γ -Hyperconnectedness and fuzzy mappings in fuzzy bitopological spaces

1 Introduction and preliminaries

In the literature, the concept of hyperconnectedness has been studied by several authors and it is studied separately for different sense and objectives. Steen and Seebach [14] have introduced and studied the concept of hyperconnected space in topological space. Later on, Ajmal et al. [1] proposed and proved further new results concerned with this space. Also, Noiri [12, 13] has shown his interest to do work in this direction. Caldas et al. [4] have introduced the notion of hyperconnectedness in fuzzy topological space. The notion of bitopological spaces have been investigated from different aspects by Tripathy and Acharjee [19]; Tripathy and Sarma [20] and others. Kandil [7] has originated the idea of fuzzy bitopological space as a generalized form of fuzzy topological space due to Chang [5]. Later on several authors have shown their concentration on fuzzy bitopological spaces. The notion of fuzzy b-locally open sets in bitopological spaces was introduced by Tripathy and Debnath [18]. The main purpose of this paper is to study a new conception in fuzzy bitopological spaces, called (i, j)-fuzzy γ-hyperconnectedness concerned with the significant work on γ-open set due to Tripathy and Debnath [17].

In particular, we obtain one result in Section 2, based on γ-hyperconnectedness which is useful for a comparison between fuzzy notion and that of crisp in the context of bitopology. Finally in this paper, we study some particular type of functions connected with γ-open set on fuzzy bitopological spaces. Also we provide the relationship between some functions applying the (i, j)-fuzzy γ-hyperconnectedness. Suitable example is properly placed for illustration, by considering parameter while it is not so easy by using normal fuzzy sets.

Throughout this paper, simply by X and Y we shall denote the fuzzy bitopological spaces (for short, fbts) (X,  τ _i , τ _j ) and (Y,  σ _i , σ _j ), where i ≠ j and i,  j = 1,  2. In a fbts, for any subset λ the γ-closure and γ-interior of λ are denoted by (i, j) - γ-cl(λ) and (i, j)- γ-int (λ) respectively.

Definition 1.1. A function f : (X, T) → (Y, R) from a topological space X to another topological space Y is called feebly continuous [2] if, for every non-empty open set V of Y f⁻¹ (V) ≠ φ implies that int (f⁻¹ (V)) ≠ φ .

Definition 1.2. A function f : (X, τ) → (Y, σ) from a fuzzy topological space X to another fuzzy topological space Y is called

fuzzy contra continuous [6] if the inverse image of every fuzzy open set of Y is a fuzzy closed set in X and

Definition 1.3. A fuzzy topological space X is said to be fuzzy extremally disconnected [11] if the closure of every fuzzy open set of X is a fuzzy open set in X.

Definition 1.4. A fuzzy subset λ of a fbts space X is said to be

(i, j)-fuzzy semi-open [9] if λ≤j-cl (i-int(λ)),

Definition 1.5. A fuzzy set λ in a fbts X is called pairwise fuzzy dense set [15] if cl _{T 1} cl _{T 2}  (λ) = cl _{T 1} cl _{T 2}  (λ) =1 _X .

Definition 1.6. A fuzzy set λ in a fbts X is called pairwise fuzzy nowhere dense set [16] if int _{T 1} cl _{T 2}  (λ) = int _{T 2} cl _{T 1}  (λ) =0 _X .

Definition 1.7. A function f : (X, τ _i , τ _j ) → (Y, σ _i , σ _j ) from a fbts X to another fbts Y said to be (i, j)-fuzzy semi continuous [9] if the inverse image of every σ _i -fuzzy open set of Y is a (i, j)-fuzzy semi-open set in X.

2 Properties of (i, j)-fuzzy γ-Hyperconnected space

In this section, we consider the (i, j)-fuzzy γ-open set to study a new notion in fbts and some of it basic properties. Finally in this section, by using this new notion we show that in a fbts the collection of all (i, j)-fuzzy γ-semi-open sets forms a fuzzy topology.

Now first we define the concept of hyperconnectedness in fuzzy bitopological space then study the new concept concerned with (i, j)-fuzzy γ-open set related to this.

Definition 2.1. A fbts (X, τ _i , τ _j ) is said to be (i, j)-fuzzy hyperconnected space if for any τ _i -fuzzy open set μ, τ _j -cl (μ) =1 _X .

Definition 2.2. A subset μ of a fbts X is said to be(i, j)-fuzzy γ-dense if (i, j)- γ-cl(μ) =1 _X  .

A subset μ in a fbts X is said to be (i, j)-fuzzy γ -nowhere dense if (i, j)- γ-int((i, j)-γ-cl(μ) =0 _X  .

Definition 2.3. A fbts (X, τ _i , τ _j ) is said to be (i, j)-fuzzy γ-hyperconnected space if every (i, j)-fuzzy γ-open set is (i, j)-fuzzy γ-dense set in X.

Example 2.4. Let X = {x, y}, τ _i  = {(x, 0.54) , (y, 0.55)} , 0 _X , 1 _X } and τ _j  = {(x, 0.47) , (y, 0.46)} , 0 _X , 1 _X }.

Here (i, j)FγO(X) = {{(x, α) , (y, β)}, where α ≥ 0.54 and β ≥ 0.55}. Thus X is a (i, j)-fuzzy γ-hyperconnected space, since for any (i, j)-fuzzy γ-open set μ, we have (i, j) γ-cl (μ) =1 _X  .

Remark 2.5. In a fbts (X, τ _i , τ _j ) the concepts of (i, j)-fuzzy hyperconnectedness and (i, j)-fuzzy γ-hyperconnected space are completely independent of each other as shown in the following example.

Example 2.6. (i) Let X = {x, y}, τ _i  = {(x, 0.53) , (y, 0.54)} , 0 _X , 1 _X } and τ _j  = {(x, 0.47) , (y, 0.46)} , 0 _X , 1 _X }.

Here (i, j)FγO (X) = {{(x, α) , (y, β)}, where α ≥ 0.53 and β ≥ 0.54}. Here X is (i, j)-fuzzy γ-hyperconnected space but not a (i, j)-fuzzy hyperconnected space, since τ _j -cl({(x, 0.53) , (y, 0.54)} ≠1 _X

(ii) Let X = {x, y}, τ _i  = {(x, 0.50) , (y, 0.51)} , 0 _X , 1 _X } and τ _j  = {(x, 0.50) , (y, 0.62} , 0 _X , 1 _X } .

Here X is (i, j)-fuzzy hyperconnected space but not a (i, j)-fuzzy γ- hyperconnected space, since (i, j)-γ-cl of every (i, j)-fuzzy γ-open set is not a 1 _X  .

Definition 2.7. A fbts (X,  τ _i , τ _j ) is said to be (i, j)- fuzzy γ-extremally disconnected if (i, j)-γ-cl of every (i, j)-fuzzy γ-open set is (i, j)-fuzzy γ-open set.

Proposition 2.8. Let (X, τ _i , τ _j ) be a fbts. If (X, τ _i , τ _j ) is a (i, j)-fuzzy γ-hyperconnected space then (X, τ _i , τ _j ) is a (i, j)-fuzzy γ-extremally disconnected space.

Proof. Let us suppose that (X, τ _i , τ _j ) be a (i, j)-fuzzy γ-hyperconnected space. Thus for any (i, j)-fuzzyγ-open set μ we get (i, j)γ-cl(μ) =1 _X which is a (i, j)-fuzzy γ-open set and thus (X, τ _i , τ _j ) is a (i, j)-fuzzy γ-extremally disconnected space.

Remark 2.9. Converse of the above proposition is not necessarily true and follows from the following example.

Example 2.10. Let X = {x}, τ _i  = {(x, 0.49) , 0 _X , 1 _X } and τ _j  = {(x, 0.48) , 0 _X , 1 _X } .

Here X is a (i, j)-fuzzy γ-extremally disconnected space but not a (i, j)-fuzzy γ-hyperconnected space because (i, j)- γ-cl of every (i, j)-fuzzy γ-open set cannot be 1 _X .

Proposition 2.11. In a (i, j) fuzzy γ-hyperconnected space, every non-empty subset is either a (i, j)-fuzzy γ-dense or (i, j)-fuzzy γ-nowhere dense set.

Proof. Let (X, τ _i , τ _j ) be a (i, j)-fuzzy γ-hyperconnected space where the subset λ is not a(i, j)-fuzzy γ-nowhere dense set in X. Now we have (i, j) γ-cl(1 _X  − (i,  j) γ-cl (λ)) = 1 _X  − (i, j) γ-int((i, j)γ-cl (λ)) ≠1 _X , since (i, j)γ-int((i, j)γ-cl(λ)) ≠0 _X , this implies (i, j)γ-cl((i, j)γ-cl((i, j) γ-cl (λ))) =1 _X . Now since (i, j) γ-cl((i, j) γ-int((i, j)γ-cl(λ))) =1 _X  ≤ (i,  j)γ-cl(λ), we have (i, j)γ-cl (λ) =1 _X . Hence λis a (i, j)-fuzzy γ-dense set.

Proposition 2.12. In a (i, j)-fuzzy γ-hyperconnected space every pair of non-empty (i, j) fuzzy γ- open subsets necessarily has a non-empty intersection.

Proof. Suppose that λ ∧ μ = 0 _X for any two non- empty (i, j) fuzzy γ- open subsets λ and μ of X. This implies that (i, j)γ-cl (λ) ∧ μ = 0 _X and λ is not (i, j)-fuzzy γ- dense set. Since λ is (i, j) fuzzy γ- open, then λ ≤ (i, j) γ-int((i, j)γ-cl(λ)) and λ is not (i, j)-fuzzyγ-nowhere dense set, which is a contradiction. Thus the intersection of any two non- empty (i, j)-fuzzy γ- open subsets is non-empty.

The converse of the above proposition is not necessarily true in general and it is clear from the following example.

Example 2.13. Considering the above Example 2.10, one can easily prove that the converse part of the above Proposition 2.12. is not true. Since the space X is not a (i, j)-fuzzy γ-hyperconnected space however the intersection of any two (i, j)-fuzzyγ -open sets is non -empty.

Definition 2.14. A subset μ in a fbts X is said to be (i, j)-fuzzyγ-regular-open set if μ = (i,  j)- γ-cl((i, j)-γ-int(μ)).

Theorem 2.15. Let (X,  τ _i , τ _j ) be a fbts. Then the following properties are equivalent:

(X,  τ _i , τ _j ) is (i, j) - fuzzy γ-hyperconnected space.

Proof. (i)→(ii) Let (X,  τ _i , τ _j ) be a (i, j)- fuzzy γ-hyperconnected space. Suppose that μ is a proper (i, j)-fuzzy γ-regular open then we have μ = (i,  j) γ-int-((i, j)-γ-cl((μ))). But this shows that ((i, j)- γ-int-((i, j)γ-cl(μ)))  ^c  = (i,  j) γ-cl(1 _X  − ((i,  j) γ-cl(μ)) = μ ^c  ≠ 1 _X since μ ≠ 0 _X which is a contradiction. Thus 1 _X and 0 _X are the only (i, j)-fuzzy γ-regular open sets of X.

(ii)→(i) Let 1 _X and 0 _X are the only (i, j)-fuzzy γ-regular open set in X. If possible, suppose that X is not (i, j)-fuzzy γ-hyperconnected. This implies that there exists a non-empty (i, j)-fuzzyγ-open subset μ of X such that (i, j)-γ-cl(μ) ≠1 _X . This implies that (i, j)-γ-cl ((i, j)-γ-int (μ)) =0 _X , since the only (i, j)-fuzzyγ-regular open in X are 1 _X and 0 _X . Therefore we have (i, j)-γ-cl(μ) =0 _X which is not possible. Thus X is a (i, j)-fuzzy γ-hyperconnected space.

Definition 2.16. A subset μ in a fbts X is said to be (i, j)-fuzzyγ-semi-open if μ ≤ (i,  j)- γ-cl((i, j)-γ-int(μ)).

Proposition 2.17. In a (i, j)- fuzzy γ-hyperconnected space X, any fuzzy subset λ is (i, j)-fuzzy γ-semi open set if (i, j)- γint(λ) ≠0 _X  .

Proof. Let X be a (i, j) - fuzzy γ-hyperconnected space and λ be any subset of X where (i, j)-γint(λ)≠0 _X  . Therefore (i, j)-γ-cl((i, j)-γ-int(μ))= 1 _X  . Thus λ≤ (i, j)-γ-cl((i, j)-γ-int(μ)). Hence, every fuzzy subset λ is a (i, j)-fuzzy γ-semi-open.

From the above Definition 2.16 of (i, j)-fuzzy γ-semi open set, we see that the collection of all (i, j)-fuzzy γ-semi-open sets fails to form the structure of topology. But concerning the context of (i, j)-fuzzy γ-hyperconnectedness we are in a position to show that the mentioned collection is forming atopology.

Proposition 2.18. In a (i, j)-fuzzy γ-hyperconnected space X, finite intersection of (i, j)-fuzzy γ-semi -open sets is a (i, j)-fuzzy γ-semi open set.

Proof. Let us suppose that λ and μ are two non- empty (i, j)-fuzzyγ-semi-open sets of X. Thus μ ≤ (i,  j)-γ-cl((i, j)-γ-int(μ)) and λ ≤ (i,  j)-γ-cl((i, j)-γ-int(λ)).

Consequently (i, j)-γ-cl(μ) = (i,  j)-γ-cl((i, j)-γ-int(μ)) =1 _X and (i, j)-γ-cl(λ) = (i,  j)-γ-cl((i, j)-γ-int(μ)) =1 _X . Since λ and μ are two non- empty (i, j)-fuzzy γ-semi-open sets of X, so λ ∧ μ ≠ 0 _X  . Therefore, (i, j)-γ-cl((i, j)-γ-int(μ ∧ λ)) = (i, j)-γ-cl((i, j)-γ-int(μ)) ∧ (i,  j)-γ-cl((i, j)-γ-int(λ)) =1 _X  . Hence the proof.

Definition 2.19. A subset μ in a fbts X is said to be (i, j)-fuzzyγ-pre-open set if μ ≤ (i,  j)-γ-int((i, j)-γ-cl(μ)).

Theorem 2.20. Let (X,  τ _i , τ _j ) be a fbts. Then the following conditions are equivalent:

(X, τ _i , τ _j ) is (i, j)-fuzzy γ-hyperconnected space.

Proof. (i)→(ii) Let us suppose that λ is any (i, j)-fuzzy γ -pre-open set. This implies that λ ≤ (i,  j) γ-int ((i, j)γ-cl(λ). Hence from (i), we get (i, j)γ-cl(λ) = (i, j) γ-cl(λ) ((i, j)γ-int ((i, j)γ-cl(λ))) =1 _X  . Therefore λ is (i, j)-fuzzy γ -dense set.

(ii)→(i) Let us suppose that λ is any (i, j)-fuzzy γ-pre-open. So λ ≤ (i,  j) γ-int ((i, j)γ-cl(((i, j)γ-cl(λ) = ((i, j)γ-cl(λ) ((i, j)γ-int ((i, j)γ-cl(λ))) = λ)). Now by the virtue of (ii), since λ is a (i, j)-fuzzy γ -pre-open set, we have (i, j)γ-cl(λ) = (i, j)γ-cl(λ) ((i, j)γ-int ((i, j)γ-cl(λ)) = 1 _X  . This demonstrates that X is a (i, j)-fuzzy γ -hyperconnected space.

3 Some results on related functions

In 1973 S.P. Arya et al. [2] have studied the concept of feebly continuous function in topological space. Then in 1981 K.K. Azad [3] has introduced fuzzy almost continuous function in fuzzy topological space. S.S. Kumar [9] has considered the (i, j)-fuzzy semi-continuous function in a fuzzy bitopological space. Recently in 2006 E. Ekici et al. [6] have set up the notion of fuzzy almost continuous function on fuzzy topological spaces. Now in this approach we study the functions concerned with (i, j)-fuzzy γ-open set and also investigate the relationships among them self.

Definition 3.1. A function f : (X, τ _i , τ _j ) → (Y, σ _i , σ _j ) from a fbts X to another fbts Y is said to be (i, j)-fuzzy γ -feebly continuous if for every (i, j)-fuzzy γ-open set λ of Y, f⁻¹ (λ) ≠0 _X implies that (i, j)- γint-(f⁻¹ (λ)) ≠0 _X  .

Definition 3.2. A function f : (X, τ _i , τ _j ) → (Y, σ _i , σ _j ) from a fbts X to another fbts Y is said to be (i, j)-fuzzy γ -semi- continuous if for every σ _i -open set λ of Y, f⁻¹ (λ) is (i, j)-fuzzy γ-semi-open set in X.

Theorem 3.3. Every (i, j)-fuzzy γ -semi continuous function is a (i, j)-fuzzy γ -feebly continuous function.

Proof. Let f be a (i, j)-fuzzy γ -semi- continuous function. Also let μ be a σ _i -open set of Y such that f⁻¹ (μ) ≠0 _X  . Now since f is a (i, j)-fuzzy γ -semi -continuous function, thus f⁻¹ (μ) is a (i, j)-fuzzy γ -semi-open set. This implies that (i, j)-γint-(f⁻¹ (μ) ≠0 _X  .

Remark 3.4. The converse part of the above result is not necessarily true in general as follows from the following example.

Example 3.5. Let X = Y = {x, y}, τ _i  = {{(x, 0.34) , (y, 0.38)} , 0 _X , 1 _X } , τ _j  = {{(x, 0.36) , (y, 0.34)} , 0 _X , 1 _X } , σ _i  = {{(x, 0.44) , (y, 0.45)} , 0 _Y , 1 _Y } , σ _j  = {{(x, 0.47) , (y, 0.46)} , 0 _Y , 1 _Y }.

Here (i, j)FγO (X) = {{(x, α) , (y, β)}, where 0 ≤ α ≤ 0.34, 0.64 < α ≤ 1 and 0 ≤ β ≤ 0.38, 0.66 < β ≤ 1}.

Now we consider a function f : X → Y such that f (x) = y and f (x) = y. Thus f is (i, j)-fuzzy γ -feebly continuous but not (i, j)-fuzzy γ -semi continuous, since the inverse image of σ _i -fuzzy open set {(x, 0.44) , (y, 0.45)} is not a (i, j)-fuzzy γ -semi-open set in X.

Proposition 3.6. If X be (i, j)-fuzzy γ-hyperconnected space then every (i, j)-fuzzy γ -feebly continuous function is (i, j)-fuzzy γ -semi-continuous.

Proof. Let f : (X, τ _i , τ _j ) → (Y, σ _i , σ _j ) be a (i, j)-fuzzy γ -feebly continuous function. Also suppose that λ is a (i, j)-fuzzy γ-open set of Y such that f⁻¹ (λ) ≠0 _X , so (i, j) γ-int(f⁻¹ (λ)) ≠0 _X  . Now since X is (i, j)-fuzzy γ-hyperconnected space which implies that f⁻¹ (λ) is a (i, j)-fuzzy γ -semi-open set. Hence the proof.

Definition 3.7. A function f : (X, τ _i , τ _j ) → (Y, σ _i , σ _j ) from a fbts X to another fbts Y is said to be

(i, j)-fuzzy-γ almost continuous function if the inverse image of every (i, j)-fuzzy regular open set of Y is (i, j)-fuzzy γ-open set in X.

Proposition 3.8. In a fbts (X, τ _i , τ _j ), every (i, j)-fuzzy-γ almost continuous function is (i, j)-fuzzy-γ-contra continuous.

Proof. Let f be any (i, j)-fuzzy-γ almost continuous function from X to Y. From the definition of (i, j)-fuzzy-γ almost continuity we have f (λ) ⊆ σ _i -int(σ _j -cl(δ)), where λ is a (i, j)-fuzzy γ-open set in X and δ is a σ _i -fuzzy open set in Y. It implies that f (λ) ⊆ σ _j -cl(δ). Therefore fis (i, j)-fuzzy-γ -contra continuous.

Remark 3.9. The converse part of the above result is not necessarily true in general as follows from the following example.

Example 3.10. Let X = {x, y}, τ _i  = {{(x, 0.34) , (y, 0.38)} , 0 _X , 1 _X } , τ _j  = {{(x, 0.35) , (y, 0.37)} , 0 _X , 1 _X }} , σ _i  = {{(x, 0.36) , (y, 0.0.345)} , 0 _Y , 1 _Y } , σ _j  = {{(x, 0.33) , (y, 0.0.25)} , 0 _Y , 1 _Y } .

Now if we define a function f :  X → Y such that f (x) = y and f (x) = y then f is a (i, j)-fuzzy-γ-contra continuous function but not a (i, j)-fuzzy-γ almost continuous, since the inverse image of the set {(x, 0.36) , (y, 0.345)} is not a (i, j)-fuzzy γ-open set in X.

Proposition 3.11. Let f : (X, τ _i , τ _j ) → (Y, σ _i , σ _j ) be a (i, j)-fuzzy-γ contra continuous function from a fbts Xto another fbts Y, where Y is a (i, j)-fuzzy hyperconnected space. Then f is a (i, j)-fuzzy-γ almost continuous.

Proof. Since f is a (i, j)-fuzzy-γ contra continuous, so we have f (λ) ⊆ σ _j -cl(δ), where λ is a (i, j)-fuzzy γ-open set in X and δ is a σ _i -fuzzy open set in Y. Again by the hypothesis, Y is a (i, j)-fuzzy -hyperconnected space, so we have f (λ) ⊆ σ _i -int (σ _j -cl (δ)) = 1 _Y .

This implies that f (λ) ⊆ σ _i -int (σ _j -cl (δ)). Hence the proof.

4 Applications

It is important to mention that the present work can be extended for the further study on digital space and image processing.

But in this paper we have shown that in a (i, j)-fuzzy-γ-hyperconnected space, finite intersection of (i, j)-fuzzy-γ semi-open sets is a (i, j)-fuzzy-γ semi-open set. Thus this result is applicable for further work on (i, j)-fuzzy-γ semi-open set.