is intuitionistic fuzzyα-closed in X. (3) Every intuitionistic fuzzy α-closed set in X is intuitionistic fuzzy preclosed in X. Definition 2.6 If A is an intuitionistic fuzzy set in intuitionistic fuzzy topological space(X, ), then (1) scl (A) = ∩ { F: A⊆ F, F is intuitionistic fuzzy semi-closed} [10]. (2) αcl (A) = ∩ { F: A ⊆ F, F is intuitionistic fuzzyα-closed} [10]. (3) pcl(A) = ∩ { F: A ⊆ F, F is intuitionistic fuzzy preclosed} [10]. (4) spcl (A) = ∩ { F: A ⊆ F, F is intuitionistic fuzzy semi-preclosed} [26]. Definition 2.7 An intuitionistic fuzzy set A of an intuitionistic fuzzy topological space (X, ) is called (1) Intuitionistic fuzzy g-closed [17] if cl(A) ⊆ O whenever A ⊆ O and O is intu- itionistic fuzzy open in X. (2) Intuitionistic fuzzy sg-closed [23] if scl(A) ⊆ O whenever A ⊆ and O is intu- itionistic fuzzy semi-open in X. (3) Intuitionistic fuzzy rg-closed [19] if cl(A) ⊆ O whenever A ⊆ O and O is intu- itionistic fuzzy regular open in X. (4) Intuitionistic fuzzy αg-closed [14] if αcl(A) ⊆ O whenever A ⊆ O and O is intuitionistic fuzzy open in X. (5) Intuitionistic fuzzy gsp-closed[15] if spcl(A) ⊆ O whenever A ⊆ OandOis intuitionistic fuzzy open in X. (6) Intuitionistic fuzzy w-closed [22] if cl(A) ⊆ O whenever A ⊆ O and O is intu- itionistic fuzzy semi-open in X. The complements of the above mentioned closed sets are their respective open sets. Remark 2.2 If (X, ) is an intuitionistic fuzzy topological space, then (1) Every intuitionistic fuzzy closed set in X is intuitionistic fuzzy g-closed but its converse may not be true, because in intuitionistic fuzzy topological space (X, ) where X = {a, b}, U = {< a, 0.5, 0.4 >,< b, 0.6, 0.4 >} and = {0, U, 1}, the intuitionistic fuzzy set A = {< a, 0.3, 0.6 >,< b, 0.4, 0.6 >} is intuitionistic fuzzy g-closed, but it is not intuitionistic fuzzy closed [17]. Fuzzy Inf. Eng. (2012) 4: 425-444 429 (2) Every intuitionistic fuzzy α-closed set in X is intuitionistic fuzzy αg-closed but its converse may not be true, because in intuitionistic fuzzy topological space (X, ) where X = {a, b}, U = {< a, 0.4, 0.6 >,< b, 0.2, 0.7 >}, V = {< a, 0.8, 0.2 >,< b, 0.8, 0.2 >} and = {0 , U, V, 1}, the intuitionistic fuzzy set A = {< a, 0.5, 0.4 >,< b, 0.4, 0.5 >} is intuitionistic fuzzy αg-closed, but it is not intuitionistic fuzzy α-closed [14]. (3) Every intuitionistic fuzzy g-closed set in X is intuitionistic fuzzy rg-closed but its converse may not be true, because in intuitionistic fuzzy topological space (X, ) where X = {a, b}, U = {< a, 0.5, 0.4 >,< b, 0.6, 0.4 >} and = {0, U, 1}, the intuitionistic fuzzy set A = {< a, 0.5, 0.4 >,< b, 0.4, 0.5 >} is intuitionistic fuzzy rg-closed, but it is not intuitionistic fuzzy g-closed [19]. Definition 2.8 If (X, ) is an intuitionistic fuzzy topological space, then (1) (X, ) is called intuitionistic fuzzy GO-connected if there is no proper intuition- istic fuzzy set A (i.e., A  0 and A  1) of X which is both intuitionistic fuzzy g-open and intuitionistic fuzzy g-closed [17]. (2) (X, ) is called intuitionistic fuzzy rg-connected if there is no proper intuition- istic fuzzy set of X which is both intuitionistic fuzzy rg-open and intuitionistic fuzzy rg-closed [19]. Definition 2.9 [10] Let X and Y be two nonempty sets and f : X→ Y be a function: (1) IfB= {< y,μ (y), γ (y) > :y ∈ Y} is an intuitionistic fuzzy set in Y, then the B B −1 preimage of B under f denoted by f (B), is the intuitionistic fuzzy set in X −1 −1 −1 defined by f (B) = {<x, f (μ ) (x), f (γ ) (x)> :x ∈ X}. B B (2) If A = {< x, λ (x), ν (x) > :x ∈ X} is an intuitionistic fuzzy set in X, then the A A image of A under f denoted by f(A) is the intuitionistic fuzzy set in Y defined by f (A) = {< y, f (λ )(y), (1− f (1−ν ))(y) >: y ∈ Y}, A A where −1 ( ) ⎪ sup λ (x), if f y  φ, −1 x∈ f (y) f (λ ) (y) = A ⎪ 0, otherwise, −1 inf ν (x), if f (y)  φ, ⎪ A −1 x∈ f (y) (1 - f (1- ν ))(y) = 1, otherwise. Definition 2.10 Let (X, ) and (Y, σ) be two intuitionistic fuzzy topological spaces and let f: X→Y be a function. Then f is said to be 430 S. S. Thakur · Jyoti Pandey Bajpai (2012) (1) Intuitionistic fuzzy continuous if the preimage of each intuitionistic fuzzy open set of Y is an intuitionistic fuzzy open set in X. (2) Intuitionistic fuzzy irresolute if the preimage of every intuitionistic fuzzy semi- open set of Y is intuitionistic fuzzy semi-open in X. (3) Intuitionistic fuzzy precontinuous if the preimage of every intuitionistic fuzzy open set of Y is intuitionistic fuzzy preopen in X. (4) Intuitionistic fuzzy almost continuous if the preimage of each intuitionistic fuzzy regular open set of Y is an intuitionistic fuzzy open set in X. (5) Intuitionistic fuzzy closed if the image of every intuitionistic fuzzy closed set of X is intuitionistic fuzzy closed in Y. (6) Intuitionistic fuzzy preclosed if the image of every intuitionistic fuzzy closed set of X is intuitionistic fuzzy preclosed in Y. (7) Intuitionistic fuzzy preregular-closed if the image of every intuitionistic fuzzy regular closed of X is intuitionistic fuzzy regular closed in X. Definition 2.11 Let (X, ) and (Y, σ) be two intuitionistic fuzzy topological spaces and let f : X→Y be a function. Then f is said to be (1) Intuitionistic fuzzy g-continuous if the preimage of every intuitionistic fuzzy closed set in Y is intuitionistic fuzzy g-closed in X [18]. (2) Intuitionistic fuzzy sg-continuous if the preimage of every intuitionistic fuzzy closed set in Y is intuitionistic fuzzy sg-closed in X [23]. (3) Intuitionistic fuzzy αg-continuous if the preimage of every intuitionistic fuzzy closed set in Y is intuitionistic fuzzy αg-closed in X [14]. (4) Intuitionistic fuzzy rg-continuous if the preimage of every intuitionistic fuzzy closed set in Y is intuitionistic fuzzy rg-closed in X [20]. (5) Intuitionistic fuzzy R map if the preimage of each intuitionistic fuzzy regular open set of Y is an intuitionistic fuzzy regular open set in X [19]. 3. Intuitionistic Fuzzy Gpr-closed Sets and Intuitionistic Fuzzy Gpr-open Sets Definition 3.1 An intuitionistic fuzzy set A of an intuitionistic fuzzy topological space (X, ) is called intuitionistic fuzzy generalized preregular closed set (briefly intuition- istic fuzzy gpr-closed ) if pcl(A) ⊆ U, whenever A ⊆ U and U is intuitionistic fuzzy regular open in X. Theorem 3.1 Every intuitionistic fuzzy rg-closed set in intuitionistic fuzzy topological space (X, ) is intuitionistic fuzzy gpr-closed set in X. Proof Let A be an intuitionistic fuzzy rg-closed set in X. Let A ⊆ U and U be intuitionistic fuzzy regular open in X. Then cl(A) ⊆ U because A is intuitionistic Fuzzy Inf. Eng. (2012) 4: 425-444 431 fuzzy rg-closed in X. Since every intuitionistic fuzzy closed set is intuitionistic fuzzy pre closed, pcl(A) ⊆ cl(A). Therefore, pcl(A) ⊆ U . Hence A is intuitionistic fuzzy gpr-closed. Remark 3.1 The converse of the Theorem 3.1 need not be true as seen from the following example. Example 3.1 Let X = {a, b, c, d, e} and intuitionistic fuzzy sets O, U, V defined as follows: O = {< a, 0.9, 0.1 >, < b, 0.8, 0.1 >,< c,0,1 >,< d,0,1 >,< e,0,1>}, U = {< a,0,1>, < b,0,1>, < c, 0.8, 0.1 >,< d, 0.7, 0.2>,< e,0,1 >}, V = {< a, 0.9, 0.1 >, < b, 0.8, 0.1>,< c, 0.8, 0.1>, < d, 0.7, 0.2>,< e,0,1 >}. Let = {0, O, U, V, 1} be an intuitionistic fuzzy topology on X. Then the intu- itionistic fuzzy set A = {< a, 0.9, 0.1 >, < b,0,1 >, < c,0,1 >, < d,0,1 >, < e,0, 1>} is intuitionistic fuzzy gpr-closed, but it is not intuitionistic fuzzy rg-closed. Theorem 3.2 Every intuitionistic fuzzy αg-closed set in intuitionistic fuzzy topologi- cal space (X, )is intuitionistic fuzzy gpr-closed set in X. Proof Let A be an intuitionistic fuzzy αg-closed set in X. Let A ⊆ U and U be intuitionistic fuzzy regular open set in X. Then, αcl(A) ⊆ U because every intuition- istic fuzzy regular open set is intuitionistic fuzzy open and A is intuitionistic fuzzy αg-closed in X. Now pcl(A) ⊆ αcl(A) ⊆ U implies that pcl(A) ⊆ U . Hence A is intuitionistic fuzzy gpr-closed. Remark 3.2 The converse of the Theorem 3.2 need not be true as seen from the following example. Example 3.2 Let X = {a, b, c} and intuitionistic fuzzy sets O, U, V defined as follows: O = {< a, 0.9, 0.1 >, < b,0,1 >,< c,0,1 >}, U = {< a,0,1>,< b, 0.8, 0.1>,< c,0,1>}, V = {< a, 0.9, 0.1 >, < b, 0.8, 0.1>,< c,0,1>}. Let = {0 , O, U, V, 1} be an intuitionistic fuzzy topology on X. Then intuition- istic fuzzy set A= {< a, 0.9, 0.1 >, < b, 0.8, 0.1 >, < c,0,1 >} is intuitionistic fuzzy gpr-closed set in X, but it is not intuitionistic fuzzyαg-closed. Remark 3.3 From the Theorems 3.1, 3.2 and Remarks 2.1, 2.2, we have the following diagram of implication. Intuitionistic fuzzy←Intuitionistic fuzzy →Intuitionistic fuzzy α-closed closed g-closed ↓↓ ↓ Intuitionistic fuzzy→Intuitionistic fuzzy←Intuitionistic fuzzy αg-closed gpr-closed rg-closed Theorem 3.3 Let (X, ) be an intuitionistic fuzzy topological space and A be an intuitionistic fuzzy regular-closed set of X. Then A is intuitionistic fuzzy gpr-closed if 432 S. S. Thakur · Jyoti Pandey Bajpai (2012) and only if (AqF) ⇒ (pcl(A)qF) for every intuitionistic fuzzy regular closed set F of X. Proof Necessity: Let A be intuitionistic fuzzy gpr-closed set. Let F be an intu- itionistic fuzzy regular closed set of X and (AqF). Then by Lemma 2.1(1), A ⊆ F c c and F intuitionistic fuzzy regular open in X. Therefore, pcl(A) ⊆ F because A is intuitionistic fuzzy gpr-closed. Hence by Lemma 2.1(1), (pcl(A)qF). Sufficiency: Let O be an intuitionistic fuzzy regular open set of X such that A ⊆ O c c c c i.e., A ⊆ ((O) ) . Then by Lemma 2.1(1), (A O ) and O is an intuitionistic fuzzy regular closed set in X. Hence by hypothesis (pcl(A) O ). Therefore by Lemma c c 2.1(1), pcl(A) ⊆ ((O) ) i.e., pcl(A) ⊆ O. Hence A is intuitionistic fuzzy gpr-closed in X. Theorem 3.4 Let A be an intuitionistic fuzzy gpr-closed set in an intuitionistic fuzzy topological space (X, ) and c(α,β) be an intuitionistic fuzzy point of X such that c(α,β) pcl(A). Then pcl(c(α,β))qA. Proof If pcl(c(α,β)) A, then by Lemma 2.1(1), pcl(c(α,β)) ⊆ A which implies that c c c A ⊆ (pcl(c(α,β))) and so pcl(A) ⊆ (pcl(c(α,β))) ⊆ (c(α,β)) , because A is intuitionis- tic fuzzy gpr-closed in X. Hence by Lemma 2.1(1), (c(α,β) (pcl(A))), a contradiction. Remark 3.4 The intersection of two intuitionistic fuzzy gpr-closed sets in an intu- itionistic fuzzy topological space (X, ) may not be intuitionistic fuzzy gpr-closed as seen from the following example. Example 3.3 Let X = {a, b, c} and intuitionistic fuzzy sets O, U, V defined as follows: O = {< a, 0.9, 0.1>, < b,0,1 >, < c,0,1>}, U = {< a,0,1 >, < b, 0.8, 0.1>, < c,0,1>}, V = {< a, 0.9, 0.1>, < b, 0.8, 0.1 >, < c,0,1 >}. Let = {0, O, U, V, 1} be an intuitionistic fuzzy topology on X. Then intuitionistic fuzzy set A = {< a, 0.9, 0.1 >,< b, 0.8, 0.1 >, < c,0,1 >} and B = { < a, 0.9, 0.1 >, < b,0,1>,< c, 0.8, 0.1>} are intuitionistic fuzzy gpr-closed sets in (X, )but A∩ B is not intuitionistic fuzzy gpr-closed. Theorem 3.5 Let A be an intuitionistic fuzzy gpr-closed set in an intuitionistic fuzzy topological space (X, ) and A⊆ B⊆ pcl(A). Then B is intuitionistic fuzzy gpr-closed in X. Proof Let A be an intuitionistic fuzzy gpr-closed set in an intuitionistic fuzzy topo- logical space (X, ) such that A ⊆ B ⊆ pcl(A). Let O be an intuitionistic fuzzy regular open set such that B ⊆ O. Then A ⊆ O and since A is intuitionistic fuzzy gpr-closed, we have pcl(A) ⊆ O.Now B ⊆ pcl(A) ⇒ pcl(B) ⊆ pcl(pcl(A)) ⊆ pcl(A) ⊆ O. Conse- quently, B is intuitionistic fuzzy gpr-closed in X. Theorem 3.6 If A is an intuitionistic fuzzy regular open and intuitionistic fuzzy gpr- closed set in intuitionistic fuzzy topological space (X, ), then A is an intuitionistic fuzzy preclosed and hence intuitionistic fuzzy clopen. Proof Suppose that A is an intuitionistic fuzzy regular open and intuitionistic fuzzy gpr-closed in X. Since A ⊆ A, we have pcl(A) ⊆ A. Also A ⊆ pcl(A). Therefore pcl(A) Fuzzy Inf. Eng. (2012) 4: 425-444 433 = A. Hence A is an intuitionistic fuzzy preclosed in X.Now A is an intuitionistic fuzzy regular open, A is intuitionistic fuzzy open. Hence A is intuitionistic fuzzy clopen, since every intuitionistic fuzzy preclosed (regular) set is intuitionistic fuzzy (regular) closed. Definition 3.2 An intuitionistic fuzzy set A of an intuitionistic fuzzy topological space (X, ) is called intuitionistic fuzzy gpr-open if and only if its complement A is intu- itionistic fuzzy gpr-closed. Theorem 3.7 Every intuitionistic fuzzy rg-open set in intuitionistic fuzzy topological space (X, ) is intuitionistic fuzzy gpr-open set in X. Proof Let A be an intuitionistic fuzzy rg-open set in X. Then A is intuitionistic fuzzy rg-closed. By Theorem 3.1, A is intuitionistic fuzzy gpr-closed in X. Therefore, A is intuitionistic fuzzy gpr-open in X. Remark 3.5 The converse of the Theorem 3.7 need not be true as seen from the following example. Example 3.4 Let X = {a, b, c, d, e} and intuitionistic fuzzy sets O, U, V defined as follows: O = {< a, 0.9, 0.1 >, < b, 0.8, 0.1>,< c,0,1 >,< d,0,1 >,< e,0,1>}, U = {< a,0,1>, < b,0,1>, < c, 0.8, 0.1 >,< d, 0.7, 0.2>,< e,0,1 >}, V = {< a, 0.9, 0.1>, < b, 0.8, 0.1>,< c, 0.8, 0.1>,<d, 0.7, 0.2>,<e,0,1>}. Let = {0 , O, U, V, 1} be an intuitionistic fuzzy topology on X. Then the intuitionistic fuzzy set A= {< a, 0.1, 0.9>, < b,1,0>, <c,1,0>, < d,1,0>, < e,1, 0>} is intuitionistic fuzzy gpr-open but it is not intuitionistic fuzzy rg-open. Theorem 3.8 Every intuitionistic fuzzyαg-open set in intuitionistic fuzzy topological space (X, ) is intuitionistic fuzzy gpr-open set in X, but not conversely. Proof Let A be an intuitionistic fuzzyαg-open set in X. Then A is intuitionistic fuzzy αg-closed. By Theorem 3.2, A is intuitionistic fuzzy gpr-closed in X. Therefore A is intuitionistic fuzzy gpr-open in X. Remark 3.6 The converse of the Theorem 3.8 need not be true as seen from the following example. Example 3.5 Let X = {a, b, c} and intuitionistic fuzzy sets O, U, V defined as follows: O = {< a, 0.9, 0.1 >, < b,0,1 >,< c,0,1 >}, U = {< a,0,1>, < b, 0.8, 0.1 >,< c,0,1>}, V = {< a, 0.9, 0.1 >, < b, 0.8, 0.1>,< c,0,1 >}. Let = {0, O, U, V, 1} be an intuitionistic fuzzy topology on X. Then intuitionistic fuzzy set A = {< a, 0.1, 0.9 >,< b, 0.1, 0.8>,< c,1,0>} is intuitionistic fuzzy gpr- open set in X, but it is not intuitionistic fuzzy αg-open. Theorem 3.9 An intuitionistic fuzzy set A of an intuitionistic fuzzy topological space (X, ) is intuitionistic fuzzy gpr-open if and only if F ⊆ pint(A) whenever F is intu- itionistic fuzzy regular closed and F ⊆A. Proof Necessity: Let A be intuitionisticfuzzy gpr-open in X. Let F be intuitionistic fuzzy regular closed in X such that F ⊆ A. Then F is intuitionistic fuzzy regular open 434 S. S. Thakur · Jyoti Pandey Bajpai (2012) c c c in X such that A ⊆ F . Now by hypothesis A is intuitionistic fuzzy gpr-closed, we c c c c c c have pcl (A ) ⊆ F . But pcl (A ) = (pint(A)) . Hence (pint(A)) ⊆ F , which implies F⊆pint(A). Sufficiency: Let O be an intuitionistic fuzzy regular open set in X such that A ⊆ c c O. Then O is an intuitionistic fuzzy regular closed in X and O ⊆ A. Therefore by c c c c hypothesis O ⊆ pint(A). This implies that pcl(A ) = (pint(A)) ⊆ O. Hence A is intuitionistic fuzzy gpr-closed and A is intuitionistic fuzzy gpr-open in X. Theorem 3.10 Let A be an intuitionistic fuzzy gpr-open set of an intuitionistic fuzzy topological space (X, ) and pint(A) ⊆ B⊆ A. Then B is intuitionistic fuzzy gpr-open. c c c c c c Proof Since pint(A) ⊆ B ⊆ A. ⇒ A ⊆ B ⊆ (pint(A)) ⇒ A ⊆ B ⊆ pcl(A )by Lemma 2.1(4) and A is intuitionistic fuzzy gpr-closed, it follows from Theorem 3.5 that B is intuitionistic fuzzy gpr-closed. Hence B is intuitionistic fuzzy gpr-open. Theorem 3.11 Let (X, ) be an intuitionistic fuzzy topological space and IFPO(X) (resp. IFGPRO(X)) be the family of all intuitionistic fuzzy preopen (resp. intuitionis- tic fuzzy gpr-open) sets of X. Then IFPO(X) ⊆ IFGPRO(X). Proof Let A ∈ IFPO(X). Then A is intuitionistic fuzzy preclosed and so intuitionistic fuzzy gpr-closed. This implies that A is intuitionistic fuzzy gpr-open. Hence IFPO(X) ⊆ IFGPRO(X). Theorem 3.12 Let (X, ) be an intuitionistic fuzzy topological space and IFPC(X) ( resp. IFRO(X)) be the family of all intuitionistic fuzzy preclosed (resp. intuitionistic fuzzy regular open) sets of X. Then IFPC(X) =IFRO(X) if and only if every intuition- istic fuzzy set of X is intuitionistic fuzzy gpr-closed in X. Proof Necessity: Suppose that IFPC(X) = IFRO(X) and A is any intuitionistic fuzzy set of X such that A ⊆ O where O is intuitionistic fuzzy regular open in X. Then by hypothesis O is intuitionistic fuzzy preclosed in X which implies that pcl(O)= O . Then pcl(A) ⊆ pcl(O)= O. Therefore A is intuitionistic fuzzy gpr-closed in X. Sufficiency: Suppose that every intuitionistic fuzzy set of X is intuitionistic fuzzy gpr-closed. Let U ∈ IFRO(X). Then since U ⊆ U and by hypothesis U is intuitionistic fuzzy gpr-closed set in X. Therefore pcl(U ) ⊆ U , hence U is intuitionistic fuzzy preclosed. That is U ∈ IFPC(X) which implies that IFRO(X) ⊆ IFPC(X). If T ∈ IFPC, then T ∈ IFPO(X) ⊆ IFRO(X) ⊆ IFPC(X). Hence T ∈ IFPO(X) ⊆ IFRO(X). Consequently, IFPC(X) ⊆ IFRO(X). Therefore IFRO(X) = IFPC(X). Theorem 3.13 Let A be an intuitionistic fuzzy g-closed set in an intuitionistic fuzzy topological space (X, ) and f : (X, ) → (Y, σ) be an intuitionistic fuzzy almost continuous and intuitionistic fuzzy preclosed mapping. Then f (A) is an intuitionistic gpr-closed set in Y. Proof Let A be an intuitionistic fuzzy g-closed set in X and f :(X, ) → (Y,σ)be an intuitionistic fuzzy almost continuous and intuitionistic fuzzy preclosed mapping. −1 Let f (A) ⊆ G where G is intuitionistic fuzzy regular open in Y . Then A ⊆ f (G) −1 and f (G) is intuitionistic fuzzy open in X, since f is intuitionistic fuzzy almost Fuzzy Inf. Eng. (2012) 4: 425-444 435 continuous. Now let A be an intuitionistic fuzzy g-closed set in X. Then cl(A) ⊆ −1 −1 f (G). Since pcl(A) ⊆ cl(A), hence pcl(A) ⊆ f (G). Thus f (pcl(A)) ⊆ G and f (pcl(A)) is an intuitionistic fuzzy preclosed set in Y , since pcl(A) is intuitionistic fuzzy preclosed in X and f is intuitionistic fuzzy preclosed mapping. It follows that pcl( f (A) ⊆ pcl(f (pcl(A))) = f (pcl(A))⊆ G. Hence pcl(f (A)) ⊆ G whenever f (A) ⊆ G and G is intuitionistic fuzzy regular open in Y . Hence f (A) is intuitionistic fuzzy gpr-closed set in Y . Corollary 3.1 Let A be an intuitionistic fuzzy g-closed set in an intuitionistic fuzzy topological space (X, ) and f : (X, )→ (Y,σ) be an intuitionistic fuzzy continuous and intuitionistic fuzzy closed mapping. Then f (A) is an intuitionistic gpr-closed set in Y. Theorem 3.14 Let A be an intuitionistic fuzzy gpr-closed set in an intuitionistic fuzzy topological space (X, ) and f : (X, )→ (Y,σ) be an intuitionistic fuzzy R-mapping and intuitionistic fuzzy preclosed mapping. Then f (A) is an intuitionistic gpr-closed set in Y. Proof Let A be an intuitionistic fuzzy gpr-closed set in X and f :(X, ) → (Y,σ) be an intuitionistic fuzzy R-mapping and intuitionistic fuzzy preclosed mapping. Let −1 f (A) ⊆ G where G is intuitionistic fuzzy regular open in Y . Then A ⊆ f (G) and −1 f (G) is intuitionistic fuzzy regular open in X, since f is intuitionistic fuzzy R- −1 mapping. Since A is an intuitionistic fuzzy gpr-closed set in X, pcl(A) ⊆ f (G). Now f (pcl(A)) is an intuitionistic fuzzy preclosed set in Y , since pcl(A) is intuitionistic fuzzy preclosed in X and f is intuitionistic fuzzy preclosed mapping. It follows that pcl(f (A) ⊆ pcl(f (pcl(A))) = f (pcl(A)) ⊆ G. Hence pcl(f (A)) ⊆ G. Hence f (A)is intuitionistic fuzzy gpr-closed set in Y . Theorem 3.15 If f : (X, ) → (Y, σ) is an intuitionistic fuzzy g-continuous and intuitionistic fuzzy preregular-closed mapping, then preimage of every intuitionistic fuzzy gpr-closed set of Y is intuitionistic fuzzy gpr-closed in X. Proof Let B be an intuitionistic fuzzy gpr-closed set in Y and f :(X, )→(Y,σ)be an intuitionistic fuzzy g-continuous and intuitionistic fuzzy preregular closed map- −1 c ping. Let f (B) ⊆ G where G is intuitionistic fuzzy regular open in X then f (G ) c c ⊆ B where G is intuitionistic fuzzy regular closed in X. Since f is intuitionis- tic fuzzy preregular closed, f (G ) is intuitionistic fuzzy regular closed in Y.Now c c c c B is intuitionistic fuzzy gpr-open in Y such that f (G ) ⊆ B where f (G ) is intu- c c c itionistic fuzzy regular closed in Y . Therefore f (G ) ⊆ pint(B ) = (pcl(B)) . Hence −1 −1 f (pcl(B) ⊆ G. Since f is intuitionistic fuzzy g-continuous, f ( pcl(B)) is intu- −1 itionistic fuzzy g-closed in X. Thus we have cl (f ( pcl(B)) ⊆ G. Therefore pcl −1 −1 −1 −1 (f (B)) ⊆ pcl( pcl (f (B))) ⊆ cl(f ( pcl(B)) ⊆ G. Hence f (B) is intuitionistic fuzzy gpr-closed in X. Corollary 3.2 If f : (X, ) → (Y, σ) is an intuitionistic fuzzy continuous and intu- itionistic fuzzy preregular-closed mapping, then preimage of every intuitionistic fuzzy gpr-closed set of Y is intuitionistic fuzzy gpr-closed in X. 436 S. S. Thakur · Jyoti Pandey Bajpai (2012) Theorem 3.16 If f : (X, ) → (Y, σ) is an intuitionistic fuzzy g-continuous and intuitionistic fuzzy preregular-closed mapping, then preimage of every intuitionistic fuzzy gpr-open set of Y is intuitionistic fuzzy gpr-open in X. Proof Proof follows from Theorem 3.15. Corollary 3.3 If f : (X, ) → (Y, σ) is an intuitionistic fuzzy continuous and in- tuitionistic fuzzy preregular-closed mapping. Then preimage of every intuitionistic fuzzy gpr-open set of Y is intuitionistic fuzzy gpr-open in X. 4. Intuitionistic Fuzzy Gpr-connectedness and Intuitionistic Fuzzy Gpr-compact- ness Definition 4.1 An intuitionistic fuzzy topological space (X, ) is called intuitionistic fuzzy gpr-connected if there is no proper intuitionistic fuzzy set of X which is both intuitionistic fuzzy gpr-open and intuitionistic fuzzy gpr-closed. Theorem 4.1 Every intuitionistic fuzzy gpr-connected space is intuitionistic fuzzy connected. Proof Let (X, ) be an intuitionistic fuzzy gpr-connected space and suppose that (X, ) is not intuitionistic fuzzy connected. Then there exists a proper intuitionistic fuzzy set A ( A  0, A  1) such that A is both intuitionistic fuzzy open and intuition- istic fuzzy closed. Since every intuitionistic fuzzy open set (resp. intuitionistic fuzzy closed set) is intuitionistic gpr-open (resp. intuitionistic fuzzy gpr-closed), X is not intuitionistic fuzzy gpr-connected, a contradiction. Remark 4.1 Converse of Theorem 4.1 may not be true as seen from the following example. Example 4.1 Let X = {a, b} and = {0, U , 1 } be an intuitionistic fuzzy topol- ogy on X, where U = {< a, 0.5, 0.5 >, < b, 0.4, 0.6 >}. Then intuitionistic fuzzy topological space (X, ) is intuitionistic fuzzy connected but not intuitionistic fuzzy gpr-connected because there exists a proper intuitionistic fuzzy set A = {<a, 0.5, 0.5 >, < b, 0.5, 0.5 >} which is both intuitionistic fuzzy gpr-closed and intuitionistic gpr-open in X. Theorem 4.2 Every intuitionistic fuzzy gpr-connected space is intuitionistic fuzzy GO-connected. Proof Let (X, ) be an intuitionistic fuzzy gpr-connected space and suppose that (X, ) is not intuitionistic fuzzy GO-connected. Then there exists a proper intuition- istic fuzzy set A (A  0, A  1 ) such that A is both intuitionistic fuzzy g-open and intuitionistic fuzzy g-closed. Since every intuitionistic fuzzy g-open set (resp. intuitionistic fuzzy g-closed set) is intuitionistic gpr-open (resp. intuitionistic fuzzy gpr-closed), X is not intuitionistic fuzzy gpr-connected, a contradiction. Theorem 4.3 Every intuitionistic fuzzy gpr-connected space is intuitionistic fuzzy rg-connected. Proof Let (X, ) be an intuitionistic fuzzy gpr-connected space and suppose that (X, ) is not intuitionistic fuzzy rg-connected. Then there exists a proper intuition- istic fuzzy set A ( A  0, A  1 ) such that A is both intuitionistic fuzzy rg-open Fuzzy Inf. Eng. (2012) 4: 425-444 437 and intuitionistic fuzzy rg-closed. Since every intuitionistic fuzzy rg-open set (resp. intuitionistic fuzzy rg-closed set) is intuitionistic fuzzy gpr-open (resp. intuitionistic fuzzy gpr-closed), X is not intuitionistic fuzzy gpr-connected, a contradiction. Theorem 4.4 An intuitionistic fuzzy topological (X, ) is intuitionistic fuzzy gpr- connected if and only if there exists no nonempty intuitionistic fuzzy gpr-open sets A and B in X such that A=B . Proof Necessity: Suppose A and B are intuitionistic fuzzy gpr-open sets such that c c A  0  B and A = B . Since A = B , B is an intuitionistic fuzzy gpr-open set which c c implies that B = A is intuitionistic fuzzy gpr-closed set and B  0, this implies that B 1 i.e., A  1 . Hence there exists a proper intuitionistic fuzzy set A ( A  0, A  1 ) such that A is both intuitionistic fuzzy gpr-open and intuitionistic fuzzy gpr-closed. But this is contradiction to the fact that X is intuitionistic fuzzy gpr-connected. Sufficiency: Let (X, ) be an intuitionistic fuzzy topological space and A be both intuitionistic fuzzy gpr-open set and intuitionistic fuzzy gpr-closed set in X such that 0  A  1 . Now take B = A . In this case, B is an intuitionistic fuzzy gpr-open set and A  1 , this implies that B =A  0 . Hence, A  1 which is a contradiction. Hence there is no proper intuitionistic fuzzy set of X which is both intuitionistic fuzzy gpr- open and intuitionistic fuzzy gpr-closed. Therefore intuitionistic fuzzy topological (X , ) is intuitionistic fuzzy gpr-connected. Definition 4.2 An intuitionistic fuzzy topological space (X, ) is said to be intuitionis- tic fuzzy preregular-T if every intuitionistic fuzzy gpr-closed set in X is intuitionistic 1/2 fuzzy preclosed in X. Definition 4.3 A collection { A :i∈ Λ} of intuitionistic fuzzy rga-open sets in intu- itionistic fuzzy topological space (X, ) is called intuitionistic fuzzy gpr-open cover of intuitionistic fuzzy set B of X if B⊆∪{ A :i∈ Λ}. Definition 4.4 An intuitionistic fuzzy topological space (X, ) is said to be intuition- istic fuzzy gpr-compact if every intuitionistic fuzzy gpr-open cover of X has a finite subcover. Definition 4.5 An intuitionistic fuzzy set B of intuitionistic fuzzy topological space (X, ) is said to be intuitionistic fuzzy gpr-compact relative to X if for every collection { A :i∈ Λ} of intuitionistic fuzzy gpr-open subset of X such that B⊆∪{ A :i∈ Λ }, i i there exists finite subset Λ ofΛ such that B⊆∪{ A :i∈ Λ }. o i o Definition 4.6 A crisp subset B of an intuitionistic fuzzy topological space (X, )is said to be intuitionistic fuzzy gpr-compact if B is intuitionistic fuzzy gpr-compact as an intuitionistic fuzzy subspace of X. Theorem 4.5 An intuitionistic fuzzy gpr-closed crisp subset of intuitionistic fuzzy gpr-compact space is intuitionistic fuzzy gpr-compact relative to X. Proof Let A be an intuitionistic fuzzy gpr-closed crisp subset of intuitionistic fuzzy gpr-compact space (X, ). Then A is intuitionistic fuzzy gpr-open in X. Let M be a cover of A by intuitionistic fuzzy gpr-open sets in X. Then the family {M, A } is intuitionistic fuzzy gpr-open cover of X. Since X is intuitionistic fuzzy gpr-compact, 438 S. S. Thakur · Jyoti Pandey Bajpai (2012) it has a finite subcover say {G , G , G ,··· , G }. If this subcover contains A ,we 1 2 3 n discard it. Otherwise leave the subcover as it is. Thus we obtain a finite intuitionistic fuzzy gpr-open subcover of A. Therefore A is intuitionistic fuzzy gpr-compact relative to X. Theorem 4.6 If f : (X, ) → (Y, σ) is an intuitionistic fuzzy g-continuous and intu- itionistic fuzzy preregular-closed surjection and X is intuitionistic fuzzy gpr-compact, then Y is intuitionistic fuzzy gpr-compact. Proof Obvious. 5. Intuitionistic Fuzzy Gpr-continuous and Intuitionistic Fuzzy Gpr-irresolute Mappings Definition 5.1 A mapping f : (X, ) →(Y, σ) is intuitionistic fuzzy gpr-continuous if preimage of every intuitionistic fuzzy closed set of Y is intuitionistic fuzzy gpr-closed set in X. Theorem 5.1 A mapping f : (X, ) →(Y, σ) is intuitionistic fuzzy gpr-continuous if and only if the preimage of every intuitionistic fuzzy open set of Y is intuitionistic fuzzy gpr-open in X. −1 c −1 c Proof It is obvious because f (U )=(f (U )) for every intuitionistic fuzzy set U of Y . Remark 5.1 Every intuitionistic fuzzy continuous mapping is intuitionistic fuzzy gpr-continuous, but converse may not be true as seen from the following example. Example 5.1 Let X = {a, b}, Y ={x, y} and intuitionistic fuzzy sets U and V are defined as follows: U = {< a, 0.5, 0.5 >, < b, 0.4, 0.6 >}, V = {< x, 0.5, 0.5 >, < y, 0.5, 0.5>}. Let = {0, U , 1 } and s = {0, V , 1 } be intuitionistic fuzzy topologies on X and Y respectively. Then the mapping f:(X, ) →(Y,σ) defined by f (a)= x and f (b)= y is intuitionistic fuzzy gpr-continuous but not intuitionistic fuzzy continuous. Remark 5.2 Every intuitionistic fuzzy rg-continuous mapping is intuitionistic fuzzy gpr-continuous, but converse may not be true as seen from the following example. Example 5.2 Let X = {a, b, c, d, e} and Y = {p, q, r, s, t } and intuitionistic fuzzy sets O, U, V and W defined as follows: O = {< a, 0.9, 0.1>, < b, 0.8, 0.1 >,< c, 0, 1>,< d,0,1 >, < e,0,1 >}, U = {< a,0,1 >, < b,0,1>, < c, 0.8, 0.1>,< d, 0.7, 0.2>< e,0,1>}, V = {< a, 0.9, 0.1>, < b, 0.8, 0.1 >, < c, 0.8, 0.1>,< d, 0.7, 0.2 >,< e,0,1 >}, W = {< p, 0.9, 0.1 >, < q,0,1 >, < r,0,1>, < s,0,1 >,< t,0,1 >}. Let = {0, O, U, V, 1 } and σ = {0, W , 1 } be intuitionistic fuzzy topologies on X and Y respectively. Then the mapping f:(X, ) →(Y,σ) defined by f (a)= p, f (b)= q, f (c)= r, f (d)= s and f (e)=t is intuitionistic fuzzy rg-continuous but not intuitionistic fuzzy gpr-continuous. Fuzzy Inf. Eng. (2012) 4: 425-444 439 Remark 5.3 Every intuitionistic fuzzyαg-continuous mapping is intuitionistic fuzzy gpr-continuous, but converse may not be true as seen from the following example. Example 5.3 Let X = {a, b, c}, Y ={x, y, z} and intuitionistic fuzzy sets O, U and V are defined as follows: O ={< a, 0.9, 0.1>, < b,0,1 >,< c,0,1 >}, U = {< a,0,1>, < b, 0.8, 0.1>,< c,0,1>}, V = {< x, 0.9, 0.1>, < y, 0.8, 0.1>,< z,0,1>}. Let = {0, O, U, 1 } and σ = {0, V , 1 } be intuitionistic fuzzy topologies on X and Y respectively. Then the mapping f:(X, ) →(Y,σ) defined by f (a)= xf (b)= y and f (c)= z is intuitionistic fuzzy gpr-continuous, but it is not intuitionistic fuzzy αg-continuous. Remark 5.4 From the above discussion and known results, we have the following diagram of implications: Intuitionistic fuzzy←Intuitionistic fuzzy →Intuitionistic fuzzy α-continuous continuous g-continuous ↓↓ ↓ Intuitionistic fuzzy→Intuitionistic fuzzy←Intuitionistic fuzzy αg-continuous gpr-continuous rg-continuous Theorem 5.2 If f : (X, ) →(Y, σ) is intuitionistic fuzzy gpr-continuous, then for each intuitionistic fuzzy point c(α, β) of X and each intuitionistic fuzzy open set V of Y such that f (c(α,β))⊆ V, there exists an intuitionistic fuzzy gpr-open set U of X such that c(α,β)⊆ U and f (U )⊆ V. Proof Let c(α, β) be intuitionistic fuzzy point of X and V be an intuitionistic fuzzy −1 open set of Y such that f (c(α, β)) ⊆ V . Put U = f (V ). Then by hypothesis U is −1 intuitionistic fuzzy gpr-open set of X such that c(α, β) ⊆ U and f (U)= f ( f (V )) ⊆ V . Theorem 5.3 Let f : (X, ) →(Y, σ) be intuitionistic fuzzy gpr-continuous mapping. Then for each intuitionistic fuzzy point c(α,β) of X and each intuitionistic fuzzy open set V of Y such that f (c(α,β))qV, there exists an intuitionistic fuzzy gpr-open set U of X such that c(α,β)qU and f (U ) ⊆ V. Proof Let c(α, β) be intuitionistic fuzzy point of X and V be an intuitionistic fuzzy −1 open set of Y such that f (c(α, β))qV . Put U = f (V ). Then by hypothesis, U is −1 intuitionistic fuzzy gpr-open set of X such that c(α,β)qU and f (U)= f ( f (V))⊆ V . Theorem 5.4 A mapping f from an intuitionistic fuzzy preregular-T space (X, ) 1/2 to an intuitionistic fuzzy topological space (Y,σ) is intuitionistic fuzzy precontinuous if and only if it is intuitionistic fuzzy gpr-continuous. 440 S. S. Thakur · Jyoti Pandey Bajpai (2012) Proof Necessity: Let f :(X, ) →(Y,σ) be intuitionistic fuzzy pre-continuous map- −1 ping. Let U be intuitionistic fuzzy closed set in Y . Then f (U ) is intuitionistic fuzzy preclosed in X. Since every intuitionistic fuzzy preclosed set is intuitionistic −1 fuzzy gpr-closed, f (U ) is intuitionistic fuzzy gpr-closed in X which implies that f is intuitionistic fuzzy gpr-continuous. Sufficiency: Let f :(X, ) →(Y,σ) be intuitionistic fuzzy gpr-continuous map- −1 ping. Let U be intuitionistic fuzzy closed set in Y . Then f (U ) is intuitionistic fuzzy gpr-closed in X. Since X is intuitionistic fuzzy preregular-T , therefore ev- 1/2 −1 ery intuitionistic fuzzy gpr-closed set is intuitionistic fuzzy preclosed. Hence f (U ) is intuitionistic fuzzy preclosed in X which implies that f is intuitionistic fuzzy pre- continuous. Theorem 5.5 If f : (X, ) →(Y, σ) is intuitionistic fuzzy gpr-continuous and g : (Y, σ)→(Z,μ) is intuitionistic fuzzy continuous, then gof : (X, )→(Z,μ) is intuitionistic fuzzy gpr-continuous. −1 Proof Let A be an intuitionistic fuzzy closed set in Z. Then g (A) is intuitionistic −1 fuzzy closed in Y , because g is intuitionistic fuzzy continuous. Therefore (gof ) (A) −1 −1 = f (g (A)) is intuitionistic fuzzy gpr-closed in X, because f is intuitionistic fuzzy gpr-continuous. Hence gof is intuitionistic fuzzy gpr-continuous. Remark 5.5 The composition of two intuitionistic fuzzy gpr-continuous mappings need not be intuitionistic fuzzy gpr-continuous as seen from the following example. Example 5.4 Let X = {a, b, c} and intuitionistic fuzzy sets O, U, V, W and T are defined as follows: O = {< a, 0.9, 0.1 >, < b,0,1 >, < c,0,1 >}, U = {< a,0,1 >, < b, 0.8, 0.1>, < c,0,1 >}, V = {< a,0,1>, < b,0,1>, <c, 0.9, 0.1>}, W = {< a, 0.9, 0.1 >, < b, 0.8, 0.1 >,< c,0,1>}, T = {< a,0,1>, < b, 0.8,.0.1 > ,< c, 0.9, 0.1>}. Let ={ 0, O, U, W, 1} , σ = {0, V, T, 1 } and μ = {0, T, 1 } be intuitionistic fuzzy topologies on X. Then the mapping f :(X, ) → (X,σ) defined by f (a) = b, f (b) = c and f (c) = a and mapping g :(X,σ) →(X, μ) defined by g(a) = b, g (b) = c and g (c) = c are intuitionistic fuzzy gpr-continuous, but composition mapping gof :(X, ) →(X,μ) is not intuitionistic fuzzy gpr-continuous. Theorem 5.6 If f : (X, ) → (Y, σ) is intuitionistic fuzzy gpr-continuous and g : (Y, σ)→ (Z,μ) is intuitionistic fuzzy g-continuous and (Y,σ) is intuitionistic fuzzy-(T ) 1/2 space, then gof : (X, ) → (Z,μ) is intuitionistic fuzzy gpr-continuous. −1 Proof Let A be an intuitionistic fuzzy closed set in Z. Then g (A) is intuitionistic −1 fuzzy g-closed in Y . Since Y is intuitionistic fuzzy-(T ) space, then g (A) is in- 1/2 −1 −1 −1 tuitionistic fuzzy closed in Y . Hence (gof ) (A)= f (g (A)) is intuitionistic fuzzy gpr-closed in X. Hence gof is intuitionistic fuzzy gpr-continuous. Fuzzy Inf. Eng. (2012) 4: 425-444 441 Theorem 5.7 An intuitionistic fuzzy gpr-continuous image of an intuitionistic fuzzy gpr-compact space is intuitionistic fuzzy compact. Proof Let f :(X, ) → (Y,σ) be intuitionistic fuzzy gpr-continuous mapping from an intuitionistic fuzzy gpr-compact space (X, ) onto an intuitionistic fuzzy topolog- ical space (Y,σ). Let {A : i ∈ Λ} be an intuitionistic fuzzy open cover of Y . Then −1 { f (A ): i ∈ Λ} is an intuitionistic fuzzy gpr-open cover of X. Since X is intuition- −1 −1 −1 istic fuzzy gpr-compact there is a finite subfamily { f (A ), f (A ),··· , f (A )} i i i 1 2 n −1 n −1 ˜ ˜ ˜ of { f (A ): i ∈ Λ} such that ∪ f (A ) = 1. Since f is onto f (1) = 1 and i i j=1 n −1 n −1 n n f (∪ f (A )) = ∪ f ( f (A )) = ∪ A . it follows that ∪ A = 1 and the i i i i j j j j j=1 j=1 j=1 j=1 family {A , A ,··· , A } is an intuitionistic fuzzy finite subcover of {A : i ∈ Λ}. i i i i 1 2 n Hence (Y,σ) is intuitionistic fuzzy compact. Theorem 5.8 If f : (X, ) →(Y, σ) is intuitionistic fuzzy gpr-continuous surjection and X is intuitionistic fuzzy gpr-connected, then Y is intuitionistic fuzzy connected. Proof Suppose that Y is not intuitionistic fuzzy connected. Then there exists a proper intuitionistic fuzzy set G of Y which is both intuitionistic fuzzy open and intuitionistic −1 fuzzy closed. Therefore f ( G) is a proper intuitionistic fuzzy set of X, which is both intuitionistic fuzzy gpr-open and intuitionistic fuzzy gpr-closed, because f is intuitionistic fuzzy gpr-continuous surjection. Hence X is not intuitionistic fuzzy gpr-connected, which is a contradiction. Definition 5.2 A mapping f : (X, ) →(Y, σ) is intuitionistic fuzzy gpr-irresolute if preimage of every intuitionistic fuzzy gpr-closed set of Y is intuitionistic fuzzy gpr- closed set in X. Theorem 5.9 A mapping f : (X, )→(Y,σ) is intuitionistic fuzzy gpr-irresolute if and only if the preimage of every intuitionistic fuzzy open set of Y is intuitionistic fuzzy gpr-open in X. −1 c −1 c Proof It is obvious, because f (U )= f (U )) for every intuitionistic fuzzy set U of Y . Remark 5.6 Since every intuitionistic fuzzy closed set is intuitionistic fuzzy gpr- closed, it is clear that every intuitionistic fuzzy gpr-irresolute mapping is intuition- istic fuzzy gpr-continuous, but converse may not be true as seen from the following examples. Example 5.5 Let X = {a, b}, Y = {x, y} and let = {0, U , 1 } and σ = {0, 1 } be intuitionistic fuzzy topologies on X and Y respectively where U = {< a, 0.7, 0.3>,< b, 0.5, 0.5>}. Then the mapping f :(X, )→(Y ,σ) defined by f (a) = x and f (b) = y is intuitionistic fuzzy gpr-continuous but not intuitionistic fuzzy gpr-irresolute. Example 5.6 Let X = {a, b}, Y ={x, y} and let = {0, 1} and σ = {0, V , 1 } be intuitionistic fuzzy topologies on X and Y respectively where V= {< x, 0.5, 0.5>,<y, 0.3, 0.7>}. Then the mapping f :(X, ) → (Y , σ) defined by f (a) = x and f (b) = y is intuitionistic fuzzy gpr-irresolute but not intuitionistic fuzzy continuous. Remark 5.7 Example 5.5 and Example 5.6 assert that concept of intuitionistic fuzzy gpr-irresolute and intuitionistic fuzzy continuous mappings are independent. 442 S. S. Thakur · Jyoti Pandey Bajpai (2012) Theorem 5.10 Let f : (X, ) →(Y, σ) be bijective intuitionistic fuzzy regular-open and intuitionistic fuzzy gpr-continuous. Then f is intuitionistic fuzzy gpr-irresolute. −1 Proof Let A be intuitionistic fuzzy gpr-closed in Y and let f (A) ⊆ G where G is intuitionistic fuzzy regular open in X. Then A ⊆ f (G). Since f (G) is intuitionistic fuzzy regular open in Y and A is intuitionistic fuzzy gpr-closed in Y , then pcl(A) ⊆ −1 f (G) and f (pcl(A)) ⊆ G. Since f is intuitionistic fuzzy gpr-continuous and cl(A) −1 is intuitionistic fuzzy closed in Y , f (cl(A)) is intuitionistic fuzzy gpr-closed in X, −1 −1 −1 therefore pcl (f (cl(A))) ⊆ G and so pcl (f (A)) ⊆ G. Hence f (A) is intuitionistic fuzzy gpr-closed in X. Therefore f is intuitionistic fuzzy gpr-irresolute. Theorem 5.11 Let f : (X, ) →(Y, σ) be intuitionistic fuzzy gpr-irresolute and g: (Y,σ)→(Z,μ) is intuitionistic fuzzy gpr-continuous mapping. Then gof : (X, )→(Z, μ) is intuitionistic fuzzy gpr-continuous. −1 Proof Let A be intuitionistic fuzzy closed in Z. Then g (A) is intuitionistic fuzzy −1 gpr-closed in Y , because g is intuitionistic fuzzy gpr-continuous. Therefore (gof ) (A) −1 −1 = f (g (A)) is intuitionistic fuzzy gpr-closed in X, because f is intuitionistic fuzzy gpr-irresolute. Hence gof is intuitionistic fuzzy gpr-continuous. Theorem 5.12 If f : (X, )→(Y, σ)and g: (Y, σ) →(Z, μ) be two intuitionisticfuzzy gpr-irresolute mapping, then gof: (X, )→(Z,μ) is intuitionistic fuzzy gpr-irresolute. −1 Proof Let A be an intuitionistic fuzzy gpr-closed set in Z. Then g (A) is intuition- stic fuzzy gpr-closed in Y because g is intuitionistic fuzzy gpr-irresolute. Therefore −1 −1 −1 (go f ) (A)= f (g (A)) is intuitionistic fuzzy gpr-closed in X because f is intuition- istic fuzzy gpr-irresolute. Hence gof is intuitionistic fuzzy gpr-irresolute. Theorem 5.13 Let f : (X, ) →(Y, σ) be intuitionistic fuzzy gpr-irresolute mapping and if B is fuzzy gpr-compact relative to X. Then image f (B) is intuitionistic fuzzy gpr-compact relative to Y. Proof Let {A : i∈ Λ} be an intuitionistic fuzzy gpr-open set of Y such that f (B) ⊆ −1 ∪{ A : i∈ Λ}. Then B ⊆∪ { f (A ): i∈ Λ}. By using the assumption, there exists i i −1 a finite subset Λ of Λ such that B⊆∪ { f (A ): i∈ Λ }. Therefore f (B)⊆∪{ A : o i 0 i i∈ Λ } which shows that f (B) is intuitionistic fuzzy gpr-compact relative to Y . Corollary 5.1 An intuitionistic fuzzy gpr-irresolute image of an intuitionistic fuzzy gpr-compact space is intuitionistic fuzzy gpr-compact. Proof Let f :(X, ) →(Y , σ) be intuitionistic fuzzy gpr-irresolute mapping from an intuitionistic fuzzy gpr-compact space (X, ) onto an intuitionistic fuzzy topological space (Y , σ). Let { A : i∈ Λ } be an intuitionistic fuzzy gpr-open cover of Y . Then −1 f (A ): i∈ Λ} is an intuitionistic fuzzy gpr-open cover of X. Since X is intuition- −1 −1 −1 istic fuzzy gpr-compact there is a finite subfamily { f (A ), f (A ),··· , f (A )} i i i 1 2 n −1 n −1 ˜ ˜ ˜ of { f (A ): i ∈ Λ} such that ∪ f (A ) = 1. Since f is onto f (1) = 1 and i i j=1 n −1 n −1 n n f (∪ f (A )) = ∪ f ( f (A )) = ∪ A . it follows that ∪ A = 1 and the i i i i j j j j j=1 j=1 j=1 j=1 family {A , A ,··· , A } is an intuitionistic fuzzy finite subcover of {A : i ∈ Λ}. i i i i 1 2 n Hence (Y,σ) is intuitionistic fuzzy gpr-compact. Fuzzy Inf. Eng. (2012) 4: 425-444 443 Theorem 5.14 Let (X× Y, × σ) be the intuitionistic fuzzy product space of intu- itionistic fuzzy topological spaces (X, ) and (Y,σ). Then the projection mapping P: X×Y→X is an intuitionistic fuzzy gpr-irresolute. −1 Proof Let F be any intuitionistic fuzzy gpr-closed set of X. Then F× 1= P (F)) is intuitionistic fuzzy gpr-closed and hence P is an intuitionistic fuzzy gpr-irresolute. Theorem 5.15 If the product space (X × Y, × σ ) of two intuitionistic fuzzy topo- logical spaces (X, ) and (Y, σ) is intuitionistic fuzzy gpr-compact, then each factor space is intuitionistic fuzzy gpr-compact. Proof Let (X × Y , × σ ) be intuitionistic fuzzy gpr-compact. Then by Corollary 3.3, we obtain that the intuitionistic fuzzy gpr-irresolute image of p(X × Y)= X is intuitionistic fuzzy gpr-compact. Theorem 5.16 If f : (X, ) →(Y, σ) is intuitionistic fuzzy gpr-irresolute surjection and (X, ) is intuitionistic fuzzy gpr-connected, then (Y,σ) is intuitionistic fuzzy gpr- connected. Proof Suppose Y is not intuitionistic fuzzy gpr-connected. Then there exists a proper intuitionistic fuzzy set G of Y which is both intuitionistic fuzzy gpr-open and intu- −1 itionistic fuzzy gpr-closed. Therefore f (G) is a proper intuitionistic fuzzy set of X, which is both intuitionistic fuzzy gpr-open and intuitionistic fuzzy gpr-closed, be- cause f is intuitionistic fuzzy gpr-irresolute surjection. Hence X is not intuitionistic fuzzy gpr-connected, which is a contradiction. Hence Y is intuitionistic fuzzy gpr- connected. Theorem 5.17 If the product space (X×Y, ×σ) of two nonempty intuitionistic fuzzy topological spaces (X, ) and (Y, σ) is intuitionistic fuzzy gpr-connected, then each factor space is intuitionistic fuzzy gpr-connected. Proof If (X× Y , ×σ) is intuitionistic fuzzy gpr-connected, then mapping P : X× Y → X is intuitionistic fuzzy gpr-irresolute. Hence by Theorem 5.16 the intuitionistic fuzzy gpr-irresolute image P(X×Y)= X of an intuitionistic fuzzy gpr-connected space X × Y is an intuitionistic fuzzy gpr-connected. 6. Conclusion The theory of g-closed sets plays an important role in the general topology. Since its inception many weak forms of g-closed sets have been introduced in general topology as well as in fuzzy topology and in intutionistic fuzzy topology. The present paper investigated in new weak form of intutionistic fuzzy g-closed sets called intutionistic fuzzy gpr-closed sets, which contains the classes of intutionistic fuzzy g-closed sets, intutionistic fuzzyαg-closed and intutionistic fuzzy rg-closed sets. Several properties and applications of intutionistic fuzzy gpr-closed sets are studied. Many examples are given to justify the results. Acknowledgments The authors would like to thank the referees for their valuable suggestions to improve the paper. 444 S. S. Thakur · Jyoti Pandey Bajpai (2012) References 1. Atanassova K, Stoeva S (1983) Intuitionistic fuzzy sets. In Polish Symposium on Interval and Fuzzy Mathematics, Poznan: 23-26 2. Atnassova K (1986) Intuitionistic fuzzy sets. Fuzzy Sets and Systems 20(1): 87-96 3. Balachandran K, Sundaram P, Maki H (1991) On generalized continuous map in topological spaces. Mem. Fac. Sci. Kochi ˆ Univ.Ser. A Math. 12: 5-13 4. Bhattacharyya, Lahiri (1987) Semi-generalized closed set in topology. Indian J. Math. 29: 376-382 5. Bayhan Sadik (2001) On separation axioms in intuitionistic topological spaces. Int. J. Math. Math. Sci. 27(10): 621-630 6. Chang C L (1968) Fuzzy topological spaces. J. Math. Anal. Appl. 24: 182-190 7. Coker D (1997) An introduction to intuitionistic fuzzy topological spaces. Fuzzy Sets and Systems 88: 81-89 8. Coker D, Demirci M (1995) On intuitionistic fuzzy points. Notes on IFS 2(1): 78-83 9. Coker D, A Es Hyder (1995) On fuzzy compactness in intuitionistic fuzzy topological spaces. J. Fuzzy Math. 3(4): 899-909 10. Gurcay H, Coker D, Es A Haydar (1997) On fuzzy continuity in intuitionistic fuzzy topological spaces. J. Fuzzy Math. 5(2): 365-378 11. Gnanambal Y (1997) On generalized preregular closed sets in topological spaces. Indian J. Pure. Appl. Math. 28(3): 351-360 12. Levine N (1970) Generalized closed sets in topology. Rend. Circ. Mat. Palermo 19(2): 571-599 13. Palniappan N, Rao K C (1983) Regular generalized closed sets. Kyungpook Math. J. 33: 211-219 14. Sakthivel K (2010) Intuitionistic fuzzy alpha generalized continuous mappings and intuitionistic fuzzy alpha generalized irresolute mappings. Appl. Math. Sci. 4(37): 1831-1842 15. Santi R, Jayanthi D (2010) Intuitionistic fuzzy generalized semi-precontinuous mappings. Int. J. Contemp. Math. Sci. 5(30): 1455-1469 16. Thakur S S, Malviya R (1995) Generalized closed sets in fuzzy topology. Math. Notae 38: 137-140 17. Thakur S S, Chaturvedi R (2008) Generalized closed set in intuitionistic fuzzy topology. J. Fuzzy Math. 16(3): 559-572 18. Thakur S S, Chaturvedi R (2006) Generalized continuity in intuitionistic fuzzy topological spaces. Notes on IFS 12(1): 38-44 19. Thakur S S, Chaturvedi R (2006) Regular generalized closed sets in intuitionistic fuzzy topological spaces. Stud. Cercet. Stiint. Ser. Mat. Univ. Bacu 16: 257-272 20. Thakur S S, Chaturvedi R (2007) Intuitionistic fuzzy rg-continuous mapping. J. Indian Acad. Math. 29(2): 467-473 21. Thakur S S, Chaturvedi R (2006) Intuitionistic fuzzy rg-irresolute mapping. Varhmihir J. Math. Sci. 6(1): 199-204 22. Thakur S S, Bajpai Pandy Jyoti (2010) Intuitionistic fuzzy w-closed sets and intuitionistic fuzzy w- continuity. International Journal of Contemporary Advanced Mathematics (IJCM) 1(1): 1-15 23. Thakur S S, Bajpai pandey Jyoti (2010) Intuitionistic fuzzy sg-continuous mappings. International Journal of Applied Mathematical Analysis and Application 5(1): 45-51 24. Turnali N, Coker D (2000) Fuzzy connectedness in intuitionistic fuzzy topological spaces. Fuzzy Sets and Systems 116(3): 369-375 25. Yalvac T H (1988) Semi-interior and semi-closure of fuzzy sets. J. Math. Anal. Appl. 133: 356- 364 26. Jun Y B, Song S Z (2005) Intuitionistic fuzzy semi-preopen sets and intuitionistic fuzzy semi- precontinuous mappings. J. App. Math. Comput. 19(1-2): 467-474 27. Zadeh L A (1965) Fuzzy sets. Information and Control 18: 338-353 http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Fuzzy Information and Engineering Taylor & Francis

On Intutionistic Fuzzy Gpr-closed Sets

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Fuzzy Inf. Eng. (2012) 4: 425-444 DOI 10.1007/s12543-012-0125-x ORIGINAL ARTICLE S. S. Thakur · Jyoti Pandey Bajpai Received: 13 January 2011/ Revised: 25 October 2012/ Accepted: 15 November 2012/ © Springer-Verlag Berlin Heidelberg and Fuzzy Information and Engineering Branch of the Operations Research Society of China Abstract In this paper, a new class of intuitionistic fuzzy closed sets called in- tuitionistic fuzzy generalized preregular closed sets (briefly intuitionistic fuzzy gpr- closed sets) and intuitionistic fuzzy generalized preregular open sets (briefly intuition- istic fuzzy gpr-open sets) are introduced and their properties are studied. Further the notion of intuitionistic fuzzy preregular T -spaces and intuitionistic fuzzy general- 1/2 ized preregular continuity (briefly intuitionistic fuzzy gpr-continuity) are introduced and studied. Keywords Intuitionistic fuzzy gpr-closed sets · Gpr-open sets· Gpr-connectedness · Gpr-compactness· Gpr-continuous and gpr-irresolute mappings 1. Introduction In 1970, Levine [12] introduced the concept of g-closed sets in general topology. Many researchers like Balchandran [3], Bhattacharya and Lahiri [4], Palaninappan and Rao [13] etc. have worked on g-closed sets, their generalization in general topol- ogy. After the introduction of fuzzy sets by Zadeh [27] in 1965 and fuzzy topol- ogy by Chang [6] in 1967, several researchers were conducted on generalization of the notion of fuzzy sets and fuzzy topology. The concept of intuitionistic fuzzy sets was introduced by Atanassov [1] as a generalization of fuzzy sets. In 1997, Coker [7] introduced the concept of intuitionistic fuzzy topological spaces. In 2008, Thakur and Chaturvedi extended the concepts of fuzzy g-closed sets [17] and fuzzy g-continuity[18] in intuitionistic fuzzy topological spaces. Recently, many general- izations of intuitionistic fuzzy g-closed sets like intuitionistic fuzzy rg-closed sets [19], intuitionistic fuzzy sg-closed sets [23], intuitionistic fuzzy w-closed sets [22], intuitionistic fuzzy αg-closed sets [14], intuitionistic fuzzy gsp-closed sets [15] have been appeared in the literature. S. S. Thakur () · Jyoti Pandey Bajpai () Department of Applied Mathematics, Jabalpur Engineering College, Jabalpur, India email: samajh singh@redifmail.com yk1305@gmail.com 426 S. S. Thakur · Jyoti Pandey Bajpai (2012) In the present paper, we extend the concepts of gpr-closed sets due to Gnanam- bal [11] in intuitionistic fuzzy topology which is the weaker form of intuitionistic fuzzy ag-closed sets [14] and intuitionistic fuzzy rg-closed sets [19]. In Section 3, we define the concepts of intuitionistic fuzzy gpr-closed sets and intuitionistic fuzzy gpr-open sets and obtain some of their properties and characterizations. In Section 4, we introduce the concepts of intuitionistic fuzzy gpr-connectedness and intuitionistic fuzzy gpr-compactness. Furthermore we discuss the concepts of intuitionistic fuzzy gpr-continuous mappings and intuitionistic fuzzy gpr-irresolute mappings in Section 2. Preliminaries Let X be a nonempty fixed set. An intuitionistic fuzzy set [1] A in X is an object having the form A = {< x,μ (x),γ (x) >: x ∈ X}, where the functions μ : X → A A A [0, 1] and γ : X → [0, 1] denotes the degree of membership μ (x) and the degree A A of nonmembership γ (x) of each element x ∈ X to the set A respectively and 0 ≤ μ (x)+γ (x) ≤ 1 for each x ∈ X. The intuitionistic fuzzy sets 0 = {< x, 0, 1 >: x ∈ X} A A and 1 = {< x, 1, 0 >: x ∈ X} are respectively called empty and whole intuitionistic fuzzy set on X. An intuitionistic fuzzy set A = {< x,μ (x),γ (x) >: x ∈ X} is called A A a subset of an intuitionistic fuzzy set B = {< x,μ (x),γ (x) >: x ∈ X} (for short B B A ⊆ B)if μ (x) ≤ μ (x) and γ (x) ≥ γ (x) for each x ∈ X. The complement of A B A B an intuitionistic fuzzy set A = {< x,μ (x),γ (x) >: x ∈ X} is the intuitionistic fuzzy A A set A = {< x,γ (x),μ (x) >: x ∈ X}. The intersection (respectively union) of any A A arbitrary family of intuitionistic fuzzy sets A = {< x,μ (x),γ (x) >: x ∈ X, i ∈∧} i A A i i of X are the intuitionistic fuzzy set ∩ A = {< x, ∧ μ (x), ∨ γ (x) >: x ∈ X } (resp. i Ai Ai ∪A ={<x,∨μ (x),∧γ (x)>: x∈ X }). Two intuitionistic fuzzy sets A = {<x,μ (x), i Ai Ai A γ (x)>: x ∈ X } and B = {<x,μ (x), γ (x)>: x ∈ X } are said to be q-coincident (A B A B B q for short) if and only if ∃ an element x∈ X such that μ (x)>γ (x) or γ (x)<μ (x). A B A B A family of intuitionistic fuzzy sets on a nonempty set X is called an intuitionistic fuzzy topology [7] on X if the intuitionistic fuzzy sets 0 and 1∈ , and is closed under arbitrary union and finite intersection. The ordered pair (X, ) is called an intuitionistic fuzzy topological space and each intuitionistic fuzzy set in is called an intuitionistic fuzzy open set. The complement of an intuitionistic fuzzy open set in X is known as intuitionistic fuzzy closed set. The intersection of all intuitionistic fuzzy closed sets containing A is called the closure of A, which is denoted by cl(A). The union of all intuitionistic fuzzy open subsets of A is called the interior of A.It is denoted by int(A)[7]. Lemma 2.1 [7] Let A and B be any two intuitionistic fuzzy sets of an intuitionistic fuzzy topological space (X, ). Then (1) (A B)⇔ A⊆ B . (2) A is an intuitionistic fuzzy closed set in X ⇔ cl (A) = A. (3) A is an intuitionistic fuzzy open set in X ⇔ int(A) = A. c c (4) cl (A ) = (int (A)) . Fuzzy Inf. Eng. (2012) 4: 425-444 427 c c (5) int (A ) = (cl (A)) . (6) A⊆ B⇒ int (A) ⊆ int (B). (7) A⊆ B⇒ cl (A) ⊆ cl (B). (8) cl (A∪ B)=cl(A) ∪ cl(B). (9) int(A∩ B) = int (A) ∩ int(B). Definition 2.1 [8] Let X be a nonempty set and c∈X a fixed element in X. If α∈(0, 1] andβ∈[0, 1) are two real numbers such that α+β≤1, then (1) c(α,β) = < x,α,1- β> is called an intuitionistic fuzzy point in X, where α denotes the degree of membership of c(α,β) and β denotes the degree of non- membership of c(α,β). (2) c(β) = <x, 0,β> is called a vanishing intuitionistic fuzzy point in X, where β denotes the degree of nonmembership of c( β). Definition 2.2 [9] A family {G :i∈∧} of intuitionistic fuzzy sets in X is called an intuitionistic fuzzy open cover of X if ∪{G :i∈∧} = 1 and a finite subfamily of an intuitionistic fuzzy open cover {G :i∈∧} of X which also an intuitionistic fuzzy open cover of X is called a finite subcover of {G :i∈∧}. Definition 2.3 [9] An intuitionistic fuzzy topological space (X, ) is called intuition- istic fuzzy compact if every intuitionistic fuzzy open cover of X has a finite subcover. Definition 2.4 [10] An intuitionistic fuzzy set A of an intuitionistic fuzzy topological space (X, ) is called (1) Intuitionistic fuzzy semi-open if A ⊆ cl(int(A)) and intuitionistic fuzzy semi- closed if int(cl(A)) ⊆ A. (2) Intuitionistic fuzzy α-open if A ⊆ int(cl(int(A))) and intuitionistic fuzzy α- closed if cl(int(cl(A))) ⊆ A. (3) Intuitionistic fuzzy preopen if A ⊆ int(cl(A)) and intuitionistic fuzzy preclosed if cl(int(A)) ⊆ A. (4) Intuitionistic fuzzy regular open if A = int(cl(A)) and intuitionistic fuzzy reg- ular closed if A = cl(int(A)). Definition 2.5 [26] An intuitionistic fuzzy set A in intuitionistic topological spaces (X, ) is said to be (1) Intuitionistic fuzzy semi-preopen if there exists an intuitionistic fuzzy preopen set B such that B ⊆ A ⊆ cl(B). (2) Intuitionistic fuzzy semi-preclosed if there exists an intuitionistic fuzzy pre- closed set B such that int(B) ⊆ A ⊆ B. Remark 2.1 [10] If (X, ) is an intuitionistic fuzzy topological space, then 428 S. S. Thakur · Jyoti Pandey Bajpai (2012) (1) Every intuitionistic fuzzy regular closed set in X is intuitionistic fuzzy closed in X. (2) Every intuitionistic fuzzy closed set in X is intuitionistic fuzzyα-closed in X. (3) Every intuitionistic fuzzy α-closed set in X is intuitionistic fuzzy preclosed in X. Definition 2.6 If A is an intuitionistic fuzzy set in intuitionistic fuzzy topological space(X, ), then (1) scl (A) = ∩ { F: A⊆ F, F is intuitionistic fuzzy semi-closed} [10]. (2) αcl (A) = ∩ { F: A ⊆ F, F is intuitionistic fuzzyα-closed} [10]. (3) pcl(A) = ∩ { F: A ⊆ F, F is intuitionistic fuzzy preclosed} [10]. (4) spcl (A) = ∩ { F: A ⊆ F, F is intuitionistic fuzzy semi-preclosed} [26]. Definition 2.7 An intuitionistic fuzzy set A of an intuitionistic fuzzy topological space (X, ) is called (1) Intuitionistic fuzzy g-closed [17] if cl(A) ⊆ O whenever A ⊆ O and O is intu- itionistic fuzzy open in X. (2) Intuitionistic fuzzy sg-closed [23] if scl(A) ⊆ O whenever A ⊆ and O is intu- itionistic fuzzy semi-open in X. (3) Intuitionistic fuzzy rg-closed [19] if cl(A) ⊆ O whenever A ⊆ O and O is intu- itionistic fuzzy regular open in X. (4) Intuitionistic fuzzy αg-closed [14] if αcl(A) ⊆ O whenever A ⊆ O and O is intuitionistic fuzzy open in X. (5) Intuitionistic fuzzy gsp-closed[15] if spcl(A) ⊆ O whenever A ⊆ OandOis intuitionistic fuzzy open in X. (6) Intuitionistic fuzzy w-closed [22] if cl(A) ⊆ O whenever A ⊆ O and O is intu- itionistic fuzzy semi-open in X. The complements of the above mentioned closed sets are their respective open sets. Remark 2.2 If (X, ) is an intuitionistic fuzzy topological space, then (1) Every intuitionistic fuzzy closed set in X is intuitionistic fuzzy g-closed but its converse may not be true, because in intuitionistic fuzzy topological space (X, ) where X = {a, b}, U = {< a, 0.5, 0.4 >,< b, 0.6, 0.4 >} and = {0, U, 1}, the intuitionistic fuzzy set A = {< a, 0.3, 0.6 >,< b, 0.4, 0.6 >} is intuitionistic fuzzy g-closed, but it is not intuitionistic fuzzy closed [17]. Fuzzy Inf. Eng. (2012) 4: 425-444 429 (2) Every intuitionistic fuzzy α-closed set in X is intuitionistic fuzzy αg-closed but its converse may not be true, because in intuitionistic fuzzy topological space (X, ) where X = {a, b}, U = {< a, 0.4, 0.6 >,< b, 0.2, 0.7 >}, V = {< a, 0.8, 0.2 >,< b, 0.8, 0.2 >} and = {0 , U, V, 1}, the intuitionistic fuzzy set A = {< a, 0.5, 0.4 >,< b, 0.4, 0.5 >} is intuitionistic fuzzy αg-closed, but it is not intuitionistic fuzzy α-closed [14]. (3) Every intuitionistic fuzzy g-closed set in X is intuitionistic fuzzy rg-closed but its converse may not be true, because in intuitionistic fuzzy topological space (X, ) where X = {a, b}, U = {< a, 0.5, 0.4 >,< b, 0.6, 0.4 >} and = {0, U, 1}, the intuitionistic fuzzy set A = {< a, 0.5, 0.4 >,< b, 0.4, 0.5 >} is intuitionistic fuzzy rg-closed, but it is not intuitionistic fuzzy g-closed [19]. Definition 2.8 If (X, ) is an intuitionistic fuzzy topological space, then (1) (X, ) is called intuitionistic fuzzy GO-connected if there is no proper intuition- istic fuzzy set A (i.e., A  0 and A  1) of X which is both intuitionistic fuzzy g-open and intuitionistic fuzzy g-closed [17]. (2) (X, ) is called intuitionistic fuzzy rg-connected if there is no proper intuition- istic fuzzy set of X which is both intuitionistic fuzzy rg-open and intuitionistic fuzzy rg-closed [19]. Definition 2.9 [10] Let X and Y be two nonempty sets and f : X→ Y be a function: (1) IfB= {< y,μ (y), γ (y) > :y ∈ Y} is an intuitionistic fuzzy set in Y, then the B B −1 preimage of B under f denoted by f (B), is the intuitionistic fuzzy set in X −1 −1 −1 defined by f (B) = {<x, f (μ ) (x), f (γ ) (x)> :x ∈ X}. B B (2) If A = {< x, λ (x), ν (x) > :x ∈ X} is an intuitionistic fuzzy set in X, then the A A image of A under f denoted by f(A) is the intuitionistic fuzzy set in Y defined by f (A) = {< y, f (λ )(y), (1− f (1−ν ))(y) >: y ∈ Y}, A A where −1 ( ) ⎪ sup λ (x), if f y  φ, −1 x∈ f (y) f (λ ) (y) = A ⎪ 0, otherwise, −1 inf ν (x), if f (y)  φ, ⎪ A −1 x∈ f (y) (1 - f (1- ν ))(y) = 1, otherwise. Definition 2.10 Let (X, ) and (Y, σ) be two intuitionistic fuzzy topological spaces and let f: X→Y be a function. Then f is said to be 430 S. S. Thakur · Jyoti Pandey Bajpai (2012) (1) Intuitionistic fuzzy continuous if the preimage of each intuitionistic fuzzy open set of Y is an intuitionistic fuzzy open set in X. (2) Intuitionistic fuzzy irresolute if the preimage of every intuitionistic fuzzy semi- open set of Y is intuitionistic fuzzy semi-open in X. (3) Intuitionistic fuzzy precontinuous if the preimage of every intuitionistic fuzzy open set of Y is intuitionistic fuzzy preopen in X. (4) Intuitionistic fuzzy almost continuous if the preimage of each intuitionistic fuzzy regular open set of Y is an intuitionistic fuzzy open set in X. (5) Intuitionistic fuzzy closed if the image of every intuitionistic fuzzy closed set of X is intuitionistic fuzzy closed in Y. (6) Intuitionistic fuzzy preclosed if the image of every intuitionistic fuzzy closed set of X is intuitionistic fuzzy preclosed in Y. (7) Intuitionistic fuzzy preregular-closed if the image of every intuitionistic fuzzy regular closed of X is intuitionistic fuzzy regular closed in X. Definition 2.11 Let (X, ) and (Y, σ) be two intuitionistic fuzzy topological spaces and let f : X→Y be a function. Then f is said to be (1) Intuitionistic fuzzy g-continuous if the preimage of every intuitionistic fuzzy closed set in Y is intuitionistic fuzzy g-closed in X [18]. (2) Intuitionistic fuzzy sg-continuous if the preimage of every intuitionistic fuzzy closed set in Y is intuitionistic fuzzy sg-closed in X [23]. (3) Intuitionistic fuzzy αg-continuous if the preimage of every intuitionistic fuzzy closed set in Y is intuitionistic fuzzy αg-closed in X [14]. (4) Intuitionistic fuzzy rg-continuous if the preimage of every intuitionistic fuzzy closed set in Y is intuitionistic fuzzy rg-closed in X [20]. (5) Intuitionistic fuzzy R map if the preimage of each intuitionistic fuzzy regular open set of Y is an intuitionistic fuzzy regular open set in X [19]. 3. Intuitionistic Fuzzy Gpr-closed Sets and Intuitionistic Fuzzy Gpr-open Sets Definition 3.1 An intuitionistic fuzzy set A of an intuitionistic fuzzy topological space (X, ) is called intuitionistic fuzzy generalized preregular closed set (briefly intuition- istic fuzzy gpr-closed ) if pcl(A) ⊆ U, whenever A ⊆ U and U is intuitionistic fuzzy regular open in X. Theorem 3.1 Every intuitionistic fuzzy rg-closed set in intuitionistic fuzzy topological space (X, ) is intuitionistic fuzzy gpr-closed set in X. Proof Let A be an intuitionistic fuzzy rg-closed set in X. Let A ⊆ U and U be intuitionistic fuzzy regular open in X. Then cl(A) ⊆ U because A is intuitionistic Fuzzy Inf. Eng. (2012) 4: 425-444 431 fuzzy rg-closed in X. Since every intuitionistic fuzzy closed set is intuitionistic fuzzy pre closed, pcl(A) ⊆ cl(A). Therefore, pcl(A) ⊆ U . Hence A is intuitionistic fuzzy gpr-closed. Remark 3.1 The converse of the Theorem 3.1 need not be true as seen from the following example. Example 3.1 Let X = {a, b, c, d, e} and intuitionistic fuzzy sets O, U, V defined as follows: O = {< a, 0.9, 0.1 >, < b, 0.8, 0.1 >,< c,0,1 >,< d,0,1 >,< e,0,1>}, U = {< a,0,1>, < b,0,1>, < c, 0.8, 0.1 >,< d, 0.7, 0.2>,< e,0,1 >}, V = {< a, 0.9, 0.1 >, < b, 0.8, 0.1>,< c, 0.8, 0.1>, < d, 0.7, 0.2>,< e,0,1 >}. Let = {0, O, U, V, 1} be an intuitionistic fuzzy topology on X. Then the intu- itionistic fuzzy set A = {< a, 0.9, 0.1 >, < b,0,1 >, < c,0,1 >, < d,0,1 >, < e,0, 1>} is intuitionistic fuzzy gpr-closed, but it is not intuitionistic fuzzy rg-closed. Theorem 3.2 Every intuitionistic fuzzy αg-closed set in intuitionistic fuzzy topologi- cal space (X, )is intuitionistic fuzzy gpr-closed set in X. Proof Let A be an intuitionistic fuzzy αg-closed set in X. Let A ⊆ U and U be intuitionistic fuzzy regular open set in X. Then, αcl(A) ⊆ U because every intuition- istic fuzzy regular open set is intuitionistic fuzzy open and A is intuitionistic fuzzy αg-closed in X. Now pcl(A) ⊆ αcl(A) ⊆ U implies that pcl(A) ⊆ U . Hence A is intuitionistic fuzzy gpr-closed. Remark 3.2 The converse of the Theorem 3.2 need not be true as seen from the following example. Example 3.2 Let X = {a, b, c} and intuitionistic fuzzy sets O, U, V defined as follows: O = {< a, 0.9, 0.1 >, < b,0,1 >,< c,0,1 >}, U = {< a,0,1>,< b, 0.8, 0.1>,< c,0,1>}, V = {< a, 0.9, 0.1 >, < b, 0.8, 0.1>,< c,0,1>}. Let = {0 , O, U, V, 1} be an intuitionistic fuzzy topology on X. Then intuition- istic fuzzy set A= {< a, 0.9, 0.1 >, < b, 0.8, 0.1 >, < c,0,1 >} is intuitionistic fuzzy gpr-closed set in X, but it is not intuitionistic fuzzyαg-closed. Remark 3.3 From the Theorems 3.1, 3.2 and Remarks 2.1, 2.2, we have the following diagram of implication. Intuitionistic fuzzy←Intuitionistic fuzzy →Intuitionistic fuzzy α-closed closed g-closed ↓↓ ↓ Intuitionistic fuzzy→Intuitionistic fuzzy←Intuitionistic fuzzy αg-closed gpr-closed rg-closed Theorem 3.3 Let (X, ) be an intuitionistic fuzzy topological space and A be an intuitionistic fuzzy regular-closed set of X. Then A is intuitionistic fuzzy gpr-closed if 432 S. S. Thakur · Jyoti Pandey Bajpai (2012) and only if (AqF) ⇒ (pcl(A)qF) for every intuitionistic fuzzy regular closed set F of X. Proof Necessity: Let A be intuitionistic fuzzy gpr-closed set. Let F be an intu- itionistic fuzzy regular closed set of X and (AqF). Then by Lemma 2.1(1), A ⊆ F c c and F intuitionistic fuzzy regular open in X. Therefore, pcl(A) ⊆ F because A is intuitionistic fuzzy gpr-closed. Hence by Lemma 2.1(1), (pcl(A)qF). Sufficiency: Let O be an intuitionistic fuzzy regular open set of X such that A ⊆ O c c c c i.e., A ⊆ ((O) ) . Then by Lemma 2.1(1), (A O ) and O is an intuitionistic fuzzy regular closed set in X. Hence by hypothesis (pcl(A) O ). Therefore by Lemma c c 2.1(1), pcl(A) ⊆ ((O) ) i.e., pcl(A) ⊆ O. Hence A is intuitionistic fuzzy gpr-closed in X. Theorem 3.4 Let A be an intuitionistic fuzzy gpr-closed set in an intuitionistic fuzzy topological space (X, ) and c(α,β) be an intuitionistic fuzzy point of X such that c(α,β) pcl(A). Then pcl(c(α,β))qA. Proof If pcl(c(α,β)) A, then by Lemma 2.1(1), pcl(c(α,β)) ⊆ A which implies that c c c A ⊆ (pcl(c(α,β))) and so pcl(A) ⊆ (pcl(c(α,β))) ⊆ (c(α,β)) , because A is intuitionis- tic fuzzy gpr-closed in X. Hence by Lemma 2.1(1), (c(α,β) (pcl(A))), a contradiction. Remark 3.4 The intersection of two intuitionistic fuzzy gpr-closed sets in an intu- itionistic fuzzy topological space (X, ) may not be intuitionistic fuzzy gpr-closed as seen from the following example. Example 3.3 Let X = {a, b, c} and intuitionistic fuzzy sets O, U, V defined as follows: O = {< a, 0.9, 0.1>, < b,0,1 >, < c,0,1>}, U = {< a,0,1 >, < b, 0.8, 0.1>, < c,0,1>}, V = {< a, 0.9, 0.1>, < b, 0.8, 0.1 >, < c,0,1 >}. Let = {0, O, U, V, 1} be an intuitionistic fuzzy topology on X. Then intuitionistic fuzzy set A = {< a, 0.9, 0.1 >,< b, 0.8, 0.1 >, < c,0,1 >} and B = { < a, 0.9, 0.1 >, < b,0,1>,< c, 0.8, 0.1>} are intuitionistic fuzzy gpr-closed sets in (X, )but A∩ B is not intuitionistic fuzzy gpr-closed. Theorem 3.5 Let A be an intuitionistic fuzzy gpr-closed set in an intuitionistic fuzzy topological space (X, ) and A⊆ B⊆ pcl(A). Then B is intuitionistic fuzzy gpr-closed in X. Proof Let A be an intuitionistic fuzzy gpr-closed set in an intuitionistic fuzzy topo- logical space (X, ) such that A ⊆ B ⊆ pcl(A). Let O be an intuitionistic fuzzy regular open set such that B ⊆ O. Then A ⊆ O and since A is intuitionistic fuzzy gpr-closed, we have pcl(A) ⊆ O.Now B ⊆ pcl(A) ⇒ pcl(B) ⊆ pcl(pcl(A)) ⊆ pcl(A) ⊆ O. Conse- quently, B is intuitionistic fuzzy gpr-closed in X. Theorem 3.6 If A is an intuitionistic fuzzy regular open and intuitionistic fuzzy gpr- closed set in intuitionistic fuzzy topological space (X, ), then A is an intuitionistic fuzzy preclosed and hence intuitionistic fuzzy clopen. Proof Suppose that A is an intuitionistic fuzzy regular open and intuitionistic fuzzy gpr-closed in X. Since A ⊆ A, we have pcl(A) ⊆ A. Also A ⊆ pcl(A). Therefore pcl(A) Fuzzy Inf. Eng. (2012) 4: 425-444 433 = A. Hence A is an intuitionistic fuzzy preclosed in X.Now A is an intuitionistic fuzzy regular open, A is intuitionistic fuzzy open. Hence A is intuitionistic fuzzy clopen, since every intuitionistic fuzzy preclosed (regular) set is intuitionistic fuzzy (regular) closed. Definition 3.2 An intuitionistic fuzzy set A of an intuitionistic fuzzy topological space (X, ) is called intuitionistic fuzzy gpr-open if and only if its complement A is intu- itionistic fuzzy gpr-closed. Theorem 3.7 Every intuitionistic fuzzy rg-open set in intuitionistic fuzzy topological space (X, ) is intuitionistic fuzzy gpr-open set in X. Proof Let A be an intuitionistic fuzzy rg-open set in X. Then A is intuitionistic fuzzy rg-closed. By Theorem 3.1, A is intuitionistic fuzzy gpr-closed in X. Therefore, A is intuitionistic fuzzy gpr-open in X. Remark 3.5 The converse of the Theorem 3.7 need not be true as seen from the following example. Example 3.4 Let X = {a, b, c, d, e} and intuitionistic fuzzy sets O, U, V defined as follows: O = {< a, 0.9, 0.1 >, < b, 0.8, 0.1>,< c,0,1 >,< d,0,1 >,< e,0,1>}, U = {< a,0,1>, < b,0,1>, < c, 0.8, 0.1 >,< d, 0.7, 0.2>,< e,0,1 >}, V = {< a, 0.9, 0.1>, < b, 0.8, 0.1>,< c, 0.8, 0.1>,<d, 0.7, 0.2>,<e,0,1>}. Let = {0 , O, U, V, 1} be an intuitionistic fuzzy topology on X. Then the intuitionistic fuzzy set A= {< a, 0.1, 0.9>, < b,1,0>, <c,1,0>, < d,1,0>, < e,1, 0>} is intuitionistic fuzzy gpr-open but it is not intuitionistic fuzzy rg-open. Theorem 3.8 Every intuitionistic fuzzyαg-open set in intuitionistic fuzzy topological space (X, ) is intuitionistic fuzzy gpr-open set in X, but not conversely. Proof Let A be an intuitionistic fuzzyαg-open set in X. Then A is intuitionistic fuzzy αg-closed. By Theorem 3.2, A is intuitionistic fuzzy gpr-closed in X. Therefore A is intuitionistic fuzzy gpr-open in X. Remark 3.6 The converse of the Theorem 3.8 need not be true as seen from the following example. Example 3.5 Let X = {a, b, c} and intuitionistic fuzzy sets O, U, V defined as follows: O = {< a, 0.9, 0.1 >, < b,0,1 >,< c,0,1 >}, U = {< a,0,1>, < b, 0.8, 0.1 >,< c,0,1>}, V = {< a, 0.9, 0.1 >, < b, 0.8, 0.1>,< c,0,1 >}. Let = {0, O, U, V, 1} be an intuitionistic fuzzy topology on X. Then intuitionistic fuzzy set A = {< a, 0.1, 0.9 >,< b, 0.1, 0.8>,< c,1,0>} is intuitionistic fuzzy gpr- open set in X, but it is not intuitionistic fuzzy αg-open. Theorem 3.9 An intuitionistic fuzzy set A of an intuitionistic fuzzy topological space (X, ) is intuitionistic fuzzy gpr-open if and only if F ⊆ pint(A) whenever F is intu- itionistic fuzzy regular closed and F ⊆A. Proof Necessity: Let A be intuitionisticfuzzy gpr-open in X. Let F be intuitionistic fuzzy regular closed in X such that F ⊆ A. Then F is intuitionistic fuzzy regular open 434 S. S. Thakur · Jyoti Pandey Bajpai (2012) c c c in X such that A ⊆ F . Now by hypothesis A is intuitionistic fuzzy gpr-closed, we c c c c c c have pcl (A ) ⊆ F . But pcl (A ) = (pint(A)) . Hence (pint(A)) ⊆ F , which implies F⊆pint(A). Sufficiency: Let O be an intuitionistic fuzzy regular open set in X such that A ⊆ c c O. Then O is an intuitionistic fuzzy regular closed in X and O ⊆ A. Therefore by c c c c hypothesis O ⊆ pint(A). This implies that pcl(A ) = (pint(A)) ⊆ O. Hence A is intuitionistic fuzzy gpr-closed and A is intuitionistic fuzzy gpr-open in X. Theorem 3.10 Let A be an intuitionistic fuzzy gpr-open set of an intuitionistic fuzzy topological space (X, ) and pint(A) ⊆ B⊆ A. Then B is intuitionistic fuzzy gpr-open. c c c c c c Proof Since pint(A) ⊆ B ⊆ A. ⇒ A ⊆ B ⊆ (pint(A)) ⇒ A ⊆ B ⊆ pcl(A )by Lemma 2.1(4) and A is intuitionistic fuzzy gpr-closed, it follows from Theorem 3.5 that B is intuitionistic fuzzy gpr-closed. Hence B is intuitionistic fuzzy gpr-open. Theorem 3.11 Let (X, ) be an intuitionistic fuzzy topological space and IFPO(X) (resp. IFGPRO(X)) be the family of all intuitionistic fuzzy preopen (resp. intuitionis- tic fuzzy gpr-open) sets of X. Then IFPO(X) ⊆ IFGPRO(X). Proof Let A ∈ IFPO(X). Then A is intuitionistic fuzzy preclosed and so intuitionistic fuzzy gpr-closed. This implies that A is intuitionistic fuzzy gpr-open. Hence IFPO(X) ⊆ IFGPRO(X). Theorem 3.12 Let (X, ) be an intuitionistic fuzzy topological space and IFPC(X) ( resp. IFRO(X)) be the family of all intuitionistic fuzzy preclosed (resp. intuitionistic fuzzy regular open) sets of X. Then IFPC(X) =IFRO(X) if and only if every intuition- istic fuzzy set of X is intuitionistic fuzzy gpr-closed in X. Proof Necessity: Suppose that IFPC(X) = IFRO(X) and A is any intuitionistic fuzzy set of X such that A ⊆ O where O is intuitionistic fuzzy regular open in X. Then by hypothesis O is intuitionistic fuzzy preclosed in X which implies that pcl(O)= O . Then pcl(A) ⊆ pcl(O)= O. Therefore A is intuitionistic fuzzy gpr-closed in X. Sufficiency: Suppose that every intuitionistic fuzzy set of X is intuitionistic fuzzy gpr-closed. Let U ∈ IFRO(X). Then since U ⊆ U and by hypothesis U is intuitionistic fuzzy gpr-closed set in X. Therefore pcl(U ) ⊆ U , hence U is intuitionistic fuzzy preclosed. That is U ∈ IFPC(X) which implies that IFRO(X) ⊆ IFPC(X). If T ∈ IFPC, then T ∈ IFPO(X) ⊆ IFRO(X) ⊆ IFPC(X). Hence T ∈ IFPO(X) ⊆ IFRO(X). Consequently, IFPC(X) ⊆ IFRO(X). Therefore IFRO(X) = IFPC(X). Theorem 3.13 Let A be an intuitionistic fuzzy g-closed set in an intuitionistic fuzzy topological space (X, ) and f : (X, ) → (Y, σ) be an intuitionistic fuzzy almost continuous and intuitionistic fuzzy preclosed mapping. Then f (A) is an intuitionistic gpr-closed set in Y. Proof Let A be an intuitionistic fuzzy g-closed set in X and f :(X, ) → (Y,σ)be an intuitionistic fuzzy almost continuous and intuitionistic fuzzy preclosed mapping. −1 Let f (A) ⊆ G where G is intuitionistic fuzzy regular open in Y . Then A ⊆ f (G) −1 and f (G) is intuitionistic fuzzy open in X, since f is intuitionistic fuzzy almost Fuzzy Inf. Eng. (2012) 4: 425-444 435 continuous. Now let A be an intuitionistic fuzzy g-closed set in X. Then cl(A) ⊆ −1 −1 f (G). Since pcl(A) ⊆ cl(A), hence pcl(A) ⊆ f (G). Thus f (pcl(A)) ⊆ G and f (pcl(A)) is an intuitionistic fuzzy preclosed set in Y , since pcl(A) is intuitionistic fuzzy preclosed in X and f is intuitionistic fuzzy preclosed mapping. It follows that pcl( f (A) ⊆ pcl(f (pcl(A))) = f (pcl(A))⊆ G. Hence pcl(f (A)) ⊆ G whenever f (A) ⊆ G and G is intuitionistic fuzzy regular open in Y . Hence f (A) is intuitionistic fuzzy gpr-closed set in Y . Corollary 3.1 Let A be an intuitionistic fuzzy g-closed set in an intuitionistic fuzzy topological space (X, ) and f : (X, )→ (Y,σ) be an intuitionistic fuzzy continuous and intuitionistic fuzzy closed mapping. Then f (A) is an intuitionistic gpr-closed set in Y. Theorem 3.14 Let A be an intuitionistic fuzzy gpr-closed set in an intuitionistic fuzzy topological space (X, ) and f : (X, )→ (Y,σ) be an intuitionistic fuzzy R-mapping and intuitionistic fuzzy preclosed mapping. Then f (A) is an intuitionistic gpr-closed set in Y. Proof Let A be an intuitionistic fuzzy gpr-closed set in X and f :(X, ) → (Y,σ) be an intuitionistic fuzzy R-mapping and intuitionistic fuzzy preclosed mapping. Let −1 f (A) ⊆ G where G is intuitionistic fuzzy regular open in Y . Then A ⊆ f (G) and −1 f (G) is intuitionistic fuzzy regular open in X, since f is intuitionistic fuzzy R- −1 mapping. Since A is an intuitionistic fuzzy gpr-closed set in X, pcl(A) ⊆ f (G). Now f (pcl(A)) is an intuitionistic fuzzy preclosed set in Y , since pcl(A) is intuitionistic fuzzy preclosed in X and f is intuitionistic fuzzy preclosed mapping. It follows that pcl(f (A) ⊆ pcl(f (pcl(A))) = f (pcl(A)) ⊆ G. Hence pcl(f (A)) ⊆ G. Hence f (A)is intuitionistic fuzzy gpr-closed set in Y . Theorem 3.15 If f : (X, ) → (Y, σ) is an intuitionistic fuzzy g-continuous and intuitionistic fuzzy preregular-closed mapping, then preimage of every intuitionistic fuzzy gpr-closed set of Y is intuitionistic fuzzy gpr-closed in X. Proof Let B be an intuitionistic fuzzy gpr-closed set in Y and f :(X, )→(Y,σ)be an intuitionistic fuzzy g-continuous and intuitionistic fuzzy preregular closed map- −1 c ping. Let f (B) ⊆ G where G is intuitionistic fuzzy regular open in X then f (G ) c c ⊆ B where G is intuitionistic fuzzy regular closed in X. Since f is intuitionis- tic fuzzy preregular closed, f (G ) is intuitionistic fuzzy regular closed in Y.Now c c c c B is intuitionistic fuzzy gpr-open in Y such that f (G ) ⊆ B where f (G ) is intu- c c c itionistic fuzzy regular closed in Y . Therefore f (G ) ⊆ pint(B ) = (pcl(B)) . Hence −1 −1 f (pcl(B) ⊆ G. Since f is intuitionistic fuzzy g-continuous, f ( pcl(B)) is intu- −1 itionistic fuzzy g-closed in X. Thus we have cl (f ( pcl(B)) ⊆ G. Therefore pcl −1 −1 −1 −1 (f (B)) ⊆ pcl( pcl (f (B))) ⊆ cl(f ( pcl(B)) ⊆ G. Hence f (B) is intuitionistic fuzzy gpr-closed in X. Corollary 3.2 If f : (X, ) → (Y, σ) is an intuitionistic fuzzy continuous and intu- itionistic fuzzy preregular-closed mapping, then preimage of every intuitionistic fuzzy gpr-closed set of Y is intuitionistic fuzzy gpr-closed in X. 436 S. S. Thakur · Jyoti Pandey Bajpai (2012) Theorem 3.16 If f : (X, ) → (Y, σ) is an intuitionistic fuzzy g-continuous and intuitionistic fuzzy preregular-closed mapping, then preimage of every intuitionistic fuzzy gpr-open set of Y is intuitionistic fuzzy gpr-open in X. Proof Proof follows from Theorem 3.15. Corollary 3.3 If f : (X, ) → (Y, σ) is an intuitionistic fuzzy continuous and in- tuitionistic fuzzy preregular-closed mapping. Then preimage of every intuitionistic fuzzy gpr-open set of Y is intuitionistic fuzzy gpr-open in X. 4. Intuitionistic Fuzzy Gpr-connectedness and Intuitionistic Fuzzy Gpr-compact- ness Definition 4.1 An intuitionistic fuzzy topological space (X, ) is called intuitionistic fuzzy gpr-connected if there is no proper intuitionistic fuzzy set of X which is both intuitionistic fuzzy gpr-open and intuitionistic fuzzy gpr-closed. Theorem 4.1 Every intuitionistic fuzzy gpr-connected space is intuitionistic fuzzy connected. Proof Let (X, ) be an intuitionistic fuzzy gpr-connected space and suppose that (X, ) is not intuitionistic fuzzy connected. Then there exists a proper intuitionistic fuzzy set A ( A  0, A  1) such that A is both intuitionistic fuzzy open and intuition- istic fuzzy closed. Since every intuitionistic fuzzy open set (resp. intuitionistic fuzzy closed set) is intuitionistic gpr-open (resp. intuitionistic fuzzy gpr-closed), X is not intuitionistic fuzzy gpr-connected, a contradiction. Remark 4.1 Converse of Theorem 4.1 may not be true as seen from the following example. Example 4.1 Let X = {a, b} and = {0, U , 1 } be an intuitionistic fuzzy topol- ogy on X, where U = {< a, 0.5, 0.5 >, < b, 0.4, 0.6 >}. Then intuitionistic fuzzy topological space (X, ) is intuitionistic fuzzy connected but not intuitionistic fuzzy gpr-connected because there exists a proper intuitionistic fuzzy set A = {<a, 0.5, 0.5 >, < b, 0.5, 0.5 >} which is both intuitionistic fuzzy gpr-closed and intuitionistic gpr-open in X. Theorem 4.2 Every intuitionistic fuzzy gpr-connected space is intuitionistic fuzzy GO-connected. Proof Let (X, ) be an intuitionistic fuzzy gpr-connected space and suppose that (X, ) is not intuitionistic fuzzy GO-connected. Then there exists a proper intuition- istic fuzzy set A (A  0, A  1 ) such that A is both intuitionistic fuzzy g-open and intuitionistic fuzzy g-closed. Since every intuitionistic fuzzy g-open set (resp. intuitionistic fuzzy g-closed set) is intuitionistic gpr-open (resp. intuitionistic fuzzy gpr-closed), X is not intuitionistic fuzzy gpr-connected, a contradiction. Theorem 4.3 Every intuitionistic fuzzy gpr-connected space is intuitionistic fuzzy rg-connected. Proof Let (X, ) be an intuitionistic fuzzy gpr-connected space and suppose that (X, ) is not intuitionistic fuzzy rg-connected. Then there exists a proper intuition- istic fuzzy set A ( A  0, A  1 ) such that A is both intuitionistic fuzzy rg-open Fuzzy Inf. Eng. (2012) 4: 425-444 437 and intuitionistic fuzzy rg-closed. Since every intuitionistic fuzzy rg-open set (resp. intuitionistic fuzzy rg-closed set) is intuitionistic fuzzy gpr-open (resp. intuitionistic fuzzy gpr-closed), X is not intuitionistic fuzzy gpr-connected, a contradiction. Theorem 4.4 An intuitionistic fuzzy topological (X, ) is intuitionistic fuzzy gpr- connected if and only if there exists no nonempty intuitionistic fuzzy gpr-open sets A and B in X such that A=B . Proof Necessity: Suppose A and B are intuitionistic fuzzy gpr-open sets such that c c A  0  B and A = B . Since A = B , B is an intuitionistic fuzzy gpr-open set which c c implies that B = A is intuitionistic fuzzy gpr-closed set and B  0, this implies that B 1 i.e., A  1 . Hence there exists a proper intuitionistic fuzzy set A ( A  0, A  1 ) such that A is both intuitionistic fuzzy gpr-open and intuitionistic fuzzy gpr-closed. But this is contradiction to the fact that X is intuitionistic fuzzy gpr-connected. Sufficiency: Let (X, ) be an intuitionistic fuzzy topological space and A be both intuitionistic fuzzy gpr-open set and intuitionistic fuzzy gpr-closed set in X such that 0  A  1 . Now take B = A . In this case, B is an intuitionistic fuzzy gpr-open set and A  1 , this implies that B =A  0 . Hence, A  1 which is a contradiction. Hence there is no proper intuitionistic fuzzy set of X which is both intuitionistic fuzzy gpr- open and intuitionistic fuzzy gpr-closed. Therefore intuitionistic fuzzy topological (X , ) is intuitionistic fuzzy gpr-connected. Definition 4.2 An intuitionistic fuzzy topological space (X, ) is said to be intuitionis- tic fuzzy preregular-T if every intuitionistic fuzzy gpr-closed set in X is intuitionistic 1/2 fuzzy preclosed in X. Definition 4.3 A collection { A :i∈ Λ} of intuitionistic fuzzy rga-open sets in intu- itionistic fuzzy topological space (X, ) is called intuitionistic fuzzy gpr-open cover of intuitionistic fuzzy set B of X if B⊆∪{ A :i∈ Λ}. Definition 4.4 An intuitionistic fuzzy topological space (X, ) is said to be intuition- istic fuzzy gpr-compact if every intuitionistic fuzzy gpr-open cover of X has a finite subcover. Definition 4.5 An intuitionistic fuzzy set B of intuitionistic fuzzy topological space (X, ) is said to be intuitionistic fuzzy gpr-compact relative to X if for every collection { A :i∈ Λ} of intuitionistic fuzzy gpr-open subset of X such that B⊆∪{ A :i∈ Λ }, i i there exists finite subset Λ ofΛ such that B⊆∪{ A :i∈ Λ }. o i o Definition 4.6 A crisp subset B of an intuitionistic fuzzy topological space (X, )is said to be intuitionistic fuzzy gpr-compact if B is intuitionistic fuzzy gpr-compact as an intuitionistic fuzzy subspace of X. Theorem 4.5 An intuitionistic fuzzy gpr-closed crisp subset of intuitionistic fuzzy gpr-compact space is intuitionistic fuzzy gpr-compact relative to X. Proof Let A be an intuitionistic fuzzy gpr-closed crisp subset of intuitionistic fuzzy gpr-compact space (X, ). Then A is intuitionistic fuzzy gpr-open in X. Let M be a cover of A by intuitionistic fuzzy gpr-open sets in X. Then the family {M, A } is intuitionistic fuzzy gpr-open cover of X. Since X is intuitionistic fuzzy gpr-compact, 438 S. S. Thakur · Jyoti Pandey Bajpai (2012) it has a finite subcover say {G , G , G ,··· , G }. If this subcover contains A ,we 1 2 3 n discard it. Otherwise leave the subcover as it is. Thus we obtain a finite intuitionistic fuzzy gpr-open subcover of A. Therefore A is intuitionistic fuzzy gpr-compact relative to X. Theorem 4.6 If f : (X, ) → (Y, σ) is an intuitionistic fuzzy g-continuous and intu- itionistic fuzzy preregular-closed surjection and X is intuitionistic fuzzy gpr-compact, then Y is intuitionistic fuzzy gpr-compact. Proof Obvious. 5. Intuitionistic Fuzzy Gpr-continuous and Intuitionistic Fuzzy Gpr-irresolute Mappings Definition 5.1 A mapping f : (X, ) →(Y, σ) is intuitionistic fuzzy gpr-continuous if preimage of every intuitionistic fuzzy closed set of Y is intuitionistic fuzzy gpr-closed set in X. Theorem 5.1 A mapping f : (X, ) →(Y, σ) is intuitionistic fuzzy gpr-continuous if and only if the preimage of every intuitionistic fuzzy open set of Y is intuitionistic fuzzy gpr-open in X. −1 c −1 c Proof It is obvious because f (U )=(f (U )) for every intuitionistic fuzzy set U of Y . Remark 5.1 Every intuitionistic fuzzy continuous mapping is intuitionistic fuzzy gpr-continuous, but converse may not be true as seen from the following example. Example 5.1 Let X = {a, b}, Y ={x, y} and intuitionistic fuzzy sets U and V are defined as follows: U = {< a, 0.5, 0.5 >, < b, 0.4, 0.6 >}, V = {< x, 0.5, 0.5 >, < y, 0.5, 0.5>}. Let = {0, U , 1 } and s = {0, V , 1 } be intuitionistic fuzzy topologies on X and Y respectively. Then the mapping f:(X, ) →(Y,σ) defined by f (a)= x and f (b)= y is intuitionistic fuzzy gpr-continuous but not intuitionistic fuzzy continuous. Remark 5.2 Every intuitionistic fuzzy rg-continuous mapping is intuitionistic fuzzy gpr-continuous, but converse may not be true as seen from the following example. Example 5.2 Let X = {a, b, c, d, e} and Y = {p, q, r, s, t } and intuitionistic fuzzy sets O, U, V and W defined as follows: O = {< a, 0.9, 0.1>, < b, 0.8, 0.1 >,< c, 0, 1>,< d,0,1 >, < e,0,1 >}, U = {< a,0,1 >, < b,0,1>, < c, 0.8, 0.1>,< d, 0.7, 0.2>< e,0,1>}, V = {< a, 0.9, 0.1>, < b, 0.8, 0.1 >, < c, 0.8, 0.1>,< d, 0.7, 0.2 >,< e,0,1 >}, W = {< p, 0.9, 0.1 >, < q,0,1 >, < r,0,1>, < s,0,1 >,< t,0,1 >}. Let = {0, O, U, V, 1 } and σ = {0, W , 1 } be intuitionistic fuzzy topologies on X and Y respectively. Then the mapping f:(X, ) →(Y,σ) defined by f (a)= p, f (b)= q, f (c)= r, f (d)= s and f (e)=t is intuitionistic fuzzy rg-continuous but not intuitionistic fuzzy gpr-continuous. Fuzzy Inf. Eng. (2012) 4: 425-444 439 Remark 5.3 Every intuitionistic fuzzyαg-continuous mapping is intuitionistic fuzzy gpr-continuous, but converse may not be true as seen from the following example. Example 5.3 Let X = {a, b, c}, Y ={x, y, z} and intuitionistic fuzzy sets O, U and V are defined as follows: O ={< a, 0.9, 0.1>, < b,0,1 >,< c,0,1 >}, U = {< a,0,1>, < b, 0.8, 0.1>,< c,0,1>}, V = {< x, 0.9, 0.1>, < y, 0.8, 0.1>,< z,0,1>}. Let = {0, O, U, 1 } and σ = {0, V , 1 } be intuitionistic fuzzy topologies on X and Y respectively. Then the mapping f:(X, ) →(Y,σ) defined by f (a)= xf (b)= y and f (c)= z is intuitionistic fuzzy gpr-continuous, but it is not intuitionistic fuzzy αg-continuous. Remark 5.4 From the above discussion and known results, we have the following diagram of implications: Intuitionistic fuzzy←Intuitionistic fuzzy →Intuitionistic fuzzy α-continuous continuous g-continuous ↓↓ ↓ Intuitionistic fuzzy→Intuitionistic fuzzy←Intuitionistic fuzzy αg-continuous gpr-continuous rg-continuous Theorem 5.2 If f : (X, ) →(Y, σ) is intuitionistic fuzzy gpr-continuous, then for each intuitionistic fuzzy point c(α, β) of X and each intuitionistic fuzzy open set V of Y such that f (c(α,β))⊆ V, there exists an intuitionistic fuzzy gpr-open set U of X such that c(α,β)⊆ U and f (U )⊆ V. Proof Let c(α, β) be intuitionistic fuzzy point of X and V be an intuitionistic fuzzy −1 open set of Y such that f (c(α, β)) ⊆ V . Put U = f (V ). Then by hypothesis U is −1 intuitionistic fuzzy gpr-open set of X such that c(α, β) ⊆ U and f (U)= f ( f (V )) ⊆ V . Theorem 5.3 Let f : (X, ) →(Y, σ) be intuitionistic fuzzy gpr-continuous mapping. Then for each intuitionistic fuzzy point c(α,β) of X and each intuitionistic fuzzy open set V of Y such that f (c(α,β))qV, there exists an intuitionistic fuzzy gpr-open set U of X such that c(α,β)qU and f (U ) ⊆ V. Proof Let c(α, β) be intuitionistic fuzzy point of X and V be an intuitionistic fuzzy −1 open set of Y such that f (c(α, β))qV . Put U = f (V ). Then by hypothesis, U is −1 intuitionistic fuzzy gpr-open set of X such that c(α,β)qU and f (U)= f ( f (V))⊆ V . Theorem 5.4 A mapping f from an intuitionistic fuzzy preregular-T space (X, ) 1/2 to an intuitionistic fuzzy topological space (Y,σ) is intuitionistic fuzzy precontinuous if and only if it is intuitionistic fuzzy gpr-continuous. 440 S. S. Thakur · Jyoti Pandey Bajpai (2012) Proof Necessity: Let f :(X, ) →(Y,σ) be intuitionistic fuzzy pre-continuous map- −1 ping. Let U be intuitionistic fuzzy closed set in Y . Then f (U ) is intuitionistic fuzzy preclosed in X. Since every intuitionistic fuzzy preclosed set is intuitionistic −1 fuzzy gpr-closed, f (U ) is intuitionistic fuzzy gpr-closed in X which implies that f is intuitionistic fuzzy gpr-continuous. Sufficiency: Let f :(X, ) →(Y,σ) be intuitionistic fuzzy gpr-continuous map- −1 ping. Let U be intuitionistic fuzzy closed set in Y . Then f (U ) is intuitionistic fuzzy gpr-closed in X. Since X is intuitionistic fuzzy preregular-T , therefore ev- 1/2 −1 ery intuitionistic fuzzy gpr-closed set is intuitionistic fuzzy preclosed. Hence f (U ) is intuitionistic fuzzy preclosed in X which implies that f is intuitionistic fuzzy pre- continuous. Theorem 5.5 If f : (X, ) →(Y, σ) is intuitionistic fuzzy gpr-continuous and g : (Y, σ)→(Z,μ) is intuitionistic fuzzy continuous, then gof : (X, )→(Z,μ) is intuitionistic fuzzy gpr-continuous. −1 Proof Let A be an intuitionistic fuzzy closed set in Z. Then g (A) is intuitionistic −1 fuzzy closed in Y , because g is intuitionistic fuzzy continuous. Therefore (gof ) (A) −1 −1 = f (g (A)) is intuitionistic fuzzy gpr-closed in X, because f is intuitionistic fuzzy gpr-continuous. Hence gof is intuitionistic fuzzy gpr-continuous. Remark 5.5 The composition of two intuitionistic fuzzy gpr-continuous mappings need not be intuitionistic fuzzy gpr-continuous as seen from the following example. Example 5.4 Let X = {a, b, c} and intuitionistic fuzzy sets O, U, V, W and T are defined as follows: O = {< a, 0.9, 0.1 >, < b,0,1 >, < c,0,1 >}, U = {< a,0,1 >, < b, 0.8, 0.1>, < c,0,1 >}, V = {< a,0,1>, < b,0,1>, <c, 0.9, 0.1>}, W = {< a, 0.9, 0.1 >, < b, 0.8, 0.1 >,< c,0,1>}, T = {< a,0,1>, < b, 0.8,.0.1 > ,< c, 0.9, 0.1>}. Let ={ 0, O, U, W, 1} , σ = {0, V, T, 1 } and μ = {0, T, 1 } be intuitionistic fuzzy topologies on X. Then the mapping f :(X, ) → (X,σ) defined by f (a) = b, f (b) = c and f (c) = a and mapping g :(X,σ) →(X, μ) defined by g(a) = b, g (b) = c and g (c) = c are intuitionistic fuzzy gpr-continuous, but composition mapping gof :(X, ) →(X,μ) is not intuitionistic fuzzy gpr-continuous. Theorem 5.6 If f : (X, ) → (Y, σ) is intuitionistic fuzzy gpr-continuous and g : (Y, σ)→ (Z,μ) is intuitionistic fuzzy g-continuous and (Y,σ) is intuitionistic fuzzy-(T ) 1/2 space, then gof : (X, ) → (Z,μ) is intuitionistic fuzzy gpr-continuous. −1 Proof Let A be an intuitionistic fuzzy closed set in Z. Then g (A) is intuitionistic −1 fuzzy g-closed in Y . Since Y is intuitionistic fuzzy-(T ) space, then g (A) is in- 1/2 −1 −1 −1 tuitionistic fuzzy closed in Y . Hence (gof ) (A)= f (g (A)) is intuitionistic fuzzy gpr-closed in X. Hence gof is intuitionistic fuzzy gpr-continuous. Fuzzy Inf. Eng. (2012) 4: 425-444 441 Theorem 5.7 An intuitionistic fuzzy gpr-continuous image of an intuitionistic fuzzy gpr-compact space is intuitionistic fuzzy compact. Proof Let f :(X, ) → (Y,σ) be intuitionistic fuzzy gpr-continuous mapping from an intuitionistic fuzzy gpr-compact space (X, ) onto an intuitionistic fuzzy topolog- ical space (Y,σ). Let {A : i ∈ Λ} be an intuitionistic fuzzy open cover of Y . Then −1 { f (A ): i ∈ Λ} is an intuitionistic fuzzy gpr-open cover of X. Since X is intuition- −1 −1 −1 istic fuzzy gpr-compact there is a finite subfamily { f (A ), f (A ),··· , f (A )} i i i 1 2 n −1 n −1 ˜ ˜ ˜ of { f (A ): i ∈ Λ} such that ∪ f (A ) = 1. Since f is onto f (1) = 1 and i i j=1 n −1 n −1 n n f (∪ f (A )) = ∪ f ( f (A )) = ∪ A . it follows that ∪ A = 1 and the i i i i j j j j j=1 j=1 j=1 j=1 family {A , A ,··· , A } is an intuitionistic fuzzy finite subcover of {A : i ∈ Λ}. i i i i 1 2 n Hence (Y,σ) is intuitionistic fuzzy compact. Theorem 5.8 If f : (X, ) →(Y, σ) is intuitionistic fuzzy gpr-continuous surjection and X is intuitionistic fuzzy gpr-connected, then Y is intuitionistic fuzzy connected. Proof Suppose that Y is not intuitionistic fuzzy connected. Then there exists a proper intuitionistic fuzzy set G of Y which is both intuitionistic fuzzy open and intuitionistic −1 fuzzy closed. Therefore f ( G) is a proper intuitionistic fuzzy set of X, which is both intuitionistic fuzzy gpr-open and intuitionistic fuzzy gpr-closed, because f is intuitionistic fuzzy gpr-continuous surjection. Hence X is not intuitionistic fuzzy gpr-connected, which is a contradiction. Definition 5.2 A mapping f : (X, ) →(Y, σ) is intuitionistic fuzzy gpr-irresolute if preimage of every intuitionistic fuzzy gpr-closed set of Y is intuitionistic fuzzy gpr- closed set in X. Theorem 5.9 A mapping f : (X, )→(Y,σ) is intuitionistic fuzzy gpr-irresolute if and only if the preimage of every intuitionistic fuzzy open set of Y is intuitionistic fuzzy gpr-open in X. −1 c −1 c Proof It is obvious, because f (U )= f (U )) for every intuitionistic fuzzy set U of Y . Remark 5.6 Since every intuitionistic fuzzy closed set is intuitionistic fuzzy gpr- closed, it is clear that every intuitionistic fuzzy gpr-irresolute mapping is intuition- istic fuzzy gpr-continuous, but converse may not be true as seen from the following examples. Example 5.5 Let X = {a, b}, Y = {x, y} and let = {0, U , 1 } and σ = {0, 1 } be intuitionistic fuzzy topologies on X and Y respectively where U = {< a, 0.7, 0.3>,< b, 0.5, 0.5>}. Then the mapping f :(X, )→(Y ,σ) defined by f (a) = x and f (b) = y is intuitionistic fuzzy gpr-continuous but not intuitionistic fuzzy gpr-irresolute. Example 5.6 Let X = {a, b}, Y ={x, y} and let = {0, 1} and σ = {0, V , 1 } be intuitionistic fuzzy topologies on X and Y respectively where V= {< x, 0.5, 0.5>,<y, 0.3, 0.7>}. Then the mapping f :(X, ) → (Y , σ) defined by f (a) = x and f (b) = y is intuitionistic fuzzy gpr-irresolute but not intuitionistic fuzzy continuous. Remark 5.7 Example 5.5 and Example 5.6 assert that concept of intuitionistic fuzzy gpr-irresolute and intuitionistic fuzzy continuous mappings are independent. 442 S. S. Thakur · Jyoti Pandey Bajpai (2012) Theorem 5.10 Let f : (X, ) →(Y, σ) be bijective intuitionistic fuzzy regular-open and intuitionistic fuzzy gpr-continuous. Then f is intuitionistic fuzzy gpr-irresolute. −1 Proof Let A be intuitionistic fuzzy gpr-closed in Y and let f (A) ⊆ G where G is intuitionistic fuzzy regular open in X. Then A ⊆ f (G). Since f (G) is intuitionistic fuzzy regular open in Y and A is intuitionistic fuzzy gpr-closed in Y , then pcl(A) ⊆ −1 f (G) and f (pcl(A)) ⊆ G. Since f is intuitionistic fuzzy gpr-continuous and cl(A) −1 is intuitionistic fuzzy closed in Y , f (cl(A)) is intuitionistic fuzzy gpr-closed in X, −1 −1 −1 therefore pcl (f (cl(A))) ⊆ G and so pcl (f (A)) ⊆ G. Hence f (A) is intuitionistic fuzzy gpr-closed in X. Therefore f is intuitionistic fuzzy gpr-irresolute. Theorem 5.11 Let f : (X, ) →(Y, σ) be intuitionistic fuzzy gpr-irresolute and g: (Y,σ)→(Z,μ) is intuitionistic fuzzy gpr-continuous mapping. Then gof : (X, )→(Z, μ) is intuitionistic fuzzy gpr-continuous. −1 Proof Let A be intuitionistic fuzzy closed in Z. Then g (A) is intuitionistic fuzzy −1 gpr-closed in Y , because g is intuitionistic fuzzy gpr-continuous. Therefore (gof ) (A) −1 −1 = f (g (A)) is intuitionistic fuzzy gpr-closed in X, because f is intuitionistic fuzzy gpr-irresolute. Hence gof is intuitionistic fuzzy gpr-continuous. Theorem 5.12 If f : (X, )→(Y, σ)and g: (Y, σ) →(Z, μ) be two intuitionisticfuzzy gpr-irresolute mapping, then gof: (X, )→(Z,μ) is intuitionistic fuzzy gpr-irresolute. −1 Proof Let A be an intuitionistic fuzzy gpr-closed set in Z. Then g (A) is intuition- stic fuzzy gpr-closed in Y because g is intuitionistic fuzzy gpr-irresolute. Therefore −1 −1 −1 (go f ) (A)= f (g (A)) is intuitionistic fuzzy gpr-closed in X because f is intuition- istic fuzzy gpr-irresolute. Hence gof is intuitionistic fuzzy gpr-irresolute. Theorem 5.13 Let f : (X, ) →(Y, σ) be intuitionistic fuzzy gpr-irresolute mapping and if B is fuzzy gpr-compact relative to X. Then image f (B) is intuitionistic fuzzy gpr-compact relative to Y. Proof Let {A : i∈ Λ} be an intuitionistic fuzzy gpr-open set of Y such that f (B) ⊆ −1 ∪{ A : i∈ Λ}. Then B ⊆∪ { f (A ): i∈ Λ}. By using the assumption, there exists i i −1 a finite subset Λ of Λ such that B⊆∪ { f (A ): i∈ Λ }. Therefore f (B)⊆∪{ A : o i 0 i i∈ Λ } which shows that f (B) is intuitionistic fuzzy gpr-compact relative to Y . Corollary 5.1 An intuitionistic fuzzy gpr-irresolute image of an intuitionistic fuzzy gpr-compact space is intuitionistic fuzzy gpr-compact. Proof Let f :(X, ) →(Y , σ) be intuitionistic fuzzy gpr-irresolute mapping from an intuitionistic fuzzy gpr-compact space (X, ) onto an intuitionistic fuzzy topological space (Y , σ). Let { A : i∈ Λ } be an intuitionistic fuzzy gpr-open cover of Y . Then −1 f (A ): i∈ Λ} is an intuitionistic fuzzy gpr-open cover of X. Since X is intuition- −1 −1 −1 istic fuzzy gpr-compact there is a finite subfamily { f (A ), f (A ),··· , f (A )} i i i 1 2 n −1 n −1 ˜ ˜ ˜ of { f (A ): i ∈ Λ} such that ∪ f (A ) = 1. Since f is onto f (1) = 1 and i i j=1 n −1 n −1 n n f (∪ f (A )) = ∪ f ( f (A )) = ∪ A . it follows that ∪ A = 1 and the i i i i j j j j j=1 j=1 j=1 j=1 family {A , A ,··· , A } is an intuitionistic fuzzy finite subcover of {A : i ∈ Λ}. i i i i 1 2 n Hence (Y,σ) is intuitionistic fuzzy gpr-compact. Fuzzy Inf. Eng. (2012) 4: 425-444 443 Theorem 5.14 Let (X× Y, × σ) be the intuitionistic fuzzy product space of intu- itionistic fuzzy topological spaces (X, ) and (Y,σ). Then the projection mapping P: X×Y→X is an intuitionistic fuzzy gpr-irresolute. −1 Proof Let F be any intuitionistic fuzzy gpr-closed set of X. Then F× 1= P (F)) is intuitionistic fuzzy gpr-closed and hence P is an intuitionistic fuzzy gpr-irresolute. Theorem 5.15 If the product space (X × Y, × σ ) of two intuitionistic fuzzy topo- logical spaces (X, ) and (Y, σ) is intuitionistic fuzzy gpr-compact, then each factor space is intuitionistic fuzzy gpr-compact. Proof Let (X × Y , × σ ) be intuitionistic fuzzy gpr-compact. Then by Corollary 3.3, we obtain that the intuitionistic fuzzy gpr-irresolute image of p(X × Y)= X is intuitionistic fuzzy gpr-compact. Theorem 5.16 If f : (X, ) →(Y, σ) is intuitionistic fuzzy gpr-irresolute surjection and (X, ) is intuitionistic fuzzy gpr-connected, then (Y,σ) is intuitionistic fuzzy gpr- connected. Proof Suppose Y is not intuitionistic fuzzy gpr-connected. Then there exists a proper intuitionistic fuzzy set G of Y which is both intuitionistic fuzzy gpr-open and intu- −1 itionistic fuzzy gpr-closed. Therefore f (G) is a proper intuitionistic fuzzy set of X, which is both intuitionistic fuzzy gpr-open and intuitionistic fuzzy gpr-closed, be- cause f is intuitionistic fuzzy gpr-irresolute surjection. Hence X is not intuitionistic fuzzy gpr-connected, which is a contradiction. Hence Y is intuitionistic fuzzy gpr- connected. Theorem 5.17 If the product space (X×Y, ×σ) of two nonempty intuitionistic fuzzy topological spaces (X, ) and (Y, σ) is intuitionistic fuzzy gpr-connected, then each factor space is intuitionistic fuzzy gpr-connected. Proof If (X× Y , ×σ) is intuitionistic fuzzy gpr-connected, then mapping P : X× Y → X is intuitionistic fuzzy gpr-irresolute. Hence by Theorem 5.16 the intuitionistic fuzzy gpr-irresolute image P(X×Y)= X of an intuitionistic fuzzy gpr-connected space X × Y is an intuitionistic fuzzy gpr-connected. 6. Conclusion The theory of g-closed sets plays an important role in the general topology. Since its inception many weak forms of g-closed sets have been introduced in general topology as well as in fuzzy topology and in intutionistic fuzzy topology. 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Journal

Fuzzy Information and EngineeringTaylor & Francis

Published: Dec 1, 2012

Keywords: Intuitionistic fuzzy gpr-closed sets; Gpr-open sets; Gpr-connectedness; Gpr-compactness; Gpr-continuous and gpr-irresolute mappings

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