On Some Types Of Continuous Fuzzy Functions ∗
Erdal Ekici
†Received 3 January 2003
Abstract
In this paper, by using operations, some characterizations and some properties of fuzzy continuous functions and its weaker and stronger forms including fuzzy weakly continuous, fuzzyθ-continuous, fuzzy stronglyθ-continuous, fuzzy almost stronglyθ-continuous, fuzzy weaklyθ-continuous, fuzzy almost continuous, fuzzy super continuous, fuzzyδ-continuous, are presented.
1 Introduction
Several types of fuzzy continuous functions and its weaker and stronger forms occur in the literature. The aim of this paper is to give some characterizations and some properties of fuzzy continuous functions and its weaker and stronger forms including fuzzy weakly continuous, fuzzyθ-continuous, fuzzy stronglyθ-continuous, fuzzy almost stronglyθ-continuous, fuzzy weaklyθ-continuous, fuzzy almost continuous, fuzzy super continuous, fuzzyδ-continuous.
The class of fuzzy sets on a universeX will be denoted byIX and fuzzy sets on X will be denoted by Greek letters as µ, ρ, η, etc. A family τ of fuzzy sets inX is called a fuzzy topology forX iff(1)∅,X ∈τ, (2)µ∧ρ∈τ wheneverµ,ρ∈τ and (3) V{µα:α∈I}∈τ whenever eachµα∈τ (α∈I). Moreover, the pair (X,τ) is called a fuzzy topological space. Every member ofτ is called an open fuzzy set [8].
A fuzzy set inX is called a fuzzy point iffit takes the value 0 for ally∈X except one, say, x∈X. If its value at xisα(0<α≤1) we denote this fuzzy point by xα, where the pointxis called its support [8]. For any fuzzy pointxεand any fuzzy setµ, we write xε∈µiffε≤µ(x).
Letf :X→Y be a fuzzy function from a fuzzy topological space (X,τ) to a fuzzy topological space (Y,υ). The functionf is called fuzzy continuous ifffor eachxε∈X and each fuzzy open setρcontainingf(xε), there exists a fuzzy open setµcontaining xεsuch thatf(µ)≤ρ[9].
Byint(µ) andcl(µ), we mean the interior of µand the closure ofµ.
Let f : X →Y be a fuzzy function from a fuzzy topological space X to a fuzzy topological spaceY. Then the functiong:X →X×Y defined byg(xε) = (xε, f(xε)) is called the graph function off and it will be denoted bygrf [1].
∗Mathematics Subject Classifications: 54A40, 03E72.
†Department of Mathematics, Cumhuriyet University, Sivas 58140, Turkey
21
2 Some Types of Continuous Fuzzy Functions
There are some useful definitions.
DEFINITION 1. Let (X,τ) be a fuzzy topological space. A mappingα:IX →IX is called an operation onIX if for eachµ∈IX\{∅},int(µ)≤µαand∅α=∅whereµα denotes the value of αin µ[5].
DEFINITION 2. Let (X,τ) be a fuzzy topological space and letαbe an operation on IX. α is called a monotonous operation if for eachµ, ρ ∈ IX and µ ≤ ρ, then µα≤ρα[5].
DEFINITION 3. Let (X,τ) and (Y,υ) be fuzzy topological spaces and letϕ,ψ be operations onIX,IY respectively. A functionf from (X,τ) into (Y,υ) is called fuzzy ϕψ-continuous if for each xε ∈X and each fuzzy open set ρcontaining f(xε), there exists a fuzzy open set µcontainingxεsuch thatf(µϕ)≤ρψ.
The following table provides us a list of fuzzyϕψ-continuous function with opera- tionsϕandψ.
Operations Fuzzyϕψ-continuity 1.ϕ=ψ=i f. continuity [9]
2.ϕ=i, ψ=cl f. weakly continuity [1]
3.ϕ=ψ=cl f. θ-continuity [3, 7]
4.ϕ=cl, ψ=i f. stronglyθ-continuity [4, 6]
5.ϕ=cl, ψ=int◦cl f. almost stronglyθ-continuity [7]
6.ϕ=int◦cl, ψ=cl f. weaklyθ-continuity [7]
7.ϕ=i, ψ=int◦cl f. almost continuity [1]
8.ϕ=int◦cl, ψ=i f. super continuity [6]
9.ϕ=ψ=int◦cl f. δ-continuity [2, 10]
DEFINITION 4. Let (X,τ) be a fuzzy topological space and let (xαεα) be a net in X. (xαεα) is calledϕ-converges toxε if for each open setµ containingxε, there exists an indexα0∈J such thatxαεα∈µϕ for allα≥α0. We will denote byxαεα →ϕ xε.
DEFINITION 5. Suppose that (X,τ) is a fuzzy topological space and ϕ is an operation onIX. Let (X,τ) be a fuzzy topological space and let (xαεα) be a net inX. Then (xαεα) is calledϕ-eventually in the fuzzy setµ≤Xif there exists an indexα0∈J such thatxαεα∈µϕ for allα≥α0.
The following theorem gives us the characterizations of fuzzyϕψ-continuous func- tion.
THEOREM 1. Suppose that (X,τ) and (Y,υ) are fuzzy topological spaces andϕ,ψ are operations onIX,IY respectively. For a functionf : (X,τ)→(Y,υ), the following statements are equivalent.
i-)f is fuzzyϕψ-continuous.
ii-) For eachxε∈X and for each net (xαεα) inX, ifxαεα→ϕ xε, thenf(xαεα)→ψ f(xε).
iii-) For eachxε∈X and for each net (xαεα) inX, ifxαεα→ϕ xε, then the netf(xαεα) isψ-eventually inρfor all fuzzy open setρcontainingf(xε).
PROOF. (i)⇒(ii). Letf(xε)∈ρ∈υ. Since f is fuzzyϕψ-continuous, there exists an open set µ containing xε such that f(µϕ)≤ρψ. Since xαεα →ϕ xε, there exists an index α0 ∈ J such that xαεα ∈ µϕ for all α ≥ α0. Thus f(xαεα) ∈ f(µϕ) ≤ ρψ and f(xαεα)∈ρψ for allα≥α0. We obtain thatf(xαεα)→ψ f(xε).
(ii)⇒(iii). Obvious.
(iii)⇒(i). Suppose that (i) is not true. There would then exist a pointxε and an open set ρcontainingf(xε) such thatµϕf−1(ρψ) for each µ∈τ wherexε∈µ. Let xεµ ∈µϕand xεµ ∈/f−1(ρψ) for eachµ∈τ wherexε∈µ. Then for the neighborhood net (xεµ),xεµ
→ϕ xε, but (f(xεµ)) is notψ-eventually inρ. This is a contradiction.
EXAMPLE 1. Suppose that (X,τ) and (Y,υ) are fuzzy topological spaces. For a function f : (X,τ)→ (Y,υ), the following statements are equivalent with operations ϕ=i,ψ=cl.
i-)f is fuzzy weakly continuous.
ii-) For each xε ∈ X and for each net xαεα inX, ifxαεα → xε, then for each open set µ containingf(xε), there exists an indexα0 ∈J such thatf(xαεα)∈cl(µ) for all α≥α0.
iii-) For eachxε∈X and for each netxαεα in X, if xαεα →xε, then the netf(xαεα) is eventually incl(ρ) for all fuzzy open setρcontainingf(xε).
THEOREM 2. Let f : X →Y be a fuzzy function from fuzzy topological space (X,τ) to fuzzy topological space (Y,υ) and letϕ, ψbe operations on IX,IY, respec- tively. If f is fuzzyϕψ-continuous and ϕis a monotonous operation onIX, then the restriction function f |µ:µ→Y for any fuzzy setµ≤X is a fuzzyϕψ-continuous.
PROOF. Letxε∈µandf |µ (xε)∈ρ∈υ. Sincefis fuzzyϕψ-continuous, it follows that there exists a fuzzy open setη containingxεsuch thatf(ηϕ)≤ρψ. From here we obtain thatf(ηϕ)∧µ≤ρψ∧µ. Sincef |µ(µϕ) =f(µϕ)∧µ,f |µ(µϕ)≤ρψ∧µ≤ρψ. Sinceϕis a monotonous operation onIX, it follows thatf |µ((η∧µ)ϕ)≤f |µ(ηϕ)≤ ρψ. Thus, we obtain that the restriction functionf |µ is fuzzyϕψ-continuous.
THEOREM 3. Suppose that (X,τ) and (Y,υ) are fuzzy topological spaces andϕ, ψ are operations on IX, IY respectively. Let f : X → Y be any fuzzy function. If {µα :α ∈J}is an open cover of X and fα =f |µα is fuzzy ϕψ-continuous for each α∈J, thenf is fuzzyϕψ-continuous.
PROOF. Letxε ∈ X, f(xε)∈ ρ∈υ. Since {µα : α∈J}is an open cover ofX, there exists an index αsuch thatxε∈µαand fα(xε)∈ρ∈υ. Since eachfα is fuzzy ϕψ-continuous, there exists an open set xε ∈ η such that (η∧µα)ϕ ≤fα−1(ρψ) and hence (η∧µα)ϕ≤fα−1(ρψ) =f−1(ρψ)∧µα. Since{µα:α∈J}is an open cover ofX, (η∧µα)ϕ≤f−1(ρψ) andxε∈η∧µα∈τ. Thus,f is fuzzyϕψ-continuous.
DEFINITION 6. Let (T
α∈JXα,τ) be a product space and letψbe an operation on IQα∈JXα and onIXαfor allα∈J. ψis called a productive operation if (T
α∈Jµα)ψ≤ T
α∈Jµψα for allT
α∈Jµα≤T
α∈JXα,µα≤Xα.
DEFINITION 7. Let (Y,υ) be a fuzzy topological space and letψbe an operation onIY. (Y,υ) is called fuzzyψ-hyperconnected space ifµψ =Y for all fuzzy open set µ=∅.
EXAMPLE 2. Let (T
α∈JXα,τ) be a product space. Take ψ = cl. It is known that the operation ψ = cl is productive, since cl(T
α∈Jµα) ≤ T
α∈Jcl(µα) for all T
α∈Jµα≤T
α∈JXα,µα≤Xα.
Let the fuzzy topological space (Y,υ) beψ-hyperconnected. It means thatcl(µ) =Y for all fuzzy open set µ=∅.
THEOREM 4. Suppose that (X,τ) and (Y,υ) are fuzzy topological spaces andϕ, ψ are operations on IX, IY, respectively. Letf : X → Y be any fuzzy function and letgrf : (X,τ)→(X×Y,τp) be graph function off andψbe a productive operation onIX×Y. Ifgrf isϕψ-continuous, thenf isϕψ-continuous.
PROOF. Letxε∈X andf(xε)∈ρ∈υ. Thengrf(xε) = (xε, f(xε))∈X×ρ∈τp. Since grf is ϕψ-continuous and ψis a productive operation, there exists an open set fuzzy setη containingxε such thatgrf(ηϕ) =ηϕ×f(ηϕ)≤(X×ρ)ψ ≤X×ρψ and hencef(ηϕ)≤ρψ. Thus,f isϕψ-continuous.
THEOREM 5. Suppose thatf :X →Y is a fuzzy function from fuzzy topological space (X,τ) to fuzzy topological space (Y,υ) and ϕ, ψ are operations on IX, IY, respectively. Let grf : (X,τ) → (X ×Y,τp) be graph function of f and ψ be a productive operation on IX×Y and let X×Y be fuzzyψ-hyperconnected space. grf is fuzzyϕψ-continuous function if and only iff is fuzzyϕψ-continuous function.
PROOF.⇒: Obvious from the above theorem.
⇐: Let xε ∈ X and let Z
j∈J(µj×ηj) ≤ X ×Y be a fuzzy open set such that xε∈(grf)−1(Z
j∈J(µj×ηj)). SinceX×Y is fuzzyψ-hyperconnected space, it follows that (Z
j∈J(µj×ηj))ψ =X×Y. Hence for all open fuzzy setρ containingxε, ρϕ≤ (grf)−1((Z
j∈J(µj×ηj))ψ) = (grf)−1(X×Y) =X. Thus,grf is fuzzyϕψ-continuous function.
THEOREM 6. Suppose that (Xα,τα) is fuzzy topological space andψis an opera- tion onIXα for allα. Let (T
α∈JXα,τp) be a product space and letψbe a productive operation onIQα∈JXα. Let (X,τ) be a fuzzy topological space, letϕbe an operation onIX and letf : (X,τ)→(T
α∈JXα,τp) be any fuzzy function. Iff ϕψ-continuous, thenpα◦f is fuzzyϕψ-continuous wherepα is projection function for eachα∈J.
PROOF. Let xε ∈ X and (pα◦f)(xε) ∈ ρα ∈ τα. Then f(xε) ∈ p−α1(ρα) = ρα×(T
β=αXβ)∈τp. Sincef isϕψ-continuous, there exists an open setµcontaining xε such thatf(µϕ)≤(ρα×T
β=αXβ)ψ. Since ψis a productive operation, f(µϕ)≤ ρψα×(T
β=αXβ)ψ =ρψα×T
β=αXβψ =ρψα×T
β=αXβ =p−α1(ρψα) and hence µϕ ≤ (pα◦f)−1(ρψα) and we obtain thatpα◦f is fuzzy ϕψ-continuous for eachα∈J.
THEOREM 7. Suppose that (X,τ) and (Y,υ) are fuzzy topological spaces andϕ,ψ are operations on IX,IY respectively. Letf :X →Y be any fuzzy function and let ß be a base ofυand letψbe a monotonous operation onIY. f is fuzzyϕψ-continuous iff for eachxε∈X and eachρ∈ß containingf(xε), there exists an open setµcontaining xεsuch thatf(µϕ)≤ρψ.
PROOF. (⇒:) Obvious.
(⇐:) Letxε∈X andf(xε)∈η ∈υ. Since ß is a base ofυ, there exists an open set ξ∈ß containingf(xε) such thatξ≤η. Now from hypothesis, there exists an open set
γ containingxε such that f(γϕ)≤ξψ. Since ψis a monotonous operation on IY and ξ≤η, f(γϕ)≤ξψ ≤ηψ. Hence, f isϕψ-continuous.
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