on Some Relations of fuzzy z-open set in fuzzy topology space on fuzzy set
Assist. Prof. Dr. Munir Abdul Khalik Al- khafaj ,Gumana sabry mouhsen
Department of mathematics,College of Education, Al-Mustansiriy University,Baghdad. Iraq
Abstract
The aim of this paper is to introduce and study a fuzzy Z-open set (resp .fuzzy b-open ,fuzzy
𝛽 –open, fuzzy 𝛽 ∗ -open, fuzzy e-open )sets in fuzzy topology space on fuzzy set and devote
to study some properties and the relation between them.
1. INTRODUCTION
The concept of fuzzy sets was introduced by Zadah [1], chang [2] introducedthe definition
of fuzzy topology space,[3] Chakrabarty ,M.K. and Ahsanullal,T,M.G. " Fuzzy topological
space on fuzzy sets And Tolerance topology" Fuzzy sets and system 45,103108(1992).
Chaudhuri , A .k. and Das ,P. ,"Some Resulte on Fuzzy Topological On Fuzzy Sets" , Fuzzy
sets and systems 56 , 331-336 , (1993) [4] , EI.Magharabi, MohammedA.AL- juhani New
Types Of Fuction By M-open Sets Of Taibahuniversity For Science Joumal In (2013) [5]
In this paper , we introduce some types of fuzzy open sets in fuzzy topological space on fuzzy set
and study some relations and given some counter examples.
2. Fuzzy Topological Space On Fuzzy Set
DEFINITION 2.1[1]
Let X be a non-empty set , a fuzzy set 𝐴̃ in X is characterized by a function 𝜇𝐴̃ : X→ I , where
I =[0,1] which is written 𝐴̃={(x, 𝜇𝐴̃ (x)) : x ∈ X, 0 ≤ 𝜇𝐴̃ (x)≤ 1},The collection of all fuzzy
sets in X will be denoted by 𝐼 𝑋 , that is 𝐼 𝑋 ={𝐴̃:𝐴̃ is fuzzy sets in X} where 𝜇𝐴̃ is called the
membership function.
Definition 2.2[3]
A collection 𝑇̃of a fuzzy subsets of 𝐴̃ , such that 𝑇̃ ⊆ P(𝐴̃) is said to be fuzzy topology on 𝐴̃, if
it satisfied the following conditions :
̃ ∈ 𝑇̃
1) 𝐴̃, ∅
2) If 𝐵̃ ,𝐶̃ ∈ 𝑇̃, then 𝐵̃ ∩ 𝐶̃ ∈ 𝑇̃
3) If 𝐵̃𝛼 ∈ 𝑇̃, then ∪𝛼 𝐵̃𝛼 ∈ 𝑇̃ ,α ∈ 𝛬
(𝐴̃, 𝑇̃) is said to be fuzzy topological space and every member of 𝑇̃ is called fuzzy open set in
𝐴̃ and its complement is a fuzzy closed set .
1
Definition 2.3[4]
Let 𝐵̃ be a fuzzy set in a fuzzy topological space(𝐴̃, 𝑇̃) then :
• The interior of B is denoted by Int (𝐵̃) and defined by
Int(𝐵̃ )= max {µ𝐺̃ (x) : 𝐺̃ is a fuzzy open set in 𝐴̃, µ𝐺̃ (x) ≤ µ𝐵̃ (x) }
• The closure of Β is denoted by CL(𝐵̃ ) and defined by
CL (𝐵̃ )= min { 𝜇𝐹̃ : 𝐹̃ is a fuzzy closed set in 𝐴̃ , µ𝐵̃ (x) ≤ µ𝐹̃ (x) }
Definition 2.4 [8]
A fuzzy set 𝐵̃in a fuzzy topological space (𝐴̃,𝑇̃) is said to be
• 𝐹𝑢𝑧𝑧𝑦 𝛿 − 𝑜𝑝𝑒𝑛 𝑠𝑒𝑡 𝑖𝑓 𝜇(𝐼𝑛𝑡(𝑐𝑙(𝐵̃))) (x) ≤ 𝜇𝐵̃ (x) .
• Fuzzy 𝛿 − 𝑐𝑙𝑜𝑠𝑒𝑑 𝑠𝑒𝑡 𝑖𝑓 𝜇𝐵̃ (x) ≤ 𝜇(𝑐𝑙(𝐼𝑛𝑡(𝐵̃))) (x) .
Definition 2.5 [7]
̃ in a topological space (𝐴̃,𝑇̃) is said to be
A fuzzy set 𝐷
• Fuzzy Z-open set if 𝜇𝐷̃ (𝑥) ≤ max {𝜇(𝑐𝑙(𝐼𝑛𝑡
̃ ))
𝛿(𝐷
•Fuzzy Z-closed set if min {𝜇(𝑐𝑙(𝐼𝑛𝑡
̃ ))
𝛿(𝐷
(x) , 𝜇(𝐼𝑛𝑡(𝑐𝑙(𝐷̃))) (x)}.
(x) , 𝜇(𝑐𝑙(𝑖𝑛𝑡(𝐷̃))) (x)} ≤ 𝜇𝐷̃ (𝑥) .
Definition 2.6:
̃ be a fuzzy set in a fuzzy topological space(𝐴̃ , 𝑇̃) then :
Let 𝐷
̃ is denoted by Z Int (𝐷
̃ ) and defined by
• The Z- interior of 𝐷
̃ ) = max {𝜇𝐺̃ (𝑥): 𝐺̃ is a fuzzy z- open set in 𝐴̃ , 𝜇𝐺̃ (x) ≤ 𝜇𝐷̃ (x)}
ZInt(𝐷
̃ is denoted by Z CL(𝐷
̃ )and defined by
• The Z- closure of 𝐷
̃ ) = min {𝜇𝑠̃ (x): 𝑆̃is a fuzzy Z-closed set in à , 𝜇𝐷̃ (x)≤ 𝜇𝑠̃ (x) }
Zcl (𝐷
2
Definition 2.7[3]
Let 𝐵̃, 𝐶̃ be a fuzzy set in a fuzzy topological space (𝐴̃, 𝑇̃) then :
• A fuzzy point 𝑥𝑟 is said to be quasi coincident with the fuzzy set 𝐵̃ if there exist 𝑥 ∈ X
such that 𝜇 𝑥𝑟 (x)+ µ𝐵̃ (x) > 𝜇𝐴̃ (𝑥) and denoted by 𝑥𝑟 𝑞 𝐵̃ , if 𝜇 𝑥𝑟 (x) + µ𝐵̃ (x) ≤
𝜇𝐴̃ (𝑥) ∀𝑥 ∈ 𝑋 then 𝑥𝑟 is not quasi coincident with a fuzzy set 𝐵̃ and is denoted by 𝑥𝑟 𝑞⏞ 𝐵̃ .
• A fuzzy set 𝐵̃ is said to be quasi coincident ( overlap ) with a fuzzy set 𝐶̃ if there exist x ∈
𝑋such that 𝜇𝐵̃ (x) +𝜇𝐶̃ (x) > 𝜇𝐴̃ (x) and denoted by 𝐵̃ 𝑞 𝐶̃ , if 𝜇𝐶̃ (x) + µ𝐵̃ (x) ≤ 𝜇𝐴̃ (𝑥) ∀𝑥 ∈
𝑋 then 𝐵̃ is not quasi coincident with a fuzzy set 𝐶̃ and is denoted by 𝐵̃ ⏞
𝑞 𝐶̃ .
Remark 2.8:
1. The intersection of two fuzzy Z-open set is not necessary a fuzzy Z-open set.
2. The union of two fuzzy Z-closed set is not necessary a fuzzy Z-closed set .
As shown by the following example :Example 2.9 :
̃ ,𝐸̃ ,𝐹̃ ,𝐺̃ ,𝐻
̃ be fuzzy subset of à where :
Let X={a , b } and 𝐵̃,𝐶̃ ,𝐷
à ={ (a,0.6) ,(b,0.8)}
, 𝐵̃={ (a,0.0) ,(b,0.6)}
𝐶̃ = { (a,0.6) ,(b,0.0)}
̃ = { (a,0.6) ,(b,0.6)}
,𝐷
𝐸̃ = { (a,0.4) ,(b,0.4)}
, 𝐹̃ ={ (a,0.0) ,(b,0.3)}
𝐺̃ = { (a,0.0),(b,0.4)}
̃ = { (a,0.4) ,(b,0.6)}
,𝐻
𝑘̃ = { (a,0.6),(b,0.4)}
̃ = { (a,0.6),(b,0.3)}
,𝑀
̃ = { (a,0.4),(b,0.3)}
𝑁
̃ = { (a,0.6),(b,0.2)}
,𝑊
̃ ,Ã ,𝐵̃ ,𝐶̃ , 𝐷
̃ 𝐻
̃ ,𝐸̃ , 𝐹̃ ,𝐺,
̃ , 𝑘̃, 𝑀
̃, 𝑁
̃ } be a fuzzy topological on Ã
And 𝑇̃= { ∅
̃ are fuzzy Z-open sets but 𝐵̃ ∩ 𝑊
̃ is not fuzzy Z-open set.
𝐵̃ and 𝑊
3
Proposition 2.10:
̃ ∈ p(Ã),
Let (Ã,𝑇̃) be a fuzzy topological space and 𝐷
Then 𝜇𝑧𝑖𝑛𝑡(𝐷̃) (x) = 𝜇(𝑧𝑐𝑙(𝐷̃𝑐)) 𝑐 (x) ∀ x ∈X
Proof :
Since 𝜇𝑧𝑖𝑛𝑡(𝐷̃) (x) ≤ 𝜇𝐷̃ (x)
̃ ) is a fuzzy Z-open set
And Zint (𝐷
Then 𝜇𝐷̃𝑐 (x) ≤ 𝜇𝑧𝑖𝑛𝑡(𝐷̃)𝐶 (x) and 𝜇𝑧𝑐𝑙(𝐷̃)𝐶 (𝑥) ≤ 𝜇𝑧𝑖𝑛𝑡(𝐷̃)𝐶 (x) ,
Hence𝜇𝑧𝑖𝑛𝑡(𝐷̃) (x) ≤ 𝜇(𝑧𝑐𝑙(𝐷̃𝑐)) 𝑐 (x) ………..(*)
̃ )is a fuzzy Z-closed set
Since 𝜇𝐷̃𝐶 (x) ≤ 𝜇𝑧𝑐𝑙(𝐷̃)𝐶 (x) and 𝑧𝑐𝑙(𝐷
Then𝜇(𝑧𝑐𝑙(𝐷̃𝑐)) 𝑐 (x) ≤ 𝜇𝐷̃ (x)
Hence 𝜇(𝑧𝑐𝑙(𝐷̃𝑐)) 𝑐 (x) ≤ 𝜇𝑧𝑖𝑛𝑡(𝐷̃) (x)………..(**)
From (*) and (**) we get 𝜇𝑧𝑖𝑛𝑡(𝐷̃) (x) =𝜇(𝑧𝑐𝑙(𝐷̃𝑐)) 𝑐 (x).
Definition 2.11
Let (Ã,𝑇̃) be a fuzzy topological space 𝜇𝐵̃ be a fuzzy set of à a fuzzy point x ∈ à is said to
be a fuzzy Z- limit point of 𝐵̃
if for every fuzzy Z-open set 𝐺̃ we have
̃
(𝐺̃ –x)∩ 𝐵̃= ∅
We shall call the set of all fuzzy Z- limit point of à the Z-derived set of à and denoted by
d(Ã).
Proposition 2.12
Let (Ã,𝑇̃) be a fuzzy topological space 𝐵̃ ⊂ 𝐶̃ ⊂ Ã then:
1) Cl (𝐵̃ )= 𝐵̃ ∪ d (𝐵̃)
2) 𝐵̃ is Z-closed set iff d(𝐵̃)
3) Cl (𝐵̃ ) ⊂ d (𝐵̃)
4
Proof:
(⟹ ) Let x ∉ 𝜇𝑐𝑙(𝐵̃) then there exist a Z-closed set µF̃ in 𝜇𝐴̃ such that 𝜇𝐴̃ ⊂ µF̃ and x ∉ µF̃
Hence 𝜇𝑉̃ = x-µF̃ is Z-open set such that x ∈ 𝜇𝑉̃ and min {𝜇𝑉̃ , 𝜇𝐴̃ } = 𝜇∅̃
Therefor x ∉ 𝜇𝐵̃ and x ∉ 𝜇𝑑(𝐵̃) , then x ∉ 𝑚𝑎𝑥 {𝜇𝐵̃ , 𝜇𝑑(𝐵̃) }
Thus max{ 𝜇𝐵̃ , 𝜇𝑐𝑙(𝐵̃) } ⊂ 𝜇𝑑(𝐵̃) .
(⟸) x ∉ 𝑚𝑎𝑥 {𝜇𝐵̃ , 𝜇𝑑(𝐵̃) } implies that there exist a Z-open set 𝜇𝑉̃ in 𝜇𝐴̃ such that x ∈
𝜇𝑉̃ and min {𝜇𝑉̃ , 𝜇𝐴̃ } = 𝜇∅̃ .
Hence µF̃ = 𝜇𝐴̃ -𝜇𝑉̃ is Z-closed set in 𝜇𝐴̃ such that 𝜇𝐴̃ ⊂ µF̃ and x ∉ µF̃
Hence x ∉ 𝜇𝑐𝑙(𝐴̃)
Thus 𝜇𝑐𝑙(𝐴̃) ⊂ max { 𝜇𝐴̃ , 𝜇𝑐𝑙(𝐴̃) }
Therefor 𝜇𝑐𝑙(𝐵̃) = 𝑎𝑥 {𝜇𝐵̃ , 𝜇𝑑(𝐵̃) }
For (2) and (3) the proof is easy.
Proposition 2.13
Let (Ã,𝑇̃) be a fuzzy topological space and 𝐵̃ ⊂ Ã , then cl (𝐵̃) is the smallest Z-closed set
containing 𝐵̃
Proof :
Suppose that 𝐶̃ is Z-closed set such that 𝜇𝐵̃ ⊂ µC̃ .since 𝜇𝑐𝑙(𝐵̃) = 𝜇𝐵̃ ∪ 𝜇𝑑(𝐵̃) by proposition
2.11(1), and 𝜇𝐵̃ ⊂ µC̃ then 𝜇𝑐𝑙(𝐵̃) = 𝜇𝐵̃ ∪ 𝜇𝑑(𝐵̃) ⊆ µC̃ ∪ 𝜇𝑐𝑙(𝐶̃) ⊂ µC̃ .
Thus 𝜇𝑐𝑙(𝐵̃) ⊂ µC̃ .
Therefor 𝜇𝑐𝑙(𝐵̃) is the smallest Z-closed set containing 𝐵̃.
5
Proposition 2.14
Let (Ã,𝑇̃) be a fuzzy topological space , if 𝐺̃ be a fuzzy open set in à and 𝐵̃ is a fuzzy Zopen sets in à , then 𝐺̃ ∩ 𝐵̃ is a fuzzy Z-open set
Proof :
Since 𝜇𝐺̃ (x) = 𝜇𝑖𝑛𝑡(𝐺̃) (x) and 𝜇(𝐵̃) (𝑋)= 𝜇𝑧𝑖𝑛𝑡(𝐵̃) (𝑋) then
Min {𝜇𝑖𝑛𝑡(𝐺̃) (x) ,𝜇𝑧𝑖𝑛𝑡(𝐵̃) (x) } ≤ µ zint (min { 𝜇𝐺̃ (𝑥), 𝜇𝐵̃ (𝑥)}) (x)
And since 𝜇𝑧𝑖𝑛𝑡 (min { 𝜇𝐺̃ (𝑥), 𝜇𝐵̃ (x)}) ≤ min { 𝜇𝐺̃ (𝑥), 𝜇𝐵̃ (𝑥)}
Then we get
Min { { 𝜇𝐺̃ (𝑥), 𝜇𝐵̃ (𝑥 )}= 𝜇𝑧𝑖𝑛𝑡 (min {𝜇𝐺̃ (𝑥), 𝜇𝐵̃ (𝑥)} ) (x)
Thus 𝐺̃ ∩ 𝐵̃ is a fuzzy Z-open set in Ã.
Proposition 2.15 :
Let (Ã,𝑇̃) be a fuzzy topological space , if 𝐶̃ be a fuzzy Z-open set in 𝐴̃ and 𝐵̃ , is a
maximal fuzzy open set in 𝐴̃ , then 𝐶̃ ∩ 𝐵̃ is a fuzzy Z-open set in 𝐴̃
Proof : obvious
Proposition 2.16:
Let (𝐴̃, 𝑇̃) is a fuzzy topological space , if 𝐵̃ is a fuzzy set in 𝐴̃ , then 𝐺̃ and is a fuzzy Zopen set in 𝐴̃ , then 𝐺̃ ⏞
𝑞 𝐵̃ ⇔ 𝐺̃ ⏞
𝑞 z-cl(𝐵̃ ) .
Proof
(⇒) Let 𝐺̃ ⏞
𝑞 z-cl(𝐵̃ ) , since 𝐵̃ ≤ z-cl(𝐵̃ )
Then 𝐺̃ ⏞
𝑞 𝐵̃
by remark( 1.1.11)
(⟸)suppose that 𝐺̃ ⏞
𝑞 𝐵̃
Then 𝜇𝐺̃ (x) ≤ 𝜇𝐵̃𝐶 (x) by proposition (1.1.10)
And µzint (𝐺̃ ) (x) ≤ 𝜇𝑧𝑖𝑛𝑡 (𝐵̃𝐶 ) (x) , since (𝐺̃ ) is a fuzzy Z-open
Hence 𝐺̃ ⏞
𝑞 z-cl(𝐵̃ ).
6
Definition 2.17 [5 ,6,9,10]
̃ 𝑇̃) is said to be :A fuzzy set 𝐵̃ of a fuzzy topological space (𝐴,
1) Fuzzy b- open (Fuzzy b- closed) set if
𝜇𝐵̃ ≤ max { 𝜇𝑖𝑛𝑡(𝑐𝑙(𝐵̃) (x) , 𝜇𝑐𝑙 (𝑖𝑛𝑡(𝐵̃) (x) }.
𝜇𝐵̃ ≤ min { 𝜇𝑖𝑛𝑡(𝑐𝑙(𝐵̃) (x) , 𝜇𝑐𝑙 (𝑖𝑛𝑡(𝐵̃) (x) }.
The family of all fuzzy B open set ( fuzzy B closed ) sets in 𝐴̃ will be denoted by
FBO(𝐴̃) ( FBO(𝐴̃) ) .
2) Fuzzy 𝛽- open (Fuzzy 𝛽 -closed ) set if
𝜇𝐵̃ ≤ { 𝜇𝑐𝑙(𝑖𝑛𝑡(𝑐𝑙(𝐵̃)) }
{𝜇𝑖𝑛𝑡(𝑐𝑙(𝑖𝑛𝑡(𝐵̃))) } ≤ 𝜇𝐵̃
The family of all fuzzy 𝛽-open (fuzzy𝛽- closed ) set in 𝐴̃ will be denoted by
F𝛽O (𝐴̃) (F𝛽𝐶( 𝐴̃) ).
3) Fuzzy e-open (Fuzzy e-closed) set if
𝜇𝐵̃ ≤ max { 𝜇𝑐𝑙𝑖𝑛𝑡𝛿(𝐵̃) (x) , 𝜇𝑖𝑛𝑡 (𝑐𝑙𝛿(𝐵̃) (x) }.
𝜇𝐵̃ ≥ min { 𝜇𝑐𝑙𝑖𝑛𝑡𝛿(𝐵̃) (x) , 𝜇𝑖𝑛𝑡 (𝑐𝑙𝛿(𝐵̃) (x) }.
The family of all fuzzy e-open (fuzzy e-closed )set in 𝐴̃ will be denoted by FEO(𝐴̃)
(F𝐸𝐶(𝐴̃ )).
4) Fuzzy𝛽 ∗ - open (Fuzzy 𝛽 ∗ -closed ) set if
𝜇𝐵̃ ≤ max{ 𝜇𝑐𝑙(𝑖𝑛𝑡(𝑐𝑙(𝐵̃) (x) , 𝜇𝑖𝑛𝑡(𝑐𝑙𝛿) (𝐵̃) (x) }
𝜇𝐵̃ ≥ min{ 𝜇𝑐𝑙(𝑖𝑛𝑡(𝑐𝑙(𝐵̃) (x), 𝜇𝑖𝑛𝑡(𝑐𝑙𝛿) (𝐵̃ ) (x) } }
The family of all Fuzzy 𝛽 ∗ -open (Fuzzy 𝛽 ∗ -closed ) set in 𝐴̃ will be denoted by (F𝛽 ∗ O (𝐴̃))
(F𝛽 ∗ 𝐶(𝐴̃) ).
7
Proposition 2.18:
̃ 𝑇̃) be a fuzzy topological space then:
Let (𝐴,
1. Every fuzzy Z-open set (resp. fuzzy Z-closed set) is fuzzy b-open set (resp. fuzzy bclosed set).
2. Every fuzzy Z-open set (resp. fuzzy Z-closed set ) is fuzzy 𝛽- open set (resp. fuzzy 𝛽
-closed set ).
3. Every fuzzy Z-open set (resp. fuzzy Z-closed set ) is fuzzy e - open set (resp. fuzzy e
-closed set ).
4. Every fuzzy Z-open set (resp. fuzzy Z-closed set ) is fuzzy 𝛽 ∗ open set (resp. fuzzy
𝛽 ∗ -closed set )
proof :Obvious .
Remark 2.19 :
The converse of proposition (2.13) is not true in general as following examples show:
Example 2.20
̃ 𝐶̃ ,𝐷
̃ ,𝐸̃ ,𝐹̃ are fuzzy subset in 𝐴̃ where:
Let X={a ,b } and 𝐵,
𝐴̃ ={ (a,0.9),(b,0.9) }
𝐵̃ ={ (a,0.3),(b,0.3) }
𝐶̃ ={ (a,0.2),(b,0.2) }
̃ ={ (a,0.5),(b,0.5) }
𝐷
𝐸̃ ={ (a,0.4),(b,0.4) }
𝐹̃ ={ (a,0.6),(b,0.5) }
̃ ,𝐴̃, 𝐵,
̃ 𝐷,
̃ 𝐶,
̃ 𝐸̃ } be a fuzzy topology on 𝐴̃
Let 𝑇̃ ={∅
Then the fuzzy set 𝐹̃ is a fuzzy b-open set but not fuzzy Z-open set
8
Example 2.21:
̃ 𝐶̃ ,𝐷
̃ ,𝐸̃ ,𝐹̃ are fuzzy subset in 𝐴̃ where:
Let X={a,b} and 𝐵,
𝐴̃ ={ (a,0.8),(b,0.8) }
𝐵̃ ={ (a,0.4),(b,0.4) }
𝐶̃ ={ (a,0.2),(b,0.2) }
̃ ={ (a,0.5),(b,0.4) }
𝐷
𝐸̃ ={ (a,0.6),(b,0.4) }
𝐹̃ ={ (a,0.5),(b,0.5) }
̃ ,𝐴̃, 𝐵,
̃ 𝐷,
̃ 𝐶,
̃ 𝐸̃ } be a fuzzy topology on 𝐴̃
Let 𝑇̃ ={∅
Then the fuzzy set 𝐹̃ is a fuzzy 𝛽-open and fuzzy 𝛽 ∗ -open but not fuzzy Z-open set.
Example 2.22:
̃ 𝐶̃ ,𝐷
̃ ,𝐸̃ ,𝐹̃ are fuzzy subset in 𝐴̃ where:
Let X={a,b} and 𝐵,
𝐴̃ ={ (a,0.6),(b,0.6) }
𝐵̃ ={ (a,0.1),(b,0.3) }
𝐶̃ ={ (a,0.4),(b,0.4) }
̃ ={ (a,0.4),(b,0.3) }
𝐷
𝐸̃ ={ (a,0.2),(b,0.2) }
𝐹̃ ={ (a,0.5),(b,0.2) }
𝐺̃ ={(a,0.2),(b,0.3) }
̃ ={(a,0.1),(b,0.2) }
𝐻
̃ ,𝐴̃, 𝐵,
̃ 𝐷,
̃ 𝐶,
̃ 𝐸̃ ,𝐺̃ , 𝐻
̃ } be a fuzzy topology on 𝐴̃
Let 𝑇̃ ={∅
Then the fuzzy set 𝐹̃ is fuzzy e-open set but not fuzzy Z-open set
9
Remark 2.23
This diagram explain the relation between fuzzy (b-open, 𝛽-open, 𝛽 ∗ - open ,e-open)sets with
fuzzy Z-open set.
Fffuzzy
Z-open set
Fuzzy
Fuzzy
b-open
set
𝛽-open
set
Fuzzy
𝛽 ∗- open
set
Fuzzy
e-open
set
Diagram (1)
Proposition(2.24) :
̃ 𝑇̃ ) be a fuzzy topological space then:
Let (𝐴,
1. Every fuzzy b-open set (resp. fuzzy b-closed set) is fuzzy 𝛽-open set (resp. fuzzy 𝛽 –
closed set) .
2. Every fuzzy 𝛽-open set (resp. fuzzy 𝛽-closed set) is fuzzy 𝛽 ∗ -open set (resp. fuzzy 𝛽 ∗ open set).
3. Every fuzzy 𝛽 ∗ -open set (resp. fuzzy 𝛽 ∗ -closed set) is fuzzy e-open set (resp. fuzzy eclosed set).
4. Every fuzzy b-open set (resp. fuzzy b-closed set) is fuzzy e -open set (resp. fuzzy e–
closed set) .
Proof: Obvious
Remark(2.25) :
The converse of proposition (1.4.2) is not true in general as following examples show:
10
Example 2.26
̃ 𝐻
̃ 𝐶̃ ,𝐷
̃ ,𝐸̃ ,𝐹̃ ,𝐺,
̃, 𝑤
Let X={a,b} and 𝐵,
̃are fuzzy subset in 𝐴̃ where:
𝐴̃ ={ (a,0.7),(b,0.7) }
𝐵̃ ={ (a,0.0),(b,0.4) }
𝐶̃ ={ (a,0.4),(b,0.0) }
̃ ={ (a,0.4),(b,0.4) }
𝐷
𝐸̃ ={ (a,0.2),(b,0.2) }
𝐹̃ ={ (a,0.0),(b,0.2) }
𝐺̃ ={ (a,0.2),(b,0.4) }
̃ ={ (a,0.2),(b,0.0) }
𝐻
̃ ={ (a,0.4),(b,0.2) }
𝑊
̃ = { (a,0.0),(b,0.5) }
𝐾
̃ ,𝐴̃, 𝐵,
̃ 𝐷,
̃ 𝐶,
̃ 𝐸̃ , 𝐹̃ , 𝐺̃ , 𝐻
̃, 𝑊
̃ } be a fuzzy topology on 𝐴̃
Let 𝑇̃ ={∅
̃ is fuzzy 𝛽 ∗ -open (resp. fuzzy b-open ) but not fuzzy e-open set.
Then the fuzzy set 𝐾
Example 2.27:
̃ 𝐶̃ ,𝐷
̃ ,𝐸̃ ,𝐹̃ are fuzzy subset in 𝐴̃ where:
Let X={a,b} and 𝐵,
𝐴̃ ={ (a,0.5),(b,0.5) }
𝐵̃ ={ (a,0.2),(b,0.1) }
𝐶̃ ={ (a,0.3),(b,0.2) }
̃ ={ (a,0.2),(b,0.2) }
𝐷
𝐸̃ ={ (a,0.3),(b,0.1) }
̃ ,𝐴̃, 𝐵,
̃ 𝐷,
̃ 𝐶,
̃ 𝐸̃ } be a fuzzy topology on 𝐴̃
𝐹̃ ={ (a,0.1),(b,0.3) }, Let 𝑇̃ ={∅
Then the fuzzy 𝛽- open set but not fuzzy b-open set.
11
Example 2.28:
̃ 𝐻
̃ 𝐶̃ ,𝐷
̃ ,𝐸̃ ,𝐹̃ , 𝐺,
̃are fuzzy subset in 𝐴̃ where:
Let X={a,b} and 𝐵,
𝐴̃ ={ (a,0.6),(b,0.6) }
𝐵̃ ={ (a,0.6),(b,0.1) }
𝐶̃ ={ (a,0.0),(b,0.2) }
̃ ={ (a,0.5),(b,0.2) }
𝐷
𝐸̃ ={ (a,0.0),(b,0.1) }
𝐹̃ ={ (a,0.6),(b,0.2) }
𝐺̃ ={ (a,0.5),(b,0.1) }
̃ ={ (a,0.0),(b,0.4) }
𝐻
̃ ,𝐴̃, 𝐵,
̃ 𝐷,
̃ 𝐶,
̃ 𝐸̃ , 𝐹̃ , 𝐺̃ } be a fuzzy topology on 𝐴̃
Let 𝑇̃ ={∅
̃ is fuzzy 𝛽 ∗ -open set but not fuzzy 𝛽-open set.
Then the fuzzy set 𝐻
Example 2.29
̃ 𝐶̃ ,𝐷
̃ ,𝐸̃ ,𝐹̃ are fuzzy subset in 𝐴̃ where:
Let X={a,b} and 𝐵,
𝐴̃ ={ (a,0.7),(b,0.7) }
𝐵̃ ={ (a,0.5),(b,0.5) }
𝐶̃ ={ (a,0.4),(b,0.4) }
̃ ={ (a,0.2),(b,0.2) }
𝐷
𝐸̃ ={ (a,0.3),(b,0.3) }
̃ ,𝐴̃, 𝐵,
̃ 𝐷̃} be a fuzzy topology on 𝐴̃ Then the 𝐸̃ fuzzy e-open set but not fuzzy
̃ 𝐶,
Let 𝑇̃ ={∅
b-open set.
12
Remark 2.30
This diagram explain the relation between some type of fuzzy (b-open, 𝛽-open, 𝛽 ∗ - open ,eopen)sets with fuzzy Z-open set
Fuzzy
z-open set
Fuzzy
e-open set
Fuzzy‫ببببب ببببب‬
b-open set
‫ببعبعبع‬
Fuzzy
𝛽-open set
Fuzzy
𝛽 ∗ -open set
Diagram (2)
13
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