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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 [email protected] [email protected] 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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Fuzzy Information and Engineering – Taylor & 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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