International Journal of Scientific & Engineering Research, Volume 6, Issue 3, March ISSN

International Journal of Scientific & Engineering Research, Volume 6, Issue 3, March-2015 969 Soft Generalized Separation Axioms in Soft Generalized Topological Spaces Jyothis Thomas and Sunil Jacob John Abstract: In this paper, the concepts of soft generalized μ-t 0, soft generalized μ-t 1, soft generalized Hausdorff, soft generalized regular, soft generalized normal and soft generalized completely regular spaces in Soft Generalized Topological Spaces are defined and studied. Further some of its properties and characterizations are established. Keywords: Soft set, Soft Generalized Topological Space, soft generalized μ-t 0, soft generalized μ-t 1, soft generalized Hausdorff, soft generalized regular, soft generalized normal, soft generalized completely regular spaces. 1 INTRODUCTION S everal set theories can be considered as mathematical tools sets theory and not sets. Some other studies on GTS s can be for dealing with uncertainties, namely, the theory of fuzzy listed as [19], [20], [21]. sets [1], the theory of intuitionistic fuzzy sets [2], [3], the This paper is organized as follows. In the second section, theory of vague sets [4], the theory of interval mathematics [5] we give as a preliminaries, some well-known results in soft set and the theory of rough sets [6]. Molodtsov [7] in 1999, introduced the concept of soft set theory as a general mathematical of soft generalized separation axioms in SGTS s and discuss theory and SGTS s. In section three, we introduce the concept tool to deal with uncertainties while modeling the problems some of its properties and characterizations. We also investigate the behavior some soft generalized separation axioms with incomplete information. Many researchers improved the concept of soft sets. Maji et. al. [8] defined operations of soft under the soft continuous, soft open and soft closed mappings. sets. Pie and Miao [9] showed that soft sets are a class of special information systems. Cagman and Enginoglu [10] rede- 2. PRELIMINARIES fined the operations of the soft sets and constructed a In this section, we recall the basic definitions and results of uni-int decision making method by using these new operations and developed soft set theory. Aktas and Cagman Throughout this paper U denotes initial universe, E denotes soft set theory and SGTS s which will be needed in the sequel. [11] compared soft sets to fuzzy sets and rough sets. Babitha the set of all possible parameters, (U) is the power set of U and and Sunil [12] introduced the soft set relation and discussed A is a nonempty subset of E. related concepts such as equivalent soft set relation, partition Definition 2.1. [14] A soft set F A on the universe U is defined and composition. Kharal and Ahmad [13] defined soft images by the set of ordered pairs F A= {(e, f A(e)) / e E, f A(e) P (U)}, and soft inverse images of soft sets. They also applied these where f A : E P (U) such that f A(e) = if e A. Here f A is notions to the problem of medical diagnosis in medical systems. Topological structure of soft sets also was studied by f A(e) may be arbitrary. Some of them may be empty, some may called an approximate function of the soft set F A. The value of many researchers. Cagman [14] studied the concepts of soft have nonempty intersection. The set of all soft sets over U with topological spaces and some related concepts. Varol et al. [15] E as the parameter set will be denoted by S(U) E or simply S(U). interpreted a classical topology as a soft set over the power set Definition 2.2. [14] Let F A S(U). If f A(e) = for all e E, then P (X) and characterized also some other categories related to F is called an empty soft set, denoted by F. f A(e) = means crisp topology and fuzzy topology as subcategories of the category of soft sets. Therefore we do not display such elements in the soft sets as it that there is no element in U related to the parameter e in E. General Topology was developed by many researchers. A Csaszar [16] introduced the theory of generalized topo- Definition 2.3. [14] Let F A S(U). If f A(e) = U for all e A, then is meaningless to consider such parameters. logical spaces. Jyothis and Sunil [17], [18] introduced the concept of Soft Generalized Topological Space (SGTS) and studied then the A-universal soft set is called an universal soft set, de- F is called an A-universal soft set, denoted by F A. If A = E, the Soft μ-compactness in SGTSs. The generalized topology is noted by F E. different from topology by its axioms. According to Csaszar, a Definition 2.4. [14] Let F A, F B S(U). Then F B is a soft subset of collection of subsets of X is a generalized topology on X if and F A (or F A is a soft superset of F B), denoted by F B F A, if f B(e) only if it contains the empty set and arbitrary union of its f A(e), for all e E. members. But the soft generalized topology is based on soft Definition 2.5. [14] Let F A, F B S(U). Then F B and F A are soft equal, denoted by F B = F A, if f B(e) = f A(e), for all e E. Definition 2.6. [14] Let F A, F B S(U). Then, the soft union of Sunil Jacob John, Department of Mathematics, National Institute of Technology, Calicut, Calicut 673 601, India, sunil@nitc.ac.in F A and F B, denoted by F A F B, is defined by the approximate Jyothis Thomas, Department of Mathematics, National Institute of Technology, Calicut, Calicut 673 601, India, jyothistt@gmail.com Definition 2.7. [14] Let F A, F B S(U). Then, the soft function f (A B)(e) = f A(e) f B(e). intersec-

International Journal of Scientific & Engineering Research, Volume 6, Issue 3, March-2015 970 tion of F A and F B, denoted by F A F B, is defined by the approximate function f (A B)(e) = f A(e) f B(e). F A and F B are said to be soft disjoint if F A F B = F. Definition 2.8. [14] Let F A, F B S(U). Then, the soft difference of F A and F B, denoted by F A F B, is defined by the approximate function f (A B)(e) = f A(e) f B(e). Definition 2.9. [14] Let F A S(U). Then, the soft complement of F A, denoted by F, is defined by the approximate function f (e) = (f A(e)) c, where (f A(e)) c is the complement of the set f A(e), that is, (f A(e)) c = U f A(e) for all e E. Clearly (F ) = F A, F = F E, and F = F. Definition 2.10. [14] Let F A S(U). The soft power set of F A, denoted by (F A), is defined by (F A) = {F Ai / F Ai FA, i J} Theorem 2.11. [14] Let F A, F B, F C S(U). Then, (1) F A F = F E (2) F A F = F. (3) F A FB F F. (4) F A F B = F FA F (5) F A F B = F B F A. (6) F A F B = F B F A. (7) (F A F B) c = F F. (8) (F A F B) c = F F. (9) F A (F B F C) = (F A F B) (F A F C). (10) F A (F B F C) = (F A F B) (F A F C). Definition 2.12. [13] Let S(U) E and S(V) K be the families of all soft sets over U and V respectively. Let φ : U V and χ : E K be two mappings. The soft mapping : S(U) E S(V) K is defined as: (1) Let F A be a soft set in S(U) E. The image of F A under the soft mapping is the soft set over V, denoted by (F A) and is defined by φ (f )(k) = { ( ) φ(f (e)), if χ (k) A ; for, otherwise all k K. (2) Let G be a soft set in S(V) K. The inverse image of G under the soft mapping φ is the soft set over U, denoted by φ (G ) and is defined by φ (g )(e) = { φ (g (χ(e))), if χ(e) B;, otherwise for all e E. The soft mapping is called soft injective, if φ and χ are injective. The soft mapping is called soft surjective, if φ and χ are surjective. The soft mapping is called soft bijective iff is soft injective and soft surjective. Theorem 2.13. [13] Let S(U) E and S(V) K be the families of all soft sets over U and V respectively. Let F A, F B, F Ai S(U) E and G A, G B, G Bi S(V) K. For a soft mapping : S(U) E S(V) K the following statements are true: (1) If F B F A, then (F B) (F A). (2) (F ) = F. (3) ( i J F Ai) = i J((F Ai)). (4) ( i J F Ai) i J ((F Ai)), equality holds if is soft injective. (5) F A 1 ((F A)), equality holds if is soft injective. (6) ( 1(F A)) F A, equality holds if is soft surjective. (7) If G B G A, then 1(G B) 1(G A). (8) 1(F ) = F. (9) 1(G ) = ( 1(G B)) c (10) 1( i J G Bi) = i J( 1(G Bi)). (11) 1( i J G Bi) = i J( 1(G Bi)). SOFT GENERALIZED TOPOLOGİCAL SPACES Definition 2.14. [17] Let F A S(U). A Soft Generalized Topology (SGT) on F A, denoted by μ or μ is a collection of soft subsets of F A having the following properties: (i) F μ and (ii) The soft union of any number of soft sets in μ belong to μ. The pair (F A, μ) is called a Soft Generalized Topological Space (SGTS). Observe that F A μ must not hold. Definition 2.15. [17] Let F A S(U) and μ be the collection of all possible soft subsets of F A, then μ is a SGT on F A, and is called the discrete SGT on F A. Definition 2.16. [17] A soft generalized topology μ on F A is said to be strong if F A μ. Definition 2.17. [17] Let (F A, μ) be a SGTS. Then, every element of μ is called a soft μ open set. Note that F is a soft μ open set. Definition 2.18. [17] Let (F A, μ) be a SGTS and F B F A. Then the collection μ = {F D F B / F D μ} is called a Subspace Soft Generalized Topology (SSGT) on F B. The pair (F, μ ) is called a Soft Generalized Topological Subspace (SGTSS) of F A. Definition 2.19. [17] Let (F A, μ) be a SGTS and F B F A. Then F B is said to be a soft μ-closed set if its soft complement F is a soft μ-open set. Theorem 2.20. [17] Let (F A, μ) be a SGTS. Then the following conditions hold: (1) The universal soft set F E is soft μ closed. (2) Arbitrary soft intersections of the soft μ closed sets are soft μ closed. Definition 2.21. [17] Let (F A, μ) be a SGTS and F B F A. Then the soft μ-closure of F B, denoted by c(f B) or F is defined as the soft intersection of all soft μ-closed super sets of F B. Note that F is the smallest soft μ-closed set that containing F B. Theorem 2.22. [17] Let (F A, μ) be a SGTS and F B F A. F B is a soft μ-closed set iff F B = F. Theorem 2.23. [17] Let (F A, μ) be a SGTS and F G, F H F A. Then (1) F G F (2) (F ) = F (3) F G F H F F (4) F F F F (5) F F F F (6) α F every soft μ-open set F G containing α soft intersect F H Remark 2.24. Converse of theorem 2.23.(6) is not true in gen-

International Journal of Scientific & Engineering Research, Volume 6, Issue 3, March-2015 971 eral as shown in the following example. Example 2.25. Let U ={u 1, u 2, u 3}, E = {x 1, x 2, x 3}, A = {x 1, x 2} E and F A = {(x 1, {u 1, u 2}), (x 2, {u 2, u 3})} and μ = {F, F A, F P, F Q, F R}. Then (F A, μ) is a SGTS where F P = {(x 1, {u 2})}, F Q = {(x 1, {u 2}), (x 2, {u 3})}, F R = {(x 1, {u 1, u 2 }), (x 2, {u 2})}. The soft μ-closed sets are: F = F E, F = {(x 1, {u 3}), (x 2, {u 1}), (x 3, U)}, F = {(x 1, {u 1, u 3}), (x 2, U), (x 3, U)}, F = {(x 1, {u 1, u 3}), (x 2, {u 1, u 2}), (x 3, U)}, F = {(x 1, {u 3}), (x 2, {u 1, u 3}), (x 3, U)}. Let F B = {(x 1, {u 1}), (x 2, {u 2, u 3})}. Then F = F = {(x 1, {u 1, u 3}), (x 2, U), (x 3, U)}. Take α = (x 1, {u 1, u 2}) then, F R is a soft μ-open set containing α. Now F R F B = {(x 1, {u 1}), (x 2, {u 2})} F. But α F. i.e, We can find a soft μ-open set F R containing α soft intersect F B and α F. Definition 2.26. [17] Let (F A, μ) be a SGTS and α F A. If there is a soft μ-open set F B such that α F B, then F B is called a soft μ-open neighborhood or soft μ-nbd of α. The set of all soft μ- nbds of α, denoted by ψ(α), is called the family of soft μ-nbds of α. i.e, ψ(α) = {F B / F B μ, α F B}. Definition 2.27. [17] Let (F A, μ) and (F B, η) be two SGTS s and : (F A, μ) (F B, η) be a soft function. Then 1. is said to be soft (μ, η)-continuous (briefly, soft continuous), if for each soft η-open subset F G of F B, the inverse image 1(F G) is a soft μ-open subset of F A. 2. is said to be soft (μ, η)-open, if for each soft μ-open subset F G of F A, the image (F G) is a soft η-open subset of F B. 3. is said to be soft (μ, η)-closed, if for each soft μ- closed subset F G of F A, the image (F G) is a soft η- closed subset of F B. 3. SOFT GENERALİZED SEPARATION AXIOMS IN SGTSs Definition 3.1. Let (F A, μ) be a SGTS and α, β F A such that α β. If there exists soft μ-open sets F G and F H such that α F G and β F G or β F H and α F H, then (F A, μ) is called a soft generalized μ-t 0 space. Theorem 3.2. Let (F A, μ) be a SGTS and α, β F A such that α β. If there exists soft μ-open sets F G and F H such that α F G and β F or β F H and α F, then (F A, μ) is a soft generalized μ-t 0 space. Proof: Let α, β F A such that α β and F G, F H μ such that α F G and β F or β F H and α F. If α F then α (F ) = F H. Similarly if β F then β (F ) = F G. Hence F G, F H μ such that α F G and β F G or β F H and α F H. Hence (F A, μ) is a soft generalized μ-t 0 space. Example 3.3. A discrete SGTS (F E, μ) is a soft generalized μ-t 0 space, since every {α} is a soft μ-open set. Theorem 3.4. Let : (F A, μ) (F B, η) be a soft (μ, ) continuous soft bijective function. If (F B, ) is a soft generalized η-t 0 space, then (F A, μ) is also a soft generalized μ-t 0 space. Proof: Let (F B, ) be a soft generalized η-t 0 space. Suppose α, β F A such that α β Since is soft injective, γ, δ F B such that γ = (α), δ = (β) and γ δ. Since (F B, ) is a soft generalized η-t 0 space, F G, F H such that γ F G and δ F G or δ F H and γ F H. This implies that (α) F G and (β) F G or (β) F H and (α) F H α 1(F G) and β 1(F G) or β 1(F H) and α 1(F H). Since is a soft (μ, ) continuous function, 1(F G) and 1(F H) are soft μ-open sets. Hence (F A, μ) is a soft generalized μ-t 0 space. Theorem 3.5. Let : (F A, μ) (F B, ) be a soft (μ, ) open soft bijective function. If (F A, μ) is a soft generalized μ-t 0 space, then (F B, ) is a soft generalized η-t 0 space. Proof: Suppose that (F A, μ) is a soft generalized μ-t 0 space. Let α, β F B such that α β Since is a soft bijective function γ, δ F A such that α = (γ), β = (δ) and γ δ. Since (F A, μ) is a soft generalized μ-t 0 space, F G, F H μ such that γ F G and δ F G or δ F H and γ F H. This implies that (γ) (F G) and (δ) (F G) or (δ) (F H) and (γ) (F H) α (F G) and β (F G) or β (F H) and α (F H). Since is a soft (μ, ) open function, both (F G) and (F H) are soft - open sets. Hence (F B, ) is a soft generalized -T 0 space. Definition 3.6. Let (F A, μ) be a SGTS and α, β F A such that α β If there exists soft μ-open sets F G and F H such that α F G and β F G and β F H and α F H, then (F A, μ) is called a soft generalized μ-t 1 space. Theorem 3.7. Let (F Ẽ, μ) be a SGTS. If for each α F E, {α} is a soft μ-closed set, then (F E, μ) is a soft generalized μ-t 1 space. Proof: Let α and β be two points of F Ẽ such that α β. Given that {α} and {β} are soft μ-closed sets. Then {α} Theorem 2.28. [17] Let (F A, μ) and (F B, η) be two SGTS s and φ c and {β} c are soft χ μ-open sets. Clearly α {β} : (F A, μ) (F B, η) be a soft function. Then is soft continuous c and β {β} c and β {α} c and α {α} if and only if for every soft η-closed subset F H of F B, the soft set c. Hence (F Ẽ, μ) is a soft generalized μ-t 1 space. φ 1 χ (F H) is soft μ-closed in F A. Theorem 3.8. Let (F A, μ) be a SGTS and α, β F A such that α β If F G, F H μ such that α F G and β F and β F H and α F then (F A, μ) is a soft generalized μ-t 1 space. Proof: The proof is similar to the proof of theorem 3.2. Theorem 3.9. Every soft generalized μ-t 1 space is a soft generalized μ-t 0 space. Proof: Let (F A, μ) be a soft generalized μ-t 1 space and α, β F A such that α β. So there exists soft μ-open sets F G and F H such that α F G and β F G and β F H and α F H. Obviously then we have α F G and β F G or β F H and α F H. Hence (F A, μ) is a soft generalized μ-t 0 space. Theorem 3.10. Let (F A, μ) be a soft generalized μ-t 1 space and α F A. Then for each soft μ-open set F G with α F G, {α} F G. Proof: Since α F G for each soft μ-open set F G, α F G. Then it is obvious that {α} F G Theorem 3.11. Let (F A, μ) be a SGTS and α, β F A such that α β. If F G, F H μ such that α F G and {β} F G = F and β F H and {α} F H = F, then (F A, μ) is a soft generalized μ-t 1 space. Proof: Similar to the theorem 3.8. Theorem 3.12. Let : (F A, μ) (F B, ) be a soft (μ, ) continuous soft bijective function. If (F B, ) is a soft generalized η-t 1

International Journal of Scientific & Engineering Research, Volume 6, Issue 3, March-2015 972 space, then (F A, μ) is also a soft generalized μ-t 1 space. Proof: Proof is similar to that of soft generalized μ-t 0 space. Theorem 3.13. Let : (F A, μ) (F B, ) be a soft (μ, )-open soft bijective function. If (F A, μ) is a soft generalized μ-t 1 space, then (F B, ) is also a soft generalized η-t 1 space. Definition 3.14. Let (F A, μ) be a SGTS. If for all α 1, α 2 F A with α 1 α 2, there exists F G ψ(α1) and F H ψ(α2) such that F G F H = F, then (F A, μ) is called a soft generalized μ-t 2 space or soft generalized Hausdorff space (SGHS). Theorem 3.15. If (F A, μ) be a SGHS and F B is a non-soft empty soft subset of F A containing finite number of points, then F B is soft μ-closed. Proof: Suppose that (F A, μ) is a SGHS. Let us take F B = {α}. Now we show that F B is soft μ-closed. If β is a point of F A different from α, then since (F A, μ) is a SGHS, F G, F H μ such that α F G, β F H and F G F H = F. Now F G F H = F F G {β} = F α cannot belong to the soft μ-closure of the soft set {β}. As a result, the soft μ-closure of the soft set {α} is {α} itself. Hence F B is soft μ-closed. Proof: Let (F A, μ) be a SGHS and α, β F A such that α β Then F G, F H μ such that α F G, β F H and F G F H = F. Since F G F H = F, α F H, and β F G. Thus F G, F H μ such that α F G, β F G and β F H and α F H. Hence (F A, μ) is a soft generalized μ-t 1 space. Theorem 3.18. Let (F A, μ) be a SGHS and α F A. Then {α} = F P for each soft μ-open set F P with α F P. Proof: Assume that there exists a β F A such that α β and β F P; F P μ, α F P. Since (F A, μ) is a SGHS, F G, F H μ such that α F G, β F H and F G F H = F. Now F G F H = F F G {β} = F. This is a contradiction to the assumption that β F P. Hence {α} = F P for each soft μ-open set F P with α F P. Theorem 3.19. Let : (F A, μ) (F B, ) be a soft bijective soft (μ, η) open function. If (F A, μ) is a SGHS then (F B, ) is a SGHS. Proof: Let α, β F B such that α β Since is a soft bijective function, γ, δ F A such that α = (γ), β = (δ) and γ δ. Since (F A, μ) is a SGHS, F G, F H μ such that γ F G, δ F H and F G F H = F. This implies that (γ) (F G), (δ) (F H) α (F G) and β (F H). Since is a soft (μ, η)- open function, (F G) and (F H) are soft η-open sets. Again since is a soft bijective, (F G) (F H) = (F G F H) = (F ) = F. Hence the proof. Theorem 3.20. Let : (F A, μ) (F B, ) be a soft bijective soft (μ, ) continuous function. If (F B, ) is a SGHS, then (F A, μ) is also a SGHS. Proof: Let α, β F A such that α β. Since is a soft bijective function, γ, δ F B such that α = 1(γ), β = 1(δ) and γ δ. Since (F B, ) is a SGHS, F G, F H such that γ F G, δ F H and F G F H = F. Then 1(F G) and 1(F H) are soft μ-open sets, because is soft (μ, ) continuous. Also F G F H = F 1(F G F H) = 1(F ) 1(F G) 1(F H) = F. Now γ FG and δ F H 1(γ) 1(F G) and 1(δ) 1(F H) α 1(F G) and β 1(F H). Hence (F A, μ) is a SGHS. Definition 3.21. Let (F A, μ) be a SGTS. If for every point α F A and every soft μ-closed set F M such that α F M, there exists two soft μ-open sets F G and F H such that α F G, F M F H and F G F H = F, then (F A, μ) is called a soft generalized regular space (SGRS). Theorem 3.22. Let (F A, μ) be a SGTS and let F K be a soft μ- closed set and α F A such that α F K. If (F A, μ) is a SGRS, then there exists soft μ-open set F G such that α F G and F G F K = Theorem 3.16. A soft generalized subspace of a SGHS is a F. SGHS. Proof: Let F K be a soft μ-closed set and α F A such that α F Proof: Let (F A, μ) be a SGHS and (F B, μ ) is a SGTSS of F A. Let α, β F B such that α β Then, since F B F A, α and β in F A such that α 1 β 1 and {α} {α 1} and {β {β 1}. Since (F A, μ) is SGHS, F G, F H K. Since (F A, μ) is a SGRS, F G, F H μ such that α F G, F K F H and F G F H = F. Now F G F H = F F G F K = F. Hence the proof. μ such that α 1 F G, β 1 F H and F G F H = F. Theorem 3.23. Let (F Then clearly α F G F B and β F H F B and (F G F B) (F H A, μ) be a SGRS and α F A. Then F B) = (F G F H) F B = F. i.e., there exist two soft disjoint soft (i) For a soft μ-closed set F K, α F K iff {α} F K = F μ -open sets F G F B and F H F B containing α and (ii) For a soft μ-open set F H, {α} F H = F α F H. β respectively. Hence (F B, μ ) is a SGH sub space. Proof: (i) Suppose that (F A, μ) be a SGRS and α F A. Let F K be Theorem 3.17. Every SGHS is a soft generalized μ-t 1 space. a soft μ-closed set such that α F K. Then by theorem 3.22. F G μ such that α F G and F G F K = F. Since {α} F G, we have {α} F K = F. The converse part is obvious. (ii) Obvious. Theorem 3.24. Let (F A, μ) be a SGRS and α F A. Then for each soft μ-closed set F K such that {α} F K, there exists soft μ-open sets F G and F H such that {α} F G, F K F H and F G F K = F. Proof: Assume that (F A, μ) is a SGRS and α F A. Let F K be a soft μ-closed set such that {α} F K. Then α F K. Since (F A, μ) is a SGRS, F G, F H μ such that α F G, F K F H and F G F H = F. Since α F G, {α} F G. Hence F G and F H are soft μ-open sets such that {α} F G, F K F H and F G F H = F. Theorem 3.25. Let (F A, μ) is a SGRS. Then for every α F A and every soft μ-open set F D with α F D, there exists a soft μ-open set F H such that α F H F F D. Proof: Suppose that (F A, μ) is a SGRS. Let α F A, and F D be any soft μ-open set such that α F D. Then F is a soft μ-closed set such that α F Since (F A, μ) is a SGRS, there exists soft μ- open sets F G and F H such that α F H, F F G and F G F H = F. Now F G F H = F F H F. Also F F G F F D. This implies that α F H F (F ) F F D. Theorem 3.26. Let (F A, μ) and (F B, ) be SGTS s and : (F A, μ) (F B, ) be a soft bijective, soft (μ, ) continuous and soft (μ, ) closed map. If (F B, ) is a SGRS, then (F A, μ) is also a SGRS.

International Journal of Scientific & Engineering Research, Volume 6, Issue 3, March-2015 973 Proof: Assume that (F B, ) is a SGRS. Let α F A and F K be any soft μ-closed set in F A such that α F K. Since is a soft bijective function, δ F B such that (α) = δ α = 1(δ). Also (F K) is a soft -closed set in F B, since is a closed map. Now α F K (α) (F K) δ (F K). Since (F B, ) is a SGRS, F G, F H such that δ F G, (F K) F H and F G F H = F. Then 1(F G) and 1(F H) are soft μ-open sets, since is a soft (μ, ) continuous map. Now δ F G 1(δ) 1(F G) α φ χ 1(F G); (F K) F H 1((F K)) 1(F H) F K 1(F H) and 1(F G) 1(F H) = 1(F G F H) = 1(F ) = F, since is a soft bijective map. Hence (F A, μ) is a SGRS. Theorem 3.27. Let (F A, μ) be a SGTS. Then (F A, μ) is a SGRS iff for each α F A and a soft μ-closed set F K such that α F K, there exists a soft μ-open set F G such that α F G and F F K = F. Proof: Suppose that (F A, μ) is a SGRS. Let α F A and F K be a soft μ-closed set such that α F K. Then there exists soft μ-open sets F G and F H such that α F G, F K F H and F G F H = F. Now F G F H = F F G F and F K F H F F. Thus F G F F. This implies that F (F = ) F F. Therefore F F K = F. Conversely, suppose α F A and F K be a soft μ-closed set such that α F K. Then by hypothesis there exists a soft μ- open set F G such that α F G and F F K = F. Now F F K = F F K (F ). Also F G F (F ) F F G ((F ) ). Therefore F G (F ) = F. Thus F G, (F ) μ such that α F G, F K (F ), F G (F ) = F (F A, μ) is a SGRS. Definition 3.29. Let (F A, μ) be a SGTS. If for every pair of soft disjoint soft μ-closed sets F M and F N, there exists two soft μ- open sets F G and F H such that F M F G, F N F H and F G F H = F. Then (F A, μ) is called soft generalized normal space (SGNS). Theorem 3.30. A SGTS (F A, μ) is a SGNS iff for any soft μ- closed set F K and a soft μ-open set F D containing F K, there exists soft μ-open set F G such that F K F G and F F D. Proof: Let (F A, μ) be a SGNS and F K be a soft μ-closed set and F D be a soft μ-open set such that F K F D. Then F K and F are soft disjoint soft μ-closed sets. Since (F A, μ) is a SGNS, F G, F H μ such that F K F G, F F H and F G F H = F. Now F G F H = F F G F F F = F. Also F F H F (F ) = F D. Hence F K F G and F F D. Conversely, suppose that F M and F N are two soft μ- closed sets with soft empty soft intersection. Then, since F M F N = F, F M F. i.e., F M is a soft μ-closed set and F is a soft μ- open set containing F M. So by hypothesis, there exists soft μ- open set F G such that F M F G and F F. Now F F F N (F ). Also F G F (F ) F (F ) ((F ) ) F G ((F ) ) F G (F ) = F. Thus F G and (F ) are soft μ-closed sets such that F M F G, F N (F ) and F G (F ) = F. Hence (F A, μ) is a SGNS. Theorem 3.31. Let : (F A, μ) (F B, ) be a soft bijective function which is both soft (μ, ) continuous and soft (μ, η) open. If (F A, μ) is a SGNS then (F B, ) is also a SGNS. Proof: Let F M and F N be a pair of soft η-closed sets in (F B, ) such that F M F N = F. Since is a soft (μ, ) continuous function, 1(F M) and 1(F N) are soft μ-closed sets in (F A, μ). Also 1(F M) 1(F N) = 1(F M F N) = 1(F ) = F. Again since (F A, μ) is a SGNS, F G, F H μ such that 1(F M) F G, 1(F N) F H and F G F H = F F M (F G) and F N (F H), since is soft surjective. Since is soft (μ, η) open, (F G) and (F H) are soft η-open sets. Also (F G) (F H) = (F G F H) = (F ) = F, since is soft bijective. Hence (F G) and (F H) are soft η-open sets such that F M (F G), F N (F H) and (F G) (F H) = F. Hence (F B, ) is a SGNS. Theorem 3.32. If (F E, μ) is a SGNS, then for every pair of soft μ-open sets F D and F P whose soft union is F E, then there exists Theorem 3.28. Let : (F A, soft μ-closed sets F M and F N such that F M F D, F N F P and F M F N = F E. μ) (F B, ) be a soft (μ, ) continuous, soft (μ, ) open, soft bijective function. If (F A, μ) is a SGRS of soft μ-open sets such that F D F P = F E. Then F and F are Proof: Suppose that (F E, μ) is a SGNS. Let F D and F P be a pair then (F B, ) is also a SGRS. soft μ-closed sets. Also F F = (F D F P) c = F = F. Since (F E, μ) is a SGNS, F Proof: Let α F B and F K a soft -closed set such that α F G, F H μ such that F F G, F F H and F G K. F Since is soft bijective, δ F A such that α = (δ). Again H = F. Take F M = F and F N = F. Then F M and F N are soft μ- closed sets. Also F M F N = F F = (F G F H) since is soft (μ, ) continuous, 1(F K) is a soft μ -closed set c = F E. Since F F G, F F H F F D and F F P. Thus there exists soft μ- such that δ 1(F K). Since (F A, μ) is a SGRS, F G, F H μ such closed sets F M and F N such that F M F D, F N F P and F M F N = that δ F G, 1(F K) F H and F G F H = F. Then (δ) (F G), F E. ( 1(F K)) (F H) and (F G F H) = (F ) α (F G), F K (F H) and (F G) (F H) = F, since is soft bijective. Definition 3.33. Let (F A, μ) be a SGTS. If for any point α F A Moreover (F G) and (F H) are soft η-open sets, because is and a soft μ-closed set F B such that α F B, there exists soft continuous function : F A F [0,1] such that (α) = (0, F [0,1](0)) (μ, )-open. Hence (F B, ) is a SGRS. and (F B) = (1, F [0,1](1)) where F [0,1] is a SGHS. Then (F A, μ) is called a soft generalized completely regular space (SGCRS). Theorem 3.34. Every SGCRS is a SGRS. Proof: Let (F A, μ) be a SGCRS. Let α F A and F K is a soft μ- closed set such that α F K. Since (F A, μ) is a SGCRS, there exists a soft continuous function : F A F [0,1] where F [0,1] is a SGHS and (α) = (0, F [0,1](0)) and (F K) = (1, F [0,1] (1)). Let β = (0, F [0,1](0)) and γ = (1, F [0,1](1)). Since F [0,1] is a SGHS, there exists two soft sets F G and F H such that β F G, γ F H and F G F H = F. Let F P = 1 (F G) and F Q = 1 (F H). Since is a soft continuous function, F P and F Q are soft μ-open sets. Then clearly α F P, F K F Q and F P F Q = F. Hence (F A, μ) is a SGRS. 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