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

Abstract

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.

Introduction

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.

Preliminaries

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:

• (O1) $\tau \left(\lambda \right)\le \underset{-}{1}-{\tau }^{\ast }\left(\lambda \right)$ for each λ∈I^X.

• (O2) τ(λ₁∧λ₂)≥τ(λ₁)∧τ(λ₂) and τ^∗(λ₁∧λ₂)≤τ^∗(λ₁)∨τ^∗(λ₂) for each λ₁, λ₂∈I^X.

• (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

$${\mathcal{T}}_{(r,s)}=\{\lambda \in I^X|\tau \left(\lambda \right)\ge r,{\tau }^{\ast }\left(\lambda \right)\le s\}.$$

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:

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

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

• (C1) $C_{\tau ,{\tau }^{\ast }}(\underset{-}{0},r,s)=\underset{-}{0}$,

• (C2) λ≤C_τ,τ^∗(λ,r,s),

• (C3) C_τ,τ^∗(λ,r,s)∨C_τ,τ^∗(μ,r,s)=C_τ,τ^∗(λ∨μ,r,s),

• (C4) C_τ,τ^∗(λ,r₁,s₁)≤C_τ,τ^∗(λ,r₂,s₂) if r₁≤r₂and s₁≥s₂,

• (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:

$$I_{\tau ,{\tau }^{\ast }}(\lambda ,r,s)=\vee \{\mu \in I^X\mid \mu \le \lambda ,\tau (\mu )\ge r,{\tau }^{\ast }(\mu )\le s\}.$$

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

• (I1) $I_{\tau ,{\tau }^{\ast }}(\underset{-}{1}-\lambda ,r,s)=\underset{-}{1}-C_{\tau ,{\tau }^{\ast }}(\lambda ,r,s)$,

• (I2) $I_{\tau ,{\tau }^{\ast }}(\underset{-}{1},r,s)=\underset{-}{1}$,

• (I3) I_τ,τ^∗(λ,r,s)≤λ,

• (I4) I_τ,τ^∗(λ,r,s)∧I_τ,τ^∗(μ,r,s)=I_τ,τ^∗(λ∧μ,r,s),

• (I5) I_τ,τ^∗(λ,r₁,s₁)≥I_τ,τ^∗(λ,r₂,s₂) if r₁≤r₂and s₁≥s₂,

• (I6) I_τ,τ^∗(I_τ,τ^∗(λ,r,s),r,s)=I_τ,τ^∗(λ,r,s),

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

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

• (3) C _τ,τ ^∗(λ,r,s)≤D _τ,τ ^∗(λ,r,s),

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

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

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

• (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.

Double Fuzzy weakly preopen functions

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

$$f\left(\lambda \right)\le P{I}_{{\tau }_2,{\tau }_2^{\ast }}\left(f\right(C_{{\tau }_1,{\tau }_1^{\ast }}(\lambda ,r,s)),r,s)$$

for each 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:

$$\begin{array}{ll}{\lambda }_1\left(a\right)& =0.5,{\lambda }_1\left(b\right)=0.3,{\lambda }_1\left(c\right)=0.2,\\ {\lambda }_2\left(x\right)& =0.9,{\lambda }_2\left(y\right)=1,{\lambda }_2\left(z\right)=0.7,\\ {\lambda }_3\left(x\right)& =0.2,{\lambda }_3\left(y\right)=0.2,{\lambda }_3\left(z\right)=0.3.\end{array}$$

Define τ₁and τ₂ as follows:

$${\tau }_1\left(\lambda \right)=\left\{\begin{array}{c}1\mathit{if}\lambda =\underset{-}{0},\underset{-}{1};\\ \frac{1}{3}\mathit{if}\lambda ={\lambda }_1;\\ 0\text{otherwise.}\end{array}\right.,{\tau }_1^{\ast }\left(\lambda \right)=\left\{\begin{array}{c}0\mathit{if}\lambda =\underset{-}{0},\underset{-}{1};\\ \frac{1}{4}\mathit{if}\lambda ={\lambda }_1;\\ 1\text{otherwise.}\end{array}\right.$$

$${\tau }_2\left(\lambda \right)=\left\{\begin{array}{c}1\mathit{if}\lambda =\underset{-}{0},\underset{-}{1};\\ \frac{1}{3}\mathit{if}\lambda ={\lambda }_2;\\ \frac{2}{3}\mathit{if}\lambda ={\lambda }_3;\\ 0\text{otherwise.}\end{array}\right.,{\tau }_2^{\ast }\left(\lambda \right)=\left\{\begin{array}{c}0\mathit{if}\lambda =\underset{-}{0},\underset{-}{1};\\ \frac{1}{4}\mathit{if}\lambda ={\lambda }_2;\\ \frac{1}{3}\mathit{if}\lambda ={\lambda }_3;\\ 1\text{otherwise.}\end{array}\right.$$

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:

$$\begin{array}{cc}{\lambda }_1\left(a\right)& =0.5,{\lambda }_1\left(b\right)=0.3,{\lambda }_1\left(c\right)=0.2;\\ {\lambda }_2\left(x\right)& =0.9,{\lambda }_2\left(y\right)=1,{\lambda }_2\left(z\right)=0.7;\\ {\lambda }_3\left(x\right)& =0.2,{\lambda }_3\left(y\right)=0.9,{\lambda }_3\left(z\right)=0.3.\end{array}$$

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

$${\tau }_1\left(\lambda \right)=\left\{\begin{array}{c}1\mathit{if}\lambda =\underset{-}{0},\underset{-}{1}\\ \frac{1}{2}\mathit{if}\lambda ={\lambda }_1;\\ 0\text{otherwise.}\end{array}\right.,{\tau }_1^{\ast }\left(\lambda \right)=\left\{\begin{array}{c}0\mathit{if}\lambda =\underset{-}{0},\underset{-}{1}\\ \frac{1}{2}\mathit{if}\lambda ={\lambda }_1;\\ 1\text{otherwise.}\end{array}\right.$$

$${\tau }_2\left(\lambda \right)=\left\{\begin{array}{c}1\mathit{if}\lambda =\underset{-}{0},\underset{-}{1}\\ \frac{1}{2}\mathit{if}\lambda ={\lambda }_2;\\ \frac{1}{3}\mathit{if}\lambda ={\lambda }_3;\\ 0\text{otherwise.}\end{array}\right.,{\tau }_2^{\ast }\left(\lambda \right)=\left\{\begin{array}{c}0\mathit{if}\lambda =\underset{-}{0},\underset{-}{1}\\ \frac{1}{2}\mathit{if}\lambda ={\lambda }_2;\\ \frac{1}{3}\mathit{if}\lambda ={\lambda }_3;\\ 1\text{otherwise.}\end{array}\right.$$

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) f is double fuzzy weakly preopen,

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

• (4) $f^{-1}\left(P{C}_{{\tau }_2,{\tau }_2^{\ast }}\right(\nu ,r,s\left)\right)\le D_{{\tau }_1,{\tau }_1^{\ast }}\left(f^{-1}\right(\nu ),r,s)$ 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

$$\begin{array}{l}P{I}_{{\tau }_2,{\tau }_2^{\ast }}\left(f\right(\gamma ),r,s)\le P{I}_{{\tau }_2,{\tau }_2^{\ast }}\left(f\right(C_{{\tau }_1,{\tau }_1^{\ast }}(\gamma ,r,s)),r,s)\\ \le P{I}_{{\tau }_2,{\tau }_2^{\ast }}\left(f\right(\lambda ),r,s).\end{array}$$

Since f is double fuzzy weakly preopen,

$$f\left(\gamma \right)\le P{I}_{{\tau }_2,{\tau }_2^{\ast }}\left(f\right(C_{{\tau }_1,{\tau }_1^{\ast }}(\gamma ,r,s)),r,s)\le P{I}_{{\tau }_2,{\tau }_2^{\ast }}\left(f\right(\lambda ),r,s).$$

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

$$\begin{array}{l}f\left(\mu \right)\le f\left(T_{{\tau }_1,{\tau }_1^{\ast }}\right(C_{{\tau }_1,{\tau }_1^{\ast }}(\mu ,r,s),r,s\left)\right)\\ \le P{I}_{{\tau }_2,{\tau }_2^{\ast }}\left(f\right(C_{{\tau }_1,{\tau }_1^{\ast }}(\mu ,r,s)),r,s).\end{array}$$

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

$$\begin{array}{ll}\underset{-}{1}-D_{{\tau }_1,{\tau }_1^{\ast }}\left(f^{-1}\right(\nu ),r,s)& =T_{{\tau }_1,{\tau }_1^{\ast }}(\underset{-}{1}-f^{-1}(\nu ),r,s)\\ =T_{{\tau }_1,{\tau }_1^{\ast }}\left(f^{-1}\right(\underset{-}{1}-\nu ),r,s)\\ \le f^{-1}\left(P{I}_{{\tau }_2,{\tau }_2^{\ast }}\right(\underset{-}{1}-\nu ,r,s\left)\right)\\ =f^{-1}(\underset{-}{1}-P{C}_{{\tau }_2,{\tau }_2^{\ast }}(\nu ,r,s\left)\right)\\ =\underset{-}{1}-\left(f^{-1}\right(P{C}_{{\tau }_2,{\tau }_2^{\ast }}(\nu ,r,s)\left)\right).\end{array}$$

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) f is double fuzzy weakly preopen,

• (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) f is double fuzzy weakly preopen;

• (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) $P{C}_{{\tau }_2,{\tau }_2^{\ast }}\left(f\right(I_{{\tau }_1,{\tau }_1^{\ast }}(\nu ,r,s)),r,s)\le f\left(\nu \right)$ 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,

$$f(\underset{-}{1}-\nu )=\underset{-}{1}-f\left(\nu \right)\le P{I}_{{\tau }_2,{\tau }_2^{\ast }}\left(f\right(C_{{\tau }_1,{\tau }_1^{\ast }}(\underset{-}{1}-\nu ,r,s)),r,s),$$

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) f is double fuzzy weakly preopen;

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

• (5) $f\left(\lambda \right)\le P{I}_{{\tau }_2,{\tau }_2^{\ast }}\left(f\right(C_{{\tau }_1,{\tau }_1^{\ast }}(\lambda ,r,s)),r,s)$, 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),

$$\begin{array}{ll}f\left(I_{{\tau }_1,{\tau }_1^{\ast }}\right(\nu ,r,s\left)\right)& \le P{I}_{{\tau }_2,{\tau }_2^{\ast }}\left(f\right(C_{{\tau }_1,{\tau }_1^{\ast }}\left(I_{{\tau }_1,{\tau }_1^{\ast }}\right(\nu ,r,s),r,s)),r,s)\\ \le P{I}_{{\tau }_2,{\tau }_2^{\ast }}\left(f\right(C_{{\tau }_1,{\tau }_1^{\ast }}(\nu ,r,s)),r,s)\\ =P{I}_{{\tau }_2,{\tau }_2^{\ast }}\left(f\right(\nu ),r,s)\end{array}$$

(2)⇒(3) It is clear.

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

$$\begin{array}{l}f\left(\lambda \right)\le f\left(I_{{\tau }_1,{\tau }_1^{\ast }}\right(C_{{\tau }_1,{\tau }_1^{\ast }}(\lambda ,r,s),r,s\left)\right)\\ \le P{I}_{{\tau }_2,{\tau }_2^{\ast }}\left(f\right(C_{{\tau }_1,{\tau }_1^{\ast }}(\lambda ,r,s)),r,s).\end{array}$$

(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

$$f\left(\lambda \right)\le P{I}_{{\tau }_2,{\tau }_2^{\ast }}\left(f\right(C_{{\tau }_1,{\tau }_1^{\ast }}(\lambda ,r,s)),r,s).$$

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.

Proof

Let λ∈I^X; τ₁(λ)≥r and ${\tau }_1^{\ast }\left(\lambda \right)\le s$. Then, we have

$$f\left(\lambda \right)\le f\left(C_{{\tau }_1,{\tau }_1^{\ast }}\right(\lambda ,r,s\left)\right)=P{I}_{{\tau }_2,{\tau }_2^{\ast }}\left(f\right(C_{{\tau }_1,{\tau }_1^{\ast }}(\lambda ,r,s)),r,s).$$

□

The converse of the above theorem need not be true in general as in the following Example.

Example 3.3

Let X={a,b,c} and Y={x,y,z}. Define fuzzy sets λ₁, λ₂ as follows:

$$\begin{array}{ll}{\lambda }_1\left(a\right)& =0,{\lambda }_1\left(b\right)=0.2,{\lambda }_1\left(c\right)=0.7;\\ {\lambda }_2\left(x\right)& =0,{\lambda }_2\left(y\right)=0.2,{\lambda }_2\left(x\right)=0.2.\end{array}$$

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

$${\tau }_1\left(\lambda \right)=\left\{\begin{array}{c}1\mathit{if}\lambda =\underset{-}{1},\underset{-}{0};\\ \frac{1}{3}\mathit{if}\lambda ={\lambda }_1;\\ 0\text{otherwise}.\end{array}\right.,{\tau }_1^{\ast }\left(\lambda \right)=\left\{\begin{array}{c}0\mathit{if}\lambda =\underset{-}{1},\underset{-}{0};\\ \frac{1}{3}\mathit{if}\lambda ={\lambda }_1;\\ 1\text{otherwise}.\end{array}\right.$$

$${\tau }_2\left(\lambda \right)=\left\{\begin{array}{c}1\mathit{if}\lambda =\underset{-}{1},\underset{-}{0};\\ \frac{1}{3}\mathit{if}\lambda ={\lambda }_2;\\ 0\text{otherwise}.\end{array}\right.,{\tau }_2^{\ast }\left(\lambda \right)=\left\{\begin{array}{c}0\mathit{if}\lambda =\underset{-}{1},\underset{-}{0};\\ \frac{1}{3}\mathit{if}\lambda ={\lambda }_2;\\ 1\text{otherwise}.\end{array}\right.$$

Then the function $f:(X,{\tau }_1,{\tau }_1^{\ast })\to (Y,{\tau }_2,{\tau }_2^{\ast })$ defined as f(a)=x, f(b)=y and f(c)=z is double fuzzy weakly preopen but it isn’t double fuzzy contra-preclosed.

Definition 3.10

An I-dfts (X,τ,τ^∗) is said to be (r,s)-fuzzy regular space if for each λ∈I^X; τ(λ)≥r and τ^∗(λ)≤s is a union of (r,s)-fo sets μ _i ∈I^Xsuch that C_τ,τ^∗(μ _i ,r,s)≤λ for each i∈J.

Theorem 3.11

Let (X,τ,τ^∗) be (r,s)-regular fuzzy topological space. Then, $f:(X,{\tau }_1,{\tau }_1^{\ast })\to (Y,{\tau }_2,{\tau }_2^{\ast })$ is double fuzzy weakly preopen if and only if f is double fuzzy preopen.

Proof

The sufficiency is clear. For the necessity, let λ∈I^X, r∈I₀, s∈I₁; $\lambda \ne \underset{-}{0}$, τ₁(λ)≥r and ${\tau }_1^{\ast }\left(\lambda \right)\le s$. For each x _t ≤λ, let $x_t\le {\mu }_{x_t}\le C_{{\tau }_1,{\tau }_1^{\ast }}({\mu }_{x_t},r,s)\le \lambda$. Hence we obtain that $\lambda =\vee \{{\mu }_{x_t}\mid x_t\le \lambda \}=\vee \left\{C_{{\tau }_1,{\tau }_1^{\ast }}\right({\mu }_{x_t},r,s)\mid x_t\le \lambda \}$ and,

$$\begin{array}{ll}f\left(\lambda \right)& =\vee \left\{f\right({\mu }_{x_t})\mid x_t\le \lambda \}\\ \le \vee \left\{P{I}_{{\tau }_2,{\tau }_2^{\ast }}\right(f\left(C_{{\tau }_1,{\tau }_1^{\ast }}\right({\mu }_{x_t},r,s\left)\right),r,s)\mid x_t\le \lambda \}\\ \le P{I}_{{\tau }_2,{\tau }_2^{\ast }}\left(f\right(\vee \left\{C_{{\tau }_1,{\tau }_1^{\ast }}\right({\mu }_{x_t},r,s)\mid x_t\}),r,s)\\ =P{I}_{{\tau }_2,{\tau }_2^{\ast }}\left(f\right(\lambda ),r,s).\end{array}$$

Thus f is double fuzzy preopen. □

Theorem 3.12

If $f:(X,{\tau }_1,{\tau }_1^{\ast })\to (Y,{\tau }_2,{\tau }_2^{\ast })$ is double fuzzy almost open function, then it is double fuzzy weakly preopen.

Proof

Let λ∈I^X; τ₁(λ)≥r and τ 1∗(λ)≤s. Since f is double fuzzy almost open and $I_{{\tau }_1,{\tau }_1^{\ast }}\left(C_{{\tau }_1,{\tau }_1^{\ast }}\right(\lambda ,r,s),r,s)$ is (r,s)-fro, then

$$\begin{array}{cc}{I}_{{\tau }_2,{\tau }_2^{\ast }}\left(f\right(I_{{\tau }_1,{\tau }_1^{\ast }}\left(C_{{\tau }_1,{\tau }_1^{\ast }}\right(\lambda ,r,s),r,s)),r,s)& =f\left(I_{{\tau }_1,{\tau }_1^{\ast }}\right(C_{{\tau }_1,{\tau }_1^{\ast }}\\ \times (\lambda ,r,s),r,s\left)\right)\end{array}$$

and hence

$$\begin{array}{ll}f\left(\lambda \right)& \le f\left(I_{{\tau }_1,{\tau }_1^{\ast }}\right(C_{{\tau }_1,{\tau }_1^{\ast }}(\lambda ,r,s),r,s)\\ \le I_{{\tau }_2,{\tau }_2^{\ast }}\left(f\right(C_{{\tau }_1,{\tau }_1^{\ast }}(\lambda ,r,s)),r,s)\\ \le P{I}_{{\tau }_2,{\tau }_2^{\ast }}\left(f\right(C_{{\tau }_1,{\tau }_1^{\ast }}(\lambda ,r,s)),r,s).\end{array}$$

This shows that f is double fuzzy weakly preopen. □

Definition 3.11

Let (X,τ,τ^∗) be an I-dfts, r∈I₀and s∈I₁. The two fuzzy sets λ, μ∈I^Xare said to be (r,s)-fuzzy separated iff $\lambda \bar{q}{C}_{\tau ,{\tau }^{\ast }}(\mu ,r,s)$ and $\mu \bar{q}{C}_{\tau ,{\tau }^{\ast }}(\lambda ,r,s)$. A fuzzy set which cannot be expressed as a union of two (r,s)-fuzzy separated sets is said to be (r,s)-fuzzy connected.

Definition 3.12

Let (X,τ,τ^∗) an I-dfts. The fuzzy sets λ, μ∈I^Xsuch that $\lambda \ne \underset{-}{0}$, $\mu \ne \underset{-}{0}$, are said to be fuzzy (r,s)-pre-separated if $\lambda \bar{q}P{C}_{\tau ,{\tau }^{\ast }}(\mu ,r,s)$ and $\mu \bar{q}P{C}_{\tau ,{\tau }^{\ast }}(\lambda ,r,s)$ or equivalently if there exist two (r,s)-fpo sets ν, γ such that λ≤ν, μ≤γ, $\lambda \bar{q}\gamma$ and $\mu \bar{q}\nu$. An I-dfts which can not be expressed as a union of two fuzzy (r,s)-pre-separated sets is said to be fuzzy (r,s)-pre-connected space.

Theorem 3.13

If $f:(X,{\tau }_1,{\tau }_1^{\ast })\to (Y,{\tau }_2,{\tau }_2^{\ast })$ is an injective double fuzzy weakly preopen and strongly double fuzzy continuous function from the space $(X,{\tau }_1,{\tau }_1^{\ast })$ onto an (r,s)-fuzzy pre-connected space $(Y,{\tau }_2,{\tau }_2^{\ast })$, then $(X,{\tau }_1,{\tau }_1^{\ast })$ is (r,s)-fuzzy connected.

Proof

Let $(X,{\tau }_1,{\tau }_1^{\ast })$ be not (r,s)-fuzzy connected. Then there exist (r,s)-fuzzy separated sets β, γ∈I^X such that $\beta \vee \gamma =\underset{-}{1}$. Since β and γ are (r,s)-fuzzy separated, there exists λ, μ∈I^X; τ₁(λ)≥r, τ₁(μ)≥r and ${\tau }_1^{\ast }\left(\lambda \right)\le s$, ${\tau }_1^{\ast }\left(\mu \right)\le s$ such that β≤λ, γ≤μ, $\beta \bar{q}\mu$ and $\gamma \bar{q}\lambda$. Hence we have f(β)≤f(λ), f(γ)≤f(μ), $f\left(\beta \right)\bar{q}f\left(\mu \right)$ and $f\left(\gamma \right)\bar{q}f\left(\lambda \right)$. Since f is double fuzzy weakly preopen and double fuzzy strongly continuous function, from Theorem 3.10 we have f(λ) and f(μ) are (r,s)-fpo sets. Therefore, f(β) and f(γ) are (r,s)-fuzzy pre-separated and

$$\underset{-}{1}=f\left(\underset{-}{1}\right)=f(\beta \vee \gamma )=f\left(\beta \right)\vee f\left(\gamma \right)$$

which is contradiction with $(Y,{\tau }_2,{\tau }_2^{\ast })$ is (r,s)-fuzzy pre-connected. Thus $(X,{\tau }_1,{\tau }_1^{\ast })$ is (r,s)-fuzzy connected. □

Double Fuzzy weakly preclosed functions

Definition 4.13

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

$$P{C}_{{\tau }_2,{\tau }_2^{\ast }}\left(f\right(I_{{\tau }_1,{\tau }_1^{\ast }}(\lambda ,r,s)),r,s)\le f\left(\lambda \right)$$

for each λ∈I^X, r∈I₀ and s∈I₁; ${\tau }_1(\underset{-}{1}-\lambda )\ge r$ and ${\tau }_1^{\ast }(\underset{-}{1}-\lambda )\le s$.

Remark 4.3

Clearly, every double fuzzy preclosed function is double fuzzy weakly preclosed, but the converse need not be true in general, as the next example shows.

Example 4.4

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

$$\begin{array}{ll}{\lambda }_1\left(x\right)& =0.4,{\lambda }_1\left(y\right)=0.3;\\ {\lambda }_2\left(a\right)& =0.5,{\lambda }_2\left(b\right)=0.6.\end{array}$$

Let

$${\tau }_1\left(\lambda \right)=\left\{\begin{array}{c}1\mathit{if}\lambda =\underset{-}{1},\underset{-}{0};\\ \frac{1}{2}\mathit{if}\lambda ={\lambda }_2;\\ 0\text{otherwise.}\end{array}\right.,{\tau }_1^{\ast }\left(\lambda \right)=\left\{\begin{array}{c}0\mathit{if}\lambda =\underset{-}{1},\underset{-}{0};\\ \frac{1}{2}\mathit{if}\lambda ={\lambda }_2;\\ 1\text{otherwise.}\end{array}\right.$$

$${\tau }_2\left(\lambda \right)=\left\{\begin{array}{c}1\mathit{if}\lambda =\underset{-}{1},\underset{-}{0};\\ \frac{1}{2}\mathit{if}\lambda ={\lambda }_1;\\ 0\text{otherwise.}\end{array}\right.,{\tau }_2^{\ast }\left(\lambda \right)=\left\{\begin{array}{c}0\mathit{if}\lambda =\underset{-}{1},\underset{-}{0};\\ \frac{1}{2}\mathit{if}\lambda ={\lambda }_1;\\ 1\text{otherwise.}\end{array}\right.$$

Then the function $f:(X,{\tau }_1,{\tau }_1^{\ast })\to (Y,{\tau }_2,{\tau }_2^{\ast })$ defined by f(a)=x, f(b)=y is double fuzzy weakly preclosed but is not double fuzzy preclosed.

Theorem 4.14

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

• (1) f is double fuzzy weakly preclosed;

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

• (4) $P{C}_{{\tau }_2,{\tau }_2^{\ast }}\left(f\right(I_{{\tau }_1,{\tau }_1^{\ast }}(\lambda ,r,s),r,s)\le f(\lambda )$ for each (r,s)-fpc set λ∈I ^X, r∈I ₀and s∈I ₁;

• (5) $P{C}_{{\tau }_2,{\tau }_2^{\ast }}\left(f\right(I_{{\tau }_1,{\tau }_1^{\ast }}(\lambda ,r,s),r,s)\le f(\lambda )$ for each (r,s)-fα c λ∈I ^X, r∈I ₀and s∈I ₁.

Proof

Straightforward. □

Theorem 4.15

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

• (1) f is double fuzzy weakly preclosed;

• (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 (r,s)-fro set λ∈I ^X, r∈I ₀and s∈I ₁;

• (3) For each ν∈I ^Y, μ∈I ^X, r∈I ₀and s∈I ₁; τ ₁(μ)≥r and ${\tau }_1^{\ast }\left(\mu \right)\le s$ with f ⁻¹(ν)≤μ, there exists (r,s)-fpo set γ∈I ^Ywith ν≤γ and $f^{-1}\left(\gamma \right)\le C_{{\tau }_1,{\tau }_1^{\ast }}(\mu ,r,s)$;

• (4) For each fuzzy point y _s ∈ P(Y) and each μ∈I ^X, r∈I ₀and s∈I ₁such that τ ₁(μ)≥r and ${\tau }_1^{\ast }\left(\mu \right)\le s$ with f ⁻¹(y _s )≤μ, there exists (r,s)-fpo set γ∈I ^Y; y _s ≤γ and $f^{-1}\left(\gamma \right)\le C_{{\tau }_1,{\tau }_1^{\ast }}(\mu ,r,s)$;

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

• (6) $P{C}_{{\tau }_2,{\tau }_2^{\ast }}\left(f\right(I_{{\tau }_1,{\tau }_1^{\ast }}\left(D_{{\tau }_1,{\tau }_1^{\ast }}\right(\lambda ,r,s),r,s)),r,s)\le f\left(D_{{\tau }_1,{\tau }_1^{\ast }}\right(\lambda ,r,s\left)\right)$ for each λ∈I ^X, r∈I ₀and s∈I ₁;

• (7) $P{C}_{{\tau }_2,{\tau }_2^{\ast }}\left(f\right(\lambda ),r,s)\le f\left(C_{{\tau }_1,{\tau }_1^{\ast }}\right(\lambda ,r,s\left)\right)$ for each (r,s)-fpo set λ∈I ^X, r∈I ₀and s∈I ₁.

Proof

We will prove (2)⇒(3) and (1)⇒(6).

(2)⇒(3): Let ν∈I^Y, r∈I₀, s∈I₁ and let μ∈I^X; τ₁(μ)≥r and ${\tau }_1^{\ast }\left(\mu \right)\le s$ with f⁻¹(ν)≤μ. Then $f^{-1}\left(\nu \right)\bar{q}{C}_{{\tau }_1,{\tau }_1^{\ast }}(\underset{-}{1}-C_{{\tau }_1,{\tau }_1^{\ast }}(\mu ,r,s),r,s)$ and consequently, $\nu \bar{q}f\left(C_{{\tau }_1,{\tau }_1^{\ast }}\right(\underset{-}{1}-C_{{\tau }_1,{\tau }_1^{\ast }}(\mu ,r,s),r,s)$. Since $\underset{-}{1}-C_{{\tau }_1,{\tau }_1^{\ast }}(\mu ,r,s)$ is (r,s)-fro, $\nu \bar{q}P{C}_{{\tau }_2,{\tau }_2^{\ast }}\left(f\right(\underset{-}{1}-C_{{\tau }_1,{\tau }_1^{\ast }}(\mu ,r,s)),r,s)$ by (2). Let $\gamma =\underset{-}{1}-P{C}_{{\tau }_2,{\tau }_2^{\ast }}\left(f\right(\underset{-}{1}-C_{{\tau }_1,{\tau }_1^{\ast }}(\mu ,r,s)),r,s)$. Then γ is (r,s)-fpo with ν≤γ and

$$\begin{array}{ll}{f}^{-1}\left(\gamma \right)& \le \underset{-}{1}-f^{-1}\left(P{C}_{{\tau }_2,{\tau }_2^{\ast }}\right(\underset{-}{1}-C_{{\tau }_1,{\tau }_1^{\ast }}(\mu ,r,s),r,s\left)\right)\\ \le \underset{-}{1}-f^{-1}f(\underset{-}{1}-C_{{\tau }_1,{\tau }_1^{\ast }}(\mu ,r,s\left)\right)\\ \le C_{{\tau }_1,{\tau }_1^{\ast }}(\mu ,r,s).\end{array}$$

(1)⇒(6): Let ν∈I^Y, r∈I₀ and s∈I₁; ${\tau }_2(\underset{-}{1}-\nu )\ge r$, ${\tau }_2^{\ast }(\underset{-}{1}-\nu )\le s$ and $y_s\le \underset{-}{1}-f\left(\nu \right)$. Since $f^{-1}\left(y_s\right)\le \underset{-}{1}-\nu$, there exists (r,s)-fpo γ∈I^Y with y _s ≤γ and $f^{-1}\left(\gamma \right)\le C_{{\tau }_1,{\tau }_1^{\ast }}(\underset{-}{1}-\nu ,r,s)=\underset{-}{1}-I_{{\tau }_1,{\tau }_1^{\ast }}(\nu ,r,s)$ by (6). Therefore $\gamma \bar{q}f\left(I_{{\tau }_1,{\tau }_1^{\ast }}\right(\nu ,r\left)\right)$, so that $y_s\le \underset{-}{1}-P{C}_{{\tau }_2,{\tau }_2^{\ast }}\left(f\right(I_{{\tau }_1,{\tau }_1^{\ast }}(\nu ,r,s)),r,s)$. □

Theorem 4.16

If $f:(X,{\tau }_1,{\tau }_1^{\ast })\to (Y,{\tau }_2,{\tau }_2^{\ast })$ is double fuzzy weakly preclosed, then for each y _s ∈ P(Y) and each $\mu \in {\mathbf{Q}}_{{\tau }_1,{\tau }_1^{\ast }}\left(f^{-1}\right(y_s),r,s)$, there exists (r,s)-fpo set γ∈I^Y; $\gamma \in {\mathbf{Q}}_{{\tau }_2,{\tau }_2^{\ast }}(y_s,r,s)$, such that $f^{-1}\left(\gamma \right)\le C_{{\tau }_1,{\tau }_1^{\ast }}(\mu ,r,s)$.

Proof

Let $\mu \in {\mathbf{Q}}_{{\tau }_1,{\tau }_1^{\ast }}\left(f^{-1}\right(y_s,r,s)$. Then μ(x) + s>1 and hence there exists t∈(0,1) such that μ(x)>t>1−s. Then $\mu \in {\mathbf{Q}}_{{\tau }_1,{\tau }_1^{\ast }}\left(f^{-1}\right(y_t),r,s)$. By Theorem 3.7-6 there exists (r,s)-fpo set γ∈I^Y; y _t ≤γ such that $f^{-1}\left(\gamma \right)\le C_{{\tau }_1,{\tau }_1^{\ast }}(\mu ,r,s)$. Now, γ(y)>t and hence γ(y)>1−s. Thus γ is (r,s)-fpo neighborhood of y _s . □

Definition 4.14

Let (X,τ,τ^∗) be an I-dfts. A fuzzy set λ∈I^Xis called (r,s)-fuzzy pre-Q-neighborhood of x _t if there exists (r,s)-fpo set μ∈I^Xsuch that x _t qμ≤λ. We denote the set of all (r,s)-fuzzy pre-Q-neighborhood of x _t by PQ_τ,τ^∗(x _t ,r,s).

Theorem 4.17

In an I-dfts (X,τ,τ^∗). A fuzzy point x _t ∈P C_τ,τ^∗(λ,r) if and only if for every μ ∈ PQ_τ,τ^∗(x _t ,r,s), μqλ is hold.

Proof

Straightforward. □

Theorem 4.18

If $f:(X,{\tau }_1,{\tau }_1^{\ast })\to (Y,{\tau }_2,{\tau }_2^{\ast })$ is double fuzzy weakly preclosed and if for each ν∈I^X, r∈I₀and s∈I₁; ${\tau }_1(\underset{-}{1}-\nu )\ge r$, ${\tau }_1^{\ast }(\underset{-}{1}-\nu )\le s$ and each $f^{-1}\left(y_s\right)\le \underset{-}{1}-\nu$ there exists $\mu \in {\mathbf{Q}}_{{\tau }_1,{\tau }_1^{\ast }}\left(f^{-1}\right(y_s),r,s)$ such that $f^{-1}\left(y_s\right)\le \mu \le C_{{\tau }_1,{\tau }_1^{\ast }}(\mu ,r,s)\le \underset{-}{1}-\nu$. Then f is double fuzzy preclosed.

Proof

Let ν∈I^X, r∈I₀and s∈I₁; ${\tau }_1(\underset{-}{1}-\nu )\ge r$, ${\tau }_1^{\ast }(\underset{-}{1}-\nu )\le s$ and let $y_s\le \underset{-}{1}-f\left(\nu \right)$. Then $f^{-1}\left(y_s\right)\le \underset{-}{1}-\nu$, and hence there exists $\mu \in {\mathbf{Q}}_{{\tau }_1,{\tau }_1^{\ast }}\left(f^{-1}\right(y_s),r,s)$ such that $f^{-1}\left(y_s\right)\le \mu \le C_{{\tau }_1,{\tau }_1^{\ast }}(\mu ,r,s)\le \underset{-}{1}-\nu$. Since f is double fuzzy weakly preclosed by using Theorem 3.12, there exists (r,s)-fuzzy pre-Q-neighborhood γ∈I^Ywith y _s ≤γ and $f^{-1}\left(\gamma \right)\le C_{{\tau }_1,{\tau }_1^{\ast }}(\mu ,r,s)$. Therefore, we obtain $f^{-1}\left(\gamma \right)\bar{q}\nu$ and hence $\gamma \bar{q}f\left(\nu \right)$, this shows that $y_s\notin P{C}_{{\tau }_2,{\tau }_2^{\ast }}\left(f\right(\nu ),r,s)$. Therefore, f(ν) is (r,s)-fpc and f is double fuzzy preclosed function. □

Definition 4.15

A function $f:(X,{\tau }_1,{\tau }_1^{\ast })\to (Y,{\tau }_2,{\tau }_2^{\ast })$ is said to be double fuzzy contra-open (resp. double fuzzy contra-closed) if ${\tau }_2(\underset{-}{1}-f(\lambda \left)\right)\ge r$ and ${\tau }_2^{\ast }(\underset{-}{1}-f(\lambda \left)\right)\le s$(resp. τ₂(f(λ))≥r and ${\tau }_2^{\ast }\left(f\right(\lambda \left)\right)\le s$) for each λ∈I^X, r∈I₀and s∈I₁; τ₁(λ)≥r and ${\tau }_1^{\ast }\left(\lambda \right)\le s$ (resp. ${\tau }_1(\underset{-}{1}-\lambda )\ge r$ and ${\tau }_1^{\ast }(\underset{-}{1}-\lambda )\le s$).

Theorem 4.19

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

Proof

Let λ∈I^X, r∈I₀ and s∈I₁ such that ${\tau }_1(\underset{-}{1}-\lambda )\ge r$ and ${\tau }_1^{\ast }(\underset{-}{1}-\lambda )\le s$. Then,

$$\begin{array}{l}P{C}_{{\tau }_2,{\tau }_2^{\ast }}\left(f\right(I_{{\tau }_1,{\tau }_1^{\ast }}(\lambda ,r,s)),r,s)\le f\left(I_{{\tau }_1,{\tau }_1^{\ast }}\right(\lambda ,r,s\left)\right)\\ \le f\left(\lambda \right).\end{array}$$

□

Theorem 4.20

If $f:(X,{\tau }_1,{\tau }_1^{\ast })\to (Y,{\tau }_2,{\tau }_2^{\ast })$ is double fuzzy weakly preclosed, then for every ν∈I^Yand every λ∈I^X, r∈I₀, s∈I₁such that τ₁(λ)≥r and ${\tau }_1^{\ast }\left(\lambda \right)\le s$ with f⁻¹(ν)≤λ, there exists (r,s)-fpc set γ∈I^Ysuch that ν≤γ and $f^{-1}\left(\gamma \right)\le C_{{\tau }_1,{\tau }_1^{\ast }}(\lambda ,r,s)$.

Proof

Let ν∈I^Yand let λ∈I^X, r∈I₀and s∈I₁such that τ₁(λ)≥r and ${\tau }_1^{\ast }\left(\lambda \right)\le s$ with f⁻¹(ν)≤λ. Put $\gamma =P{C}_{{\tau }_2,{\tau }_2^{\ast }}\left(f\right(I_{{\tau }_1,{\tau }_1^{\ast }}\left(C_{{\tau }_1,{\tau }_1^{\ast }}\right(\lambda ,r,s),r,s)),r,s)$, then γ is (r,s)-fpc set in I^Ysuch that ν≤γ since $\nu \le f\left(\lambda \right)\le f\left(I_{{\tau }_1,{\tau }_1^{\ast }}\right(C_{{\tau }_1,{\tau }_1^{\ast }}(\lambda ,r,s),r,s\left)\right)\le P{C}_{{\tau }_2,{\tau }_2^{\ast }}\left(f\right(I_{{\tau }_1,{\tau }_1^{\ast }}\left(C_{{\tau }_1,{\tau }_1^{\ast }}\right(\lambda ,r,s),r,s)),r,s)=\gamma$. And since f is double fuzzy weakly preclosed, $f^{-1}\left(\gamma \right)\le C_{{\tau }_1,{\tau }_1^{\ast }}(\lambda ,r,s)$. □

Corollary 4.21

If $f:(X,{\tau }_1,{\tau }_1^{\ast })\to (Y,{\tau }_2,{\tau }_2^{\ast })$ is double fuzzy weakly preclosed, then for every y _s ∈ P(Y) and every λ∈I^X, r∈I₀and s∈I₁such that τ₁(λ)≥r and ${\tau }_1^{\ast }\left(\lambda \right)\le s$ with f⁻¹(y _s )≤λ, there exists (r,s)-fpc set γ∈I^Y; y _s ≤γ such that $f^{-1}\left(\gamma \right)\le C_{{\tau }_1,{\tau }_1^{\ast }}(\lambda ,r,s)$.

Definition 4.16

A fuzzy set λ∈I^Xis called (r,s)-fuzzy θ-compact if for each family {μ _i ∣i∈J} in $\{\mu \in I^X\mid \mu \in {\mathbf{Q}}_{\tau ,{\tau }^{\ast }}(\lambda ,r,s\left)\right\}$ satisfy $\left(\underset{i\in J}{\vee }{\mu }_i\right)\left(x\right)\ge \lambda \left(x\right)$ for each x∈X, there exist a finite subset J₀of J such that $\lambda \le I_{\tau ,{\tau }^{\ast }}(\vee \{C_{\tau ,{\tau }^{\ast }}({\mu }_i,r,s)\mid i\in J_0\},r,s)$.

Theorem 4.22

If $f:(X,{\tau }_1,{\tau }_1^{\ast })\to (Y,{\tau }_2,{\tau }_2^{\ast })$ is double fuzzy weakly preclosed with all fibers (r,s)-fuzzy θ-closed, then f(λ) is (r,s)-fpc for each (r,s)-fuzzy θ-compact λ∈I^X, r∈I₀and s∈I₁.

Proof

Let λ be (r,s)-fuzzy θ-compact and let $y_s\le \underset{-}{1}-f\left(\lambda \right)$. Then $f^{-1}\left(y_s\right)\bar{q}\lambda$ and for each x _t ≤λ there is ${\mu }_{x_t}\in {\mathbf{Q}}_{{\tau }_1,{\tau }_1^{\ast }}(x_t,r,s)$ with $x_t\le {\mu }_{x_t}$ and $C_{{\tau }_1,{\tau }_1^{\ast }}({\mu }_{x_t},r,s)\bar{q}{f}^{-1}\left(y_s\right)$. Clearly $\{{\mu }_{x_t}\mid x_t\le \lambda ,{\mu }_{x_t}\in {\mathbf{Q}}_{{\tau }_1,{\tau }_1^{\ast }}(\lambda ,r,s\left)\right\}$ satisfy $\left(\underset{i\in J}{\vee }{\mu }_i\right)\left(x\right)\ge \lambda \left(x\right)$ for each x∈X and since λ is (r,s)-fuzzy θ-compact, there is $\{{\mu }_{x_1},{\mu }_{x_2},{\mu }_{x_3},.,{\mu }_{x_n}\}\subseteq \{{\mu }_{x_t}\mid x_t\le \lambda ,{\mu }_{x_t}\in {\mathbf{Q}}_{{\tau }_1,{\tau }_1^{\ast }}(\lambda ,r,s\left)\right\}$ such that $\lambda \le I_{{\tau }_1,{\tau }_1^{\ast }}(\xi ,r,s)$, where $\xi =\vee \left\{C_{{\tau }_1,{\tau }_1^{\ast }}\right({\mu }_{x_i},r,s)\mid i=1,2,\dots ,n\}$. Since f is double fuzzy weakly preclosed, by using Theorem 3.12 there exists $\gamma \in {\mathbf{PQ}}_{{\tau }_1,{\tau }_1^{\ast }}(y_s,r,s)$ with

$$\begin{array}{l}{f}^{-1}\left(y_s\right)\le f^{-1}\left(\gamma \right)\le C_{{\tau }_1,{\tau }_1^{\ast }}(\underset{-}{1}-\xi ,r,s)=\underset{-}{1}-I_{{\tau }_1,{\tau }_1^{\ast }}(\xi ,r,s)\\ \le \underset{-}{1}-\mathrm{\lambda .}\end{array}$$

Therefore y _s ≤γ and $\gamma \bar{q}f\left(\lambda \right)$. Thus $y_s\le \underset{-}{1}-P{C}_{{\tau }_2,{\tau }_2^{\ast }}\left(f\right(\lambda ),r,s)$. Thus f(λ) is (r,s)-fpc set. □

Definition 4.17

Let (X,τ,τ^∗) be an I-dfts. The fuzzy sets λ, μ∈I^Xare (r,s)-fuzzy strongly separated if there exist ν, γ∈I^Xsuch that τ(ν)≥r and τ^∗(ν)≤s, τ(γ)≥r with λ≤ν, μ≤γ and $C_{\tau ,{\tau }^{\ast }}(\nu ,r,s)\bar{q}{C}_{\tau ,{\tau }^{\ast }}(\gamma ,r,s)$.

Definition 4.18

An I-dfts (X,τ,τ^∗) is called (r,s)-fuzzy pre T₂if for each $x_{t_1}$, $x_{t_2}$ with different supports there exists (r,s)-fpo sets λ, μ∈I^Xsuch that $x_{t_1}\le \lambda \le x_{1-t_2}$, $x_{t_2}\le \mu \le x_{1-t_1}$ and $\lambda \bar{q}\mu$.

Theorem 4.23

If $f:(X,{\tau }_1,{\tau }_1^{\ast })\to (Y,{\tau }_2,{\tau }_2^{\ast })$ is double fuzzy weakly preclosed surjection and all fibers are (r,s)-fuzzy strongly separated, then $(Y,{\tau }_1,{\tau }_1^{\ast })$ is (r,s)-fuzzy pre-T₂.

Proof

Let $y_{s_1},y_{s_2}\in \mathbf{P}\left(Y\right)$ and let γ,ν∈I^X, r∈I₀and s∈I₁; τ₁(γ)≥r, ${\tau }_1^{\ast }\left(\gamma \right)\le s$, τ₁(ν)≥r and τ₁(ν)≤s such that $f^{-1}\left(y_{s_1}\right)\le \gamma$ and $f^{-1}\left(y_{s_2}\right)\le \nu$ respectively with $C_{{\tau }_1,{\tau }_1^{\ast }}(\gamma ,r,s)\bar{q}{C}_{{\tau }_1,{\tau }_1^{\ast }}(\nu ,r,s)$. By using Theorem 3.12-4 there are (r,s)-fpo sets λ, μ∈I^Ysuch that $y_{s_1}\le \lambda$ and $y_{s_2}\le \mu$, $f^{-1}\left(\lambda \right)\le C_{{\tau }_1,{\tau }_1^{\ast }}(\gamma ,r,s)$ and $f^{-1}\left(\mu \right)\le C_{{\tau }_1,{\tau }_1^{\ast }}(\nu ,r,s)$. Therefore $\lambda \bar{q}\mu$, because $C_{{\tau }_1,{\tau }_1^{\ast }}(\gamma ,r,s)\bar{q}{C}_{{\tau }_2,{\tau }_2^{\ast }}(\nu ,r,s)$ and f is surjective. Thus $(Y,{\tau }_2,{\tau }_2^{\ast })$ is (r,s)-fuzzy pre-T₂. □

Definition 4.19

an I-dfts (X,τ,τ^∗) is said to be (r,s)-extremally disconnected if τ(C_τ,τ^∗(λ,r,s))≥r and τ^∗(C_τ,τ^∗(λ,r,s))≤s for each λ∈I^X; τ(λ)≥r and τ^∗(λ)≤s.

Definition 4.20

an I-dfts (X,τ,τ^∗) is said to be (r,s)-fuzzy almost compact if for each (r,s)-fuzzy open cover {λ _i ∣i∈J} of X, there is a finite subset J₀of J such that $\vee \left\{C_{\tau ,{\tau }^{\ast }}\right({\lambda }_i,r,s)\mid i\in J_0\}=\underset{-}{1}$.

Definition 4.21

A fuzzy set λ in an I-dfts (X,τ,τ^∗) is said to be (r,s)-fuzzy p-compact iff for each family of (r,s)-fpo sets {μ _i ∣i∈J} satisfies $\left(\underset{i\in J}{\vee }{\mu }_i\right)\left(x\right)=\lambda \left(x\right)$ for each x∈X. There exists finite subfamily J₀of J such that $\left(\underset{i\in J_0}{\vee }P{C}_{\tau ,{\tau }^{\ast }}\right({\mu }_i,r,s\left)\right)\left(x\right)\ge \lambda \left(x\right)$ for each x∈X.

Theorem 4.24

Let $(X,{\tau }_1,{\tau }_1^{\ast })$ be (r,s)-extremally disconnected I-dfts. Let $f:(X,{\tau }_1,{\tau }_1^{\ast })\to (Y,{\tau }_2,{\tau }_2^{\ast })$ be double fuzzy open and double fuzzy preclosed injective function such that f⁻¹(y _s ) is (r,s)-fuzzy almost compact for each y _s ∈ P(Y). If λ∈I^Yis (r,s)-fuzzy P-compact. Then f⁻¹(λ) is (r,s)-fuzzy almost compact.

Proof

Let {ν _j ∣æ∈J} be (r,s)-fuzzy open cover of f⁻¹(λ). Then for each y _s ≤λ∧f(X), $f^{-1}\left(y_s\right)\le \vee \left\{C_{{\tau }_1,{\tau }_1^{\ast }}\right({\nu }_j,r,s)\mid \text{æ}\in J(y_s\left)\right\}={\gamma }_{y_s}$, for some finite subfamily J(y _s ) of J. Since $(X,{\tau }_1,{\tau }_1^{\ast })$ is (r,s)-extremally disconnected each ${\tau }_1\left(C_{{\tau }_1,{\tau }_1^{\ast }}\right({\nu }_j,r,s\left)\right)\ge r$ and ${\tau }_1^{\ast }\left(C_{{\tau }_1,{\tau }_1^{\ast }}\right({\nu }_j,r,s\left)\right)\le s$, hence ${\tau }_1\left({\gamma }_{y_s}\right)\ge r$ and ${\tau }_1^{\ast }\left({\gamma }_{y_s}\right)\le s$. So by Corollary 4.21 there exists (r,s)-fpc set ${\mu }_{y_s};y_s\le {\mu }_{y_s}$ such that $f^{-1}\left({\mu }_{y_s}\right)\le C_{{\tau }_1,{\tau }_1^{\ast }}({\gamma }_{y_s},r,s)$. Then, $\{{\mu }_{y_s}\mid y_s\le \lambda \wedge f(X\left)\right\}\vee \{\underset{-}{1}-f(X\left)\right\}$ is (r,s)-fuzzy preclosed cover of λ, $\lambda \le \vee \left\{C_{{\tau }_2,{\tau }_1}\right({\mu }_{y_s},r,s)\mid y_s\le \lambda \wedge f(X\left)\right\}\vee \left\{C_{{\tau }_2,{\tau }_2^{\ast }}\right(\underset{-}{1}-f\left(X\right),r,s\left)\right\}$ for some finite fuzzy subset K of λ∧f(X). Hence,

$$\begin{array}{ll}{f}^{-1}\left(\lambda \right)& \le \underset{y_s\in K}{\vee }{f}^{-1}\left(C_{{\tau }_2,{\tau }_2^{\ast }}\right({\mu }_{y_s},r,s\left)\right)\\ \vee \left\{f^{-1}\right(C_{{\tau }_2,{\tau }_2^{\ast }}(\underset{-}{1}-f(X),r,s)\left)\right\}\\ \le \underset{y_s\in K}{\vee }{C}_{{\tau }_1,{\tau }_1^{\ast }}\left(f^{-1}\right({\mu }_{y_s}),r,s)\\ \vee \left\{C_{{\tau }_1,{\tau }_1^{\ast }}\right(f^{-1}(\underset{-}{1}-f(X\left)\right),r,s\left)\right\}\\ \le \underset{y_s\in K}{\vee }{C}_{{\tau }_1,{\tau }_1^{\ast }}\left(f^{-1}\right({\mu }_{y_s}),r,s)\end{array}$$

so $f^{-1}\left(\lambda \right)\le \underset{\text{æ}\in J\left(y_s\right),y_s\in K}{\vee }{C}_{{\tau }_1,{\tau }_1^{\ast }}({\nu }_{\text{æ}},r,s)$. Therefore f⁻¹ (λ) is (r,s)-fuzzy almost compact. □