Open Access

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

SpringerPlus20121:19

DOI: 10.1186/2193-1801-1-19

Received: 23 May 2012

Accepted: 20 July 2012

Published: 25 September 2012

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 0 and 1 , we denote the smallest and the greatest fuzzy subsets on X. For a fuzzy subset λI X , 1 λ denotes its complement. Given a function f ~ : X Y , f(λ) and f−1(λ) define the direct image and the inverse image of f, defined by f ( λ ) ( y ) = f ( x ) = y λ ( x ) and f−1(ν)(x)=ν(f(x)), for each λI X , νI Y , and xX, 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 xX such that λ(x) + μ(x)>1. Negation of such a statement is denoted as λ q ̄ μ . 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)

    τ ( λ ) 1 τ ( λ ) for each λI X .

     
  2. (O2)

    τ(λ1λ2)≥τ(λ1)τ(λ2) and τ(λ1λ2)≤τ(λ1)τ(λ2) for each λ1, λ2I X .

     
  3. (O3)

    τ ( i Γ λ i ) i Γ τ ( λ i ) and τ ( i Γ λ i ) i Γ τ ( λ i ) for each λ i I X , iΓ.

     

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 1 λ is an (r,s)-fo set. Let ( X , τ 1 , τ 1 ) and ( Y , τ 2 , τ 2 ) be two I-dfts’s. A function f ~ : X Y is said to be a double fuzzy continuous iff τ1(f−1(ν))≥τ2(ν) and τ 1 ( f 1 ( ν ) ) τ 2 ( ν ) for each νI 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 , T r , s ) , where
T ( r , s ) = { λ I X | τ ( λ ) r , τ ( λ ) 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 X instead 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 rI0, sI1 and λI X , we define an operator Cτ,τ:I X ×I0×I1I X as follows:
C τ , τ ( λ , r , s ) = { μ I X λ μ , τ ( 1 μ ) r , τ ( 1 μ ) s } .
For λ, μI X , r1r2I0and s1s2I1, the operator Cτ,τsatisfies the following statements:
  1. (C1)

    C τ , τ ( 0 , r , s ) = 0 ,

     
  2. (C2)

    λCτ,τ(λ,r,s),

     
  3. (C3)

    Cτ,τ(λ,r,s)Cτ,τ(μ,r,s)=Cτ,τ(λμ,r,s),

     
  4. (C4)

    Cτ,τ(λ,r1,s1)≤Cτ,τ(λ,r2,s2) if r1r2and s1s2,

     
  5. (C5)

    Cτ,τ(Cτ,τ(λ,r,s),r,s)=Cτ,τ(λ,r,s).

     

Theorem 2.2

[(Çoker and Demirci1996; Lee and Im2001)] Let (X τ τ) be an I-dfts. Then for each rI0, sI1 and λI X , we define an operator Iτ,τ:I X ×I0×I1I X as follows:
I τ , τ ( λ , r , s ) = { μ I X μ λ , τ ( μ ) r , τ ( μ ) s } .
For λ μI X , r r1r2I0and s s1s2I1, the operator Iτ,τsatisfies the following statements:
  1. (I1)

    I τ , τ ( 1 λ , r , s ) = 1 C τ , τ ( λ , r , s ) ,

     
  2. (I2)

    I τ , τ ( 1 , r , s ) = 1 ,

     
  3. (I3)

    Iτ,τ(λ,r,s)≤λ,

     
  4. (I4)

    Iτ,τ(λ,r,s)Iτ,τ(μ,r,s)=Iτ,τ(λμ,r,s),

     
  5. (I5)

    Iτ,τ(λ,r1,s1)≥Iτ,τ(λ,r2,s2) if r1r2and s1s2,

     
  6. (I6)

    Iτ,τ(Iτ,τ(λ,r,s),r,s)=Iτ,τ(λ,r,s),

     
  7. (I7)

    If Iτ,τ(Cτ,τ(λ,r,s),r,s)=λ, then C τ , τ ( I τ , τ ( 1 λ , r , s ) , r , s ) = 1 λ .

     

Definition 2.2

Let (X,τ,τ) be an I-dfts. For λI X , rI0and sI1.
  1. (1)
    λ is called (r,s)-fuzzy preopen ((r,s)-fpo, for short) if λI 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 1 λ 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 τ , τ ( λ , r , s ) = { ν I X ν λ , ν is ( r , s ) fpo } .
    The (r,s)-fuzzy preclosure of λ, denoted by P Cτ,τ(λ,r,s) is defined by
    P C τ , τ ( λ , r , s ) = { ν I X λ ν , ν is ( r , s ) 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 1 λ is (r,s)-fro set.

     
  3. (3)

    λ is called (r,s)-fuzzy α-open ((r,s)-fα o, for short) if λI 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 1 λ is (r,s)-fα o set.

     

Theorem 2.3

Let (X,τ,τ) be an I-dfts. For λI X , rI0and sI1.
  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)≤λP C τ,τ (λ,r,s)≤C τ,τ (λ,r,s),

     
  3. (3)

    1 P I τ , τ ( λ , r , s ) = P C τ , τ ( 1 λ , r , s ) and P I τ , τ ( 1 λ , r , s ) = 1 P C τ , τ ( λ , r , s ) .

     

Definition 2.3

Let f : ( X , τ 1 , τ 1 ) ( Y , τ 2 , τ 2 ) be a function from an I-dfts ( X , τ 1 , τ 1 ) into an I-dfts ( Y , τ 2 , τ 2 ) . The function f is called:
  1. (1)

    double fuzzy preclosed if f(λ) is (r,s)-fpc set in I Y for each λI X , rI 0and sI 1; τ 1 ( 1 λ ) r , τ 1 ( 1 λ ) s ,

     
  2. (2)

    double fuzzy open if τ 2(f(λ))≥τ 1(λ) and τ 2 ( f ( λ ) ) τ 1 ( λ ) for each λI X , rI 0and sI 1,

     
  3. (3)

    double fuzzy almost open if τ 2(f(λ))≥r and τ 2 ( f ( λ ) ) s for each (r,s)-fro set λI X , rI 0and sI 1.

     

Definition 2.4

Let f : ( X , τ 1 , τ 1 ) ( Y , τ 2 , τ 2 ) be a function from an I-dfts ( X , τ 1 , τ 1 ) into an I-dfts ( Y , τ 2 , τ 2 ) . The function f is called:
  1. (1)

    double fuzzy weakly open if f ( λ ) I τ 2 , τ 2 ( f ( C τ 1 , τ 1 ( λ , r , s ) ) , r , s ) for each λI X , rI 0and sI 1; τ 1(λ)≥r and τ 1 ( λ ) s ,

     
  2. (2)

    double fuzzy α-open if f(λ) is (r,s)-fα o in I Y for each λI X , rI 0and sI 1; τ 1(λ)≥r and τ 1 ( λ ) s .

     

Definition 2.5

Let (X,τ,τ) be an I-dfts, μI X , x t P(X), rI0and sI1where 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 . We denote the set of all (r,s)-fuzzy open Q-neighborhood of x t by Qτ,τ(x t ,r,s).

Definition 2.6

Let (X,τ,τ) be an I-dfts, λI X , x t P(X), rI0and sI1. x t is called (r,s)-fuzzy θ-cluster point of λ if for every μQτ,τ(x t ,r,s), we have Cτ,τ(μ,r,s). We denote D τ , τ ( λ , r , s ) = { x t P ( X ) x t is ( r , s ) -fuzzy θ -cluster point of λ } . Where Dτ,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, μI X and r, sI0, we have the following:
  1. (1)

    D τ , τ ( λ , r , s ) = { μ I X λ I τ , τ ( μ , r , s ) , τ ( 1 μ ) r , τ ( 1 μ ) 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 Tτ,τ(λ,r,s) is defined by T τ , τ ( λ , r , s ) = { ν I X C τ , τ ( ν , r , s ) λ , τ ( ν ) r , τ ( ν ) 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 λI X , rI 0and sI 1,

     
  2. (2)

    T τ,τ (λ,r,s)=I τ,τ (λ,r,s) for each λI X , rI 0and sI 1; τ(λ)≥r and τ (λ)≤s.

     

Double Fuzzy weakly preopen functions

Definition 3.7

A function f : ( X , τ 1 , τ 1 ) ( Y , τ 2 , τ 2 ) is said to be double fuzzy weakly preopen if
f ( λ ) P I τ 2 , τ 2 ( f ( C τ 1 , τ 1 ( λ , r , s ) ) , r , s )

for each λI X , rI0 and sI1; τ1(λ)≥r and τ 1 ( λ ) 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:
λ 1 ( a ) = 0 . 5 , λ 1 ( b ) = 0 . 3 , λ 1 ( c ) = 0 . 2 , λ 2 ( x ) = 0 . 9 , λ 2 ( y ) = 1 , λ 2 ( z ) = 0 . 7 , λ 3 ( x ) = 0 . 2 , λ 3 ( y ) = 0 . 2 , λ 3 ( z ) = 0 . 3 .
Define τ1and τ2 as follows:
τ 1 λ = 1 if λ = 0 , 1 ; 1 3 if λ = λ 1 ; 0 otherwise. , τ 1 λ = 0 if λ = 0 , 1 ; 1 4 if λ = λ 1 ; 1 otherwise.
τ 2 λ = 1 if λ = 0 , 1 ; 1 3 if λ = λ 2 ; 2 3 if λ = λ 3 ; 0 otherwise. , τ 2 λ = 0 if λ = 0 , 1 ; 1 4 if λ = λ 2 ; 1 3 if λ = λ 3 ; 1 otherwise.

Then the mapping f : ( X , τ 1 , τ 1 ) ( Y , τ 2 , τ 2 ) defined by f(a)=z, f(b)=x and f(c)=y is double fuzzy weakly preopen but not double fuzzy preopen. Where τ 1 ( λ ) 1 3 , τ 1 ( λ ) 1 3 and f(λ) is not ( 1 3 , 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:
λ 1 ( a ) = 0 . 5 , λ 1 ( b ) = 0 . 3 , λ 1 ( c ) = 0 . 2 ; λ 2 ( x ) = 0 . 9 , λ 2 ( y ) = 1 , λ 2 ( z ) = 0 . 7 ; λ 3 ( x ) = 0 . 2 , λ 3 ( y ) = 0 . 9 , λ 3 ( z ) = 0 . 3 .
Let ( τ 1 , τ 1 ) and ( τ 2 , τ 2 ) defined as follows:
τ 1 λ = 1 if λ = 0 , 1 1 2 if λ = λ 1 ; 0 otherwise. , τ 1 λ = 0 if λ = 0 , 1 1 2 if λ = λ 1 ; 1 otherwise.
τ 2 λ = 1 if λ = 0 , 1 1 2 if λ = λ 2 ; 1 3 if λ = λ 3 ; 0 otherwise. , τ 2 λ = 0 if λ = 0 , 1 1 2 if λ = λ 2 ; 1 3 if λ = λ 3 ; 1 otherwise.

Then the mapping f : ( X , τ 1 , τ 1 ) ( Y , τ 2 , τ 2 ) 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 ( λ 1 ) ≦̸ I τ 2 , τ 2 ( f ( C τ 1 , τ 1 ( λ 1 , r , s ) ) , r , s ) .

Theorem 3.5

For a function f : ( X , τ 1 , τ 1 ) ( Y , τ 2 , τ 2 ) . The following statements are equivalent:
  1. (1)

    f is double fuzzy weakly preopen,

     
  2. (2)

    f ( T τ 1 , τ 1 ( λ , r , s ) ) P I τ 2 , τ 2 ( f ( λ ) , r , s ) for each λI X , rI 0and sI 1,

     
  3. (3)

    T τ 1 , τ 1 ( f 1 ( ν ) , r , s ) f 1 ( P I τ 2 , τ 2 ( ν , r , s ) ) for each νI Y , rI 0and sI 1,

     
  4. (4)

    f 1 ( P C τ 2 , τ 2 ( ν , r , s ) ) D τ 1 , τ 1 ( f 1 ( ν ) , r , s ) for each νI Y , rI 0and sI 1.

     

Proof

(1)(2) Let λI X and x p T τ 1 , τ 1 ( λ , r , s ) . Then there exists γ Q τ 1 , τ 1 ( x p , r , s ) such that γ C τ 1 , τ 1 ( γ , r , s ) λ . Thus f ( γ ) f ( C τ 1 , τ 1 ( γ , r , s ) ) f ( λ ) and hence
P I τ 2 , τ 2 ( f ( γ ) , r , s ) P I τ 2 , τ 2 ( f ( C τ 1 , τ 1 ( γ , r , s ) ) , r , s ) P I τ 2 , τ 2 ( f ( λ ) , r , s ) .
Since f is double fuzzy weakly preopen,
f ( γ ) P I τ 2 , τ 2 ( f ( C τ 1 , τ 1 ( γ , r , s ) ) , r , s ) P I τ 2 , τ 2 ( f ( λ ) , r , s ) .

and hence f ( x p ) P I τ 2 , τ 2 ( f ( λ ) , r , s ) . This shows that x p f 1 ( P I τ 2 ( f ( λ ) , r , s ) ) . Thus T τ 1 ( λ , r , s ) f 1 ( P I τ 2 , τ 2 (f(λ),r,s)) and so, f ( T τ 1 , τ 1 ( λ , r , s ) ) P I τ 2 , τ 2 ( f ( λ ) , r , s ) .

(2)(1) Let μI X ; τ1(μ)≥r and τ 1 ( μ ) s . Since μ T τ 1 , τ 1 ( C τ 1 , τ 1 ( μ , r , s ) , r , s ) , then
f ( μ ) f ( T τ 1 , τ 1 ( C τ 1 , τ 1 ( μ , r , s ) , r , s ) ) P I τ 2 , τ 2 ( f ( C τ 1 , τ 1 ( μ , r , s ) ) , r , s ) .

Hence f is double fuzzy weakly preopen.

(2)(3) Let νI Y . By using (2), f ( T τ 1 , τ 1 ( f 1 ( ν ) , r , s ) ) P I τ 2 , τ 2 ( ν , r , s ) . Therefore, T τ 1 , τ 1 ( f 1 ( ν ) , r , s ) f 1 ( P I τ 2 , τ 2 ( ν , r , s ) ) .

(3)(2) Trivial.

(3)(4) Let νI Y . Using (3), we have
1 D τ 1 , τ 1 ( f 1 ( ν ) , r , s ) = T τ 1 , τ 1 ( 1 f 1 ( ν ) , r , s ) = T τ 1 , τ 1 ( f 1 ( 1 ν ) , r , s ) f 1 ( P I τ 2 , τ 2 ( 1 ν , r , s ) ) = f 1 ( 1 P C τ 2 , τ 2 ( ν , r , s ) ) = 1 ( f 1 ( P C τ 2 , τ 2 ( ν , r , s ) ) ) .

Therefore, we obtain f 1 ( P C τ 2 , τ 2 ( ν , r , s ) ) D τ 1 , τ 1 ( f 1 ( ν ) , r,s).

(4)(3) Similarly we obtain, 1 f 1 ( P I τ 2 , τ 2 ( ν , r , s ) ) 1 T τ 1 , τ 1 ( f 1 ( ν ) , r , s ) , for every νI Y , rI0 and sI1, i.e., T τ 1 , τ 1 ( f 1 ( ν ) , r , s ) f 1 ( P I τ 2 , τ 2 ( ν , r , s ) ) . □

Theorem 3.6

For the function f : ( X , τ 1 , τ 1 ) ( Y , τ 2 , τ 2 ) . The following statements are equivalent:
  1. (1)

    f is double fuzzy weakly preopen,

     
  2. (2)

    For each x t P(X) and each μI X ; τ 1(μ)≥r and τ 1 ( μ ) s with x t μ, there exists (r,s)-fpo set γ such that f(x t )≤γ and γ f ( C τ 1 , τ 1 ( μ , r , s ) ) .

     

Proof

(1)(2) Let x t P(X) and μI X such that τ1(μ)≥r, τ 1 ( μ ) s and x t μ. Since f is double fuzzy weakly preopen, then f ( μ ) P I τ 2 , τ 2 ( f ( C τ 1 , τ 1 ( μ , r , s ) ) , r , s ) . Let γ = P I τ 2 , τ 2 ( f ( C τ 1 , τ 1 ( μ , r , s ) ) , r , s ) . Hence γ f ( C τ 1 , τ 1 (μ,r,s)), with f(x t )≤γ.

(2)(1) Let μI X ; τ1(μ)≥r, τ 1 ( μ ) s and y s f(μ). It follows from (2) that γ f ( C τ 1 , τ 1 ( μ , r , s ) ) for some (r,s)-fpo γI Y and y s γ. Hence we have, y s γ P I τ 2 , τ 2 ( f ( C τ 1 , τ 1 ( μ , r , s ) ) , r , s ) . This shows that f ( μ ) P I τ 2 , τ 2 ( f ( C τ 1 , τ 1 ( μ , r , s ) ) , r , s ) , i.e. f is double fuzzy weakly preopen function. □

Theorem 3.7

Let f : ( X , τ 1 , τ 1 ) ( Y , τ 2 , τ 2 ) be a bijective function. Then the following statements are equivalent:
  1. (1)

    f is double fuzzy weakly preopen;

     
  2. (2)

    P C τ 2 , τ 2 ( f ( λ ) , r , s ) f ( C τ 1 , τ 1 ( λ , r , s ) ) for each λI X , rI 0and sI 1; τ 1(λ)≥r and τ 1 ( λ ) s ;

     
  3. (3)

    P C τ 2 , τ 2 ( f ( I τ 1 , τ 1 ( ν , r , s ) ) , r , s ) f ( ν ) for each νI X , rI 0and sI 1; τ 1 ( 1 ν ) r and τ 1 ( 1 ν ) s .

     

Proof

(1)(2) Let νI X ; τ1(ν)≥r and τ 1 ( ν ) s . Then we have,
f ( 1 ν ) = 1 f ( ν ) P I τ 2 , τ 2 ( f ( C τ 1 , τ 1 ( 1 ν , r , s ) ) , r , s ) ,

and so 1 f ( ν ) 1 P C τ 2 , τ 2 ( f ( I τ 1 , τ 1 ( ν , r , s ) ) , r , s ) . Hence P C τ 2 , τ 2 ( f ( I τ 1 , τ 1 ( ν , r , s ) ) , r , s ) f ( ν ) .

(2)(3) Let λI X ; τ1(λ)≥r and τ 1 ( λ ) s . Since C τ 1 , τ 1 ( λ , r , s ) is (r,s)-fc set and λ I τ 1 , τ 1 ( C τ 1 , τ 1 ( λ , r , s ) , r,s) by (3) we have P C τ 2 , τ 2 ( f ( λ ) , r , s ) P C τ 2 , τ 2 ( f ( I τ 1 , τ 1 ( λ , r , s ) ) , r , s ) f ( C τ 1 , τ 1 ( λ , r , s ) ) .

(3)(2) Trivial.

(2)(1) Trivial. □

Theorem 3.8

For a function f : ( X , τ 1 , τ 1 ) ( Y , τ 2 , τ 2 ) . The following statements are equivalent:
  1. (1)

    f is double fuzzy weakly preopen;

     
  2. (2)

    f ( I τ 1 , τ 1 ( ν , r , s ) ) P I τ 2 , τ 2 ( f ( ν ) , r , s ) for each νI X , rI 0and sI 1; τ 1(ν)≥r and τ 1 ( ν ) s ;

     
  3. (3)

    f ( I τ 1 , τ 1 ( C τ 1 , τ 1 ( λ , r , s ) , r , s ) ) P I τ 2 , τ 2 ( f ( C τ 1 , τ 1 ( λ , r , s ) ) , r , s ) for each λI X , rI 0and sI 1; τ 1(λ)≥r and τ 1 ( λ ) s ;

     
  4. (4)

    f ( λ ) P I τ 2 , τ 2 ( f ( C τ 1 , τ 1 ( λ , r , s ) ) , r , s ) , for each (r,s)-fpo set λI X ;

     
  5. (5)

    f ( λ ) P I τ 2 , τ 2 ( f ( C τ 1 , τ 1 ( λ , r , s ) ) , r , s ) , for each (r,s)-fα o set λI X .

     

Proof

(1)(2) Let νI X , rI0 and sI1; τ 1 ( 1 ν ) r and τ 1 ( 1 ν ) s . By (1),
f ( I τ 1 , τ 1 ( ν , r , s ) ) P I τ 2 , τ 2 ( f ( C τ 1 , τ 1 ( I τ 1 , τ 1 ( ν , r , s ) , r , s ) ) , r , s ) P I τ 2 , τ 2 ( f ( C τ 1 , τ 1 ( ν , r , s ) ) , r , s ) = P I τ 2 , τ 2 ( f ( ν ) , r , s )

(2)(3) It is clear.

(3)(4) Let λ be (r,s)-fpo set. Hence by (3),
f ( λ ) f ( I τ 1 , τ 1 ( C τ 1 , τ 1 ( λ , r , s ) , r , s ) ) P I τ 2 , τ 2 ( f ( C τ 1 , τ 1 ( λ , r , s ) ) , r , s ) .

(4)(5) and (5)(1) are clear. □

Definition 3.8

A function f : ( X , τ 1 , τ 1 ) ( Y , τ 2 , τ 2 ) is said to be double fuzzy strongly continuous, if f ( C τ 1 , τ 1 ( λ , r , s ) ) f ( λ ) for each λI X , rI0and sI1.

Theorem 3.9

If f : ( X , τ 1 , τ 1 ) ( Y , τ 2 , τ 2 ) is double fuzzy weakly preopen and double fuzzy strongly continuous function, then f is double fuzzy preopen.

Proof

Let λI X such that τ1(λ)≥r and τ 1 ( λ ) s . Since f is double fuzzy weakly preopen
f ( λ ) P I τ 2 , τ 2 ( f ( C τ 1 , τ 1 ( λ , r , s ) ) , r , s ) .

However, since f is double fuzzy strongly continuous, then f ( λ ) P I τ 2 , τ 2 ( f ( λ ) , r , s ) and therefore f(λ) is (r,s)-fpo. □

Definition 3.9

A function f : ( X , τ 1 , τ 1 ) ( Y , τ 2 , τ 2 ) is said to be double fuzzy contra-preclosed if f(λ) is (r,s)-fpo for each λI X , rI0and sI1; τ 1 ( 1 λ ) r and τ 1 ( 1 λ ) s .

Theorem 3.10

If f : ( X , τ 1 , τ 1 ) ( Y , τ 2 , τ 2 ) is double fuzzy contra-preclosed, then f is double fuzzy weakly preopen function.

Proof

Let λI X ; τ1(λ)≥r and τ 1 ( λ ) s . Then, we have
f ( λ ) f ( C τ 1 , τ 1 ( λ , r , s ) ) = P I τ 2 , τ 2 ( f ( C τ 1 , τ 1 ( λ , 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:
λ 1 ( a ) = 0 , λ 1 ( b ) = 0 . 2 , λ 1 ( c ) = 0 . 7 ; λ 2 ( x ) = 0 , λ 2 ( y ) = 0 . 2 , λ 2 ( x ) = 0 . 2 .
Let ( τ 1 , τ 1 ) and ( τ 2 , τ 2 ) defined as follows:
τ 1 λ = 1 if λ = 1 , 0 ; 1 3 if λ = λ 1 ; 0 otherwise . , τ 1 λ = 0 if λ = 1 , 0 ; 1 3 if λ = λ 1 ; 1 otherwise .
τ 2 λ = 1 if λ = 1 , 0 ; 1 3 if λ = λ 2 ; 0 otherwise . , τ 2 λ = 0 if λ = 1 , 0 ; 1 3 if λ = λ 2 ; 1 otherwise .

Then the function f : ( X , τ 1 , τ 1 ) ( Y , τ 2 , τ 2 ) 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 λI X ; τ(λ)≥r and τ(λ)≤s is a union of (r,s)-fo sets μ i I X such that Cτ,τ(μ i ,r,s)≤λ for each iJ.

Theorem 3.11

Let (X,τ,τ) be (r,s)-regular fuzzy topological space. Then, f : ( X , τ 1 , τ 1 ) ( Y , τ 2 , τ 2 ) is double fuzzy weakly preopen if and only if f is double fuzzy preopen.

Proof

The sufficiency is clear. For the necessity, let λI X , rI0, sI1; λ 0 , τ1(λ)≥r and τ 1 ( λ ) s . For each x t λ, let x t μ x t C τ 1 , τ 1 ( μ x t , r , s ) λ . Hence we obtain that λ = { μ x t x t λ } = { C τ 1 , τ 1 ( μ x t , r , s ) x t λ } and,
f ( λ ) = { f ( μ x t ) x t λ } { P I τ 2 , τ 2 ( f ( C τ 1 , τ 1 ( μ x t , r , s ) ) , r , s ) x t λ } P I τ 2 , τ 2 ( f ( { C τ 1 , τ 1 ( μ x t , r , s ) x t } ) , r , s ) = P I τ 2 , τ 2 ( f ( λ ) , r , s ) .

Thus f is double fuzzy preopen. □

Theorem 3.12

If f : ( X , τ 1 , τ 1 ) ( Y , τ 2 , τ 2 ) is double fuzzy almost open function, then it is double fuzzy weakly preopen.

Proof

Let λI X ; τ1(λ)≥r and τ 1(λ)≤s. Since f is double fuzzy almost open and I τ 1 , τ 1 ( C τ 1 , τ 1 ( λ , r , s ) , r , s ) is (r,s)-fro, then
I τ 2 , τ 2 ( f ( I τ 1 , τ 1 ( C τ 1 , τ 1 ( λ , r , s ) , r , s ) ) , r , s ) = f ( I τ 1 , τ 1 ( C τ 1 , τ 1 × ( λ , r , s ) , r , s ) )
and hence
f ( λ ) f ( I τ 1 , τ 1 ( C τ 1 , τ 1 ( λ , r , s ) , r , s ) I τ 2 , τ 2 ( f ( C τ 1 , τ 1 ( λ , r , s ) ) , r , s ) P I τ 2 , τ 2 ( f ( C τ 1 , τ 1 ( λ , r , s ) ) , r , s ) .

This shows that f is double fuzzy weakly preopen. □

Definition 3.11

Let (X,τ,τ) be an I-dfts, rI0and sI1. The two fuzzy sets λ, μI X are said to be (r,s)-fuzzy separated iff λ q ̄ C τ , τ ( μ , r , s ) and μ q ̄ C τ , τ ( λ , 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, μI X such that λ 0 , μ 0 , are said to be fuzzy (r,s)-pre-separated if λ q ̄ P C τ , τ ( μ , r , s ) and μ q ̄ P C τ , τ ( λ , r , s ) or equivalently if there exist two (r,s)-fpo sets ν, γ such that λν, μγ, λ q ̄ γ and μ q ̄ ν . 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 , τ 1 , τ 1 ) ( Y , τ 2 , τ 2 ) is an injective double fuzzy weakly preopen and strongly double fuzzy continuous function from the space ( X , τ 1 , τ 1 ) onto an (r,s)-fuzzy pre-connected space ( Y , τ 2 , τ 2 ) , then ( X , τ 1 , τ 1 ) is (r,s)-fuzzy connected.

Proof

Let ( X , τ 1 , τ 1 ) be not (r,s)-fuzzy connected. Then there exist (r,s)-fuzzy separated sets β, γI X such that β γ = 1 . Since β and γ are (r,s)-fuzzy separated, there exists λ, μI X ; τ1(λ)≥r, τ1(μ)≥r and τ 1 ( λ ) s , τ 1 ( μ ) s such that βλ, γμ, β q ̄ μ and γ q ̄ λ . Hence we have f(β)≤f(λ), f(γ)≤f(μ), f ( β ) q ̄ f ( μ ) and f ( γ ) q ̄ f ( λ ) . 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
1 = f ( 1 ) = f ( β γ ) = f ( β ) f ( γ )

which is contradiction with ( Y , τ 2 , τ 2 ) is (r,s)-fuzzy pre-connected. Thus ( X , τ 1 , τ 1 ) is (r,s)-fuzzy connected. □

Double Fuzzy weakly preclosed functions

Definition 4.13

A function f : ( X , τ 1 , τ 1 ) ( Y , τ 2 , τ 2 ) is said to be double fuzzy weakly preclosed function if
P C τ 2 , τ 2 ( f ( I τ 1 , τ 1 ( λ , r , s ) ) , r , s ) f ( λ )

for each λI X , rI0 and sI1; τ 1 ( 1 λ ) r and τ 1 ( 1 λ ) 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:
λ 1 ( x ) = 0 . 4 , λ 1 ( y ) = 0 . 3 ; λ 2 ( a ) = 0 . 5 , λ 2 ( b ) = 0 . 6 .
Let
τ 1 λ = 1 if λ = 1 , 0 ; 1 2 if λ = λ 2 ; 0 otherwise. , τ 1 λ = 0 if λ = 1 , 0 ; 1 2 if λ = λ 2 ; 1 otherwise.
τ 2 λ = 1 if λ = 1 , 0 ; 1 2 if λ = λ 1 ; 0 otherwise. , τ 2 λ = 0 if λ = 1 , 0 ; 1 2 if λ = λ 1 ; 1 otherwise.

Then the function f : ( X , τ 1 , τ 1 ) ( Y , τ 2 , τ 2 ) 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 , τ 1 , τ 1 ) ( Y , τ 2 , τ 2 ) . The following statements are equivalent.
  1. (1)

    f is double fuzzy weakly preclosed;

     
  2. (2)

    P C τ 2 , τ 2 ( f ( λ ) , r , s ) f ( C τ 1 , τ 1 ( λ , r , s ) ) for each λI X , rI 0and sI 1; τ 1(λ)≥r and τ 1 ( λ ) s ;

     
  3. (3)

    P C τ 2 , τ 2 ( f ( I τ 1 , τ 1 ( λ , r , s ) , r , s ) f ( λ ) for each λI X , rI 0and sI 1; τ 1 ( 1 λ ) r and τ 1 ( 1 λ ) s ;

     
  4. (4)

    P C τ 2 , τ 2 ( f ( I τ 1 , τ 1 ( λ , r , s ) , r , s ) f ( λ ) for each (r,s)-fpc set λI X , rI 0and sI 1;

     
  5. (5)

    P C τ 2 , τ 2 ( f ( I τ 1 , τ 1 ( λ , r , s ) , r , s ) f ( λ ) for each (r,s)-fα c λI X , rI 0and sI 1.

     

Proof

Straightforward. □

Theorem 4.15

For a function f : ( X , τ 1 , τ 1 ) ( Y , τ 2 , τ 2 ) . The following statements are equivalent.
  1. (1)

    f is double fuzzy weakly preclosed;

     
  2. (2)

    P C τ 2 , τ 2 ( f ( λ ) , r , s ) f ( C τ 1 , τ 1 ( λ , r , s ) ) for each (r,s)-fro set λI X , rI 0and sI 1;

     
  3. (3)

    For each νI Y , μI X , rI 0and sI 1; τ 1(μ)≥r and τ 1 ( μ ) s with f −1(ν)≤μ, there exists (r,s)-fpo set γI Y with νγ and f 1 ( γ ) C τ 1 , τ 1 ( μ , r , s ) ;

     
  4. (4)

    For each fuzzy point y s P(Y) and each μI X , rI 0and sI 1such that τ 1(μ)≥r and τ 1 ( μ ) s with f −1(y s )≤μ, there exists (r,s)-fpo set γI Y ; y s γ and f 1 ( γ ) C τ 1 , τ 1 ( μ , r , s ) ;

     
  5. (5)

    P C τ 2 , τ 2 ( f ( I τ 1 , τ 1 ( C τ 1 , τ 1 ( λ , r , s ) , r , s ) ) , r , s ) f ( C τ 1 , τ 1 ( λ , r , s ) ) for each λI X , rI 0and sI 1;

     
  6. (6)

    P C τ 2 , τ 2 ( f ( I τ 1 , τ 1 ( D τ 1 , τ 1 ( λ , r , s ) , r , s ) ) , r , s ) f ( D τ 1 , τ 1 ( λ , r , s ) ) for each λI X , rI 0and sI 1;

     
  7. (7)

    P C τ 2 , τ 2 ( f ( λ ) , r , s ) f ( C τ 1 , τ 1 ( λ , r , s ) ) for each (r,s)-fpo set λI X , rI 0and sI 1.

     

Proof

We will prove (2)(3) and (1)(6).

(2)(3): Let νI Y , rI0, sI1 and let μI X ; τ1(μ)≥r and τ 1 ( μ ) s with f−1(ν)≤μ. Then f 1 ( ν ) q ̄ C τ 1 , τ 1 ( 1 C τ 1 , τ 1 ( μ , r , s ) , r , s ) and consequently, ν q ̄ f ( C τ 1 , τ 1 ( 1 C τ 1 , τ 1 ( μ , r , s ) , r , s ) . Since 1 C τ 1 , τ 1 ( μ , r , s ) is (r,s)-fro, ν q ̄ P C τ 2 , τ 2 ( f ( 1 C τ 1 , τ 1 ( μ , r , s ) ) , r , s ) by (2). Let γ = 1 P C τ 2 , τ 2 ( f ( 1 C τ 1 , τ 1 ( μ , r , s ) ) , r , s ) . Then γ is (r,s)-fpo with νγ and
f 1 ( γ ) 1 f 1 ( P C τ 2 , τ 2 ( 1 C τ 1 , τ 1 ( μ , r , s ) , r , s ) ) 1 f 1 f ( 1 C τ 1 , τ 1 ( μ , r , s ) ) C τ 1 , τ 1 ( μ , r , s ) .

(1)(6): Let νI Y , rI0 and sI1; τ 2 ( 1 ν ) r , τ 2 ( 1 ν ) s and y s 1 f ( ν ) . Since f 1 ( y s ) 1 ν , there exists (r,s)-fpo γI Y with y s γ and f 1 ( γ ) C τ 1 , τ 1 ( 1 ν , r , s ) = 1 I τ 1 , τ 1 ( ν , r , s ) by (6). Therefore γ q ̄ f ( I τ 1 , τ 1 ( ν , r ) ) , so that y s 1 P C τ 2 , τ 2 ( f ( I τ 1 , τ 1 ( ν , r , s ) ) , r , s ) . □

Theorem 4.16

If f : ( X , τ 1 , τ 1 ) ( Y , τ 2 , τ 2 ) is double fuzzy weakly preclosed, then for each y s P(Y) and each μ Q τ 1 , τ 1 ( f 1 ( y s ) , r , s ) , there exists (r,s)-fpo set γI Y ; γ Q τ 2 , τ 2 ( y s , r , s ) , such that f 1 ( γ ) C τ 1 , τ 1 ( μ , r , s ) .

Proof

Let μ Q τ 1 , τ 1 ( f 1 ( y s , r , s ) . Then μ(x) + s>1 and hence there exists t(0,1) such that μ(x)>t>1−s. Then μ Q τ 1 , τ 1 ( f 1 ( y t ) , r , s ) . By Theorem 3.7-6 there exists (r,s)-fpo set γI Y ; y t γ such that f 1 ( γ ) C τ 1 , τ 1 ( μ , 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 λI X is called (r,s)-fuzzy pre-Q-neighborhood of x t if there exists (r,s)-fpo set μI X such that x t λ. We denote the set of all (r,s)-fuzzy pre-Q-neighborhood of x t by PQτ,τ(x t ,r,s).

Theorem 4.17

In an I-dfts (X,τ,τ). A fuzzy point x t P Cτ,τ(λ,r) if and only if for every μ PQτ,τ(x t ,r,s), μqλ is hold.

Proof

Straightforward. □

Theorem 4.18

If f : ( X , τ 1 , τ 1 ) ( Y , τ 2 , τ 2 ) is double fuzzy weakly preclosed and if for each νI X , rI0and sI1; τ 1 ( 1 ν ) r , τ 1 ( 1 ν ) s and each f 1 ( y s ) 1 ν there exists μ Q τ 1 , τ 1 ( f 1 ( y s ) , r , s ) such that f 1 ( y s ) μ C τ 1 , τ 1 ( μ , r , s ) 1 ν . Then f is double fuzzy preclosed.

Proof

Let νI X , rI0and sI1; τ 1 ( 1 ν ) r , τ 1 ( 1 ν ) s and let y s 1 f ( ν ) . Then f 1 ( y s ) 1 ν , and hence there exists μ Q τ 1 , τ 1 ( f 1 ( y s ) , r , s ) such that f 1 ( y s ) μ C τ 1 , τ 1 ( μ , r , s ) 1 ν . Since f is double fuzzy weakly preclosed by using Theorem 3.12, there exists (r,s)-fuzzy pre-Q-neighborhood γI Y with y s γ and f 1 ( γ ) C τ 1 , τ 1 ( μ , r , s ) . Therefore, we obtain f 1 ( γ ) q ̄ ν and hence γ q ̄ f ( ν ) , this shows that y s P C τ 2 , τ 2 ( f ( ν ) , r , s ) . Therefore, f(ν) is (r,s)-fpc and f is double fuzzy preclosed function. □

Definition 4.15

A function f : ( X , τ 1 , τ 1 ) ( Y , τ 2 , τ 2 ) is said to be double fuzzy contra-open (resp. double fuzzy contra-closed) if τ 2 ( 1 f ( λ ) ) r and τ 2 ( 1 f ( λ ) ) s (resp. τ2(f(λ))≥r and τ 2 ( f ( λ ) ) s ) for each λI X , rI0and sI1; τ1(λ)≥r and τ 1 ( λ ) s (resp. τ 1 ( 1 λ ) r and τ 1 ( 1 λ ) s ).

Theorem 4.19

If f : ( X , τ 1 , τ 1 ) ( Y , τ 2 , τ 2 ) is double fuzzy contra-open, then f is double fuzzy weakly preclosed.

Proof

Let λI X , rI0 and sI1 such that τ 1 ( 1 λ ) r and τ 1 ( 1 λ ) s . Then,
P C τ 2 , τ 2 ( f ( I τ 1 , τ 1 ( λ , r , s ) ) , r , s ) f ( I τ 1 , τ 1 ( λ , r , s ) ) f ( λ ) .

Theorem 4.20

If f : ( X , τ 1 , τ 1 ) ( Y , τ 2 , τ 2 ) is double fuzzy weakly preclosed, then for every νI Y and every λI X , rI0, sI1such that τ1(λ)≥r and τ 1 ( λ ) s with f−1(ν)≤λ, there exists (r,s)-fpc set γI Y such that νγ and f 1 ( γ ) C τ 1 , τ 1 ( λ , r , s ) .

Proof

Let νI Y and let λI X , rI0and sI1such that τ1(λ)≥r and τ 1 ( λ ) s with f−1(ν)≤λ. Put γ = P C τ 2 , τ 2 ( f ( I τ 1 , τ 1 ( C τ 1 , τ 1 ( λ , r , s ) , r , s ) ) , r , s ) , then γ is (r,s)-fpc set in I Y such that νγ since ν f ( λ ) f ( I τ 1 , τ 1 ( C τ 1 , τ 1 ( λ , r , s ) , r , s ) ) P C τ 2 , τ 2 ( f ( I τ 1 , τ 1 ( C τ 1 , τ 1 ( λ , r , s ) , r , s ) ) , r , s ) = γ . And since f is double fuzzy weakly preclosed, f 1 ( γ ) C τ 1 , τ 1 ( λ , r , s ) . □

Corollary 4.21

If f : ( X , τ 1 , τ 1 ) ( Y , τ 2 , τ 2 ) is double fuzzy weakly preclosed, then for every y s P(Y) and every λI X , rI0and sI1such that τ1(λ)≥r and τ 1 ( λ ) s with f−1(y s )≤λ, there exists (r,s)-fpc set γI Y ; y s γ such that f 1 ( γ ) C τ 1 , τ 1 ( λ , r , s ) .

Definition 4.16

A fuzzy set λI X is called (r,s)-fuzzy θ-compact if for each family {μ i iJ} in { μ I X μ Q τ , τ ( λ , r , s ) } satisfy ( i J μ i ) ( x ) λ ( x ) for each xX, there exist a finite subset J0of J such that λ I τ , τ ( { C τ , τ ( μ i , r , s ) i J 0 } , r , s ) .

Theorem 4.22

If f : ( X , τ 1 , τ 1 ) ( Y , τ 2 , τ 2 ) 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 λI X , rI0and sI1.

Proof

Let λ be (r,s)-fuzzy θ-compact and let y s 1 f ( λ ) . Then f 1 ( y s ) q ̄ λ and for each x t λ there is μ x t Q τ 1 , τ 1 ( x t , r , s ) with x t μ x t and C τ 1 , τ 1 ( μ x t , r , s ) q ̄ f 1 ( y s ) . Clearly { μ x t x t λ , μ x t Q τ 1 , τ 1 ( λ , r , s ) } satisfy ( i J μ i ) ( x ) λ ( x ) for each xX and since λ is (r,s)-fuzzy θ-compact, there is { μ x 1 , μ x 2 , μ x 3 , . , μ x n } { μ x t x t λ , μ x t Q τ 1 , τ 1 ( λ , r , s ) } such that λ I τ 1 , τ 1 ( ξ , r , s ) , where ξ = { C τ 1 , τ 1 ( μ x i , r , s ) i = 1 , 2 , , n } . Since f is double fuzzy weakly preclosed, by using Theorem 3.12 there exists γ PQ τ 1 , τ 1 ( y s , r , s ) with
f 1 ( y s ) f 1 ( γ ) C τ 1 , τ 1 ( 1 ξ , r , s ) = 1 I τ 1 , τ 1 ( ξ , r , s ) 1 λ.

Therefore y s γ and γ q ̄ f ( λ ) . Thus y s 1 P C τ 2 , τ 2 ( f ( λ ) , 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, μI X are (r,s)-fuzzy strongly separated if there exist ν, γI X such that τ(ν)≥r and τ(ν)≤s, τ(γ)≥r with λν, μγ and C τ , τ ( ν , r , s ) q ̄ C τ , τ ( γ , 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, μI X such that x t 1 λ x 1 t 2 , x t 2 μ x 1 t 1 and λ q ̄ μ .

Theorem 4.23

If f : ( X , τ 1 , τ 1 ) ( Y , τ 2 , τ 2 ) is double fuzzy weakly preclosed surjection and all fibers are (r,s)-fuzzy strongly separated, then ( Y , τ 1 , τ 1 ) is (r,s)-fuzzy pre-T2.

Proof

Let y s 1 , y s 2 P ( Y ) and let γ,νI X , rI0and sI1; τ1(γ)≥r, τ 1 ( γ ) s , τ1(ν)≥r and τ1(ν)≤s such that f 1 ( y s 1 ) γ and f 1 ( y s 2 ) ν respectively with C τ 1 , τ 1 ( γ , r , s ) q ̄ C τ 1 , τ 1 ( ν , r , s ) . By using Theorem 3.12-4 there are (r,s)-fpo sets λ, μI Y such that y s 1 λ and y s 2 μ , f 1 ( λ ) C τ 1 , τ 1 ( γ , r , s ) and f 1 ( μ ) C τ 1 , τ 1 ( ν , r , s ) . Therefore λ q ̄ μ , because C τ 1 , τ 1 ( γ , r , s ) q ̄ C τ 2 , τ 2 ( ν , r , s ) and f is surjective. Thus ( Y , τ 2 , τ 2 ) 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(Cτ,τ(λ,r,s))≥r and τ(Cτ,τ(λ,r,s))≤s for each λI 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 iJ} of X, there is a finite subset J0of J such that { C τ , τ ( λ i , r , s ) i J 0 } = 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 iJ} satisfies ( i J μ i ) ( x ) = λ ( x ) for each xX. There exists finite subfamily J0of J such that ( i J 0 P C τ , τ ( μ i , r , s ) ) ( x ) λ ( x ) for each xX.

Theorem 4.24

Let ( X , τ 1 , τ 1 ) be (r,s)-extremally disconnected I-dfts. Let f : ( X , τ 1 , τ 1 ) ( Y , τ 2 , τ 2 ) 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 λI Y is (r,s)-fuzzy P-compact. Then f−1(λ) is (r,s)-fuzzy almost compact.

Proof

Let {ν j æJ} be (r,s)-fuzzy open cover of f−1(λ). Then for each y s λf(X), f 1 ( y s ) { C τ 1 , τ 1 ( ν j , r , s ) æ J ( y s ) } = γ y s , for some finite subfamily J(y s ) of J. Since ( X , τ 1 , τ 1 ) is (r,s)-extremally disconnected each τ 1 ( C τ 1 , τ 1 ( ν j , r , s ) ) r and τ 1 ( C τ 1 , τ 1 ( ν j , r , s ) ) s , hence τ 1 ( γ y s ) r and τ 1 ( γ y s ) s . So by Corollary 4.21 there exists (r,s)-fpc set μ y s ; y s μ y s such that f 1 ( μ y s ) C τ 1 , τ 1 ( γ y s , r , s ) . Then, { μ y s y s λ f ( X ) } { 1 f ( X ) } is (r,s)-fuzzy preclosed cover of λ, λ { C τ 2 , τ 1 ( μ y s , r , s ) y s λ f ( X ) } { C τ 2 , τ 2 ( 1 f ( X ) , r , s ) } for some finite fuzzy subset K of λf(X). Hence,
f 1 ( λ ) y s K f 1 ( C τ 2 , τ 2 ( μ y s , r , s ) ) { f 1 ( C τ 2 , τ 2 ( 1 f ( X ) , r , s ) ) } y s K C τ 1 , τ 1 ( f 1 ( μ y s ) , r , s ) { C τ 1 , τ 1 ( f 1 ( 1 f ( X ) ) , r , s ) } y s K C τ 1 , τ 1 ( f 1 ( μ y s ) , r , s )

so f 1 ( λ ) æ J ( y s ) , y s K C τ 1 , τ 1 ( ν æ , r , s ) . Therefore f−1 (λ) is (r,s)-fuzzy almost compact. □

Declarations

Acknowledgements

The author would like to thank the reviewers for their valuable comments and helpful suggestions for improvement of the original manuscript.

Authors’ Affiliations

(1)
Mathematics Department, Faculty of Science, South Valley University
(2)
Department of Mathematics, College of Science in Al-Zulfi, Majmaah University

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© Ghareeb; licensee Springer. 2012

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