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Gen. Math. Notes, Vol. 19, No. 2, December, 2013, pp. 59-70 c ISSN 2219-7184; Copyright ICSRS Publication, 2013 www.i-csrs.org Available free online at http://www.geman.in On Intuitionistic Fuzzy Soft Topology İsmail Osmanoğlu1 and Deniz Tokat2 1,2 Department of Mathematics, Faculty of Arts and Sciences Nevşehir Hacı Bektaş Veli University, Nevşehir, Turkey 1 E-mail: [email protected] 2 E-mail: [email protected] (Received: 16-9-13 / Accepted: 22-10-13) Abstract In this paper, we define subspace, separation axioms, compactness and connectedness on intuitionistic fuzzy (IF, in short) soft topological spaces. We also give base theorems of these concepts. Keywords: IF soft set, IF soft topology, IF soft subspace, IF soft separation axioms, IF soft compactness and IF soft connectedness. 1 Introduction Classical mathematical methods are not enough to solve the problems of daily life and also are not enough to meet the new requirements. Therefore, some theories such as Fuzzy set theory [14], Rough set theory [9] and Soft set theory [8] have been developed to solve these problems. Applications of these theories appear in topology and many areas of mathematics. Topological structure on concepts of fuzzy, soft, fuzzy soft, intuitionistic fuzzy and intuitionistic fuzzy soft was defined by Chang [4], Shabir and Naz [10], Tanay et al. [11], D.Coker [5] and Z. Li et al [6], respectively. Z. Li et al [6] studied the concepts of intuitionistic fuzzy (IF, in short) soft open set, IF soft interior point, IF soft closure point and IF soft base. We will examine the concepts of subspace of a IF soft topological space, IF soft separation axioms, IF soft compactness and IF soft connectedness with some base theorems. 60 2 İsmail Osmanoğlu et al. Preliminaries In this section, we present the basic definitions of soft set, IF set and IF soft set. Definition 2.1 (Soft set). [8] A pair (F, A) is called a soft set (over U ) where F is a mapping F : A → P (U ). In other words, the soft set is a parameterized family of subsets of the set U . Every set F (e), e ∈ E , from this family may be considered as the set of e-elements of the soft set (F, A), or as the set of e-approximate elements of the soft set. Definition 2.2 (IF set). [1] An intuitionistic fuzzy (IF, in short) set A over the universe U can be defined as follows A = {(x, µA (x), νA (x)) : x ∈ U } where µA : U → [0, 1], νA : U → [0, 1] with the property 0 ≤ µA (x)+νA (x) ≤ 1, ∀x ∈ U . The values µA (x) and νA (x) represent the degree of membership and nonmembership of x to A respectively. Definition 2.3. [1] Let A = {(x, µA (x), νA (x)) : x ∈ U } and B = {(x, µB (x), νB (x)) : x ∈ U } be intuitionistic fuzzy sets of U . (1) A ⊆ B if and only if µA (x) ≤ µB (x) and νA (x) ≥ νB (x) for all x ∈ U, (2) A ∩ B = {(x, min{µA (x), µB (x)}, max{νA (x), νB (x)}) : x ∈ U } , (3) A ∪ B = {(x, max{µA (x), µB (x)}, min{νA (x), νB (x)}) : x ∈ U } . Definition 2.4. [1] An intuitionistic fuzzy set A over the universe U defined as A = (x, 0, 1) : x ∈ U ) is said to be intuitionistic fuzzy null set and is denoted by 0̃. Definition 2.5. [1] An intuitionistic fuzzy set A over the universe U defined as A = (x, 1, 0) : x ∈ U ) is said to be intuitionistic fuzzy absolute set and is denoted by 1̃. Definition 2.6 (IF soft set). [7] Let U be an initial universe set and E be the set of parameters. Let IF U denote the collection of all IF subsets of U . Let A⊆E. A pair (F, A) is called an IF soft set over U where F is a mapping given by F : A → IF U . In general, for every e ∈ A, F (e) is an IF set of U and it is called IF value set of parameter e. Clearly, F (e) can be written as an IF set such that F (e) = {(x, µA (x), νA (x)) : x ∈ U }. The set of all IF soft sets over U with parameters from E is called an IF soft class and it is denoted by IF S(UE ). On Intuitionistic Fuzzy Soft Topology 61 Example 2.7. [2] Let (F, A) describe the character of the students with respect to the given parameters, for finding the best student of an academic year. Let the set of students under consideration be U = {s1 , s2 , s3 , s4 }. Let E = {good result (r), conduct (c), games and sports perf ormances (g), sincerity (s), pleasing personality (p)} be the set of parameters framed to choose the best student. Suppose Mr. X has the parameter set A = {r, c, p} ⊆ E choose the best student. Then (F, A) = {F (r) = {(s1 , 0.8, 0.1), (s2 , 0.7, 0.05), (s3 , 0.9, 0.1), (s4 , 0.7, 0.2)}, F (c) = {(s1 , 0.6, 0.2), (s2 , 0.7, 0.1), (s3 , 0.5, 0.3), (s4 , 0.3, 0.6)}, F (p) = {(s1 , 0.6, 0.2), (s2 , 0.7, 0.1), (s3 , 0.5, 0.3), (s4 , 0.3, 0.6)}} Definition 2.8. [7] Let (F, A) and (G, B) be two IF soft sets over U . ˜ (G, B) = (H, C) where C = A ∪ B and ∀e ∈ C, Union (F, A)∪ if e ∈ A − B F (e) H(e) if e ∈ B − A H(e) = F (e) ∪ G(e) if e ∈ A ∩ B ˜ (G, B) = (H, C) where C = A ∩ B and ∀e ∈ C, H(E) = Intersection (F, A)∩ F (e) ∩ G(e). ˜ Subset (F, A)⊆(G, B) where i) A ⊆ B, ii) for all e ∈ A, F (e) ⊆ G(e) Complement The complement of an intuitionistic fuzzy soft set (F, A) is denoted by (F, A)c and is defined by (F, A)c = (F c , A), where F c : A → IF U is a mapping given by F c (e) = [F (e)]c for all e ∈ A. Thus if F (e) = (x, µF (e) (x), νF (e) (x)) : x∈ U , then ∀e ∈ A, F c (e) = (F (e))c = (x, νF (e) (x), µF (e) (x)) : x ∈ U . Absolute IF soft set A soft set (F, A) over U is said to be null intuitionistic fuzzy soft set denoted by à if ∀e ∈ A, F (e) is the absolute intuitionistic fuzzy set 1̃ of U where 1̃(x) = 1, ∀x ∈ U . We would use the notation ŨA to represent the absolute intuitionistic fuzzy soft set with respect to the set of parameters A. Null IF soft set A soft set (F, A) over U is said to be null intuitionistic fuzzy soft set denoted by Φ if ∀e ∈ A, F (e) is the null intuitionistic fuzzy set 0̃ of U where 0̃(x) = 0, ∀x ∈ U . We would use the notation ΦA to represent the null intuitionistic fuzzy soft set with respect to the set of parameters A. 62 İsmail Osmanoğlu et al. Definition 2.9. [3] Let (F, A) and (G, B) be two IF soft sets over U . We define the difference of (F, A) and (G, B) as the IF soft (H, C) written as (F, A) − (G, B) = (H, C) where C = A ∩ B and ∀e ∈ C, x ∈ U, µH(e) (x) = min(µF (e) (x), νG(e) (x)), νH(e) (x) = max(νF (e) (x), µG(e) (x)). Definition 2.10. [13] Let IF S(XE ) and IF S(YK ) be two intuitionistic fuzzy soft classes, and let ϕ : X → Y and φ : E → K be mappings. Then a mapping (ϕ, φ) : IF S(XE ) → IF S(YK ) is defined as: for (F, A) ∈ IF S(XE ), the image of (F, A) under (ϕ, φ), denoted by (ϕ, φ)(F, A) = (ϕ(F ), φ(A)), is an intuitionistic fuzzy soft set in IF S(YK ) given by ( sup µF (e) (x) if ϕ−1 (y) 6= ∅ −1 −1 µϕ(F )(k) (y) = e∈φ (k)∩A,x∈ϕ (y) 0 otherwise, and ( inf νF (e) (x) if ϕ−1 (y) 6= ∅ νϕ(F )(k) (y) = e∈φ−1 (k)∩A,x∈ϕ−1 (y) 1 otherwise, for all k ∈ φ(A) and y ∈ Y . For (G, B) ∈ IF S(YK ), the inverse image of (G, B) under (ϕ, φ), denoted by (ϕ, φ)−1 (G, B) = (ϕ−1 (G), φ−1 (B)) is an intuitionistic fuzzy soft set in IF S(XE ) given by µϕ−1 (G)(e) (x) = µG(φ(e)) (ϕ(x)) and νϕ−1 (G)(e) (x) = νG(φ(e)) (ϕ(x)) for all e ∈ φ−1 (B) and x ∈ X. 3 Some Topological Structures on IF Soft Topology Definition 3.1. [6] Let τ ⊆ IF S(XE ), then τ is said to be a IF soft topology on X if the following conditions hold. i. ΦE , X̃E belong to τ . ii. The union of any number of IF soft sets in τ belongs to τ . iii. The intersection of any two IF soft sets in τ belongs to τ . τ is called a IF soft topology over X and the binary (XE , τ ) is called a IF soft topological space over X. The members of τ are said to be IF soft open sets in X. A IF soft set (F, E) over X is said to be a IF soft closed set in X, if its complement (F, E)c belongs to τ . On Intuitionistic Fuzzy Soft Topology 63 Example 3.2. Let X = {x1 , x2 }, E = {e1 , e2 } and τ = {ΦE , X̃E , (F1 , E), (F2 , E), (F3 , E)} where (F1 , E), (F2 , E), (F3 , E) are IF soft sets over X, defined as follows F1 (e1 ) = {(x1 , 0.3, 0.4), (x2 , 1, 0.3)}, F1 (e2 ) = {(x1 , 0.3, 0.2), (x2 , 1, 0.1)}, F2 (e1 ) = {(x1 , 0.9, 0.2), (x2 , 1, 0.1)}, F2 (e2 ) = {(x1 , 0.9, 0.1), (x2 , 1, 0.1)}, F3 (e1 ) = {(x1 , 0, 0.4), (x2 , 0.6, 0.3)}, F3 (e2 ) = {(x1 , 0, 0.4), (x2 , 0.6, 0.2)}. Then τ defines a IF soft topology on X and hence (XE , τ ) is a IF soft topological space over X. Definition 3.3. Let (XE , τ1 ) and (XE , τ2 ) be IF soft topological spaces. Then, the following hold. If τ2 ⊇ τ1 , then τ2 is IF soft finer than τ1 . If τ2 ⊃ τ1 , then τ2 is IF soft strictly finer than τ1 . If either τ2 ⊇ τ1 or τ2 ⊆ τ1 , then τ1 is comparable with τ2 . Definition 3.4. Let X be the universal set, E be the set of parameters. • Let τ be the collection of all IF soft sets which can be defined over X. Then τ is called the IF soft discrete topology on X and (XE , τ ) is said to be a IF soft discrete space over X. • τ = {Φ, X̃} is called the IF soft indiscrete topology on X and (XE , τ ) is said to be a IF soft indiscrete space over X. Definition 3.5. Let (XE , τ ) be a IF soft topological space over X and Y be a non-empty subset of X. Then τY = {(Y F, E) : (F, E) ∈ T } is said to be the IF soft topology on Y and (YE , τY ) is called a IF soft subspace ˜ (F, E) of (XE , τ ). Where (Y F, E) = ỸE ∩ We can easily verify that τY is, in fact, a IF soft topology on Y . Example 3.6. Any IF soft subspace of a IF soft discrete topological space is a IF soft discrete topological space. Also, any IF soft subspace of a IF soft indiscrete topological space is a IF soft indiscrete topological space 3.1 Separation Axioms Definition 3.7. A IF soft set (F, A) is said to be a IF soft point, denoted by eF , if for the element e ∈ A, F (e) 6= 0̃ and F (é) = 0̃, ∀é ∈ A − {e}. Definition 3.8. The complement of a IF soft point eF is a IF soft point ecF such that F c (e) = (F (e))c . 64 İsmail Osmanoğlu et al. Example 3.9. Let X = {x1 , x2 , x3 } and E = {e1 , e2 }. Then e2F = {(x1 , 0.8, 0.1), (x2 , 0.7, 0.05), (x3 , 0.9, 0.1)} is a IF soft point whose complement is ec2F = {(x1 , 0.1, 0.8), (x2 , 0.05, 0.7), (x3 , 0.1, 0.9)}. Definition 3.10. A IF soft point eF is said to be in a IF soft set (G, A), denoted by eF ∈ (G, A) if for the element e ∈ A, F (e) ≤ G(e). Definition 3.11. We define the separation axioms as follows. T0 Let (XE , τ ) be a IF soft topological space over X and eK , eS ∈ X̃E such that eK 6= eS . If there exist IF soft open sets (F, E) and (G, E) such that eK ∈ (F, E) and eS ∈ / (F, E) or eS ∈ (G, E) and eK ∈ / (G, E), then (XE , τ ) is called a IF soft T0 -space. T1 Let (XE , τ ) be a IF soft topological space over X and eK , eS ∈ X̃E such that eK 6= eS . If there exist IF soft open sets (F, E) and (G, E) such that eK ∈ (F, E), eS ∈ / (F, E) and eS ∈ (G, E), eK ∈ / (G, E), then (XE , τ ) is called a IF soft T1 -space. T2 (Hausdorf f ) Let (XE , τ ) be a IF soft topological space over X and eK , eS ∈ X̃E such that eK 6= eS . If there exist IF soft open sets (F, E) and ˜ (G, E) = ΦE , (G, E) such that eK ∈ (F, E), eS ∈ (G, E) and (F, E)∩ then (XE , τ ) is called a IF soft T2 -space. Regular Let (XE , τ ) be a IF soft topological space over X, (G, E) be a IF soft closed set in X and eK ∈ X̃E such that eK ∈ / (G, E). If there exist IF soft open sets (F1 , E) and (F2 , E) such that ˜ (F2 , E) = ΦE , ˜ 2 , E) and (F1 , E)∩ eK ∈ (F1 , E), (G, E)⊂(F then (XE , τ ) is called a IF soft regular space. T3 Let (XE , τ ) be a IF soft topological space over X. Then (XE , τ ) is said to be a IF soft T3 -space if it is IF soft regular and IF soft T1 -space. N ormal Let (XE , τ ) be a IF soft topological space over X, (F, E) and (G, E) ˜ (G, E) = ΦE . If there exist IF soft closed sets over X such that (F, E)∩ IF soft open sets (F1 , E) and (F2 , E) such that ˜ (F2 , E) = ΦE , ˜ 1 , E), (G, E)⊂(F ˜ 2 , E) and (F1 , E)∩ (F, E)⊂(F then (XE , τ ) is called a IF soft normal space. T4 Let (XE , τ ) be a IF soft topological space over X. Then (XE , τ ) is said to be a IF soft T4 -space if it is IF soft normal and IF soft T1 -space. On Intuitionistic Fuzzy Soft Topology 65 Proposition 3.12. Let (XE , τ ) be a IF soft topological space over X and Y be a non-empty subset of X. i. If (XE , τ ) is a IF soft T0 -space then (YE , τY ) is a IF soft T0 -space. ii. If (XE , τ ) is a IF soft T1 -space then (YE , τY ) is a IF soft T1 -space. iii. If (XE , τ ) is a IF soft T2 -space then (YE , τY ) is a IF soft T2 -space. iv. If (XE , τ ) is a IF soft regular space then (YE , τY ) is a IF soft regular space. Proof. We give here the proof of i. The others can be proved in a similar way. i. Let eK , eS ∈ ỸE such that eK 6= eS . If (XE , τ ) is a IF soft T0 -space then there exist IF soft open sets (F, E) and (G, E) such that eK ∈ (F, E), ˜ (F, E) = eS ∈ / (F, E) or y ∈ (G, E), eK ∈ / (G, E). Hence eK ∈ ỸE ∩ Y Y Y Y ( F, E), eS ∈ / ( F, E) or eS ∈ ( G, E) and eK ∈ / ( G, E). Thus (YE , τY ) is a IF soft T0 -space. Proposition 3.13. i. Every IF soft T1 -space is a IF soft T0 -space. ii. Every IF soft T2 -space is a IF soft T1 -space. iii. Every IF soft T3 -space is a IF soft T2 -space. iv. Every IF soft T4 -space is a IF soft T3 -space. Proof. We give here the proof of i. The others can be proved in a similar way. Let (XE , τ ) be a IF soft topological space over X and eK , eS ∈ X̃E such that eK 6= eS . i. If (XE , τ ) is a IF soft T1 -space then there exist IF soft open sets (F, E) and (G, E) such that eK ∈ (F, E) and eS ∈ / (F, E), and eS ∈ (G, E) and eK ∈ / (G, E). Obviously then we have eK ∈ (F, E) and eS ∈ / (F, E), or eS ∈ (G, E) and eK ∈ / (G, E). Thus (XE , T ) is a IF soft T0 -space. 66 İsmail Osmanoğlu et al. 3.2 Compactness IF soft compactness was defined by Turanlı et al in [12]. However, we will give a different definition for IF soft compactness. Definition 3.14. [12] Let (XE , τ ) and (YK , σ) be two IF soft topological spaces. (1) A IF soft function f : (XE , τ ) → (YK , σ) is called IF soft continuous if for all (G, B) ∈ σ, f −1 (G, B) ∈ τ. (2) A IF soft function f : (XE , τ ) → (YK , σ) is called IF soft open if for all (F, A) ∈ τ, f (F, A) ∈ σ. Definition 3.15. A family Ψ of IF soft sets is a cover of a IF softset (F, A) if ˜∪ ˜ {(Fi , A) : (Fi , A) ∈ Ψ, i ∈ I} . (F, A) ⊆ It is a IF soft open cover if each member of Ψ is a IF soft open set. A subcover of Ψ is a subfamily of Ψ which is also a cover. Definition 3.16. Let (XE , τ ) be IF soft topological space and (F, A) ∈ IF S( XE ). IF soft (F, A) is called compact if each IF soft open cover of (F, A) has a finite subcover. Also IF soft topological space (XE , τ ) is called compact if each IF soft open cover of X̃E has a finite subcover. Example 3.17. A IF soft topological space (XE , τ ) is compact if X is finite. Example 3.18. Let (XE , τ ) and (YK , σ) be two IF soft topological spaces and τ ⊂ σ. Then, IF soft topological space (XE , τ ) is compact if (YK , σ) is compact. Proposition 3.19. Let (G, B) be a IF soft closed set in IF soft compact space (XE , τ ). Then (G, B) is also compact. Proof. Let (Fi , A) be any open covering of (G, B). Then ˜ (∪ ˜ i∈I (Fi , A)) ∪ (G, B)c ; that is, (Fi , A) together with IF soft open set X̃E ⊆ (G, B)c is a open covering of X̃E . Therefore there exists a finite subcovering (F1 , A) , (F2 , A) , ..., (Fn , A) , (G, B)c . So ˜ (F1 , A) ∪ ˜ (F2 , A) ∪ ˜ ...∪ ˜ (Fn , A) ∪ ˜ (G, B)c . X̃E ⊆ Therefore ˜ (F1 , A) ∪ ˜ (F2 , A) ∪ ˜ ...∪ ˜ (Fn , A) ∪ ˜ (G, B)c (G, B) ⊆ which clearly implies ˜ (F1 , A) ∪ ˜ (F2 , A) ∪ ˜ ...∪ ˜ (Fn , A) (G, B) ⊆ c ˜ (G, B) = ΦE . Hence (G, B) has a finite subcovering and so since (G, B) ∩ is compact. On Intuitionistic Fuzzy Soft Topology 67 Proposition 3.20. Let (G, B) be a IF soft compact set in IF soft Hausdorff space (XE , τ ). Then (G, B) is closed.eK , eS ∈ X̃E Proof. Let eK ∈ (G, B)c . For each eS ∈ (G, B), we have eK 6= eS , so there are disjoint IF soft open sets (FeS , A) and (Fy , A) so that eK ∈ (FeS , A) and eS ∈ (HeS , A). Then {(HeS , A) : eS ∈ (G, B)} is an IF soft open cover of (G, B) Let {(HeS 1 , A) , (HeS 2 , A) , ..., (HeS n , A)} be a finite subcover. Then ˜ ni=1 (FeS i , A) is an open set containing eK and contained in (G, B)c . Thus ∩ (G, B)c is IF soft open and (G, B) is closed. Theorem 3.21. Let (XE , τ ) and (YK , σ) be IF soft topological spaces and f : (XE , τ ) → (YK , σ) continuous and onto IF soft function. If (XE , τ ) is IF soft compact, then (YK , σ) is IF soft compact, Proof. We will use Theorem 5.2. and Theorem 5.4. of [13] . Let (Fi , A) be any ˜∪ ˜ −1 (∪ ˜ i∈I (Fi , A). Then f −1 ỸK ⊆f ˜ i∈I (Fi , A)); open covering of ỸK ; i.e., YK ⊆ ˜∪ ˜ i∈I f −1 ((Fi , A)). So f −1 ((Fi , A)) is an open covering of X̃E . As and X̃E ⊆ (X, τ ) is compact, there are 1, 2, ..., n in I such that ˜ −1 ((F1 , A)) ∪ ˜ f −1 ((F2 , A)) ∪ ˜ ...∪ ˜ f −1 ((Fn , A)). X̃E ⊆f Since (ϕ, ψ) we have is surjective, ỸK = f X̃E ˜ (f −1 ((F1 , A)) ∪ ˜ ...∪ ˜ f −1 ((Fn , A))) ⊆f ˜ ...∪ ˜ f (f −1 ((Fn , A))) = f (f −1 ((F1 , A))) ∪ ˜ (F2 , A) ∪ ˜ ...∪ ˜ (Fn , A). = (F1 , A) ∪ ˜ (F1 , A) ∪ ˜ (F2 , A) ∪ ˜ ...∪ ˜ (Fn , A); i.e., ỸK is covered by a finite So we have ỸK ⊆ number of (Fi , A). Hence (Y, σ) is compact. Definition 3.22. Let (XE , τ ) and (YK , σ) be two IF soft topological spaces. A IF soft function f : (XE , τ ) → (YK , σ) is called IF soft closed if f ((F, A)) is IF soft closed set in (YK , σ), for all IF soft closed set (F, A) in (XE , τ ). Theorem 3.23. Let (XE , τ ) be a IF soft topological space and (YK , σ) be a IF soft Hausdorff space. IF soft function f is closed if IF soft function f : (XE , τ ) → (YK , σ) is continuous. Proof. Let (G, B) be any IF soft closed set in (XE , τ ). By proposition 3.19 we have (G, B) is compact. Since IF soft function f is continuous, IF softset f ((G, B)) is compact in (YK , σ). As (YK , σ) is IF soft Hausdorff space, IF soft set f ((G, B)) is closed. Then IF soft function f is closed. Definition 3.24. A family Ψ of IF soft sets has the finite intersection property if the intersection of the members of each finite subfamily of Ψ is not the null IF soft set. 68 İsmail Osmanoğlu et al. Theorem 3.25. A IF soft topological space is compact if and only if each family of IF soft closed sets with the finite intersection property has a nonnull intersection. ˜ {(Fi , A) : (Fi , A) ∈ Proof. ⇒: Let Ψ be any family of IF soft closed such that ∩ c Ψ, i ∈ I} = ΦE . Consider Ω = {(Fi , A) : (Fi , A) ∈ Ψ, i ∈ I}. So Ω is a IF soft open cover of X̃E . As IF soft topological space is compact, there ˜ ni=1 (Fi , A) = exists a finite subcovering (F1 , A)c , (F2 , A)c , ..., (F3 , A)c .Then ∩ ˜ ni=1 (Fi , A)c = X̃E − X̃E = ΦE . Thence Ψ cannot have finite intersection X̃E − ∪ property. ⇐: Assume that a IF soft topological space is not compact. Then any IF soft open cover of X̃E has not a finite subcover. Let {(Fi , A) : i ∈ I} be IF soft ˜ ni=1 (Fi , A) 6= X̃E . Therefore ∩ ˜ ni=1 (Fi , A)c 6= ΦE . Thus, open cover of X̃E . So ∪ {(Fi , A)c : i = 1, ..., n} have finite intersection property. By using hypothesis, ˜ (Fi , A)c 6= ΦE and so ∪ ˜ (Fi , A) 6= X̃E . This is a contradiction. Thus IF soft ∩ topological space is compact. 3.3 Connectedness Definition 3.26. Let (XE , τ ) be a IF soft topological space over X. A IF soft separation of X̃E is a pair (F, E), (G, E) of no-null IF soft open sets such that ˜ (G, E), (F, E)∩ ˜ (G, E) = Φ. X̃E = (F, E)∪ Definition 3.27. A IF soft topological space (XE , τ ) is said to be IF soft connected if there does not exist a IF soft separation of X̃E .Otherwise, (XE , τ ) is said to be IF soft disconnected. Example 3.28. Let us consider the IF soft topological space (XE , τ ) that is given in Example 3.2. ˜ (F2 , E) 6= ΦE , (F1 , E)∩ ˜ (F3 , E) 6= ΦE and (F2 , E)∩ ˜ (F3 , E) 6= Since (F1 , E)∩ ˜ ˜ ˜ ΦE also (F1 , E)∪(F2 , E) 6= X̃E , (F1 , E)∪(F3 , E) 6= X̃E and (F2 , E)∪(F3 , E) 6= X̃E , IF soft topological space (XE , τ ) is IF soft connected. Theorem 3.29. IF soft topological space (XE , τ ) is IF soft connected if and only if the only IF soft sets in IF S(XE ) that are both IF soft open and IF soft closed are Φ and X̃ Proof. Let (XE , τ ) be IF soft connected. Suppose to the contrary that (F, E) is both IF soft open and IF soft closed different from ΦE and X̃E . Clearly, (F, E)c ˜ (F, E)c = X̃E and is a IF soft open set different from ΦE and X̃E . Also (F, E)∪ ˜ (F, E)c = ΦE . Therefore we have (F, E), (F, E)c is a IF soft separation (F, E)∩ of X̃E . This is a contradiction. Thus the only IF soft closed and open sets in XE are ΦE and X̃E . On Intuitionistic Fuzzy Soft Topology 69 Conversely, let (F, E), (G, E) be a IF soft separation of X̃E . Let (F, E) = X̃E . Then (G, E) = ΦE . This is a contradiction. Hence, (F, E) 6= X̃E . Therefore (F, E) = (G, E)c . This shows that (F, E) is both IF soft open and IF soft closed different from ΦE and X̃E . This is a contradiction. Therefore, (XE , τ ) is soft connected. Example 3.30. Since the only IF soft sets in IF S(XE ) that are both IF soft open and IF soft closed are X̃E and ΦE , IF soft indiscrete topological space (XE , τ ) is IF soft connected. Example 3.31. IF Soft discrete topological space (XE , τ ) is IF soft disconnected. Because for at least one IF soft set (F, E) in IF S(XE ), IF soft set (F, E) is both IF soft open set and IF soft closed. Corollary 3.32. Let (XE , τ ) be a IF soft topological space over X. Then following statements are equivalent. 1. (XE , τ ) is IF soft connected. ˜ (G, E) but 2. No-null IF soft open sets (F, E), (G, E) and X̃E = (F, E)∪ ˜ (F, E)∩(G, E) 6= ΦE . 3. The only IF soft sets in IF S(XE ) that are both IF soft open and IF soft closed in XE are Φ and X̃ ˜ (G, E) and (F, E)∩ ˜ (G, E) = ΦE then (F, E) = ΦE or 4. If X̃E = (F, E)∪ (G, E) = ΦE . ˜ (G, E) and (F, E)∩ ˜ (G, E) = ΦE then (F, E) = X̃E or 5. If X̃E = (F, E)∪ (G, E) = X̃E . 4 Conclusion In this work, we introduced IF soft subspace, IF soft separation axioms, IF soft compactness and IF soft connectedness. Also we gave some basic properties of these concepts. References [1] K. Atanassov, Intuitionistic fuzzy sets, Fuzzy Sets and Systems, 20(1986), 87-96. [2] M. Bora, T.J. Neog and D.K. Sut, Some results on intuitionistic fuzzy soft sets, International Journals of Mathematics Trends and Technology, 3(2) (2012), 63-69. 70 İsmail Osmanoğlu et al. [3] M. Bora, T.J. Neog and D.K. 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