# Wθg-Closed and Wδg-Closed in [0,1]-Topological Spaces

Authors: M. E. El-Shafei; A. H. Zakari
Journal: International Journal of Mathematics and Mathematical Sciences
 (2011)

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## Abstract

We investigate various classes of generalized closed  fuzzy sets in[0,1]-topological spaces, namely, Wθg-closed fuzzy sets and Wδg-closed fuzzy sets. Also, we introduce a new separation axiom FT3/4∗ of the [0,1]-topological spaces, and we prove that every FT3/4∗-space is a FT3/4-space. Furthermore, we using the new generalized closed fuzzy sets to construct new types of fuzzy mappings.


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## Body

## 1. Introduction

In 1970, Levine [1] introduced the notion of generalized closed sets in topological spaces as a generalization of closed sets. Since then, many concepts related to generalized closed sets were defined and investigated. In 1997, Balasubramanian and Sundaram [2] introduced the concepts of generalized closed sets in fuzzy setting. Also, they studied various generalizations fuzzy continues mappings.Recently, El-Shafei and Zakari [3–5] introduced new types of generalized closed fuzzy sets in [0,1]-topological spaces and studied many of their properties. Also, they studied various generalizations fuzzy continues mappings.In the present paper, we introduce the concepts ofWθg-closed fuzzy sets and Wδg-closed fuzzy sets and study some of their properties. Also, we introduce the concept of FT3/4*-space. Moreover, we introduce and study the concepts of two new classes of fuzzy mappings, namely, fuzzy Wθg-continuous mappings and fuzzy Wδg-irresolute mappings.


## 2. Preliminaries

LetX be a set and I the unit interval. A fuzzy set in X is an element of the set of all  functions  from X into I. The family of all fuzzy sets in X is denoted by IX. A  fuzzy singleton xt is a fuzzy   set   in   X  defined    by xt(x)=t, xt(y)=0 for all y≠x, t∈(0,1]. The set of all fuzzy singletons in X is denoted by S(X). For every xt∈S(X) and μ∈IX, we define xt∈μ if and only if t≤μ(x). A fuzzy set μ is called quasicoincident with a fuzzy set ρ, denoted by μqρ, if and only if there exists x∈X such that μ(x)+ρ(x)>1. If μ is not quasicoincident with ρ, then we write μq̅ρ. By cl(μ), int(μ), μc, N(xt,τ), and NQ(xt,τ), we mean the fuzzy closure of μ, the fuzzy interior of μ, the complement of μ, the class of all open neighborhoods of xt, and the class of all open Q-neighborhoods of xt, respectively.Definition 2.1 (see [6, 7]).
A fuzzy subsetμ of a [0,1]-topological space (X,τ) is called(i)
regular open if and only ifμ=int(cl(μ)),(ii)
preopen if and only ifμ≤int(cl(μ)).
The complement of a regular open (resp. preopen) fuzzy set is called a regular closed (resp. preclosed).Definition 2.2 (see [8, 9]).
Let(X,τ) be a [0,1]-topological space, xt∈S(X), and μ∈IX. Then,(i)
theθ-closure of μ, denoted by clθ(μ), is defined byxt∈clθ(μ) if and only if cl(η)qμ for each η∈NQ(xt,τ),(ii)
theδ-closure of μ denoted by clδ(μ), is defined byxt∈clδ(μ) if and only if int(cl(η))qμ for each η∈NQ(xt,τ),(iii)
μ is called θ-closed (resp. δ-closed) if and only if μ=clθ(μ) (resp. μ=clδ(μ)).Definition 2.3 (see [9]).
Let(X,τ) be a [0,1]-topological space and μ∈IX. Then,(i)
the familyγ={ηj:j∈J}⊆τ is called an open P-cover of μ if and only if for every xt∈μ, there exists j0∈J such that xt∈ηj0,(ii)
μ is called a C-set if and only if every open P-cover of μ has a finite subcover.Definition 2.4 (see [2–4]).
Let(X,τ) be a [0,1]-topological space. A fuzzy set μ∈IX is called(i)
a generalized closed (g-closed, for short) if and only if cl(μ)≤η whenever μ≤η and η is open fuzzy set,(ii)
aθ-generalized closed (θg-closed, for short) if and only if clθ(μ)≤η whenever μ≤η and η is open fuzzy set,(iii)
aδ-generalized closed (δg-closed, for short) if and only if clδ(μ)≤η whenever μ≤η and η is open fuzzy set.Definition 2.5 (see [2, 4, 6, 10]).
A[0,1]-topological space (X,τ) is called(i)
FR1 if and only if xtq̅cl(yr) implies that there exist η∈N(xt,τ) and υ∈N(yr,τ) such that ηq̅υ,(ii)
FR2 or F-regular if and only if xtq̅λ, λ is closed fuzzy set implies that there exist η∈N(xt,τ) and υ∈τ, λ≤υ such that ηq̅υ,(iii)
FT1/2 if and only if every g-closed fuzzy set in X is closed,(iv)
FT3/4 if and only if every δg-closed fuzzy set in X is δ-closed,(v)
fuzzy weakly Hausdorff (FWT2, for short) if xtq̅yr implies that there exists regular open fuzzy set η∈N(xt,τ) such that yrq̅η,(vi)
fuzzy semiregular if and only if the collection of all regular open fuzzy sets inX forms a base for the [0,1]-topology τ,(vii)
a fuzzy partition space if and only if every open fuzzy subset is closed.Definition 2.6 (see [2–4, 11]).
A fuzzy mappingf:(X,τ)→(Y,Δ) is called(i)
fuzzy generalized continuous  (fuzzyg-continuous, for short) if and only if f-1(η) is g-closed in X for any closed fuzzy set η in Y,(ii)
fuzzyθ-generalized continuous  (fuzzy θg-continuous, for short) if and only if f-1(η) is θg-closed in X for any closed fuzzy set η in Y,(iii)
fuzzyδ-generalized continuous  (fuzzy δg-continuous, for short) if and only if f-1(η) is δg-closed in X for any closed fuzzy set η in Y,(iv)
fuzzyδ-continuous if the inverse image of every δ-open fuzzy set in Y is δ-open in X,(v)
fuzzyδ-open (fuzzy δ-open, for short) if and only if f(η) is δ-open in Y for  any δ-open fuzzy set η in X,(vi)
fuzzyδ-closed (fuzzy δ-closed, for short) if and only if f(η) is δ-closed in Y for any δ-closed fuzzy set η in X.Theorem 2.7 (see [3]).
A fuzzy subsetμ of an FR2-fts (X,τ) is θg-closed if and only if it is g-closed.Theorem 2.8 (see [3]).
Let(X,τ) be a [0,1]-topological space. Then, the following conditions are equivalent:(i)
(X,τ) is FR1-space,(ii)
for eachC-set μ∈IX, cl(μ)=clθ(μ),(iii)
for eachxt∈S(X), cl(xt)=clθ(xt).Theorem 2.9 (see [3, 4]).
Let(X,τ) be a [0,1]-topological space and μ∈IX be a preopen. Then, μ is θg-closed (resp.δg-closed) if and only if it is  g-closed.Theorem 2.10 (see [3]).
Let(X,τ) be a [0,1]-topological space and μ,η∈IX. Then,(i)
clθ(μ∨η)=clθ(μ)∨clθ(η),(ii)
clθ(μ∧η)≤clθ(μ)∧clθ(η).Theorem 2.11 (see [4]).
Let(X,τ) be a fuzzy semiregular space and μ∈IX. Then, (i)
μ is δg-closed if and only if μ is g-closed,(ii)
If, in addition,(X,τ) is FT1/2, then μ is δg-closed if and only if μ is closed.Theorem 2.12 (see [4]).
Let(X,τ) be an FR1-space and μ∈IX be a C-set. Then, μ isδg-closed if and only if it is g-closed.Theorem 2.13 (see [4]).
Let(X,τ) be a fuzzy partition space and μ∈IX. Then, μ is δg-closed if and only if it is  g-closed.Theorem 2.14 (see [4]).
Let(X,τ) be a [0,1]-topological space and μ,η∈IX. Then,(i)
clδ(μ∨η)=clδ(μ)∨clδ(η),(ii)
clδ(μ∧η)≤clδ(μ)∧clδ(η).Theorem 2.15 (see [4]).
A[0,1]-topological space (X,τ) is FT3/4-space if for every xt∈S(X) either xt is δ-open or xt is closed.Theorem 2.16 (see [4]).
Let(X,τ) be a [0,1]-topological space. Then, the following conditions are equivalent: (i)
X is an FWT2-space,(ii)
xt=clδ(xt) for each xt∈S(X).


## 3.Wθg-Closed Fuzzy Sets

In this section, we introduce the concept of weaklyθ-generalized closed fuzzy sets, and we study some of their properties.Definition 3.1.
A fuzzy subsetμ of a [0,1]-topological space  (X,τ) is said to be weakly θ-generalized closed (Wθg-closed, for short) if and only if clθ(μ)≤η whenever μ≤η and η is θ-open fuzzy set.
The complement of aWθg-closed fuzzy set is called Wθg-open.Theorem 3.2.
Let(X,τ) be a [0,1]-topological space. Then,(i)
Everyθ-closed fuzzy set is Wθg-closed,(ii)
Everyθg-closed fuzzy set is Wθg-closed.Proof.
Obvious.From the above discussion, we introduce the following diagram.(3.1)None of these implications is reversible as the following examples show.Example 3.3.
LetX={x,y} and  τ={0X,y0.7,1X}. If μ=x0.5∨y0.6, then μ is Wθg-closed fuzzy set but not θ-closed.Example 3.4.
LetX={x} and τ={0X,x0.6,1X}. If μ=x0.5, then μ is Wθg-closed, since the only θ-open superset of μ is 1X. But μ is not θg-closed, since  μ≤x0.6 and clθ(μ)=1X≰x0.6.Theorem 3.5.
A fuzzy subsetμ of a [0,1]-topological space (X,τ) is Wθg-closed if for every xt∈S(X) such that xtqclθ(μ), one has clθ(xt)qμ.Proof.
Letη be θ-open and μ≤η. If xtqclθ(μ), then by assumption, clθ(xt)qμ. Hence, there exists y∈X such that clθ(xt)(y)+μ(y)>1. Put clθ(xt)(y)=ε. Then, yε∈clθ(xt) and yεqμ. Thus, ρqxt for each ρ∈NQ(yε,τθ). Since yεqη, then ηqxt and so clθ(μ)≤η. Thus, μ is Wθg-closed.Theorem 3.6.
Let(X,τ) be a [0,1]-topological space and μ∈IX. Then, μ is Wθg-closed if there is not any θ-closed fuzzy set λ such that λq̅μ and λqclθ(μ).Proof.
Suppose thatμ is not Wθg-closed. Then, there exists θ-open fuzzy set η such that μ≤η and clθ(μ)≰η. Put λ=ηc. Then, there exists θ-closed fuzzy set λ such that λq̅μ and λqclθ(μ). This is a contradiction.Theorem 3.7.
Let(X,τ) be an FR1-space and μ∈IX be a C-set and g-closed. Then, μ is Wθg-closed.Proof.
Suppose that(X,τ) is an FR1-space and μ is a C-set in X. If μ is g-closed, then by Theorem 2.8  μ is θg-closed and hence Wθg-closed.Theorem 3.8.
Let(X,τ) be a [0,1]-topological space and μ∈IX be a preopen and g-closed. Then, μ is Wθg-closed.Proof.
It is an immediate consequence of Theorems2.9 and 3.2.Theorem 3.9.
Let(X,τ) be an FR2-space and μ∈IX be a g-closed. Then, μ is Wθg-closed.Proof.
It is an immediate consequence of Theorems2.7 and 3.2.Theorem 3.10.
A finite union ofWθg-closed fuzzy sets, is always Wθg-closed fuzzy set.Proof.
Suppose thatμ,η∈IX are Wθg-closed fuzzy sets and let υ∈τθ such that μ∨η≤υ. Since μ and η are Wθg-closed, then we have clθ(μ)∨clθ(η)≤υ and by Theorem 2.10(i) clθ(μ∨η)≤υ. Hence, μ∨η is Wθg-closed.


## 4.Wδg-Closed Fuzzy Sets

In this section, we introduce the concept of weaklyδ-generalized closed fuzzy sets, and we study some of their properties. Also, we introduce the notion of FT3/4*-space, and we prove that every FT3/4*-space is a FT3/4-space.Definition 4.1.
A fuzzy subsetμ of [0,1]-topological space (X,τ) is said to be weakly δ-generalized closed (Wδg-closed, for short) if and only if clδ(μ)≤η whenever μ≤η and η is δ-open fuzzy set.The complement of aWδg-closed fuzzy set is called Wδg-open.Theorem 4.2.
Let(X,τ) be a [0,1]-topological space. Then,(i)
Everyδ-closed fuzzy set is Wδg-closed,(ii)
Everyδg-closed fuzzy set is Wδg-closed.Proof.
Obvious.From the above discussion, we introduce the following diagram.(4.1)None of these implications is reversible as the following examples show.Example 4.3.
LetX={x} and τ={0X,x0.8,1X}. If μ=x0.7, then μ is Wδg-closed, since the only δ-open superset of μ is 1X. But μ is not δg-closed, since μ≤x0.8 and clδ(μ)=1X≰x0.8.Example 4.4.
LetX={x,y} and τ={0X,y0..8,1X}. A fuzzy subset μ=x0.2∨y0.3 is δg-closed and hence Wδg-closed, but it is not δ-closed.Theorem 4.5.
A fuzzy subsetμ of a [0,1]-topological space (X,τ) is Wδg-closed if and only if for every xt∈S(X) such that xtqclδ(μ) one has clδ(xt)qμ.Proof.
Letxtqclδ(μ) and suppose that clδ(xt)q̅μ. Since μ is Wδg-closed, then it is easy to observe that clδ(xδ)q̅clδ(μ) which implies that xtq̅clδ(μ). This is a contradiction.
The converse is similar to the proof of Theorem3.5.Theorem 4.6.
Let(X,τ) be a [0,1]-topological space and μ∈IX. Then, μ is Wδg-closed if and only if there is not any δ-closed fuzzy set λ such that λq̅μ and λqclδ(μ).Proof.
Suppose that there is aδ-closed fuzzy set λ such that λq̅μ and λqclδ(μ). Then, there exists some xt∈λ such that xtqclδ(μ). Since μ is Wδg-closed, then by using Theorem 4.5, clδ(xt)qμ and hence clδ(λ)qμ. Since λ is δ-closed, then we have λqμ. This is a contradiction.
The converse is similar to the proof of Theorem3.6.Theorem 4.7.
Let(X,τ) be a fuzzy semiregular space and μ∈IX. Then,(i)
μ is Wδg-closed if and only if it is δg-closed,(ii)
If, in addition,(X,τ) is FT1/2-space, then μ is Wδg-closed if and only if it is closed.Proof.
(i) Since(X,τ) is semiregular space, then τ=τδ, and so μ is Wδg-closed if and only if it is δg-closed.
(ii) From (i), Theorem2.11, and by FT1/2-ness, the result is given.Theorem 4.8.
Let(X,τ) be an FR1-space and μ∈IX be a C-set and g-closed. Then, μ is Wδg-closed.Proof.
Suppose that(X,τ) is an FR1-space and μ is a C-set in X. If μ is g-closed, then by Theorem 2.12  μ is δg-closed and hence Wδg-closed.Theorem 4.9.
Let(X,τ) be a [0,1]-topological space and μ∈IX be a preopen and g-closed. Then, μ is Wδg-closed.Proof.
It is an immediate consequence of Theorems2.9 and 4.2.Theorem 4.10.
Every fuzzy subset of a fuzzy partition space(X,τ) is Wδg-closed.Proof.
Let(X,τ) be a fuzzy partition space, and let μ be a fuzzy subset of X. Then, by Theorem 2.13, μ is δg-closed and hence, by Theorem 4.2, μ is Wδg-closed.Theorem 4.11.
A finite union ofWδg-closed fuzzy sets is always Wδg-closed fuzzy set.Proof.
Similar to the proof of Theorem3.10.The following example shows that the finite intersection ofWδg-closed fuzzy set may fail to be Wδg-closed fuzzy set.Example 4.12.
LetX={a,b,c,d,e}. Define λ1,λ2,λ3,λ4,λ5:X→[0,1] as follows:
(4.2)λ1(a)=1,λ1(b)=1,λ1(c)=0,λ1(d)=0,λ1(e)=0,λ2(a)=0,λ2(b)=0,λ2(c)=1,λ2(d)=0,λ2(e)=0,λ3(a)=1,λ3(b)=1,λ3(c)=1,λ3(d)=0,λ3(e)=0,λ4(a)=1,λ4(b)=0,λ4(c)=1,λ4(d)=1,λ4(e)=0,λ5(a)=0,λ5(b)=1,λ5(c)=1,λ5(d)=0,λ5(e)=1.
Consider the[0,1]-topology τ={0X,λ1,λ2,λ3,1X}. It is clear that λ4 and λ5 are Wδg-closed fuzzy sets. But λ4∧λ5=λ2 is not Wδg-closed.Definition 4.13.
A[0,1]-topological space (X,τ) is called FT3/4*-space if and only if every Wδg-closed fuzzy set is δ-closed.Theorem 4.14.
EveryFT3/4*-space is FT3/4-space.Proof.
It is an immediate consequence of Theorem4.2(ii).Theorem 4.15.
A[0,1]-topological space (X,τ) is  FT3/4*-space if for every xt∈S(X) either xt is δ-open or δ-closed.Proof.
Letμ∈IX be Wδg-closed, and let xtq̅μ. We consider the following two cases.
Case 1.
xt is δ-open. Then, xtc is δ-closed. Since xtq̅μ, then μ≤xtc. But xtc is δ-closed. Then, clδ(μ)≤xtc. This shows that xtq̅clδ(μ).Case 2.
xt is δ-closed. Then, xtc is δ-open. Since xtq̅μ, then μ≤xtc. But μ is Wδg-closed. Then, clδ(μ)≤xtc and hence xtq̅clδ(μ).Corollary 4.16.
EveryFWT2-space is FT3/4*-space.Proof.
This is an immediate consequence of Theorems2.16 and 4.15.
The converse of Corollary4.16 need not be true, in general, and as a sample, we give the following example.Example 4.17.
LetX={a,b,c}. Define λ1,λ2,λ3:X→[0,1] as follows:
(4.3)λ1(a)=1,λ1(b)=0,λ1(c)=0,λ2(a)=0,λ2(b)=1,λ2(c)=0,λ3(a)=1,λ3(b)=1,λ3(c)=0.
Consider the [0,1]-topology τ={0X,λ1,λ2,λ3,1X}. Then, X is FT3/4*-space but not FWT2-space.Theorem 4.18.
Let(X,τ) be a [0,1]-topological space. Then, the following conditions are equivalent:(i)
X is FWT2-space,(ii)
X is FT3/4* and each xt∈S(X) is Wδg-closed.Proof.
Obvious.


## 5.Wθg-Continuous and Wδg-Continuous Mappings

Definition 5.1.
A fuzzy mappingf:(X,τ)→(Y,Δ) is called(i)
fuzzyWθg-continuous if the inverse image of every closed fuzzy set in Y is Wθg-closed  fuzzy set in X,(ii)
fuzzyWδg-continuous if the inverse image of every closed fuzzy set in Y is Wδg-closed  fuzzy set in X.Theorem 5.2.
Every fuzzyθg-continuous (resp.δg-continuous) mapping is fuzzy Wθg-continuous (resp. Wδg-continuous).Proof.
Obvious.The converse of the above Theorem may not be true, in general, by the following example.Example 5.3.
LetX=Y={x} and consider a [0,1]-topology τ of Example 4.3, Δ={0Y,x0.3,1Y}. If f:(X,τ)→(Y,Δ) is the identity fuzzy mapping, then f is fuzzy Wθg-continuous but not fuzzy θg-continuous, since x0.7∈Δ' and f-1(x0.7)=x0.7≤x0.8∈τ but clθ(x0.7)=1X≰x0.8. Also, f is fuzzy Wδg-continuous but not fuzzy δg-continuous.Theorem 5.4.
Letf:(X,τ)→(Y,Δ) be fuzzy mapping and (X,τ) be fuzzy semiregular space. Then, the following conditions are equivalent:(i)
f is fuzzy Wδg-continuous,(ii)
f is fuzzy δg-continuous,(iii)
f is fuzzy g-continuous.Proof.
It follows directly from Theorems2.11 and 4.7(i).Definition 5.5.
A fuzzy mappingf:(X,τ)→(Y,Δ) is called(i)
fuzzyWθg-irresolute if the inverse image of every Wθg-closed fuzzy set in Y is Wθg-closed  fuzzy set in X,(ii)
fuzzyWδg-irresolute if the inverse image of every Wδg-closed fuzzy set in Y is Wδg-closed  fuzzy set in X.Theorem 5.6.
Letf:(X,τ)→(Y,Δ) and g:(Y,Δ)→(Z,Ω) be two fuzzy mappings. Then, (i)
g∘f is fuzzy Wθg-continuous if g is fuzzy continuous and f is fuzzy Wθg-continuous,(ii)
g∘f is fuzzy Wθg-irresolute if g is fuzzy Wθg-irresolute and f is fuzzy Wθg-irresolute,(iii)
g∘f is fuzzy Wθg-continuous if g is fuzzy Wθg-continuous and f is fuzzy Wθg-irresolute.Proof.
Obvious.Theorem 5.7.
Letf:(X,τ)→(Y,Δ) and g:(Y,Δ)→(Z,Ω) be two fuzzy mappings. Then, (i)
g∘f is fuzzy Wδg-continuous if g is fuzzy continuous and f is fuzzy Wδg-continuous,(ii)
g∘f is fuzzy Wδg-irresolute if g is fuzzy Wδg-irresolute and f is fuzzy Wδg-irresolute,(iii)
g∘f is fuzzy Wδg-continuous if g is fuzzy Wδg-continuous and f is fuzzy Wδg-irresolute,(iv)
Let(Y,Δ) be FT3/4*-space. Then, g∘f is fuzzy Wδg-continuous if g is fuzzy Wδg-continuous and f is fuzzy Wδg-continuous,(v)
Let(Y,Δ) be a fuzzy semiregular space. Then, g∘f is fuzzy Wδg-continuous if g is fuzzy g-continuous and f is fuzzy Wδg-irresolute.Proof.
Obvious.Definition 5.8.
A fuzzy mappingf:(X,τ)→(Y,Δ) is called fuzzy θ-open if and only if f(η) is θ-open in Y  for any θ-open fuzzy set η in X.Theorem 5.9.
If a fuzzy mappingf:(X,τ)→(Y,Δ) is bijective, fuzzy θ-open, and fuzzy Wθg-continuous, then  f is fuzzy Wθg-irresolute.Proof.
Letλ be Wθg-closed fuzzy set in Y, and let f-1(λ)≤η, where η∈τθ. Clearly, λ≤f(η). Since f(η)∈Δθ and λ is Wθg-closed in Y, then clθ(λ)≤f(η) and thus f-1(clθ(λ))≤η. Since f is fuzzy Wθg-continuous and clθ(λ) is closed in Y, then f-1(clθ(λ)) is Wθg-closed in X and hence clθ(f-1(clθ(λ)))≤η. Thus, clθ(f-1(λ))≤η and so f-1(λ) is Wθg-closed in X. This shows that f is fuzzy Wθg-irresolute.Theorem 5.10.
If a fuzzy mappingf:(X,τ)→(Y,Δ) is bijective, fuzzy δ-open, and fuzzy Wδg-continuous, then  f is fuzzy Wδg-irresolute.Proof.
Similar to the proof of Theorem5.9.Theorem 5.11.
If a fuzzy mappingf:(X,τ)→(Y,Δ) is fuzzy Wδg-irresolute and (X,τ) is FT3/4*-space, then f is fuzzy δ-continuous.Proof.
Letμ be a δ-closed fuzzy set in Y. By using Theorem 4.2, μ is Wδg-closed. Since f is fuzzy Wδg-irresolute, then f-1(μ) is Wδg-closed in X. Since X is FT3/4*-space, then f-1(μ) is δ-closed in X. Thus, f is fuzzy δ-continuous.Theorem 5.12.
If a mappingf:(X,τ)→(Y,Δ) is fuzzy δ-continuous and fuzzy δ-closed, and μ is Wδg-closed fuzzy set in X, then f(μ)n is Wδg-closed in X.Proof.
Letμ be Wδg-closed in X, and let f(μ)≤η, where η is δ-open fuzzy set in Y. Since μ≤f-1(η), μ is Wδg-closed fuzzy set in X and since f-1(η) is δ-open in X, then clδ(μ)≤f-1(η). Thus f(clδ(μ))≤η. Hence, clδ(f(μ))≤clδ(f(clδ(μ)))=f(clδ(μ))≤η, since f is fuzzy δ-closed. Hence, f(μ) is Wδg-closed in Y.Theorem 5.13.
Let(X,τ) be an FT3/4*-space. If a fuzzy mapping f:(X,τ)→(Y,Δ) be surjective, fuzzy Wδg-irresolute, and fuzzy δ-closed, then (Y,Δ) is also FT3/4*-space.Proof.
Letμ be Wδg-closed fuzzy set in Y. Since f is fuzzy Wδg-irresolute, then f-1(μ) is Wδg-closed in X. Since X is FT3/4*-space, then f-1(μ) is δ-closed in X. Thus, μ is δ-closed in Y, since f is surjective and fuzzy δ-closed. Hence, (Y,Δ) is FT3/4*-space.



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