Fuzzy soft topological spaces

1 Introduction

Fuzzy soft set is a hybridization of fuzzy sets and soft sets, in which soft set is defined over fuzzy set. Similar generalization have also spread to topological space. The notion of topological space is defined on crisp sets and hence is affected by different generalizations of crisp sets like fuzzy sets and soft sets. C.L. Chang [2] introduced fuzzy topological space in 1968 and subsequently Cagman et al. [1] and Shabir et al. [7] introduced soft topological space independently in 2011. In the same year B. Tanay et al. [8] introduced fuzzy soft topological spaces and studied neighborhood and interior of a fuzzy soft set and then used these to characterize fuzzy soft open sets. Recently Roy et al. [6] have obtained different conditions for a subfamily of fuzzy soft sets to be a fuzzy soft basis or fuzzy soft subbasis.

Now we recall a few definitions and results of fuzzy soft sets and fuzzy soft topological spaces as discussed in [3, 8] respectively.

Let U be an initial universe, E be a set of parameters, P ( U ) be the set of all subsets of U and FS ( U ; E ) be the family of all fuzzy soft sets over U via parameters in E.

Definition 1.1. Let A ⊂ E and F ( U ) be the set of all fuzzy sets in U. Then the pair (f, A) is called a fuzzy soft set over U, denoted by f _A , where f is a function given by f : A → F ( U ).

Definition 1.2. Let A, B ⊆ E. Then two fuzzy soft sets f _A and k _B are said to be disjoint if f ( a ) ⋀ k ( b ) = Φ ∼ , ∀ a ∈ A , b ∈ B.

Definition 1.3. Let f _E be a fuzzy soft set, FS ( f E ) be the set of all fuzzy soft subsets of f _E , τ be a subfamily of FS ( f E ) and A, B, C ⊆ E. Then τ is called a fuzzy soft topology on f _E if the following conditions are satisfied

Φ E ∼ , f E belongs to τ;

Example 1.4. Let U = {h¹, h², h³} and E = {e₁, e₂, e₃}. Consider a fuzzy soft set f E = { ( e 1 , { h 0.2 1 , h 0.8 2 , h 0.5 3 } ) , ( e 2 , { h 0.8 1 , h 0.1 2 , h 1 3 } ) , ( e 3 , { h 0.7 1 , h 0.5 2 , h 0.2 3 } ) } defined on U.

Then the subfamily τ = { Φ E ∼ , f E , { ( e 1 , { h 0.2 1 , h 0.4 2 , h 0.1 3 } ) } , { ( e 1 , { h 0.1 1 , h 0.5 2 , h 0.5 3 } ) , ( e 2 , { h 0.7 1 , h 0 2 , h 0.7 3 } ) , ( e 3 , { h 0.6 1 , h 0.1 2 , h 0.1 3 } ) } , { ( e 1 , { h 0.2 1 , h 0.6 2 , h 0.4 3 } ) , ( e 2 , { h 0.1 1 , h 0.1 2 , h 0.9 3 } ) , ( e 3 , { h 0.5 1 , h 0.5 2 , h 0.1 3 } ) } { ( e 1 , { h 0 1 , h 0.8 2 , h 0.5 3 } ) , ( e 2 , { h 0.8 1 , h 0 2 , h 0.1 3 } ) , ( e 3 , { h 0.4 1 , h 0.3 2 , h 0 3 } ) } , { ( e 1 , { h 0.2 1 , h 0.8 2 , h 0.5 3 } ) , ( e 2 , { h 0.8 1 , h 0.1 2 , h 0.9 3 } ) , ( e 3 , { h 0.5 1 , h 0.3 2 , h 0.1 3 } ) } , { ( e 1 , { h 0.1 1 , h 0.8 2 , h 0.5 3 } ) , ( e 2 , { h 0.8 1 , h 0 2 , h 0.7 3 } ) , ( e 3 , { h 0.6 1 , h 0.3 2 , h 0.1 3 } ) } { ( e 1 , { h 0.2 1 , h 0.5 2 , h 0.5 3 } ) , ( e 2 , { h 0.7 1 , h 0 2 , h 0.7 3 } ) , ( e 3 , { h 0.6 1 , h 0.1 2 , h 0.1 3 } ) } , { ( e 1 , { h 0.2 1 , h 0.8 2 , h 0.5 3 } ) , ( e 2 , { h 0.8 1 , h 0 2 , h 0.1 3 } ) , ( e 3 , { h 0.4 1 , h 0.3 2 , h 0 3 } ) } , { ( e 1 , { h 0.2 1 , h 0.6 2 , h 0.5 3 } ) , ( e 2 , { h 0.7 1 , h 0.1 2 , h 0.9 3 } ) , ( e 3 , { h 0.6 1 , h 0.5 2 , h 0.1 3 } ) } } is a fuzzy soft topology on f _E and (f _E , τ) is a fuzzy soft topological space.

Definition 1.5. Let (f _E , τ) be a fuzzy soft topological space. A fuzzy soft set g _C in FS ( f E ) is said to be a neighborhood of a fuzzy soft set h _A if and only if there exists a fuzzy soft open set k _B such that h A ⊂ ∼ k B ⊂ ∼ g C.

Definition 1.6. Let (f _E , τ) be a fuzzy soft topological space and g C ∈ FS ( f E ). Then the fuzzy soft topology τ g C = { g C ⋂ ∼ h A | h A ∈ τ } is called fuzzy soft subspace topology and (g _C , τ _{g C} ) is called fuzzy soft subspace of (f _E , τ).

Example 1.7. In Example 1.4, let g C = { ( e 1 , { h 0 1 , h 0.1 2 , h 0 3 } ) } . Then τ g C = { ( e 1 , { h 0 1 , h 0.1 2 , h 0 3 } ) } is a fuzzy soft subspace topology on τ.

Definition 1.8. Let (f _E , τ) be a fuzzy soft topological space and h _A , g _C be fuzzy soft sets in FS ( f E ) such that g C ⊂ ∼ h A. Then g _C is called an interior fuzzy soft set of h _A iff h _A is a neighborhood of g _C .

The union of all interior fuzzy soft sets of g _C is called the interior of g _C and is denoted by g C 0.

Definition 1.9. Let (f _E , τ₁) and (f _E , τ₂) be two fuzzy soft topological spaces. If each g _C  ∈ τ₁ is in τ₂, then τ₂ is called fuzzy soft finer than τ₁, or τ₁ is called fuzzy soft coarser than τ₂.

2 Fuzzy soft point and fuzzy soft neighborhood

Introduction of the concepts of fuzzy point [4] and soft point [9] are significant landmarks in the theoretical developments of fuzzy topological space and soft topological space. In continuation, we introduce fuzzy soft point and examine some of its set theoretic properties. Then fuzzy soft neighborhood and neighborhood system for a fuzzy soft point is obtained. The concept of neighborhood of a fuzzy soft set is defined in [8], but here we define fuzzy soft neighborhood of a fuzzy soft point.

Definition 2.1. A fuzzy soft set g _C is said to be a fuzzy soft point, denoted by e _{g C} , if for the element e ∈ A , g ( e ) ≠ Φ ∼ and.

Definition 2.2. The complement of a fuzzy soft point e _{g C} is a fuzzy soft point (e _{g C} )  ^c such that g C c ( e ) = 1 U - g ( e ) and.

Example 2.3. Let U = {h¹, h², h³, h⁴} , A = {e₁, e₂, e₃, e₄, e₅} ⊂ E, the set of parameters. Then e g C = { ( e 1 , { h 0.1 1 , h 0.9 2 , h 0.4 4 } ) } is a fuzzy soft point whose complement is ( e g C ) c = { ( e 1 , { h 0.9 1 , h 0.1 2 , h 1 3 , h 0.6 4 } ) }.

Definition 2.4. A fuzzy soft point e _gC is said to be in a fuzzy soft set h _A , denoted by e g C ∈ ∼ h A if for the element e ∈ A ∪ C,  g (e) ≤ h (e).

Theorem 2.5. Fuzzy soft points satisfy the following properties.

If a fuzzy soft point e g C ∈ ∼ h A then e _{g C} may or may not belong to h A c;

Following is an example in favor of Theorem 2.5 (ii).

Example 2.6. Let U = {h¹, h²} , E = {e₁, e₂}. Consider the fuzzy soft point e g C = { ( e 1 , { h 0.1 1 , h 0.2 2 } ) }, which is contained in the fuzzy soft set h A = { ( e 1 , { h 0.1 1 , h 0.9 2 } ) , ( e 2 , { h 0.2 1 , h 0.3 2 } ) }. Then h A c = { ( e 1 , { h 0.9 1 , h 0.1 2 } ) , ( e 2 , { h 0.8 1 , h 0.7 2 } ) } does not contain e g C c = { ( e 1 , { h 0.9 1 , h 0.8 2 } ) }.

Definition 2.7. A fuzzy soft set k _B in a fuzzy soft topological space (f _E , τ) is said to be a fuzzy soft neighborhood of a fuzzy soft point e _{g C} if ∃ a fuzzy soft open set h _A such that e g C ∈ ∼ h A ⊆ ∼ k B.

Example 2.8. Consider the fuzzy soft topological space (f _E , τ) in Example 1.4. Here { ( e 1 , { h 0.2 1 , h 0.5 2 , h 0.8 3 } ) } is a fuzzy soft neighborhood of the fuzzy soft point { ( e 1 , { h 0.1 1 , h 0.2 2 , h 0.1 3 } ) }.

The family of all neighborhoods of e _{g C} is called its neighborhood system and is denoted by N τ(e _{g C} ).

Theorem 2.9. A fuzzy soft set in a fuzzy soft topological space is fuzzy soft open iff it is a fuzzy soft neighborhood of each of its fuzzy soft points.

Proof. Let (f _E , τ) be a fuzzy soft topological space and e _{g C} be a fuzzy soft point in a fuzzy soft open set k _B . Then by definition, k _B is a fuzzy soft neighborhood of e _{g C} .

Conversely, let k _B be a fuzzy soft set such that it is fuzzy soft neighborhood of each of its fuzzy soft points, say e _{λ gC} . Then, for each λ∈ Λ,   ∃ a fuzzy soft open set h _λA such that e λ g C ⊂ ∼ h λ A ⊂ ∼ k B. Now k B = ⋃ λ ∈ Λ ∼ e λ g C ⇒ k B = ⋃ λ ∈ Λ ∼ h λ A ⇒ k B, being the union of arbitrary family of fuzzy soft open sets, is fuzzy soft open. □

Theorem 2.10. The neighborhood system of N τ(e _{g C} ) in a soft topological space (f _E , τ) satisfies the following properties:

For all fuzzy soft points e _{g C} , N τ(e g C ) ≠ Φ E ∼;

Proof.

Obvious, as f _E is a fuzzy soft neighborhood of each of its points.

Theorem 2.11. Let a collection N(e _{g C} ) of fuzzy soft sets called neighborhoods of of a fuzzy soft point e _{g C} in f _E be such that

For all fuzzy soft points e _{g C} , N τ(e g C ) ≠ Φ ∼;

Then ∃ a unique topology τ on f _E such that N(e _{g C} ) coincides with the family of neighborhoods N τ(e _{g C} ) of e _{g C} with respect to the topology τ.

Proof. Take τ = { h A | h A ∈ N(e _{g C} ) ,  ∀ e g C ∈ ∼ h A }. Claim that τ is a fuzzy soft topology on f _E .

We have (a) Φ E ∼ ∈ τ, since Φ E ∼ contains no fuzzy soft point. Again from condition (i) of hypothesis, N ( e g C ) ≠ Φ ∼, so ∃ some neighborhood of every point in f _E , which is a superset of each of the neighborhoods. Hence, by condition (iii) of hypothesis, f _E  ∈ τ. ( b ) Let k B , h A ∈ τ ⇒ k B ∈ N ( e g C ) and h A ∈ N ( e g C ) ⇒ k B ⋂ ∼ h A ∈ N ( e g C ) ( by condition ( iv ) ) ⇒ k B ⋂ ∼ h A ∈ τ . ( by definition of τ ) ( c ) Let k A λ ∈ τ , ∀ λ ∈ Λ ⇒ k A λ ∈ N ( e g C ) , ∀ λ ∈ Λ ⇒ ⋃ λ ∈ Λ ∼ k A λ ∈ N ( e g C ) ( by condition ( iii ) )

Hence τ is a fuzzy soft topology on f _E .

Next to show N τ(e g C ) = N(e _{g C} ), take k B ∈ N(e _{g C} ) ⇒ ∃ h A ∈ N(e _{g C} ) such that h A ⊆ ∼ k B and h A ∈ N(e _hA ). (by condition (v))

But h A ∈ N (e _{g C} )⇒ e g C ∈ ∼ h A (by condition (ii)) and h A ∈ N(e _{h A} ) ⇒ h _A  ∈ τ. (by definition of τ) Thus k _B is a τ neighborhood of e g C ⇒ N(e g C ) ⊆ N τ(e _{g C} )

On the other hand let k B ∈ N τ(e _{g C} ). So ∃ a τ open fuzzy soft set w _H such that e g C ∈ ∼ w H ⊂ ∼ k B. We know w _H is a neighborhood of each of its fuzzy soft points and so w H ∈ N(e _{g C} ). Then w H ⊂ ∼ k B implies k B ∈ N(e _{g C} ) ⇒ N τ(e g C ) ⊆ N(e _{g C} ). □

3 Fuzzy soft closure and fuzzy soft interior

In [8], the authors have defined fuzzy soft interior of a fuzzy soft set through fuzzy soft neighborhood as given in Definition 1.5. In this section, we discuss interior and closure of a fuzzy soft set in a fuzzy soft topological space via fuzzy soft open and fuzzy soft closed sets respectively.

Definition 3.1. Let g _C be a fuzzy soft set in a fuzzy soft topological space (f _E , τ). Then

The fuzzy soft closure of g _C is a fuzzy soft set defined as

fsclg C = ⋂ ∼ { h A | g C ⊆ ∼ h A and h _A is fuzzy soft closed set};

It is observed that the two definitions of fuzzy soft interior given here and in [8] are equivalent. Below we list some properties of fuzzy soft interior and fuzzy soft closure. Theorem 3.2. A fuzzy soft set g _C is fuzzy soft closed iff fsclg _C  = g _C .

Theorem 3.3. Let g _C , h _A be fuzzy soft sets in a fuzzy soft topological space (f _E , τ). Then

( fsclg C ) c = fsintg C c;

Proof. Straightforward. □

Definition 3.4. Let P ( f E ) denotes the power set of the fuzzy soft set f _E . An operator c : P ( f E ) ⟶ P ( f E ) defined by c (h _A ) = fsclh _A is called a fuzzy soft closure operator.

Remark 3.5. A fuzzy soft topological space can also be characterized using fuzzy soft closure operator as in general topology.

Theorem 3.6. The fuzzy soft set h _A is fuzzy soft closed in a subspace (g _C , τ _{g C} ) of (f _E , τ) iff h A = k B ⋂ ∼ g C for some fuzzy soft closed set k _B in f _E .

Theorem 3.7. The fuzzy soft closure of a fuzzy soft set h _A in a subspace (g _C , τ _{g C} ) of (f _E , τ) equals fscl ( h A ) ⋂ ∼ g C.

Proof. We know fsclh _A is a fuzzy soft closed set in f E ⇒ fsclh A ⋂ ∼ g C is fuzzy soft closed set in g _C . Now h A ⊂ ∼ fsclh A ⋂ ∼ g C and fuzzy soft closure of h _A in g _C is the smallest fuzzy soft closed set containing h _A , so fuzzy soft closure of h _A in g _C is contained in fsclh A ⋂ ∼ g C.

Conversely, w _H denoting the fuzzy soft closure of h _A in g _C is a fuzzy soft closed set in g C ⇒ w H = k B ⋂ ∼ g C where k _B is a fuzzy soft closed set in f _E (by theorem 8.3.6). Then k _B is fuzzy soft closed containing h A ⇒ fsclh A ⊂ ∼ k B ⇒ fsclh A ⋂ ∼ g C ⊂ ∼ k B ⋂ ∼ g C = w H. □

4 Fuzzy soft separation axioms

The concept of points play a pivotal role in the study of separation axioms of topological spaces. Due to the unavailability of point like structure in fuzzy soft topological space, there was a road block in the study of separation axioms. So, after the definition of fuzzy soft point, we are now in a position to introduce various separation axioms for fuzzy soft topological spaces and study their properties.

Definition 4.1. A fuzzy soft topological space (f _E , τ) is said to be a fuzzy soft T₀-space if for every pair of disjoint fuzzy soft points e _hA and e _gC , ∃ a fuzzy soft open set containing one but not the other.

Example 4.2. A discrete fuzzy soft topological space is a fuzzy soft T₀-space since every fuzzy soft point e _hA is a fuzzy soft open set in the discrete space.

Theorem 4.3. A fuzzy soft subspace of a fuzzy soft T₀-space is fuzzy soft T₀, i.e., the property of being a T₀-space of a fuzzy soft topological space is hereditary.

Proof. Let (g _C , τ _{g C} ) be a fuzzy soft subspace of a fuzzy soft T₀-space (f _E , τ) and let e _{k 1B} , e _{k 2B} be two distinct fuzzy soft points of g _C . Then these fuzzy soft points are also in f _E ⇒   ∃ a fuzzy soft open set h _A containing one fuzzy soft point but not the other ⇒ g C ⋂ ∼ h A, where h _A  ∈ τ is a fuzzy soft open set in τ _{g C} containing one fuzzy soft point but not the other. □

Definition 4.4. A fuzzy soft topological space (f _E , τ) is said to be a fuzzy soft T₁-space if for distinct pair of fuzzy soft points e _gC , e _kB of f _E ,  ∃ fuzzy soft open sets s _D and h _A such that e g C ∈ ∼ s D and e g C ∉ ∼ h A ; e k B ∈ ∼ h A and e k B ∉ ∼ s D .

Example 4.5. Let U = {h¹, h²} and E = {e₁, e₂}. Then the fuzzy soft topological space (f _E , τ) is fuzzy soft T₁, where f E = { ( e 1 , { h 1 1 , h 1 2 } ) , ( e 2 , { h 1 1 , h 1 2 } ) } and τ = { Φ E ∼ , f E , { ( e 1 , { h 1 1 , h 1 2 } ) } , { ( e 2 , { h 1 1 , h 1 2 } ) } }.

Theorem 4.6. If every fuzzy soft point of a fuzzy soft topological space (f _E , τ) is fuzzy soft closed then (f _E , τ) is fuzzy soft T₁.

Proof. Let e h A = { ( e j , { h α i i | i = 1 , 2 , . . . , n } ) }, e k B = { ( e m , { h β i i | i = 1 , 2 , . . . , n } ) }, where e _j , e _m are distinct parameters, be distinct fuzzy soft point of f _E .

Case(a) α _i , β _i  ≤ 0.5.

Then we can always find some γ _i and δ _i such that α _i ≤ γ _i ,  β _i  ≤ δ _i  ⇒ α _i  ≤ 1 - γ _i , β _i  ≤ 1 - δ _i  ⇒ the fuzzy soft sets e l F = { ( e j , { h γ i i | i = 1 , 2 , . . . , n } ) } and e t G = { ( e m , { h δ i i | i = 1 , 2 , . . . , n } ) } are such that their complements are disjoint fuzzy soft open sets containing e _hA and e _kB respectively.

Case(b) α _i , β _i  > 0.5.

Then we can always find some γ _i and δ _i such that γ _i ≤ α _i ,  δ _i  ≤ β _i  ⇒ α _i  ≤ 1 - γ _i , β _i  ≤ 1 - δ _i  ⇒ the fuzzy soft sets e l F = { ( e j , { h γ i i | i = 1 , 2 , . . . , n } ) } and e t G = { ( e m , { h δ i i | i = 1 , 2 , . . . , n } ) } are such that their complements are disjoint fuzzy soft open sets containing e _hA and e _kB respectively. □

Theorem 4.7. A fuzzy soft subspace of a fuzzy soft T₁-space is fuzzy soft T₁.

Definition 4.8. A fuzzy soft topological space (f _E , τ) is said to be a fuzzy soft T₂-space if and only if for distinct fuzzy soft points e _gC , e _kB of f _E , ∃ disjoint fuzzy soft open sets h _A and s _D such that e g C ∈ ∼ h A and e k B ∈ ∼ s D.

The fuzzy soft topological space in Example 4.5 is fuzzy soft T₂.

Theorem 4.9. If every fuzzy soft point of a fuzzy soft topological space (f _E , τ) is fuzzy soft closed then (f _E , τ) is fuzzy soft T₂.

Proof. Similar to that of Theorem 4.6. □

Theorem 4.10. A fuzzy soft subspace of a fuzzy soft T₂-space is fuzzy soft T₂.

Theorem 4.11. A fuzzy soft topological space (f _E , τ) is fuzzy soft T₂ if and only if for distinct fuzzy soft points e _gC , e _kB of f _E ,  ∃ a fuzzy soft open set s _D containing e _gC but not e _kB such that e k B ∉ ∼ fscls D.

Proof. Let e _gC , e _kB be distinct fuzzy soft points in a fuzzy soft T₂ space (f _E , τ).

(⇒) ∃ distinct fuzzy soft open sets h _A and w _D such that e k B ∈ ∼ h A, e g C ∈ ∼ w D. This implies e g C ∈ ∼ h A c. So, h A c is a fuzzy soft closed set containing e _gC but not e _kB and fsclh A c = h A c.

(⇐) Take a pair of distinct fuzzy soft points e _gC and e _kB of f _E , ∃ a fuzzy soft open set s _D containing e _gC but not e _kB such that e k B ∉ ∼ fscls D ⇒ e k B ∈ ∼ ( fscls D ) c ⇒ s D and (fscls _D )  ^c are disjoint fuzzy soft open set containing e _gC and e _kB respectively. □

Definition 4.12. A fuzzy soft topological space (f _E , τ) is said to be a fuzzy soft regular space if for every fuzzy soft point e _hA and fuzzy soft closed set k _B not containing e _hA ,  ∃ disjoint fuzzy soft open sets g_1A, g_2A such that e h A ∈ ∼ g 1 A and k B ⊆ ∼ g 2 A.

A fuzzy soft regular T₁-space is called a fuzzy soft T₃-space.

Example 4.13. Consider U = { h 1 , h 2 } , E = { e 1 } , f E = { ( e 1 , { h 1 1 , h 1 2 } ) , ( e 1 , { h 1 1 } ) , ( e 1 , { h 1 2 } ) } and τ = { Φ E ∼ , f E , { ( e 1 , { h 1 1 } ) } , { ( e 1 , { h 1 2 } ) } , { ( e 1 , { h 1 1 , h 1 2 } ) } }. Then (f _E , τ) is a fuzzy soft regular topological space.

Remark 4.14. It can be shown that the property of being fuzzy soft T₃ is hereditary.

Remark 4.15. Fuzzy soft T₃-space ⇒ fuzzy soft T₂-space ⇒ fuzzy soft T₁-space ⇒ fuzzy soft T₀-space.

Theorem 4.16. A fuzzy soft topological space (f _E , τ) in which every fuzzy soft point is fuzzy soft closed, is fuzzy soft regular iff for a fuzzy soft open set g _C containing a fuzzy soft point e _hA , there exists a fuzzy soft open set s _D containing e _hA , such that fscls D ⊂ ∼ g C.

Proof. Take a fuzzy soft open set g _C containing e _hA in a regular fuzzy soft topological space (f _E , τ). Then g C c is fuzzy soft closed. By hypothesis, ∃ disjoint fuzzy soft open sets s _D and w _H such that e h A ∈ ∼ s D and g C c ⊂ ∼ w H. Now, s _D and w _H are disjoint, so s D ⊂ ∼ w H c ⇒ fscls D ⊂ ∼ w H c ⇒ fscls D ⊂ ∼ g C.

Conversely, assume the hypothesis. Take a fuzzy soft closed set k _B not containing a fuzzy soft point e _hA . Then k B c is a fuzzy soft open set containing the fuzzy soft point e _hA ⇒   ∃ a fuzzy soft open set s _D containing e _hA such that fscls D ⊂ ∼ k B c ⇒ k B ⊂ ∼ ( fscls D ) c ⇒ ( fscls D ) c is a fuzzy soft open set containing k _B and s D ⋂ ∼ ( fscls D ) c = Φ E ∼. □

Definition 4.17. A fuzzy soft topological space (f _E , τ) is said to be a fuzzy soft normal space if for every pair of disjoint fuzzy soft closed sets h _A and k _B ,  ∃ disjoint fuzzy soft open sets g_1A, g_2A such that h A ⊆ ∼ g 1 A and k B ⊆ ∼ g 2 A.

A fuzzy soft normal T₁-space is called a fuzzy soft T₄-space.

The fuzzy soft topological space in Example 4.13 is also a fuzzy soft normal space.

Remark 4.18. Every fuzzy soft T₄-space is fuzzy soft T₃.

Theorem 4.19. A fuzzy soft topological space (f _E , τ) is fuzzy soft normal iff for any fuzzy soft closed set h _A and fuzzy soft open set g _C containing h _A , there exists a fuzzy soft open set s _D such that h A ⊂ ∼ s D and fscls D ⊂ ∼ g C.

Proof. Let (f _E , τ) be fuzzy soft normal space and h _A be a fuzzy soft closed set and g _C be a fuzzy soft open set containing h _A  ⇒ h _A and g C c are disjoint fuzzy soft closed sets ⇒ ∃ disjoint fuzzy soft open sets g_1A, g_2A such that h A ⊂ ∼ g 1 A and g C c ⊆ ∼ g 2 A. Now g 1 A ⊂ ∼ g 2 A c ⇒ fsclg 1 A ⊂ ∼ fsclg 2 A c = g 2 A c Also, g C c ⊂ ∼ g 2 A ⇒ g 2 A c ⊂ ∼ g C ⇒ fsclg 1 A ⊂ ∼ g C.

Conversely, let l _F and k _B be any disjoint pair fuzzy soft closed sets ⇒ l F ⊂ ∼ k B c, then by hypothesis there exists a fuzzy soft open set s _D such that l F ⊂ ∼ s D and fscls D ⊂ ∼ k B c ⇒ k B ⊂ ∼ ( fscls D ) c ⇒ s D and (fscls _D )  ^c are disjoint fuzzy soft open sets such that l F ⊂ ∼ s D and k B ⊂ ∼ ( fscls D ) c. □

Theorem 4.20. A fuzzy soft closed subspace of a fuzzy soft normal space is fuzzy soft normal.

5 Fuzzy soft connectedness

Here we introduce and study fuzzy soft connectedness of fuzzy soft topological spaces.

Definition 5.1. A fuzzy soft separation of a fuzzy soft topological space (f _E , τ) is a pair h _A , k _B of disjoint non empty fuzzy soft open sets whose union is f _E .

If there does not exist a fuzzy soft separation of f _E , then the fuzzy soft topological space is said to be fuzzy soft connected, otherwise fuzzy soft disconnected.

Example 5.2.

The discrete fuzzy soft topological space of more than one member is always fuzzy soft disconnected;

Theorem 5.3. A fuzzy soft topological space (f _E , τ) is fuzzy soft disconnected ⇔∃ a non empty proper fuzzy soft subset of f _E which is both fuzzy soft open and fuzzy soft closed.

Proof. Let k _B be a non empty proper subset of f _E which is both fuzzy soft open and fuzzy soft closed. Now h _A  = (k _B )  ^c is non empty proper subset of f _E which is also both fuzzy soft open and fuzzy soft closed ⇒fsclk _B  = k _B and fsclh _A  = h _A  ⇒ f _E can be expressed as the union of two separated fuzzy soft sets k _B , h _A and so, is fuzzy soft disconnected.

Conversely, let f _E be fuzzy soft disconnected ⇒ ∃ non empty fuzzy soft subsets k _B and h _A such that fsclk B ⋂ ∼ h A = Φ E ∼ , k B ⋂ ∼ fsclh A = Φ E ∼ and k B ⋃ ∼ h A = f E. Now k B ⊆ ∼ fsclk B and fsclk B ⋂ ∼ h A = Φ E ∼ ⇒ k B ⋂ ∼ h A = Φ E ∼ ⇒ h A = ( k B ) c. Then k B ⋃ ∼ fsclh A = f E and k B ⋂ ∼ fsclh A = Φ E ∼ ⇒ k B = ( fsclh A ) c and similarly h _A  = (fsclk _B )  ^c  ⇒ k _B , h _A are fuzzy soft open sets being the complements of fuzzy soft closed sets. Also h _A  = (k _B )  ^c ⇒ they are also fuzzy soft closed. □

Theorem 5.4. If the fuzzy soft sets h _A and k _B form a fuzzy soft separation of f _E , and if (g _C , τ _{g C} ) is a fuzzy soft connected subspace of f _E , then g C ⊂ ∼ h A or g C ⊂ ∼ k B.

Proof. Since h _A and k _B are disjoint fuzzy soft open sets, so are h A ⋂ ∼ g C and k B ⋂ ∼ g C and their union gives g _C , i.e., they would constitute a fuzzy soft separation of g _C , a contradiction.

Hence, one of h A ⋂ ∼ g C and k B ⋂ ∼ g C is empty and so g _C is entirely contained in one of them. □

Theorem 5.5. If g _C is a fuzzy soft subspace of (f _E , τ), a separation of g _C is a pair of disjoint non empty fuzzy soft sets k _B and h _A whose union is g _C , such that k B ⋂ ∼ fsclh A = Φ E ∼ and h A ⋂ ∼ fsclk B = Φ E ∼.

Proof. Suppose k _B and h _A forms a separation of g _C . Then k _B is both fuzzy soft open and fuzzy soft closed in g _C . The fuzzy soft closure of k _B in g _C is fsclk B ⋂ ∼ g C. Since k _B is fuzzy soft closed in g _C , k B = fsclk B ⋂ ∼ g C ⇒ fsclk B ⋂ ∼ h A = Φ E ∼. By similar argument fsclh A ⋂ ∼ k B = Φ E ∼.

Conversely, let k _B and h _A are disjoint non empty fuzzy soft sets whose union is g _C such that k B ⋂ ∼ fsclh A = Φ E ∼ and h A ⋂ ∼ fsclk B = Φ E ∼ ⇒ g C ⋂ ∼ fsclh A = Φ E ∼ and g C ⋂ ∼ fsclk B = Φ E ∼ ⇒ h A and k _B are fuzzy soft closed in g _C . Also h A = k B c implies both are fuzzy soft open in g _C . □

Theorem 5.6. Let g _C be a fuzzy soft connected subspace of (f _E , τ). If g C ⊂ ∼ k B ⊂ ∼ fsclg C, then k _B is also fuzzy soft connected.

Proof. Let the soft set k _B satisfies the hypothesis. If possible, let h _A and s _D form a fuzzy soft separation of k _B . Then, g C ⊂ ∼ h A or g C ⊂ ∼ s D. Let g C ⊂ ∼ h A ⇒ fsclg C ⊂ ∼ fsclh A; since fsclh _A and s _D are disjoint, fsclg _C cannot intersect s _D . This contradicts the fact that s _D is a nonempty. □

Remark 5.7. In particular fsclg _C is fuzzy soft connected if g _C is fuzzy soft connected.

Remark 5.8. A fuzzy soft topological space (f _E , τ) is fuzzy soft connected iff Φ E ∼ and f _E are the only sets which are both fuzzy soft open and fuzzy soft closed.

Theorem 5.9. Arbitrary union of fuzzy soft connected subsets of (f _E , τ) that have non empty intersection is fuzzy soft connected.

Proof. Let {(g _{A λ} , τ _{g A λ} )  | λ ∈ Λ} be a collection of fuzzy soft connected subspaces of (f _E , τ) with non empty intersection. If possible, take a fuzzy soft separation h _A ,  k _B of g C = ⋃ λ ∈ Λ ∼ g A λ. Now for each λ , h A ⋂ ∼ g A λ and k B ⋂ ∼ g A λ are disjoint fuzzy soft open sets in the subspace such that their union gives g _{A λ} . As g _{A λ} is connected for each λ, one of h A ⋂ ∼ g A λ and k B ⋂ ∼ g A λ must be empty. Suppose, h A ⋂ ∼ g A λ = Φ A ∼ ⇒ k B ⋂ ∼ g A λ = g A λ ⇒ g A λ ⊂ ∼ k B , ∀ λ ∈ Λ ⇒ ⋃ λ ∈ Λ ∼ g A λ ⊂ ∼ k B ⇒ h A ⋃ ∼ k B ⊂ ∼ k B ⇒ h A is empty, a contradiction. □

Theorem 5.10. Arbitrary union of a family of fuzzy soft connected subsets of (f _E , τ) such that one of the members of the family has non empty intersection with every member of the family, is fuzzy soft connected.

Proof. Let {(g _{A λ} , τ _{g A λ} )  | λ ∈ Λ} be a collection of fuzzy soft connected subspaces of (f _E , τ) and g _{A λ 0} be a fixed member such that g A λ 0 ⋂ ∼ g A λ ≠ Φ A ∼ for each λ ∈ Λ. Then by Theorem 5.9, h A λ = g A λ 0 ⋃ ∼ g A λ is a fuzzy soft connected for each λ ∈ Λ. Now, ⋃ λ ∈ Λ ∼ h A λ = ⋃ λ ∈ Λ ∼ ( g A λ 0 ⋃ ∼ g A λ ) = ⋃ λ ∈ Λ ∼ g A λ and ⋂ λ ∈ Λ ∼ h A λ = ⋂ λ ∈ Λ ∼ ( g A λ 0 ⋃ ∼ g A λ ) = g A λ 0 ⋂ λ ∈ Λ ∼ ( ⋃ ∼ g A λ ) ≠ Φ A ∼ . Therefore, ⋃ λ ∈ Λ ∼ h A λ = ⋃ λ ∈ Λ ∼ g A λ is fuzzy soft connected. □

Theorem 5.11. If (f _E , τ₁) is a fuzzy soft connected space and τ₂ is fuzzy soft coarser than τ₁, then (f _E , τ₂) is also fuzzy soft connected.

Proof. Assume that k _B , h _A form a fuzzy soft separation of (f _E , τ₂). Now k _B , h _A  ∈ τ₂ ⇒ k _B , h _A  ∈ τ₁ ⇒  k _B , h _A form a fuzzy soft separation of (f _E , τ₁), a contradiction. □

6 Conclusion

This paper initiates the study of fuzzy soft point. Further the study of various topological structures which depend on the concept of fuzzy soft point is hence carried out, viz., Neighborhood, separation axiom etc. It is noted that fuzzy soft point shows different behavior as compared to that of soft point (Theorem 2.5 lists the properties of a fuzzy soft point, for soft point please refer to [9]). Further it is noticed that the closure, interior, neighborhood system, separation axioms and connectedness show similar behavior in general topological spaces and fuzzy soft topological spaces.