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Research | Open | Published:

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 IX. By 0 and 1 , we denote the smallest and the greatest fuzzy subsets on X. For a fuzzy subset λIX, 1 λ denotes its complement. Given a function f ~ :XY, 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 λIX, νIY, 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:IXI, which satisfies the following properties:

  1. (O1)

    τ(λ) 1 τ (λ) for each λIX.

  2. (O2)

    τ(λ1λ2)≥τ(λ1)τ(λ2) and τ(λ1λ2)≤τ(λ1)τ(λ2) for each λ1, λ2IX.

  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 IX, 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 ~ :XY 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 νIY.

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 2Xinstead of IX, 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 λIX, we define an operator Cτ,τ:IX×I0×I1IX as follows:

C τ , τ ( λ , r , s ) = { μ I X λ μ , τ ( 1 μ ) r , τ ( 1 μ ) s } .

For λ, μIX, 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 λIX, we define an operator Iτ,τ:IX×I0×I1IXas follows:

I τ , τ ( λ , r , s ) = { μ I X μ λ , τ ( μ ) r , τ ( μ ) s } .

For λ μIX, 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 λIX, 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 λIX, 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 Yfor 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 Yfor 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, μIX, 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, λIX, 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)-fuzzyθ-cluster point ofλ}. 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, μIXand 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 λIX, 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 λIX 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 μIX; 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 νIY. 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 νIY. 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 νIY, 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 γf( C τ 1 , τ 1 (μ,r,s)).

Proof

(1)(2) Let x t P(X) and μIXsuch 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 γf( C τ 1 , τ 1 (μ,r,s)), with f(x t )≤γ.

(2)(1) Let μIX; τ1(μ)≥r, τ 1 (μ)s and y s f(μ). It follows from (2) that γf( C τ 1 , τ 1 (μ,r,s)) for some (r,s)-fpo γIYand y s γ. Hence we have, y s γP 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 νIX; 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 λIX; 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 νIX, 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 λIX, 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 λIX 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 λIX, 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 λIX; 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 λIX; τ(λ)≥r and τ(λ)≤s is a union of (r,s)-fo sets μ i IXsuch 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 λIX, 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 λIX; 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, μIXare 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, μIXsuch 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, γIX such that βγ= 1 . Since β and γ are (r,s)-fuzzy separated, there exists λ, μIX; 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 λIX, 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 Ywith 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 νIY, rI0, sI1 and let μIX; 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 νIY, 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 γIY 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 γIY; 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 γIY; 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 λIXis called (r,s)-fuzzy pre-Q-neighborhood of x t if there exists (r,s)-fpo set μIXsuch 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 νIX, 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 νIX, 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 γIYwith 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 λIX, 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 λIX, 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 νIYand every λIX, rI0, sI1such that τ1(λ)≥r and τ 1 (λ)s with f−1(ν)≤λ, there exists (r,s)-fpc set γIYsuch that νγ and f 1 (γ) C τ 1 , τ 1 (λ,r,s).

Proof

Let νIYand let λIX, 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 IYsuch that νγ since νf(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 λIX, rI0and sI1such that τ1(λ)≥r and τ 1 (λ)s with f−1(y s )≤λ, there exists (r,s)-fpc set γIY; y s γ such that f 1 (γ) C τ 1 , τ 1 (λ,r,s).

Definition 4.16

A fuzzy set λIXis 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 λIX, 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, μIXare (r,s)-fuzzy strongly separated if there exist ν, γIXsuch 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, μIXsuch 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,νIX, 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, μIYsuch 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 λIX; 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 λIYis (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)æJ( 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 λf(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 λf(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. □

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

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