Weak forms of continuity in I-double gradation fuzzy topological spaces

In this paper, we introduce and characterize double fuzzy weakly preopen and double fuzzy weakly preclosed functions between I-double gradation fuzzy topological spaces and also study these functions in relation to some other types of already known functions.

In the history of science, new theories have always been necessary in order for existing scientific theories to progress and this will continue to be true in the future. Two examples of essentially different mathematical theories that deal with the concept of uncertainty are probability theory and the theory of fuzzy sets. Whereas probability theory has a history of around 360 years, the theory of fuzzy sets is little more than 50 years old. Since the 1960s fuzzy methods have entered the scientific and technological world, good theoretical progress (e.g., fuzzy logic, fuzzy probability theory, fuzzy topology, fuzzy algebra) has been made, and there have been technical advances in various areas (e.g., fuzzy control, fuzzy expert systems, fuzzy clustering and data mining).

Chang (1968); Lowen (1976); Šostak (1985); Kubiak (1985); Samanta and Mondal (19972002) and many others contributed a lot to the field of Fuzzy Topology. In recent years Fuzzy Topology has been found to be very useful in solving many practical problems. Shihong Du et. al. (2005) are currently working to fuzzify the 9-intersection Egenhofer model Egenhofer and Franzosa (1991); Herring and Egenhofer (1991) for describing topological relations in Geographic Information Systems (GIS) query. In El-Naschie (19982000), El-Naschie has shown that the notion of Fuzzy Topology is applicable to quantum particle physics and quantum gravity in connection with String Theory and e^∞ Theory. Tang (2004) has used a slightly changed version of Chang’s fuzzy topological space to model spatial objects for GIS databases and Structured Query Language (SQL) for GIS.

In this paper, we will introduce the concepts of double fuzzy weakly preopen and double weakly preclosed functions in I-double gradation fuzzy topological spaces. Their properties and the relationships between these functions and other functions introduced previously are investigated.

Throughout this paper, let X be a nonempty set and I is the closed unit interval [0,1]. I_∘=(0,1] and I₁=[0,1). The family of all fuzzy subsets on X denoted by I^X. By \underset{-}{0} and \underset{-}{1}, we denote the smallest and the greatest fuzzy subsets on X. For a fuzzy subset λ∈I^X, \underset{-}{1}-\lambda denotes its complement. Given a function \tilde{f}:X\to Y, f(λ) and f⁻¹(λ) define the direct image and the inverse image of f, defined by f\left(\lambda \right)\left(y\right)={\vee }_{f\left(x\right)=y}\lambda \left(x\right) and f⁻¹(ν)(x)=ν(f(x)), for each λ∈I^X, ν∈I^Y, and x∈X, respectively. For fuzzy subsets λ and μ in X, we write λqμ to mean that λ is quasi coincident (q-coincident) with μ, that is, there exists at least one point x∈X such that λ(x) + μ(x)>1. Negation of such a statement is denoted as \lambda \bar{q}\mu. Notions and notations not described in this paper are standard and usual.

Definition 2.1

[(Samanta and Mondal (19972002); Garcia and Rodabaugh (2005)] An I-double gradation fuzzy topology (τ τ^∗) on X is a pair of maps τ, τ^∗:I^X→I, which satisfies the following properties:

1. (O1)

   \tau \left(\lambda \right)\le \underset{-}{1}-{\tau }^{\ast }\left(\lambda \right) for each λ∈I^X.

2. (O2)

   τ(λ₁∧λ₂)≥τ(λ₁)∧τ(λ₂) and τ^∗(λ₁∧λ₂)≤τ^∗(λ₁)∨τ^∗(λ₂) for each λ₁, λ₂∈I^X.

3. (O3)

   \tau \left(\underset{i\in \Gamma }{\vee }{\lambda }_i\right)\ge \underset{i\in \Gamma }{\wedge }\tau \left({\lambda }_i\right) and {\tau }^{\ast }\left(\underset{i\in \Gamma }{\vee }{\lambda }_i\right)\le \underset{i\in \Gamma }{\vee }{\tau }^{\ast }\left({\lambda }_i\right) for each λ _i ∈I^X, i∈Γ.

The triplet (X,τ,τ^∗) is called an I-double gradation fuzzy topological spaces (I-dfts, for short). A fuzzy set λ is called an (r,s)-fuzzy open ((r,s)-fo, for short) if τ(λ)≥r and τ^∗(λ)≤s. A fuzzy set λ is called an (r,s)-fuzzy closed ((r,s)-fc, for short) set iff \underset{-}{1}-\lambda is an (r,s)-fo set. Let (X,{\tau }_1,{\tau }_1^{\ast }) and (Y,{\tau }_2,{\tau }_2^{\ast }) be two I-dfts’s. A function \tilde{f}:X\to Y is said to be a double fuzzy continuous iff τ₁(f⁻¹(ν))≥τ₂(ν) and {\tau }_1^{\ast }\left(f^{-1}\right(\nu \left)\right)\le {\tau }_2^{\ast }\left(\nu \right) for each ν∈I^Y.

There was a question we must ask ourselve before starting to present our results, which was: Is it useful to introduce new concepts to I-double gradation fuzzy topological spaces?

We could know that double (initially, intuitionistic) fuzzy sets (and hence double fuzzy topological spaces) deal with ambiguity in a way better than fuzzy sets. In addition to that, double fuzzy topological spaces is a generalization of some other kinds of topological spaces; we can get fuzzy topological spaces in Chang’s sense (X,{\mathcal{T}}_{r,s}), where

Also, when the conditions τ^∗(λ)=1−τ(λ) and τ(λ) + τ^∗(λ)≮1 achieved in Definition 2.1, we get the definition of fuzzy topological spaces in Kubiak- Šostak’s sense Kubiak (1985); Šostak (1985). If we use 2^Xinstead of I^X, the resulting topological structure will be called double gradation fuzzifying topological spaces (A new structure mentioned for the first time in Bhaumik and Abbas 2008). Besides, we can also get the general topological spaces.

Theorem 2.1

[(Çoker and Demirci1996; Lee and Im (2001)] Let (X τ τ^∗) be an I-dfts. Then for each r∈I₀, s∈I₁ and λ∈I^X, we define an operator C_τ,τ^∗:I^X×I₀×I₁→I^X as follows:

For λ, μ∈I^X, r₁r₂∈I₀and s₁s₂∈I₁, the operator C_τ,τ^∗satisfies the following statements:

1. (C1)

   C_{\tau ,{\tau }^{\ast }}(\underset{-}{0},r,s)=\underset{-}{0},

2. (C2)

   λ≤C_τ,τ^∗(λ,r,s),

3. (C3)

   C_τ,τ^∗(λ,r,s)∨C_τ,τ^∗(μ,r,s)=C_τ,τ^∗(λ∨μ,r,s),

4. (C4)

   C_τ,τ^∗(λ,r₁,s₁)≤C_τ,τ^∗(λ,r₂,s₂) if r₁≤r₂and s₁≥s₂,

5. (C5)

   C_τ,τ^∗(C_τ,τ^∗(λ,r,s),r,s)=C_τ,τ^∗(λ,r,s).

Theorem 2.2

[(Çoker and Demirci1996; Lee and Im2001)] Let (X τ τ^∗) be an I-dfts. Then for each r∈I₀, s∈I₁ and λ∈I^X, we define an operator I_τ,τ^∗:I^X×I₀×I₁→I^Xas follows:

For λ μ∈I^X, r r₁r₂∈I₀and s s₁s₂∈I₁, the operator I_τ,τ^∗satisfies the following statements:

1. (I1)

   I_{\tau ,{\tau }^{\ast }}(\underset{-}{1}-\lambda ,r,s)=\underset{-}{1}-C_{\tau ,{\tau }^{\ast }}(\lambda ,r,s),

2. (I2)

   I_{\tau ,{\tau }^{\ast }}(\underset{-}{1},r,s)=\underset{-}{1},

3. (I3)

   I_τ,τ^∗(λ,r,s)≤λ,

4. (I4)

   I_τ,τ^∗(λ,r,s)∧I_τ,τ^∗(μ,r,s)=I_τ,τ^∗(λ∧μ,r,s),

5. (I5)

   I_τ,τ^∗(λ,r₁,s₁)≥I_τ,τ^∗(λ,r₂,s₂) if r₁≤r₂and s₁≥s₂,

6. (I6)

   I_τ,τ^∗(I_τ,τ^∗(λ,r,s),r,s)=I_τ,τ^∗(λ,r,s),

7. (I7)

   If I_τ,τ^∗(C_τ,τ^∗(λ,r,s),r,s)=λ, then C_{\tau ,{\tau }^{\ast }}\left(I_{\tau ,{\tau }^{\ast }}\right(\underset{-}{1}-\lambda ,r,s),r,s)=\underset{-}{1}-\lambda.

Definition 2.2

Let (X,τ,τ^∗) be an I-dfts. For λ∈I^X, r∈I₀and s∈I₁.

1. (1)

   λ is called (r,s)-fuzzy preopen ((r,s)-fpo, for short) if λ≤I _τ,τ ^∗(C _τ,τ ^∗(λ,r,s),r,s). A fuzzy set λ is called (r,s)-fuzzy preclosed ((r,s)-fpc, for short) iff \underset{-}{1}-\lambda is (r,s)-fpo set. The (r,s)-fuzzy preinterior of λ, denoted by P I _τ,τ ^∗(λ,r,s) is defined by

   P{I}_{\tau ,{\tau }^{\ast }}(\lambda ,r,s)=\vee \{\nu \in I^X\mid \nu \le \lambda ,\nu \text{is}(r,s)-\text{fpo}\}.

   The (r,s)-fuzzy preclosure of λ, denoted by P C_τ,τ^∗(λ,r,s) is defined by

   P{C}_{\tau ,{\tau }^{\ast }}(\lambda ,r,s)=\wedge \{\nu \in I^X\mid \lambda \le \nu ,\nu \text{is}(r,s)-\text{fpc}\}.
2. (2)

   λ is called (r,s)-fuzzy regular open ((r,s)-fro, for short) if λ=I _τ,τ ^∗(C _τ,τ ^∗(λ,r,s),r,s). A fuzzy set λ is called (r,s)-fuzzy regular closed ((r,s)-frc, for short) iff \underset{-}{1}-\lambda is (r,s)-fro set.

3. (3)

   λ is called (r,s)-fuzzy α-open ((r,s)-fα o, for short) if λ≤I _τ,τ ^∗(C _τ,τ ^∗(I _τ,τ ^∗(λ,r,s),r,s),r,s). A fuzzy setλ is called (r,s)-fuzzy α-closed ((r,s)-fα c, for short) iff \underset{-}{1}-\lambda is (r,s)-fα o set.

Theorem 2.3

Let (X,τ,τ^∗) be an I-dfts. For λ∈I^X, r∈I₀and s∈I₁.

1. (1)

   λ is (r,s)-fpo (resp. (r,s)-fpc) iff λ=P I _τ,τ ^∗(λ,r,s) (resp. λ=P C _τ,τ ^∗(λ,r,s)),

2. (2)

   I _τ,τ ^∗(λ,r,s)≤P I _τ,τ ^∗(λ,r,s)≤λ≤P C _τ,τ ^∗(λ,r,s)≤C _τ,τ ^∗(λ,r,s),

3. (3)

   \underset{-}{1}-P{I}_{\tau ,{\tau }^{\ast }}(\lambda ,r,s)=P{C}_{\tau ,{\tau }^{\ast }}(\underset{-}{1}-\lambda ,r,s) and P{I}_{\tau ,{\tau }^{\ast }}(\underset{-}{1}-\lambda ,r,s)=\underset{-}{1}-P{C}_{\tau ,{\tau }^{\ast }}(\lambda ,r,s).

Definition 2.3

Let f:(X,{\tau }_1,{\tau }_1^{\ast })\to (Y,{\tau }_2,{\tau }_2^{\ast }) be a function from an I-dfts (X,{\tau }_1,{\tau }_1^{\ast }) into an I-dfts (Y,{\tau }_2,{\tau }_2^{\ast }). The function f is called:

1. (1)

   double fuzzy preclosed if f(λ) is (r,s)-fpc set in I ^Yfor each λ∈I ^X, r∈I ₀and s∈I ₁; {\tau }_1(\underset{-}{1}-\lambda )\ge r, {\tau }_1^{\ast }(\underset{-}{1}-\lambda )\le s,

2. (2)

   double fuzzy open if τ ₂(f(λ))≥τ ₁(λ) and {\tau }_2^{\ast }\left(f\right(\lambda \left)\right)\le {\tau }_1^{\ast }\left(\lambda \right) for each λ∈I ^X, r∈I ₀and s∈I ₁,

3. (3)

   double fuzzy almost open if τ ₂(f(λ))≥r and {\tau }_2^{\ast }\left(f\right(\lambda \left)\right)\le s for each (r,s)-fro set λ∈I ^X, r∈I ₀and s∈I ₁.

Definition 2.4

Let f:(X,{\tau }_1,{\tau }_1^{\ast })\to (Y,{\tau }_2,{\tau }_2^{\ast }) be a function from an I-dfts (X,{\tau }_1,{\tau }_1^{\ast }) into an I-dfts (Y,{\tau }_2,{\tau }_2^{\ast }). The function f is called:

1. (1)

   double fuzzy weakly open if f\left(\lambda \right)\le I_{{\tau }_2,{\tau }_2^{\ast }}\left(f\right(C_{{\tau }_1,{\tau }_1^{\ast }}(\lambda ,r,s)),r,s) for each λ∈I ^X, r∈I ₀and s∈I ₁; τ ₁(λ)≥r and {\tau }_1^{\ast }\left(\lambda \right)\le s,

2. (2)

   double fuzzy α-open if f(λ) is (r,s)-fα o in I ^Yfor each λ∈I ^X, r∈I ₀and s∈I ₁; τ ₁(λ)≥r and {\tau }_1^{\ast }\left(\lambda \right)\le s.

Definition 2.5

Let (X,τ,τ^∗) be an I-dfts, μ∈I^X, x _t ∈P(X), r∈I₀and s∈I₁where P(X) is the family of all fuzzy points in X. μ is called an (r,s)-fuzzy open Q-neighborhood of x _t if τ(μ)≥r, τ^∗(μ)≤s and x _t qμ. We denote the set of all (r,s)-fuzzy open Q-neighborhood of x _t by Q_τ,τ^∗(x _t ,r,s).

Definition 2.6

Let (X,τ,τ^∗) be an I-dfts, λ∈I^X, x _t ∈P(X), r∈I₀and s∈I₁. x _t is called (r,s)-fuzzy θ-cluster point of λ if for every μ∈Q_τ,τ^∗(x _t ,r,s), we have C_τ,τ^∗(μ,r,s)qλ. We denote D_{\tau ,{\tau }^{\ast }}(\lambda ,r,s)=\vee \{x_t\in \mathbf{P}(X)\mid x_t\text{is}(r,s\left)\text{-fuzzy}\theta \text{-cluster point of}\lambda \right\}. Where D_τ,τ^∗(λ,r,s) is called (r,s)-fuzzy θ-closure of λ.

Theorem 2.4

Let (X,τ,τ^∗) an I-dfts. For λ, μ∈I^Xand r, s∈I₀, we have the following:

1. (1)

   D_{\tau ,{\tau }^{\ast }}(\lambda ,r,s)=\wedge \{\mu \in I^X\mid \lambda \le I_{\tau ,{\tau }^{\ast }}(\mu ,r,s),\tau (\underset{-}{1}-\mu )\ge r,{\tau }^{\ast }(\underset{-}{1}-\mu )\le s\},

2. (2)

   x _t is (r,s)-fuzzy θ-cluster point of λ iff x _t ∈D _τ,τ ^∗(λ,r,s).

3. (3)

   C _τ,τ ^∗(λ,r,s)≤D _τ,τ ^∗(λ,r,s),

4. (4)

   If τ(λ)≥r and τ ^∗(λ)≤s, then C _τ,τ ^∗(λ,r,s)=D _τ,τ ^∗(λ,r,s),

5. (5)

   If λ is (r,s)-fpo, then C _τ,τ ^∗(λ,r,s)=D _τ,τ ^∗(λ,r,s),

6. (6)

   If λ is (r,s)-fpo and λ=C _τ,τ ^∗(I _τ,τ ^∗(λ,r,s),r,s), then D _τ,τ ^∗(λ,r,s)=λ.

The complement of (r,s)-fuzzy θ-closed set is called (r,s)-fuzzy θ-open and the (r,s)-fuzzy θ-interior operator denoted by T_τ,τ^∗(λ,r,s) is defined by T_{\tau ,{\tau }^{\ast }}(\lambda ,r,s)=\vee \{\nu \in I^X\mid C_{\tau ,{\tau }^{\ast }}(\nu ,r,s)\le \lambda ,\tau (\nu )\ge r,{\tau }^{\ast }(\nu )\le s\}.

Remark 2.1

From Theorem 2.4 It is easy to see that:

1. (1)

   I _τ,τ ^∗(λ,r,s)≤T _τ,τ ^∗(λ,r,s) for any λ∈I ^X, r∈I ₀and s∈I ₁,

2. (2)

   T _τ,τ ^∗(λ,r,s)=I _τ,τ ^∗(λ,r,s) for each λ∈I ^X, r∈I ₀and s∈I ₁; τ(λ)≥r and τ ^∗(λ)≤s.

Definition 3.7

A function f:(X,{\tau }_1,{\tau }_1^{\ast })\to (Y,{\tau }_2,{\tau }_2^{\ast }) is said to be double fuzzy weakly preopen if

for each λ∈I^X, r∈I₀ and s∈I₁; τ₁(λ)≥r and {\tau }_1^{\ast }\left(\lambda \right)\le s.

Remark 3.2

Every double fuzzy weakly open function is double fuzzy preopen and every double fuzzy preopen function is double fuzzy weakly preopen, but the converse need not be true in general.

Example 3.1

Let X={a,b,c} and Y={x,y,z}. Fuzzy sets λ₁, λ₂ and λ₃ are defined as:

Define τ₁and τ₂ as follows:

Then the mapping f:(X,{\tau }_1,{\tau }_1^{\ast })\to (Y,{\tau }_2,{\tau }_2^{\ast }) defined by f(a)=z, f(b)=x and f(c)=y is double fuzzy weakly preopen but not double fuzzy preopen. Where {\tau }_1\left(\lambda \right)\ge \frac{1}{3}, {\tau }_1^{\ast }\left(\lambda \right)\le \frac{1}{3} and f(λ) is not (\frac{1}{3},\frac{1}{3})-fpo.

Example 3.2

Let X={a,b,c} and Y={x,y,z}. Fuzzy sets λ₁, λ₂ and λ₃ are defined as:

Let ({\tau }_1,{\tau }_1^{\ast }) and ({\tau }_2,{\tau }_2^{\ast }) defined as follows:

Then the mapping f:(X,{\tau }_1,{\tau }_1^{\ast })\to (Y,{\tau }_2,{\tau }_2^{\ast }) defined by f(a)=z, f(b)=x and f(c)=y is double fuzzy weakly preopen but not double fuzzy weakly open. Since f\left({\lambda }_1\right)\leqq ̸I_{{\tau }_2,{\tau }_2^{\ast }}\left(f\right(C_{{\tau }_1,{\tau }_1^{\ast }}({\lambda }_1,r,s)),r,s).

Theorem 3.5

For a function f:(X,{\tau }_1,{\tau }_1^{\ast })\to (Y, {\tau }_2,{\tau }_2^{\ast }). The following statements are equivalent:

1. (1)

   f is double fuzzy weakly preopen,

2. (2)

   f\left(T_{{\tau }_1,{\tau }_1^{\ast }}\right(\lambda ,r,s\left)\right)\le P{I}_{{\tau }_2,{\tau }_2^{\ast }}\left(f\right(\lambda ),r,s) for each λ∈I ^X, r∈I ₀and s∈I ₁,

3. (3)

   T_{{\tau }_1,{\tau }_1^{\ast }}\left(f^{-1}\right(\nu ),r,s)\le f^{-1}\left(P{I}_{{\tau }_2,{\tau }_2^{\ast }}\right(\nu ,r,s\left)\right) for each ν∈I ^Y, r∈I ₀and s∈I ₁,

4. (4)

   f^{-1}\left(P{C}_{{\tau }_2,{\tau }_2^{\ast }}\right(\nu ,r,s\left)\right)\le D_{{\tau }_1,{\tau }_1^{\ast }}\left(f^{-1}\right(\nu ),r,s) for each ν∈I ^Y, r∈I ₀and s∈I ₁.

Proof

(1)⇒(2) Let λ∈I^X and x_p\in T_{{\tau }_1,{\tau }_1^{\ast }}(\lambda ,r,s). Then there exists \gamma \in {\mathbf{Q}}_{{\tau }_1,{\tau }_1^{\ast }}(x_p,r,s) such that \gamma \le C_{{\tau }_1,{\tau }_1^{\ast }}(\gamma ,r,s)\le \lambda. Thus f\left(\gamma \right)\le f\left(C_{{\tau }_1,{\tau }_1^{\ast }}\right(\gamma ,r,s\left)\right)\le f\left(\lambda \right) and hence

Since f is double fuzzy weakly preopen,

and hence f\left(x_p\right)\in P{I}_{{\tau }_2,{\tau }_2^{\ast }}\left(f\right(\lambda ),r,s). This shows that x_p\in f^{-1}\left(P{I}_{{\tau }_2}\right(f\left(\lambda \right),r,s\left)\right). Thus T_{{\tau }_1}(\lambda ,r,s)\le f^{-1}(P{I}_{{\tau }_2,{\tau }_2^{\ast }}(f(λ),r,s)) and so, f\left(T_{{\tau }_1,{\tau }_1^{\ast }}\right(\lambda ,r,s\left)\right)\le P{I}_{{\tau }_2,{\tau }_2^{\ast }}\left(f\right(\lambda ),r,s).

(2)⇒(1) Let μ∈I^X; τ₁(μ)≥r and {\tau }_1^{\ast }\left(\mu \right)\le s. Since \mu \le T_{{\tau }_1,{\tau }_1^{\ast }}\left(C_{{\tau }_1,{\tau }_1^{\ast }}\right(\mu ,r,s),r,s), then

Hence f is double fuzzy weakly preopen.

(2)⇒(3) Let ν∈I^Y. By using (2), f\left(T_{{\tau }_1,{\tau }_1^{\ast }}\right(f^{-1}\left(\nu \right), r,s\left)\right)\le P{I}_{{\tau }_2,{\tau }_2^{\ast }}(\nu ,r,s). Therefore, T_{{\tau }_1,{\tau }_1^{\ast }}\left(f^{-1}\right(\nu ),r,s)\le f^{-1}\left(P{I}_{{\tau }_2,{\tau }_2^{\ast }}\right(\nu ,r,s\left)\right).

(3)⇒(2) Trivial.

(3)⇒(4) Let ν∈I^Y. Using (3), we have

Therefore, we obtain f^{-1}\left(P{C}_{{\tau }_2,{\tau }_2^{\ast }}\right(\nu ,r,s\left)\right)\le D_{{\tau }_1,{\tau }_1^{\ast }}\left(f^{-1}\right(\nu ), r,s).

(4)⇒(3) Similarly we obtain, \underset{-}{1}-f^{-1}\left(P{I}_{{\tau }_2,{\tau }_2^{\ast }}\right(\nu ,r,s\left)\right)\le \underset{-}{1}-T_{{\tau }_1,{\tau }_1^{\ast }}\left(f^{-1}\right(\nu ),r,s), for every ν∈I^Y, r∈I₀ and s∈I₁, i.e., T_{{\tau }_1,{\tau }_1^{\ast }}\left(f^{-1}\right(\nu ),r,s)\le f^{-1}\left(P{I}_{{\tau }_2,{\tau }_2^{\ast }}\right(\nu ,r,s\left)\right). □

Theorem 3.6

For the function f:(X,{\tau }_1,{\tau }_1^{\ast })\to (Y,{\tau }_2,{\tau }_2^{\ast }). The following statements are equivalent:

1. (1)

   f is double fuzzy weakly preopen,

2. (2)

   For each x _t ∈ P(X) and each μ∈I ^X; τ ₁(μ)≥r and {\tau }_1^{\ast }\left(\mu \right)\le s with x _t ≤μ, there exists (r,s)-fpo set γ such that f(x _t )≤γ and \gamma \le f\left(C_{{\tau }_1,{\tau }_1^{\ast }}\right(\mu ,r,s\left)\right).

Proof

(1)⇒(2) Let x _t ∈ P(X) and μ∈I^Xsuch that τ₁(μ)≥r, {\tau }_1^{\ast }\left(\mu \right)\le s and x _t ≤μ. Since f is double fuzzy weakly preopen, then f\left(\mu \right)\le P{I}_{{\tau }_2,{\tau }_2^{\ast }}\left(f\right(C_{{\tau }_1,{\tau }_1^{\ast }}(\mu ,r,s)),r,s). Let \gamma =P{I}_{{\tau }_2,{\tau }_2^{\ast }}\left(f\right(C_{{\tau }_1,{\tau }_1^{\ast }}(\mu ,r,s)),r,s). Hence \gamma \le f(C_{{\tau }_1,{\tau }_1^{\ast }}(μ,r,s)), with f(x _t )≤γ.

(2)⇒(1) Let μ∈I^X; τ₁(μ)≥r, {\tau }_1^{\ast }\left(\mu \right)\le s and y _s ≤f(μ). It follows from (2) that \gamma \le f\left(C_{{\tau }_1,{\tau }_1^{\ast }}\right(\mu ,r,s\left)\right) for some (r,s)-fpo γ∈I^Yand y _s ≤γ. Hence we have, y_s\le \gamma \le P{I}_{{\tau }_2,{\tau }_2^{\ast }}\left(f\right(C_{{\tau }_1,{\tau }_1^{\ast }}(\mu ,r,s)),r,s). This shows that f\left(\mu \right)\le P{I}_{{\tau }_2,{\tau }_2^{\ast }}\left(f\right(C_{{\tau }_1,{\tau }_1^{\ast }}(\mu ,r,s)),r,s), i.e. f is double fuzzy weakly preopen function. □

Theorem 3.7

Let f:(X,{\tau }_1,{\tau }_1^{\ast })\to (Y,{\tau }_2,{\tau }_2^{\ast }) be a bijective function. Then the following statements are equivalent:

1. (1)

   f is double fuzzy weakly preopen;

2. (2)

   P{C}_{{\tau }_2,{\tau }_2^{\ast }}\left(f\right(\lambda ),r,s)\le f\left(C_{{\tau }_1,{\tau }_1^{\ast }}\right(\lambda ,r,s\left)\right) for each λ∈I ^X, r∈I ₀and s∈I ₁; τ ₁(λ)≥r and {\tau }_1^{\ast }\left(\lambda \right)\le s;

3. (3)

   P{C}_{{\tau }_2,{\tau }_2^{\ast }}\left(f\right(I_{{\tau }_1,{\tau }_1^{\ast }}(\nu ,r,s)),r,s)\le f\left(\nu \right) for each ν∈I ^X, r∈I ₀and s∈I ₁; {\tau }_1(\underset{-}{1}-\nu )\ge r and {\tau }_1^{\ast }(\underset{-}{1}-\nu )\le s.

Proof

(1)⇒(2) Let ν∈I^X; τ₁(ν)≥r and {\tau }_1^{\ast }\left(\nu \right)\le s. Then we have,

and so \underset{-}{1}-f\left(\nu \right)\le \underset{-}{1}-P{C}_{{\tau }_2,{\tau }_2^{\ast }}\left(f\right(I_{{\tau }_1,{\tau }_1^{\ast }}(\nu ,r,s)),r,s). Hence P{C}_{{\tau }_2,{\tau }_2^{\ast }}\left(f\right(I_{{\tau }_1,{\tau }_1^{\ast }}(\nu ,r,s)),r,s)\le f\left(\nu \right).

(2)⇒(3) Let λ∈I^X; τ₁(λ)≥r and {\tau }_1^{\ast }\left(\lambda \right)\le s. Since C_{{\tau }_1,{\tau }_1^{\ast }}(\lambda ,r,s) is (r,s)-fc set and \lambda \le I_{{\tau }_1,{\tau }_1^{\ast }}\left(C_{{\tau }_1,{\tau }_1^{\ast }}\right(\lambda ,r,s),r,s) by (3) we have P{C}_{{\tau }_2,{\tau }_2^{\ast }}\left(f\right(\lambda ),r,s)\le P{C}_{{\tau }_2,{\tau }_2^{\ast }}\left(f\right(I_{{\tau }_1,{\tau }_1^{\ast }}(\lambda ,r,s)),r,s)\le f\left(C_{{\tau }_1,{\tau }_1^{\ast }}\right(\lambda ,r,s\left)\right).

(3)⇒(2) Trivial.

(2)⇒(1) Trivial. □

Theorem 3.8

For a function f:(X,{\tau }_1,{\tau }_1^{\ast })\to (Y,{\tau }_2,{\tau }_2^{\ast }). The following statements are equivalent:

1. (1)

   f is double fuzzy weakly preopen;

2. (2)

   f\left(I_{{\tau }_1,{\tau }_1^{\ast }}\right(\nu ,r,s\left)\right)\le P{I}_{{\tau }_2,{\tau }_2^{\ast }}\left(f\right(\nu ),r,s) for each ν∈I ^X, r∈I ₀and s∈I ₁; τ ₁(ν)≥r and {\tau }_1^{\ast }\left(\nu \right)\le s;

3. (3)

   f\left(I_{{\tau }_1,{\tau }_1^{\ast }}\right(C_{{\tau }_1,{\tau }_1^{\ast }}(\lambda ,r,s),r,s\left)\right)\le P{I}_{{\tau }_2,{\tau }_2^{\ast }}\left(f\right(C_{{\tau }_1,{\tau }_1^{\ast }}(\lambda ,r,s)),r,s) for each λ∈I ^X, r∈I ₀and s∈I ₁; τ ₁(λ)≥r and {\tau }_1^{\ast }\left(\lambda \right)\le s;

4. (4)

   f\left(\lambda \right)\le P{I}_{{\tau }_2,{\tau }_2^{\ast }}\left(f\right(C_{{\tau }_1,{\tau }_1^{\ast }}(\lambda ,r,s)),r,s), for each (r,s)-fpo set λ∈I ^X;

5. (5)

   f\left(\lambda \right)\le P{I}_{{\tau }_2,{\tau }_2^{\ast }}\left(f\right(C_{{\tau }_1,{\tau }_1^{\ast }}(\lambda ,r,s)),r,s), for each (r,s)-fα o set λ∈I ^X.

Proof

(1)⇒(2) Let ν∈I^X, r∈I₀ and s∈I₁; {\tau }_1(\underset{-}{1}-\nu )\ge r and {\tau }_1^{\ast }(\underset{-}{1}-\nu )\le s. By (1),

(2)⇒(3) It is clear.

(3)⇒(4) Let λ be (r,s)-fpo set. Hence by (3),

(4)⇒(5) and (5)⇒(1) are clear. □

Definition 3.8

A function f:(X,{\tau }_1,{\tau }_1^{\ast })\to (Y,{\tau }_2,{\tau }_2^{\ast }) is said to be double fuzzy strongly continuous, if f\left(C_{{\tau }_1,{\tau }_1^{\ast }}\right(\lambda ,r,s\left)\right)\le f\left(\lambda \right) for each λ∈I^X, r∈I₀and s∈I₁.

Theorem 3.9

If f:(X,{\tau }_1,{\tau }_1^{\ast })\to (Y,{\tau }_2,{\tau }_2^{\ast }) is double fuzzy weakly preopen and double fuzzy strongly continuous function, then f is double fuzzy preopen.

Proof

Let λ∈I^X such that τ₁(λ)≥r and {\tau }_1^{\ast }\left(\lambda \right)\le s. Since f is double fuzzy weakly preopen

However, since f is double fuzzy strongly continuous, then f\left(\lambda \right)\le P{I}_{{\tau }_2,{\tau }_2^{\ast }}\left(f\right(\lambda ),r,s) and therefore f(λ) is (r,s)-fpo. □

Definition 3.9

A function f:(X,{\tau }_1,{\tau }_1^{\ast })\to (Y,{\tau }_2,{\tau }_2^{\ast }) is said to be double fuzzy contra-preclosed if f(λ) is (r,s)-fpo for each λ∈I^X, r∈I₀and s∈I₁; {\tau }_1(\underset{-}{1}-\lambda )\ge r and {\tau }_1^{\ast }(\underset{-}{1}-\lambda )\le s.

Theorem 3.10

If f:(X,{\tau }_1,{\tau }_1^{\ast })\to (Y,{\tau }_2,{\tau }_2^{\ast }) is double fuzzy contra-preclosed, then f is double fuzzy weakly preopen function.