# FuzzyCe-I(ec,eo) and Fuzzy Completely Ce-I(rc,eo) Functions via Fuzzy e-Open Sets

Authors: V. Seenivasan; K. Kamala
Journal: The Scientific World Journal
 (2016)

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## Abstract

We introduced the notions of fuzzyCe-I(ec,eo) functions and fuzzy completely Ce-I(rc,eo) functions via fuzzy e-open sets. Some properties and several characterization of these types of functions are investigated.


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## Body

## 1. Introduction

With the introduction of fuzzy sets by Zadeh [1] and fuzzy topology by Chang [2], the theory of fuzzy topological spaces was subsequently developed by several fuzzy topologist based on the concepts of general topology. In 2014, the concept of fuzzy e-open sets and fuzzy e-continuity and separations axioms and their properties were defined by Seenivasan and Kamala [3]. In this paper, we introduce the notion of fuzzy Ce-I(ec,eo) functions, fuzzy Ce-continuous, fuzzy completely Ce-I(rc,eo) functions, and fuzzy e-kernel via fuzzy e-open sets and studied their properties and several characterizations of these types of functions are investigated. In this paper, we denote fuzzy e-open, fuzzy e-closed, and fuzzy regular closed as, eo, ec, and rc, respectively.


## 2. Preliminaries

Throughout this paper,(X,τ) and (Y,σ) (or simply X and Y) represent nonempty fuzzy topological spaces on which no separation axioms are assumed, unless otherwise mentioned.Letμ be any fuzzy set of X. The fuzzy closure of μ, fuzzy interior of μ, fuzzy δ-closure of μ, and the fuzzy δ-interior of μ are denoted by cl(μ), int(μ), clδ(μ), and intδ(μ), respectively. A fuzzy set μ of X is called fuzzy regular open [4] (resp., fuzzy regular closed) if μ=int(cl(μ)) (resp., μ=cl(int(μ))).The fuzzyδ-interior of fuzzy set μ of X is the union of all fuzzy regular open sets contained in μ. A fuzzy set μ is called fuzzy δ-open [5] if μ=intδ(μ). The complement of fuzzy δ-open set is called fuzzy δ-closed (i.e, μ=clδ(μ)). A fuzzy set μ of X is called fuzzy δ-preopen [6] (resp., fuzzy δ-semi open [7]) if μ≤int(clδ(μ)) (resp., μ≤cl(intδ(μ))). The complement of a fuzzy δ-preopen set (resp., fuzzy δ-semiopen set) is called fuzzy δ-preclosed (resp., fuzzy δ-semiclosed).Definition 1.
A fuzzy setμ of a fuzzy topological space X is called fuzzy e-open [3] if μ≤cl(intδμ)∨int(clδμ). Fuzzy e-closed if μ≥cl(intδμ)∧int(clδμ).
The intersection of all fuzzy e-closed sets containingμ is called fuzzy e-closure of μ and is denoted by fe-clμ and the union of all fuzzy e-open sets contained in μ is called fuzzy e-interior of μ and is denoted by fe-intμ.Definition 2.
A mappingf:X→Y is said to be fuzzy e∗-open [8] if the image of every fuzzy e-open set in X is fuzzy e-open set in Y.Definition 3.
A functionf:X→Y is called fuzzy e-irresolute [3]. f-1(λ) is fuzzy e-open in X for every fuzzy e-open set λ of Y.Definition 4.
A fuzzy setμ is quasicoincident [9] with a fuzzy set λ denoted by μqλ iff there exist x∈X such that μ(x)+λ(x)>1. If μ and λ are not quasicoincident, then we write μq¯λ and μ≤λ⇔μq¯1-λ.Definition 5.
A fuzzy pointxp is quasicoincident [9] with a fuzzy set λ denoted by xpqλ iff there exist x∈X such that p+λ(x)>1.Definition 6.
A fuzzy topological space(X,τ) is said to be fuzzy e-T1 [3] if for each pair of distinct points x and y of X there exist fuzzy e-open sets μ1and μ2 such that x∈μ1 and y∈μ2 and x∉μ2 and y∉μ1.Definition 7.
A fuzzy topological space(X,τ) is said to be fuzzy e-T2 [3] if for each pair of distinct points x and y of X there exists disjoint fuzzy e-open sets η and ρ such that x∈η and y∈ρ.Definition 8.
A fuzzy topological spaceX is said to be fuzzy weakly Hausdorff [10] if each element of X is an intersection of fuzzy regular closed sets.Definition 9.
A fuzzy topological spaceX is said to be fuzzy e-normal [3] if for every two disjoint fuzzy closed sets η and ρ of X there exist two disjoint fuzzy e-open sets μ and λ such that η≤μ and ρ≤λ and μ∧λ=0.Definition 10.
A fuzzy topological spaceX is said to be fuzzy strongly normal [10] if for every two disjoint fuzzy closed sets η and ρ of X there exist two disjoint fuzzy open sets μ and λ such that η≤μ and ρ≤λ.Definition 11.
A fuzzy topological spaceX is said to be fuzzy Urysohn [11] if for every distinct points x and y in X there exist fuzzy open sets μ and λ in X such that x∈μ and y∈λ and cl(μ)∧cl(λ)=0.Definition 12.
A space(X,τ) is called fuzzy S-closed [2] (resp., fuzzy e-compact [3]) if every fuzzy regular closed (resp., fuzzy e-open) cover of X has a finite subcover.Definition 13.
A functionf:X→Y is called fuzzy completely continuous [12] if f-1λ is fuzzy regular open in X for every fuzzy open set λ of Y.Definition 14.
A fuzzy filter baseξ is said to be fuzzy rc-convergent [10] to a fuzzy point xε in X if for any fuzzy regular closed set η in X containing xε there exists a fuzzy set ρ∈ξ such that ρ≤η.Definition 15.
A collection of fuzzy subsetsΔ of a fuzzy topological spaces X is said to form fuzzy filterbases [13] iff for every finite collection {λα:α=1,2,…,n}, ⋀α=1nλα≠0X.


## 3. FuzzyCe-I(ec,eo) Functions

In this section, the notion of fuzzyCe-I(ec,eo) functions is introduced and some characteristics and properties are studied.Definition 16.
A mappingφ:(X,τ)→(Y,σ) is called fuzzy Ce-I(ec,eo) if the inverse image of every fuzzy e-open set of Y is fuzzy e-closed in X.Remark 17.
The concepts of fuzzyCe-I(ec,eo) and fuzzy e-irresolute are independent notions as illustrated in the following example.Example 18.
LetX={a,b,c} and Y={x,y,z} and the fuzzy sets μ1,μ2,υ be defined as follows: (1)μ1a=0.1,μ2a=0.7,υx=0.2,μ1b=0.6,μ2b=0.5,υy=0.2,μ1c=0.4,μ2c=0.6,υz=0.4.Let τ={0,1,μ1,μ2,μ1∨μ2,μ1∧μ2} and σ={0,υ,1}. Then, the mapping φ:(X,τ)→(Y,σ) is defined by φ(a)=x,φ(b)=y,φ(c)=z. Then, φ is fuzzy Ce-I(ec,eo) but not fuzzy e-irresolute.Example 19.
LetX={a,b,c} and Y={x,y,z} and the fuzzy sets η1,η2,ρ are defined as follows:(2)η1a=0.4,η2a=0.3,ρx=0.4,η1b=0.7,η2b=0.5,ρy=0.5,η1c=0.8,η2c=0.2,ρz=1.
Letτ={0,1,η1,η2} and σ={0,ρ,1}. Then, the mapping φ:(X,τ)→(Y,σ) is defined by φ(a)=x,φ(b)=y,φ(c)=z. Then, φ is fuzzy e-irresolute but not fuzzy Ce-I(ec,eo).Definition 20.
A mappingφ:(X,τ)→(Y,σ) is called fuzzy Ce-continuous if the inverse image of every fuzzy open set of Y is fuzzy e-closed in X.Remark 21.
Every fuzzyCe-I(ec,eo) function is fuzzy Ce-continuous, but not conversely from the following example.Example 22.
LetX={a,b,c} and Y={x,y,z} and the fuzzy sets μ1,μ2,ν,υ are defined as follows: (3)μ1a=0.6,μ2a=0.4,υa=0.8,νx=0.3,μ1b=0.5,μ2b=0.7,υb=0.5,νy=0.4,μ1c=0.3,μ2c=0.5,υc=0.3,νz=0.6.Let τ={0,1,μ1,μ2,μ1∨μ2,μ1∧μ2} and σ={0,ν,1}. Then, the mapping φ:(X,τ)→(Y,σ) is defined by φ(a)=x,φ(b)=y, and φ(c)=z. Then, φ is fuzzy Ce-continuous but not fuzzy Ce-I(ec,eo) as the fuzzy set υ is fuzzy e-open in Y but φ-1(υ) is not fuzzy e-closed set in X.Theorem 23.
For a fuzzy functionφ:X→Y, if φ(xε)qμ, the inverse image of for every fuzzy e-closed set of Y is fuzzy e-open in X iff for any xε∈X; if φ(xε)qμ, then xεqfe-int(φ-1(μ)).Proof.
Letμ≤Y be a fuzzy e-closed set and φ(xε)qμ. Then, xεqφ-1(μ) and, by hypothesis, φ-1(μ)=fe-int(φ-1(μ)). We obtain, xεqfe-int(φ-1(μ)). Converse can be shown easily.Theorem 24.
For a fuzzy functionφ:X→Y, if φ(xε)qμ, for any fuzzy e-closed set μ≤Y and for any xε∈X, xεqfe-int(φ-1(μ)) iff there exists a fuzzy e-open set δ such that xεqδ and φ(δ)≤μ.Proof.
Letμ≤Y be any fuzzy e-closed set and let φ(xε)qμ. Then, xεqfe-int(φ-1(μ)). Take δ=fe-int(φ-1(μ)) then φ(δ)=φ(fe-int(φ-1(μ)))≤φ(φ-1(μ))≤μ, and δ is fuzzy e-open in X and xεqδ.
Conversely, letμ≤Y be any fuzzy e-closed set and let
φ(xε)qμ. By hypothesis, there exists fuzzy e-open set δ such that xεqδ and φ(δ)≤μ. This implies, δ≤φ-1(μ) and then xεqfe-int(φ-1(μ)).Theorem 25.
For a fuzzy functionφ:X→Y, the following statements are equivalent:(1)
fis fuzzy Ce-I(ec,eo).(2)
For every fuzzy e- closed setμ in Y, φ-1(μ) is fuzzy e-open in X.(3)
For every fuzzy open setμ, φ-1(fe-intμ) is fuzzy e-closed.(4)
For every fuzzy closed setη, φ-1(fe-cl⁡(η)) is fuzzy e-open.(5)
For eachxε∈X and each fuzzy e-closed set μ in Y containing φ(xε), there exists a fuzzy e-open set ρ in X containing xε such that φ(ρ)≤μ.(6)
For eachxε∈X and each fuzzy e-open set μ in Y noncontaining φ(xε), there exists a fuzzy e-closed set ν in X noncontaining xε such that φ-1(μ)≤ν.Proof.
(1)⇔ (2): let ρ be a fuzzy e-open set in Y. Then, 1Y-ρ is fuzzy e-closed. By (2), φ-1(1Y-ρ)=1X-φ-1(ρ) is fuzzy e-open. Thus, φ-1(ρ) is fuzzy e-closed. Converse can be shown easily.
(1)⇔ (3): let μ be a fuzzy open set. Since fe-int(μ) is fuzzy e-open, then by (1) it follows that φ-1(fe-int(μ)) is fuzzy e-closed. The converse is easy to prove.
(2)⇔ (4): let η be a fuzzy closed set. Since fe-cl(η) is fuzzy e-closed set, then by (2) it follows that φ-1(fe-cl(η)) is fuzzy e-open. The converse is easy to prove.
(2)⇔ (5): let μ be any fuzzy e-closed set in Y containing φ(xε). By (2), φ-1(μ) is fuzzy e-open set in X and xε∈φ-1(μ). Take ρ=φ-1(μ). Then, φ(ρ)≤μ. The converse can be shown easily.
(5)⇔ (6): let μ be any fuzzy e-open set in Y noncontaining φ(xε). Then, 1-μ is a fuzzy e-closed set containing φ(xε). By (5), there exists a fuzzy e-open set ρ in X containing xε such that φ(ρ)≤1-μ. Hence, ρ≤φ-1(1-μ)=1-φ-1(μ) and φ-1(μ)≤1-ρ. Take ν=1-ρ. We obtain that ν is a fuzzy e-closed set in X noncontaining xε. The converse can be shown easily.Theorem 26.
Letϕ:X→Y be a function and let φ:X→X×Y be the fuzzy graph function of ϕ, defined by φ(xε)=(xε,ϕ(xε)) for every xε∈X. If φ is fuzzy Ce-I(ec,eo), then ϕ is fuzzy Ce-I(ec,eo).Proof.
Letμ be a fuzzy e-closed set in Y; then, 1X×μ is a fuzzy e-closed set in X×Y. Since φ is fuzzy Ce-I(ec,eo), then ϕ-1(μ)=φ-1(1X×μ) is fuzzy e-open in X. Thus, ϕ is fuzzy Ce-I(ec,eo).Theorem 27.
LetYλ:λ∈Λ be a family of product spaces. If a function φ:X→∏Yλ is fuzzy Ce-I(ec,eo), then Pλ∘φ:X→Yλ is fuzzy Ce-I(ec,eo) for each λ∈Λ where Pλ is the projection of ∏‍⁡Yλ onto Yλ.Proof.
Letδ be any fuzzy e-open set in Yλ. Since Pλ is a fuzzy continuous and fuzzy open set, it is a fuzzy e-open set. Now Pλ:∏Yλ→Yλ, Pλ-1(δ) is a fuzzy e-open in ∏Yλ. Therefore, Pλ is a fuzzy e-irresolute function. Now (Pλ∘φ)-1(δ)=φ-1(Pλ-1(δ)), since φ is fuzzy Ce-I(ec,eo). Hence φ-1(Pλ-1(δ)) is a fuzzy e-closed set, since Pλ-1(δ) is a fuzzy e-open set. Hence, Pλ∘φ is fuzzy Ce-I(ec,eo).Theorem 28.
If the functionφ:∏‍Xλ→∏‍Yλ is fuzzy Ce-I(ec,eo), then φλ:Xλ→Yλ is fuzzy Ce-I(ec,eo) for each λ∈Λ.Proof.
Letλ0∈Λ be an arbitrary fixed index and let υλ0 be any fuzzy e-open set of Yλ0; then, ∏‍Yμ×υλ0 is fuzzy e-open in ∏‍Yλ, where λ0≠μ∈Λ. Since φ is fuzzy Ce-I(ec,eo) function, then φ-1∏‍Yμ×υλ0=∏‍Xμ×φλ0-1υλ0 is fuzzy e-closed in ∏‍Xλ and hence φλ0-1Vλ0 is fuzzy e-closed in Xλ0. This implies φλ0 is fuzzy Ce-I(ec,eo).Theorem 29.
Ifφ:X→Y is fuzzy Ce-I(ec,eo) and δ is fuzzy closed set of X, then φ|δ:δ→Y is fuzzy Ce-I(ec,eo).Proof.
Letλ be a fuzzy e-open set of Y; then, φ|δ-1(λ)=φ-1(λ)∧δ. Since φ-1(λ) and δ are fuzzy closed, hence φ|δ-1(λ) is fuzzy e-closed in the relative topology of δ.Definition 30.
The intersection of all fuzzy e-open setη of a fuzzy topological space (X,τ) containing μ is called the fuzzy e-kernel of μ (briefly, fe-Kμ), fe-Kμ=⋀η:μ≤η,ηis  fuzzy  e-open  set  ofX.
The following properties hold for fuzzy setsμ,λ of X:(1)
x∈fe-Kμ iff μ∧γ≠0 for any fuzzy e-closed set γ containing x.(2)
μ≤fe-Kμ and μ=fe-Kμ if μ is fuzzy e-open in X.(3)
μ≤λ; then, fe-Kμ≤fe-Kλ.Theorem 31.
For a fuzzy functionφ:X→Y, the following statements are equivalent:(1)
φ is fuzzy Ce-I(ec,eo).(2)
φ(fe-cl(μ))≤fe-Kφ(μ) for every fuzzy set μ of X.(3)
fe-cl⁡(φ-1(η))≤φ-1(fe-Kη) for every fuzzy set η of Y.Proof.
(
1
)
⇒
(
2
): let μ≤X and y∉fe-Kφ(μ). There exists a fuzzy e-closed set γ in Y, such that y∈γ and φ(μ)∧γ=0. Therefore, φ-1(φ(μ)∧γ)=0. This implies that μ∧φ-1(γ)=0 and fe-cl(μ)∧φ-1(γ)=0. Thus, φ(fe-clμ)∧γ=0 and y∉φ(fe-clμ). Hence, φ(fe-clφ(μ))≤fe-Kφ(μ).
(
2
)
⇒
(
3
): let η≤Y; then, φ-1(η)≤X. By hypothesis, φ(fe-clφ-1(η))≤fe-Kφ(φ-1(η))≤fe-Kη. Hence, fe-cl(φ-1(η))≤φ-1(fe-Kη).
(
3
)
⇒
(
1
): let η be any fuzzy e-open set of Y; we have fe-cl(φ-1(η))≤φ-1(fe-Kη)=φ-1(η), since η is fuzzy e-open and fe-cl(φ-1(η))=φ-1(η). This implies that φ-1(η) is fuzzy e-closed in X.Definition 32.
The fuzzy e-Frontier of a fuzzy setγ of a fuzzy topological space X is given by fe-Fr(γ)=fe-cl(γ)∧fe-cl(1X-γ).Theorem 33.
The fuzzy pointxε∈X such that φ:X→Y is not fuzzy Ce-I(ec,eo) is exactly the union of fuzzy e-Frontier if the inverse image of the fuzzy e-closed set in Y contains φ(xε).Proof.
Suppose thatφ is not fuzzy Ce-I(ec,eo) at the point xε∈X; then there exists a fuzzy e-closed set γ such that φ(xε)∈γ and φ(μ)∧(1Y-γ)≠0 for all fuzzy e-open set μ such that xε∈μ. It follows that μ∧φ-1(1Y-γ)≠0 and hence xε∈fe-clφ-1(1Y-γ)=fe-cl(1X-φ-1(γ)). Thus, xε∈φ-1(γ)≤fe-cl(φ-1(γ)) and hence xε∈fe-Fr(φ-1(γ)).
Conversely, suppose thatxε∈fe-Fr(φ-1(γ)), γ is fuzzy e-closed set of Y containing φ(xε), and φ is fuzzy Ce-I(ec,eo) at xε∈X. There exists fuzzy e-open set μ such that xε∈μ and μ≤φ-1(γ). Thus, xε∈fe-intφ-1(γ) and hence xε∉fe-Fr(φ-1(γ)) for each fuzzy e-closed set γ of Y containing φ(xε), a contradiction. Therefore, φ is not fuzzy Ce-I(ec,eo).Theorem 34.
The following hold for functionsϕ:X→Y and φ:Y→Z:(a)
Ifϕ:X→Y is fuzzy Ce-I(ec,eo) and φ:Y→Z is fuzzy Ce-continuous then φ∘ϕ:X→Z is fuzzy Ce-continuous.(b)
Ifϕ:X→Y is fuzzy Ce-I(ec,eo) and φ:Y→Z is fuzzy e-irresolute then φ∘ϕ:X→Z is fuzzy Ce-I(ec,eo).Theorem 35.
Ifϕ:X→Y is a fuzzy e-irresolute surjective function and φ:Y→Z is a fuzzy function such that φ∘ϕ:X→Z is fuzzy Ce-I(ec,eo), then φ is fuzzy Ce-I(ec,eo).Proof.
Letη be any fuzzy e-closed set in Z. Since φ∘ϕ is fuzzy Ce-I(ec,eo), (φ∘ϕ)-1(η) is fuzzy e-open in X. Therefore, ϕ-1(φ-1(η))=(φ∘ϕ)-1(η) is fuzzy e-open in X. Since ϕ is fuzzy e-irresolute, surjection implies ϕ(ϕ-1(φ-1(η)))=φ-1(η) is fuzzy e-open in Y. Thus, φ is fuzzy Ce-I(ec,eo).Theorem 36.
Ifϕ:X→Y is a fuzzy e∗-open surjective function and φ:Y→Z is a fuzzy function such that φ∘ϕ:X→Z is fuzzy Ce-continuous, then φ is fuzzy Ce-continuous.Proof.
Letη be any fuzzy closed set in Z. Since φ∘ϕ is fuzzy Ce-continuous, (φ∘ϕ)-1(η) is fuzzy e-open in X. Therefore, ϕ-1(φ-1(η))=(φ∘ϕ)-1(η) is fuzzy e-open in X. Since ϕ is fuzzy e∗-open, surjection implies ϕ(ϕ-1(φ-1(η)))=φ-1(η) is fuzzy e-open in Y. Thus, φ is fuzzy Ce-continuous.


## 4. Fuzzy CompletelyCe-I(rc,eo) Functions

In this section, the notion of fuzzy completelyCe-I(rc,eo) functions is introduced and the relation between other functions is studied and further some structure preservation properties are investigated.Definition 37.
A mappingφ:(X,τ)→(Y,σ) is called fuzzy completely Ce-I(rc,eo) if inverse image of every fuzzy e-open set in Y is fuzzy regular closed in X.Example 38.
LetX={x,y,z} and the fuzzy sets μ1,μ2 are defined as follows: (4)μ1x=0.4,μ2x=0.7,μ1y=0.6,μ2y=0.3,μ1z=0.1,μ2z=0.5.Let τ={0,1,μ1,μ2,μ1∨μ2,μ1∧μ2} and σ={0,μ1,μ1∧μ2,1}. Then, the mapping φ:(X,τ)→(X,σ) is defined by φ(x)=1-x. Then, φ is fuzzy completely Ce-I(rc,eo).Remark 39.
Every fuzzy completelyCe-I(rc,eo) function is fuzzy Ce-I(ec,eo) and fuzzy Ce-continuous, but the converse is not true, which can be seen in the following example.Example 40.
LetX={a,b,c} and Y={x,y,z} and the fuzzy sets μ1,μ2, and ν are defined as follows: (5)μ1a=0.6,μ2a=0.3,νx=0.4,μ1b=0.5,μ2b=0.7,νy=0.3,μ1c=0.2,μ2c=0.8,νz=0.1.Let τ={0,1,μ1,μ2,μ1∨μ2,μ1∧μ2} and σ={0,ν,1}. Then, the mapping φ:(X,τ)→(Y,σ) is defined by φ(a)=x,φ(b)=y,φ(c)=z. Then, φ is fuzzy Ce-continuous and also fuzzy Ce-I(ec,eo) but not fuzzy completely Ce-I(rc,eo) as the fuzzy set υ is fuzzy e-open in Y but φ-1(υ) is not fuzzy regular closed set in X.From the above examples, we have the following implications.None of the these implications is reversible.Theorem 41.
For a fuzzy functionφ:X→Y, if φ(xε)qμ, the inverse image of every fuzzy e-closed set of Y is fuzzy δ-open in X iff for any xε∈X if φ(xε)qμ, then xεqintδ(φ-1(μ)).Proof.
Letμ≤Y be a fuzzy e-closed set and φ(xε)qμ. Then, xεqφ-1(μ) and, by hypothesis, φ-1(μ)=intδ(φ-1(μ)). From here, xεqintδ(φ-1(μ)). The converse can be shown easily.Theorem 42.
For a fuzzy functionφ:X→Y, if φ(xε)qμ, for any fuzzy e-closed set μ≤Y and for any xε∈X, xεqintδφ-1μ iff there exists a fuzzy δ-open set θ such that xεqθ and φ(θ)≤μ.Proof.
Letμ≤Y be any fuzzy e-closed set and let φ(xε)qμ. Then, xεqintδ(φ-1(μ)). Take θ=intδ(φ-1(μ)); then, φ(θ)=φ(intδ(φ-1(μ)))≤φ(φ-1(μ))≤μ; θ is fuzzy δ-open in X and xεqθ.
Conversely, letμ≤Y be any fuzzy e-closed set and let φ(xε)qμ. By hypothesis, there exists fuzzy δ-open set θ such that xεqθ and φ(θ)≤μ. This implies θ≤φ-1(μ) and then xεqintδ(φ-1(μ)).Theorem 43.
For a fuzzy functionφ:X→Y, the following statements are equivalent:(1)
f is fuzzy completely Ce-I(rc,eo).(2)
For every fuzzy e-closed setμ in Y, φ-1(μ) is fuzzy regular open in X.(3)
For every fuzzy open setμ, φ-1(fe-intμ) is fuzzy regular closed.(4)
For every fuzzy closed setη, φ-1(fe-cl⁡(η)) is fuzzy regular open.(5)
For eachxε∈X and each fuzzy e-closed set μ in Y containing φ(xε), there exists a fuzzy regular open set ρ in X containing xε such that φ(ρ)≤μ.(6)
For eachxε∈X and each fuzzy e-open set μ in Y non containing φ(xε), there exists a fuzzy regular closed set ν in X noncontaining xε such that φ-1(μ)≤ν.Proof.
(1)⇔ (2): let ρ be a fuzzy e-open set in Y. Then, 1Y-ρ is fuzzy e-closed. By (2), φ-1(1Y-ρ)=1X-φ-1(ρ) is fuzzy regular open. Thus, φ-1(ρ) is fuzzy regular closed. Thus, φ is fuzzy completely Ce-I(rc,eo).The converse can be shown easily.
(1)⇔ (3): let μ be a fuzzy open set. Since fe-int(μ) is fuzzy e-open, then by (1) it follows that φ-1(fe-int(μ)) is fuzzy regular closed. The converse is easy to prove.
(2)⇔ (4): let η be a fuzzy closed set. Since fe-cl(η) is fuzzy e-closed set, then by (2) it follows that φ-1(fe-cl(η)) is fuzzy regular open. The converse is easy to prove.
(2)⇔ (5): let μ be any fuzzy e-closed set in Y containing φ(xε). By (2), φ-1(μ) is fuzzy regular open set in X and xε∈φ-1(μ). Take ρ=φ-1(μ). Then, φ(ρ)≤μ. The converse can be shown easily.
(5)⇔ (6): let μ be any fuzzy e-open set in Y noncontaining φ(xε). Then, 1-μ is a fuzzy e-closed set containing φ(xε). By (5), there exists a fuzzy regular open set ρ in X containing xε such that φ(ρ)≤1-μ. Hence, ρ≤φ-1(1-μ)=1-φ-1(μ) and φ-1(μ)≤1-ρ. Take ν=1-ρ. We obtain that ν is a fuzzy regular closed set in X noncontaining xε. The converse can be shown easily.Theorem 44.
Letφ1:X→Y be a function and let φ2:X→X×Y be the fuzzy graph function of φ1, defined by φ2(xε)=(xε,φ1(xε)) for every xε∈X. If φ2 is fuzzy completely Ce-I(rc,eo), then φ1 is fuzzy completely Ce-I(rc,eo).Proof.
Letμ be a fuzzy e-closed set in Y; then, 1X×μ is a fuzzy e-closed set in X×Y. Since φ2 is fuzzy completely Ce-I(rc,eo), then φ1-1(μ)=φ2-1(1X×μ) is fuzzy regular open in X. Thus, φ1 is fuzzy completely Ce-I(rc,eo).Theorem 45.
The following holds for functionsφ1:X→Y and φ2:Y→Z: (a)
Ifφ1:X→Y is fuzzy Ce-I(ec,eo) and φ2:Y→Z is fuzzy completely Ce-I(rc,eo), then φ2∘φ1:X→Z is fuzzy e-irresolute.(b)
Ifφ1:X→Y is fuzzy completely Ce-I(rc,eo) and φ2:Y→Z is fuzzy Ce-continuous, then φ2∘φ1:X→Z is fuzzy completely continuous.Definition 46.
A fuzzy filter baseξ is said to be fuzzy e-convergent to a fuzzy point xε in X if for any fuzzy e-open set η in X containing xε there exists a fuzzy set ρ∈ξ such that ρ≤η.Theorem 47.
If a fuzzy functionφ:X→Y is fuzzy completely Ce-I(rc,eo) for each fuzzy point xε∈X and each fuzzy filter base ξ in X is fuzzy rc-convergent to xε, then the fuzzy filter base φ(ξ) is fuzzy e-convergent to φ(xε).Proof.
Letxε∈X and let ξ be any fuzzy filter base in X which is fuzzy rc-converging to xε. Since φ is fuzzy completely Ce-I(rc,eo), then for any fuzzy e-open set η in Y containing φ(xε), there exists a fuzzy regular closed set ρ in X containing xε such that φ(ρ)≤η. Since ξ is fuzzy rc-converging to xε, there exists a δ∈ξ such that δ≤ρ. This means that φ(δ)≤ρ and therefore the fuzzy filter base φ(ξ) is fuzzy e-convergent to φ(xε).Theorem 48.
Ifφ:X→Y is a fuzzy completely Ce-I(rc,eo) surjection and X is fuzzy S-closed, then Y is fuzzy e-compact.Proof.
Suppose thatφ:X→Y is a fuzzy completely Ce-I(rc,eo) surjection and X is fuzzy S-closed. Let νii∈I be a fuzzy e-open cover of Y. Since φ is a fuzzy completely Ce-I(rc,eo), then {φ-1(νi)}i∈I is fuzzy regular closed cover of X and hence there exists finite set I0 of I such that X=⋁{φ-1(νi);i∈I0}. Therefore, we have Y=⋁{νi;i∈I0} and Y is fuzzy e-compact.Theorem 49.
Ifφ:X→Y is a fuzzy completely Ce-I(rc,eo) injection and Y is fuzzy e-T1, then X is fuzzy weakly Hausdorff.Proof.
SupposeY is fuzzy e-T1. For any distinct fuzzy points xε and yυ in X, there exist fuzzy e-open sets η and ρ in Y. Since φ is injective, φ(xε)∈η,φ(yυ)∉η,φ(xε)∉ρ, and φ(yυ)∈ρ. Since φ is fuzzy completely Ce-I(rc,eo), φ-1(η) and φ-1(ρ) are fuzzy regular closed sets of X such that xε∈φ-1(η),yυ∉φ-1(η),xε∉φ-1(ρ), and yυ∈φ-1(ρ). This shows that X is fuzzy weakly Hausdorff.Theorem 50.
Ifφ:X→Y is a fuzzy completely Ce-I(rc,eo) injection and Y is fuzzy e-normal, then X is fuzzy strongly normal.Proof.
Letη and ρ be disjoint nonempty fuzzy closed sets of X. Since φ is injective, φ(η) and φ(ρ) are disjoint fuzzy closed sets. Since Y is fuzzy e-normal, there exist fuzzy e-open sets μ and λ such that φ(η)≤μ and φ(ρ)≤λ and μ∧λ=0. This implies that fe-cl(μ) and fe-cl(λ) are fuzzy e-closed sets in Y. Then, since φ is fuzzy completely Ce-I(rc,eo), φ-1(fe-cl(μ)) and φ-1(fe-cl(λ)) are fuzzy regular open sets. Then, η≤φ-1(fe-cl(μ)) and ρ≤φ-1(fe-cl(λ)) and φ-1(fe-cl(μ)) and φ-1(fe-cl(λ)) are disjoint; by definition X is fuzzy strongly normal.Definition 51.
A fuzzy topological space(X,τ) is said to be fuzzy e-T0(r-T0 [14]) if for every fuzzy set λ of X can be written in the form λ=⋁i∈I⋀j∈Jλij, where λij are fuzzy e-open (fuzzy regular open) or fuzzy e-closed (fuzzy regular closed) sets of Y.Theorem 52.
Ifφ:X→Y is a fuzzy completely Ce-I(rc,eo) injection and Y is fuzzy e-T0, then X is fuzzy r-T0.Proof.
Letη be a any fuzzy set of X. Since Y is fuzzy e-T0, φ(η) is fuzzy e-open set of Y. Then, φ(η)=⋁i∈I⋀j∈Jλij, where λij are fuzzy e-open set or fuzzy e-closed sets of Y. Since φ is completely Ce-I(rc,eo) injection we have η=φ-1(φ(η))=φ-1(⋁i∈I⋀j∈Jλij)=⋁i∈I⋀j∈Jφ-1(λij), where φ-1(λij) are fuzzy regular open sets or fuzzy regular closed sets of X. Thus, X is fuzzy r-T0.Theorem 53.
Ifφ:X→Y is a fuzzy completely Ce-I(rc,eo) injection and Y is fuzzy e-T2, then X is fuzzy Urysohn.Proof.
Letxε and yυ be any two distinct fuzzy points in X. Since φ is injective, φ(xε)≠φ(yυ) in Y. Since Y is fuzzy e-T2, there exist fuzzy e-open sets η and ρ in Y such that φ(xε)∈η and φ(yυ)∈ρ and η∧ρ=0. This implies that fe-cl(η) and fe-cl(ρ) are fuzzy e-closed sets in Y. Then, since φ is fuzzy completely Ce-I(rc,eo), there exists fuzzy regular open sets δ and γ in X containing xε and yυ, respectively, such that φ(δ)≤fe-cl(η) and φ(γ)≤fe-cl(ρ). This implies that δ≤φ-1(fe-cl(η)) and γ≤φ-1(fe-cl(ρ)); we have that φ-1(fe-cl(η)) and φ-1(fe-cl(ρ)) are disjoint and hence cl(δ)∧cl(γ)=0; by definition, X is fuzzy Urysohn.



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