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Volume 2011, Article ID 853870,10pages doi:10.1155/2011/853870

Research Article

Wθg

-Closed and

Wδg-Closed in

0 ,

1 -Topological Spaces

M. E. El-Shafei

1

_{and A. H. Zakari}

2

1_{Department of Mathematics, Mansoura University, P.O. Box 35516, Mansoura 35516, Egypt}

2_{Department of Mathematics, Jazan University, P.O. Box 100, Jaza, Damad, Saudi Arabia}

Correspondence should be addressed to A. H. Zakari,d [email protected]

Received 8 June 2010; Accepted 14 September 2010

Academic Editor: Ayman Badawi

Copyrightq2011 M. E. El-Shafei and A. H. Zakari. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We investigate various classes of generalized closed fuzzy sets in0,1-topological spaces, namely, Wθg-closed fuzzy sets andWδg-closed fuzzy sets. Also, we introduce a new separation axiom FT_3/4∗ of the 0,1-topological spaces, and we prove that every FT_3/4∗ -space is a FT3/4-space.

Furthermore, we using the new generalized closed fuzzy sets to construct new types of fuzzy mappings.

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 Zakari3–5introduced 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 ofFT₃∗_/₄

-space. Moreover, we introduce and study the concepts of two new classes of fuzzy mappings,

namely, fuzzyWθg-continuous mappings and fuzzyWδg-irresolute mappings.

2. Preliminaries

LetXbe a set andIthe unit interval. A fuzzy set inXis an element of the set of all functions

fromX intoI. The family of all fuzzy sets inX is denoted by IX_{. A fuzzy singleton}_x

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fuzzy set in X defined by xtx t,xty 0 for all y /x,t ∈ 0,1. The set of all fuzzy

singletons inXis denoted bySX. For everyxt∈SXandμ∈IX, we definext∈μif and

only ift≤ μx. A fuzzy setμis called quasicoincident with a fuzzy setρ, denoted byμqρ,

if and only if there existsx∈Xsuch thatμx ρx>1. Ifμis not quasicoincident withρ,

then we writeμqρ. By clμ, intμ,μc,Nxt, τ, andNQxt, τ, we mean the fuzzy closure of

μ, the fuzzy interior ofμ, the complement ofμ, the class of all open neighborhoods ofxt, and

the class of all openQ-neighborhoods ofxt, respectively.

Definition 2.1see6,7. A fuzzy subsetμof a0,1-topological spaceX, τis called

iregular open if and only ifμintclμ,

iipreopen if and only ifμ≤intclμ.

The complement of a regular openresp. preopenfuzzy set is called a regular closed

resp. preclosed.

Definition 2.2see8,9. LetX, τbe a0,1-topological space,xt∈SX, andμ∈IX. Then,

itheθ-closure ofμ, denoted by clθμ, is defined by

xt∈clθμif and only if clηqμfor eachη∈NQxt, τ,

iitheδ-closure ofμdenoted by clδμ, is defined by

xt∈clδμif and only if intclηqμfor eachη∈NQxt, τ,

iiiμis calledθ-closedresp.δ-closedif and only ifμclθμ resp.μclδμ.

Definition 2.3see9. LetX, τbe a0,1-topological space andμ∈IX_{. Then,}

ithe familyγ{ηj :j∈J} ⊆τis called an openP-cover ofμif and only if for every

xt∈μ, there existsj0∈Jsuch thatxt∈ηj0,

iiμis called aC-set if and only if every openP-cover ofμhas a finite subcover.

Definition 2.4see2–4. LetX, τbe a0,1-topological space. A fuzzy setμ ∈ IX_{is called}

ia generalized closedg-closed, for shortif and only if clμ ≤ ηwheneverμ ≤ η

andηis open fuzzy set,

iia θ-generalized closedθg-closed, for short if and only if clθμ ≤ η whenever

μ≤ηandηis open fuzzy set,

iiia δ-generalized closed δg-closed, for shortif and only if clδμ ≤ η whenever

μ≤ηandηis open fuzzy set.

Definition 2.5see2,4,6,10. A0,1-topological spaceX, τis called

iFR1if and only ifxtqclyrimplies that there existη∈Nxt, τandυ∈Nyr, τ

such thatηqυ,

iiFR2 orF-regular if and only ifxtqλ,λ is closed fuzzy set implies that there exist

η∈Nxt, τandυ∈τ,λ≤υsuch thatηqυ,

iiiFT1/2if and only if everyg-closed fuzzy set inXis closed,

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vfuzzy weakly Hausdorﬀ FWT2, for shortifxtqyr implies that there exists regular

open fuzzy setη∈Nxt, τsuch thatyrqη,

vifuzzy semiregular if and only if the collection of all regular open fuzzy sets in X

forms a base for the0,1-topologyτ,

viia fuzzy partition space if and only if every open fuzzy subset is closed.

Definition 2.6see2–4,11. A fuzzy mappingf:X, τ → Y,Δis called

ifuzzy generalized continuousfuzzyg-continuous, for shortif and only iff−1η

isg-closed inXfor any closed fuzzy setηinY,

iifuzzy θ-generalized continuous fuzzy θg-continuous, for short if and only if

f−1ηis θg-closed inXfor any closed fuzzy setηinY,

iiifuzzy δ-generalized continuous fuzzy δg-continuous, for short if and only if

f−1ηis δg-closed inXfor any closed fuzzy setηinY,

ivfuzzyδ-continuous if the inverse image of everyδ-open fuzzy set inY isδ-open in

X,

vfuzzy δ-openfuzzy δ-open, for shortif and only iffηisδ-open inY for any

δ-open fuzzy setηinX,

vifuzzyδ-closedfuzzyδ-closed, for shortif and only iffηisδ-closed inY for any

δ-closed fuzzy setηinX.

Theorem 2.7see3. A fuzzy subsetμof anFR2-ftsX, τis θg-closed if and only if it is

g-closed.

Theorem 2.8see3. LetX, τbe a0,1-topological space. Then, the following conditions are

equivalent:

i X, τisFR1-space,

iifor eachC-setμ∈IX_{, clμ}_cl

θμ,

iiifor eachxt∈SX, clxt clθxt.

Theorem 2.9see3,4. LetX, τbe a0,1-topological space andμ∈IX_{be a preopen. Then,}_μ

is θg-closed (resp.δg-closed) if and only if it isg-closed.

Theorem 2.10see3. LetX, τbe a0,1-topological space andμ, η∈IX_{. Then,}

iclθμ∨η clθμ∨clθη,

iiclθμ∧η≤clθμ∧clθη.

Theorem 2.11see4. LetX, τbe a fuzzy semiregular space andμ∈IX_{. Then,}

iμisδg-closed if and only ifμisg-closed,

iiIf, in addition,X, τisFT1/2, thenμisδg-closed if and only ifμis closed.

Theorem 2.12see4. LetX, τbe anFR1-space andμ∈IXbe aC-set. Then,μisδg-closed if

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Theorem 2.13see4. LetX, τbe a fuzzy partition space andμ ∈IX_{. Then,}_μ_is_{δg-closed if}

and only if it isg-closed.

Theorem 2.14see4. LetX, τbe a0,1-topological space andμ, η∈IX_{. Then,}

iclδμ∨η clδμ∨clδη,

iiclδμ∧η≤clδμ∧clδη.

Theorem 2.15 see4. A 0,1-topological space X, τis FT3/4-space if for everyxt ∈ SX

eitherxtisδ-open orxtis closed.

Theorem 2.16see4. LetX, τbe a0,1-topological space. Then, the following conditions are

equivalent:

iXis anFWT2-space,

iixtclδxtfor eachxt∈SX.

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 spaceX, τ is said to be weakly

θ-generalized closedWθg-closed, for shortif and only if clθμ≤ηwheneverμ≤ηandηis

θ-open fuzzy set.

The complement of aWθg-closed fuzzy set is calledWθg-open.

Theorem 3.2. LetX, τbe a0,1-topological space. Then,

iEveryθ-closed fuzzy set isWθg-closed,

iiEvery θg-closed fuzzy set isWθg-closed.

Proof. Obvious.

From the above discussion, we introduce the following diagram.

Wθg

θ θg

3.1

None of these implications is reversible as the following examples show.

Example 3.3. LetX{x, y}andτ {0X, y0.7,1X}. Ifμx0.5∨y0.6, thenμisWθg-closed fuzzy

set but notθ-closed.

Example 3.4. LetX {x}andτ {0X, x0.6,1X}. Ifμ x0.5, thenμisWθg-closed, since the

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Theorem 3.5. A fuzzy subsetμof a0,1-topological spaceX, τisWθg-closed if for everyxt ∈

SXsuch thatxtqclθμ, one has clθxtqμ.

Proof. Letη beθ-open and μ ≤ η. Ifxtqclθμ, then by assumption, clθxtqμ. Hence, there

existsy ∈Xsuch that clθxty μy>1. Put clθxty ε. Then,yε ∈clθxtandyεqμ.

Thus,ρqxtfor eachρ ∈NQyε, τθ. Sinceyεqη, thenηqxtand so clθμ≤η. Thus,μis

Wθg-closed.

Theorem 3.6. LetX, τbe a0,1-topological space andμ∈IX_{. Then,}_μ_is_{Wθg-closed if there is}

not anyθ-closed fuzzy setλsuch thatλ q μandλqclθμ.

Proof. Suppose thatμis notWθg-closed. Then, there existsθ-open fuzzy setηsuch thatμ≤η

and clθμ/≤η. Putληc. Then, there existsθ-closed fuzzy setλsuch thatλ q μandλqclθμ.

This is a contradiction.

Theorem 3.7. LetX, τbe anFR1-space andμ∈IXbe aC-set andg-closed. Then,μisWθg-closed.

Proof. Suppose that X, τ is anFR1-space andμ is aC-set in X. If μis g-closed, then by

Theorem 2.8μisθg-closed and henceWθg-closed.

Theorem 3.8. LetX, τbe a0,1-topological space andμ∈IX_{be a preopen and}_{g-closed. Then,}_μ

isWθg-closed.

Proof. It is an immediate consequence of Theorems2.9and3.2.

Theorem 3.9. LetX, τbe anFR2-space andμ∈IX_{be a}_{g-closed. Then,}_μ_is_Wθg-closed.

Proof. It is an immediate consequence of Theorems2.7and3.2.

Theorem 3.10. A finite union ofWθg-closed fuzzy sets, is alwaysWθg-closed fuzzy set.

Proof. Suppose thatμ, η ∈IX _are_{Wθg-closed fuzzy sets and let}_υ_∈_τ

θ such thatμ∨η ≤ υ.

Since μand η are Wθg-closed, then we have clθμ∨clθη ≤ υ and byTheorem 2.10i

clθμ∨η≤υ. Hence,μ∨ηisWθg-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 ofFT₃∗_/₄-space, and we prove

that everyFT₃∗_/₄-space is aFT3/4-space.

Definition 4.1. A fuzzy subset μ of 0,1-topological space X, τ is said to be weakly

δ-generalized closed Wδg-closed, for shortif and only if clδμ ≤ η wheneverμ ≤ η and

ηisδ-open fuzzy set.

The complement of aWδg-closed fuzzy set is calledWδg-open.

Theorem 4.2. LetX, τbe a0,1-topological space. Then,

iEveryδ-closed fuzzy set isWδg-closed,

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Proof. Obvious.

From the above discussion, we introduce the following diagram.

Wδg

δ δg

4.1

None of these implications is reversible as the following examples show.

Example 4.3. LetX {x}andτ {0X, x0.8,1X}. Ifμ x0.7, thenμisWδg-closed, since the

onlyδ-open superset ofμis 1X. Butμis not δg-closed, sinceμ≤x0.8and clδμ 1X/≤x0.8.

Example 4.4. LetX {x, y}andτ {0X, y0..8,1X}. A fuzzy subsetμx0.2∨y0.3is δg-closed

and henceWδg-closed, but it is notδ-closed.

Theorem 4.5. A fuzzy subsetμof a0,1-topological spaceX, τisWδg-closed if and only if for

everyxt∈SXsuch thatxtqclδμone has clδxtqμ.

Proof. Let xtqclδμ and suppose that clδxtqμ. Since μ is Wδg-closed, then it is easy to

observe that clδxδqclδμwhich implies thatxtqclδμ. This is a contradiction.

The converse is similar to the proof ofTheorem 3.5.

Theorem 4.6. LetX, τbe a0,1-topological space andμ∈IX_{. Then,}_μ_is_{Wδg-closed if and only}

if there is not anyδ-closed fuzzy setλsuch thatλqμandλqclδμ.

Proof. Suppose that there is aδ-closed fuzzy setλ such thatλqμand λqclδμ. Then, there

exists somext ∈ λ such thatxtqclδμ. SinceμisWδg-closed, then by usingTheorem 4.5,

clδxtqμand hence clδλqμ. Sinceλisδ-closed, then we haveλqμ. This is a contradiction.

The converse is similar to the proof ofTheorem 3.6.

Theorem 4.7. LetX, τbe a fuzzy semiregular space andμ∈IX_{. Then,}

iμisWδg-closed if and only if it isδg-closed,

iiIf, in addition,X, τisFT1/2-space, thenμisWδg-closed if and only if it is closed.

Proof. iSinceX, τis semiregular space, thenτ τδ, and soμisWδg-closed if and only if

it isδg-closed.

iiFromi,Theorem 2.11, and byFT1/2-ness, the result is given.

Theorem 4.8. LetX, τbe anFR1-space andμ∈IXbe aC-set andg-closed. Then,μisWδg-closed.

Proof. Suppose that X, τ is anFR1-space andμ is aC-set in X. If μis g-closed, then by

Theorem 2.12μisδg-closed and henceWδg-closed.

Theorem 4.9. LetX, τbe a0,1-topological space andμ∈IX_{be a preopen and}_{g-closed. Then,}_μ

isWδg-closed.

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Theorem 4.10. Every fuzzy subset of a fuzzy partition spaceX, τisWδg-closed.

Proof. Let X, τ be a fuzzy partition space, and let μ be a fuzzy subset of X. Then, by

Theorem 2.13,μis δg-closed and hence, byTheorem 4.2,μisWδg-closed.

Theorem 4.11. A finite union ofWδg-closed fuzzy sets is alwaysWδg-closed fuzzy set.

Proof. Similar to the proof ofTheorem 3.10.

The following example shows that the finite intersection ofWδg-closed fuzzy set may

fail to beWδg-closed fuzzy set.

Example 4.12. LetX{a, b, c, d, e}. Defineλ1, λ2, λ3, λ4, λ5:X → 0,1as follows:

λ1a 1, λ1b 1, λ1c 0, λ1d 0, λ1e 0,

λ2a 0, λ2b 0, λ2c 1, λ2d 0, λ2e 0,

λ3a 1, λ3b 1, λ3c 1, λ3d 0, λ3e 0,

λ4a 1, λ4b 0, λ4c 1, λ4d 1, λ4e 0,

λ5a 0, λ5b 1, λ5c 1, λ5d 0, λ5e 1.

4.2

Consider the0,1-topologyτ {0X, λ1, λ2, λ3,1X}. It is clear thatλ4andλ5are

Wδg-closed fuzzy sets. Butλ4∧λ5λ2is notWδg-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 isδ-closed.

Theorem 4.14. EveryFT₃∗_/₄-space isFT3/4-space.

Proof. It is an immediate consequence ofTheorem 4.2ii.

Theorem 4.15. A0,1-topological spaceX, τis FT₃∗_/₄-space if for everyxt ∈ SXeitherxt is

δ-open orδ-closed.

Proof. Letμ∈IX_be_{Wδg-closed, and let}_x

tqμ. We consider the following two cases.

Case 1. xtisδ-open. Then,xct isδ-closed. Sincextqμ, thenμ ≤ xct. Butxct isδ-closed. Then,

clδμ≤xc_t. This shows thatxtqclδμ.

Case 2. xtisδ-closed. Then,xct isδ-open. Sincextqμ, thenμ≤xtc. ButμisWδg-closed. Then,

clδμ≤xctand hencextqclδμ.

Corollary 4.16. Every FWT2-space isFT₃∗_/₄-space.

Proof. This is an immediate consequence of Theorems2.16and4.15.

The converse ofCorollary 4.16need not be true, in general, and as a sample, we give

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Example 4.17. LetX{a, b, c}. Defineλ1, λ2, λ3:X → 0,1as follows:

λ1a 1, λ1b 0, λ1c 0,

λ2a 0, λ2b 1, λ2c 0,

λ3a 1, λ3b 1, λ3c 0.

4.3

Consider the0,1-topology τ {0X, λ1, λ2, λ3,1X}. Then,X isFT₃∗_/₄-space but not

FWT2-space.

Theorem 4.18. LetX, τbe a0,1-topological space. Then, the following conditions are equivalent:

iXis FWT2-space,

iiXisFT₃∗_/₄and eachxt∈SXisWδg-closed.

Proof. Obvious.

5. Wθg

-Continuous and

Wδg

-Continuous Mappings

Definition 5.1. A fuzzy mappingf:X, τ → Y,Δis called

ifuzzyWθg-continuous if the inverse image of every closed fuzzy set inY is

Wθg-closed fuzzy set inX,

iifuzzyWδg-continuous if the inverse image of every closed fuzzy set inY is

Wδg-closed fuzzy set inX.

Theorem 5.2. Every fuzzyθg-continuous (resp.δg-continuous) mapping is fuzzyWθg-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. Let X 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, thenf is fuzzy

Wθg-continuous but not fuzzy θg-continuous, sincex0.7 ∈Δ andf−1x0.7 x0.7 ≤ x0.8 ∈ τ but

clθx0.7 1X/≤x0.8. Also,fis fuzzyWδg-continuous but not fuzzyδg-continuous.

Theorem 5.4. Letf : X, τ → Y,Δbe fuzzy mapping andX, τbe fuzzy semiregular space.

Then, the following conditions are equivalent:

ifis fuzzyWδg-continuous,

iifis fuzzy δg-continuous,

iiifis fuzzyg-continuous.

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Definition 5.5. A fuzzy mappingf:X, τ → Y,Δis called

ifuzzy Wθg-irresolute if the inverse image of everyWθg-closed fuzzy set inY is

Wθg-closed fuzzy set inX,

iifuzzy Wδg-irresolute if the inverse image of everyWδg-closed fuzzy set inY is

Wδg-closed fuzzy set inX.

Theorem 5.6. Letf :X, τ → Y,Δandg:Y,Δ → Z,Ωbe two fuzzy mappings. Then,

ig◦fis fuzzyWθg-continuous ifgis fuzzy continuous andfis fuzzyWθg-continuous,

iig◦fis fuzzyWθg-irresolute ifgis fuzzyWθg-irresolute andfis fuzzyWθg-irresolute,

iiig ◦ 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,Δandg:Y,Δ → Z,Ωbe two fuzzy mappings. Then,

ig◦fis fuzzyWδg-continuous ifgis fuzzy continuous andfis fuzzyWδg-continuous,

iig◦fis fuzzyWδg-irresolute ifgis fuzzyWδg-irresolute andfis fuzzyWδg-irresolute,

iiig ◦ f is fuzzy Wδg-continuous if g is fuzzy Wδg-continuous and f is fuzzy

Wδg-irresolute,

ivLet Y,Δ be FT₃∗_/₄-space. Then, g ◦f is fuzzy Wδg-continuous if g is fuzzy

Wδg-continuous andfis fuzzyWδg-continuous,

vLetY,Δbe a fuzzy semiregular space. Then,g◦fis fuzzyWδg-continuous ifgis fuzzy

g-continuous andfis fuzzyWδg-irresolute.

Proof. Obvious.

Definition 5.8. A fuzzy mappingf:X, τ → Y,Δis called fuzzyθ-open if and only iffη

isθ-open inY for anyθ-open fuzzy setηinX.

Theorem 5.9. If a fuzzy mappingf :X, τ → Y,Δis bijective, fuzzyθ-open, and fuzzy

Wθg-continuous, thenfis fuzzyWθg-irresolute.

Proof. LetλbeWθg-closed fuzzy set inY, and letf−1λ≤η, whereη∈τθ. Clearly,λ≤fη.

Sincefη∈ΔθandλisWθg-closed inY, then clθλ≤fηand thusf−1clθλ≤η. Since

f is fuzzyWθg-continuous and clθλ is closed inY, thenf−1clθλisWθg-closed in X

and hence clθf−1clθλ≤η. Thus, clθf−1λ≤ηand sof−1λisWθg-closed inX. This

shows thatfis fuzzyWθg-irresolute.

Theorem 5.10. If a fuzzy mappingf:X, τ → Y,Δis bijective, fuzzyδ-open, and fuzzy

Wδg-continuous, thenfis fuzzyWδg-irresolute.

Proof. Similar to the proof ofTheorem 5.9.

Theorem 5.11. If a fuzzy mapping f : X, τ → Y,Δis fuzzy Wδg-irresolute andX, τ is

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Proof. Letμbe aδ-closed fuzzy set inY. By usingTheorem 4.2,μisWδg-closed. Sincef is

fuzzyWδg-irresolute, thenf−1μisWδg-closed inX. SinceXisFT₃∗_/₄-space, thenf−1μis

δ-closed inX. Thus,fis fuzzyδ-continuous.

Theorem 5.12. If a mappingf :X, τ → Y,Δis fuzzyδ-continuous and fuzzyδ-closed, andμ

isWδg-closed fuzzy set inX, thenfμn isWδg-closed inX.

Proof. LetμbeWδg-closed inX, and letfμ≤η, whereηisδ-open fuzzy set inY. Sinceμ≤

f−1η,μisWδg-closed fuzzy set inXand sincef−1ηisδ-open inX, then clδμ≤f−1η.

Thusfclδμ ≤ η. Hence, clδfμ ≤ clδfclδμ fclδμ ≤ η, sincef is fuzzy

δ-closed. Hence,fμisWδg-closed inY.

Theorem 5.13. LetX, τbe anFT₃∗_/₄-space. If a fuzzy mappingf :X, τ → Y,Δbe surjective,

fuzzyWδg-irresolute, and fuzzyδ-closed, thenY,Δis alsoFT₃∗_/₄-space.

Proof. Letμbe Wδg-closed fuzzy set inY. Sincef is fuzzy Wδg-irresolute, thenf−1μis

Wδg-closed inX. SinceXisFT₃∗_/₄-space, thenf−1μisδ-closed inX. Thus,μisδ-closed in

Y, sincefis surjective and fuzzyδ-closed. Hence,Y,ΔisFT₃∗_/₄-space.

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Discrete Mathematics

Journal of

Research Article

Wθg

-Closed and

Wδg-Closed in

0

,

1

-Topological Spaces

M. E. El-Shafei

and A. H. Zakari

1. Introduction

2. Preliminaries

3.

Wθg

-Closed Fuzzy Sets

4.

Wδg

-Closed Fuzzy Sets

5.

Wθg

-Continuous and

Wδg

-Continuous Mappings

References

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Mathematics

Complex Analysis

Optimization

Combinatorics

Function Spaces

The Scientific

World Journal

Discrete Mathematics

Stochastic Analysis

_{and A. H. Zakari}