Abstract:

In this paper we introduce fuzzy ? –

generalized Baire Spaces, fuzzy weakly generalized Baire space, fuzzy

generalized ?-Baire space and discuss about some of

its properties with suitable examples.

Key

words:

Fuzzy

? – open sets, fuzzy ? – generalized open sets, fuzzy ? – generalized Baire

space, fuzzy weakly generalized Baire space and fuzzy generalized ?-Baire space.

Introduction:

The theory of fuzzy sets was initiated by L.A.Zadeh in his

classical paper 9 in the year 1965 as an attempt to develop a mathematically precise

framework in which to treat systems or phenomena which cannot themselves be

characterized precisely. The potential of fuzzy notion was realized by the

researchers and has successfully been applied for investigations in all the

branches of Science and Technology. The paper of C.L.Chang 2 in 1968 paved

the way for the subsequent tremendous growth of the numerous fuzzy topological

concepts.

The concepts of ?-generalized closed sets have been studied

in classical topology in 3. In this paper we introduce the fuzzy ?–generalized,fuzzy

weakly generalized and fuzzy generalized ? – nowhere dense sets and fuzzy ?-generalized,

fuzzy weakly generalized and fuzzy generalized

? –Baire spaces with suitable examples.

Preliminaries:

Now review of

some basic notions and results used in the sequel. In this work by (X,T) or

simply by X, we will denote a fuzzy topological space due to Chang 2.

Definition 2.1 1 Let ? and ? be any

two fuzzy sets in a fuzzy topological space (X, T). Then we define:

???

: X ? 0,1 as follows: ??? (x) = max {?(x), ?(x)};

???

: X ? 0,1 as follows: ??? (x) = min { ?(x), ?(x)};

?

=

? ?(x) = 1-

?(x).

For a family

? I of fuzzy sets in (X, T), the union ? =

and intersection

? =

are

defined respectively as

, and

.

Definition 2.2 2

Let (X,T) be a fuzzy topological space. For a fuzzy set ? of

X, the interior and the closure of ? are defined respectively as

and cl

.

Definition 2.2 3

Let (X,T) be a topological space. For a fuzzy set ? of X is a

? – generalized closed set (briefly ?g-closed) if ?cl(?) ? µ whenever ? ? µ and

µ is fuzzy open in X.

Definition 2.3 8

A fuzzy set ? in a fuzzy topological space (X,T) is called

fuzzy dense if there exists no fuzzy closed set ? ? (X,T) such that ? < ?
< 1. That is cl(?) = 1.
Definition 2.4 7
A fuzzy set ? in a fuzzy topological space (X,T) is called
fuzzy nowhere dense if there exists no
non-zero fuzzy open set ? in (X,T) such that ? < cl(?). That is, int cl(?) =
0.
Definition 2.5 7
Let (X,T) be a fuzzy topological space. A fuzzy set ? in (X,T)
is called fuzzy first category set if
, where
's are fuzzy nowhere dense sets in (X,T). A fuzzy
set which is not fuzzy first category set is called a fuzzy second category set
in (X,T).
Definition 2.6 6
A fuzzy topological space (X,T) is called fuzzy first category space if
, where
's are fuzzy
nowhere dense sets in (X,T). A topological space which is not of fuzzy first
category is said to be of fuzzy second category space.
Definition 2.7 7
Let (X,T) be a fuzzy topological space. Then (X,T) is called
a fuzzy Baire space if
, where
) 's are fuzzy nowhere dense sets in (X,T).
Fuzzy
?-generalized nowhere dense sets, Fuzzy weakly generalized nowhere dense sets,
Fuzzy generalized ?-nowhere dense sets: We introduce fuzzy ? – generalized
nowhere dense sets, fuzzy weakly generalized nowhere dense sets, fuzzy
generalized ?-nowhere dense sets with suitable examples.
Definition 3.1
A fuzzy set ? in a fuzzy topological space (X,T) is called
fuzzy ? – generalized nowhere dense if there exists no non-zero fuzzy ? –
generalized open set ? in (X,T) such that ? < ?-cl(?). That is, ?-int ?-cl(?)
= 0.
Example
3.1
Let
X = {a,b,c}. The fuzzy sets ?, µ and ? are defined on X as follows:
? : X
0,1 defined as ?(a) = 0.8 ;
?(b) = 0.7; ?(c) = 0.8.
µ : X 0,1 defined as µ(a) = 0.9 ; µ(b) = 0.7;
µ(c) = 0.7.
? : X 0,1 defined as ?(a) = 0.8; ?(b) = 0.7 ; ?(c)
= 0.7
Then T = {0, ?, µ, ?, (? v µ),1}
is fuzzy topology on X. The fuzzy sets ?, µ, ?, and ?Vµ are fuzzy ?-open sets.
Now 1-? ? ??µ where ??µ
is fuzzy ?-open then ?-cl(1-?)? ??µ,
1-µ ? ??µ where ??µ
is fuzzy ?-open then ?-cl(1-µ)? ??µ,
1-? ? ??µ
where ??µ is fuzzy ?-open then ?-cl(1-?) ? ?Vµ,
1-?Vµ ? ??µ
where ??µ is fuzzy ?-open
then ?-cl(1-(??µ))?
??µ.
1-?, 1-µ,
1-?,1-??µ's are fuzzy generalized closed sets.
Now ?-int
?-cl(1-?) = ?-int (1-?) = 0,
?-int ?-cl(1-µ)
= ?-int(1-µ) = 0,
?-int ?-cl(1-?)
= ?-int(1-?) = 0,
?-int
?-cl1-?Vµ = ?-int(1-(?Vµ))= 0
Therefore
1-?, 1-µ, 1-? ,1-(??µ)'s are fuzzy ?-generalized nowhere
dense sets.
Example
3.2
Let
X = {a,b,c}. The fuzzy sets ?, µ and ? are defined on X as follows:
? : X
0,1 defined as ?(a) = 0.8 ; ?(b) = 0.6.
µ : X 0,1 defined as µ(a) = 0.9 ; µ(b) =
0.7.
? : X 0,1 defined as ?(a) = 0.9; ?(b) =
0.9.
Then T = {0, ?, µ, ?,1} is fuzzy
topology on X. The fuzzy sets ?, µ, and ? are fuzzy ?-open sets.
Now 1-? ? ? where ??µ is fuzzy
?-open then ?-cl(1-?)? ?,
1-µ ?
? where ??µ is fuzzy ?-open then ?-cl(1-µ)? ?,
1-?
? ? where ??µ
is fuzzy ?-open then ?-cl(1-?) ? ?,
Now we define the fuzzy sets ?,? and ? on X as follows:
? : X ? 0,1 defined as ?(a) =
0.8 ; ?(b) = 0.8.
? : X 0,1 defined as ?(a) = 0.9 ; ?(b) = 0.8.
? : X 0,1 defined as ?(a) = 0.8 ; ?(b) =
0.7.
The
fuzzy subsets ?, ? and ? are not fuzzy ? – generalized nowhere dense sets.
Since ?-int ?-cl(?) = 1?0, ?-int ?-cl(?) =1? 0 and ?-int
?-cl(?) =1? 0. Therefore ?, ? and ? are not of fuzzy
?-generalized nowhere dense set.
Definition 3.3
A fuzzy set ? in a fuzzy
topological space (X,T) is called fuzzy weakly generalized nowhere dense if
there exists no non-zero fuzzy weakly generalized open set ? in (X,T) such that
? < cl(?). That is, int cl(?) = 0.
Example 3.3
Let
X = {a,b,c}. The fuzzy sets ?, µ and ? are defined on X as follows:
? : X
0,1 defined as ?(a) = 0.9 ; ?(b) = 0.7; ?(c) = 0.6.
µ : X 0,1 defined as µ(a) = 0.8 ; µ(b) = 0.9;
µ(c) = 0.5.
? : X 0,1 defined as ?(a) = 0.9; ?(b) = 0.8 ; ?(c)
= 0.8
Then T = {0, ?, µ, ?, (? ? µ), (? ? µ), (µ??), (µ??),1} is fuzzy topology on X. The
fuzzy sets ?,
µ, ?, and ? ?
µ, ? ?
µ, µ??,
µ??
are
fuzzy open sets.
Now (1-?)<µ??
? cl int (1-?) = cl(0)? µ??,
(1-µ)<µ??
? cl int (1-µ) = cl(0)? µ??,
(1-?)<µ??
? cl int (1-?)
= cl(0)? µ??,
(1-??µ)<µ??
? cl int (1-??µ)
= cl(0)? µ??,
(1-??µ)<µ??
? cl int (1-??µ)
= cl(0)? µ??,
(1-µ??)<µ??
? cl int (1-µ??)
= cl(0)? µ??,
(1-µ??)<µ??
? cl int (1-µ??)
= cl(0)? µ??.
Where 1- ? ,1-µ ,1-?,
1-??µ, 1-??µ, 1-µ??,
1-µ??'s are fuzzy weakly generalized closed
set.
Now Int cl (1-?)
= int (1-?) = 0,
Int cl (1- µ) =
int (1- µ) = 0,
Int cl (1- ?)
= int (1- ?) = 0,
Int cl (1-??µ) = int (1-??µ) = 0,
Int cl (1-??µ)
= int (1-??µ) = 0,
Int cl (1- µ??)
= int (1- µ??) = 0,
Int cl (1- µ??)
= int (1- µ??) = 0
Therefore 1- ?,1-µ,1-?,1-(??µ),1-(??µ),1-(µ??),1-(µ??)'s
are fuzzy weakly generalized nowhere dense set.
Example
3.4
Let
X = {a,b,c}. The fuzzy sets ?, µ and ? are defined on X as follows:
? : X
0,1 defined as ?(a) = 0.8 ; ?(b) = 0.5.
µ : X 0,1 defined as µ(a) = 0.9 ; µ(b) =
0.6.
? : X 0,1 defined as ?(a) = 0.9; ?(b) = 0.8.
Then T = {0, ?,
µ, ?,1} is fuzzy topology on X. The fuzzy sets ?, µ, and ? are fuzzy open sets.
Now 1-? < ?
? cl int (1-?) = 0 ? ?,
1-µ < ?
? cl int (1-µ) = 0 ? ?,
1-?
< ? ? cl int (1-?)
= 0 ? ?,
Now we define
the fuzzy sets ?, ? and ? on X as follows:
? : X 0,1 defined as ?(a) = 0.8 ; ?(b) =
0.6.
? : X 0,1 defined as ?(a) = 0.8 ; ?(b) =
0.7.
? : X 0,1 defined as ?(a) = 0.9 ; ?(b) =
0.7.
The
fuzzy subsets ?, ? and ? are not fuzzy weakly generalized nowhere dense sets.
Since int cl(?) = 1?0, int cl(?) =1? 0 and int cl(?) =1? 0. Therefore ?, ? and ? are not of
fuzzy weakly generalized nowhere dense set.
Definition 3.5
A fuzzy set ? in a fuzzy topological space (X,T) is called
fuzzy generalized ? – nowhere dense if there exists no non-zero fuzzy
generalized ? –open set ? in (X,T) such that ? < cl(?). That is, int cl(?) =
0.
Example
3.5
Let
X = {a,b,c}. The fuzzy sets ?, µ and ? are defined on X as follows:
? : X
0,1 defined as ?(a) = 0.9 ; ?(b) = 0.9; ?(c) = 0.8.
µ : X 0,1 defined as µ(a) = 0.8 ; µ(b) =
0.7; µ(c) = 0.6.
? : X 0,1 defined as ?(a) = 0.9; ?(b) = 0.8 ; ?(c)
= 0.6
Then T = {0, ?, µ, ?, (? v µ),1}
is fuzzy topology on X. The fuzzy sets ?, µ, ?'s are fuzzy open sets.
Now 1-? < ? ?
µ is open ? cl (1-?) ? ?,
1-µ < ? ?
µ is open ? cl (1-?) ? ?,
1-? < ? ?
µ is open ? cl (1-?) ? ?.
1-?, 1-µ, 1-?'s
are fuzzy generalized ? –
closed set.
Now Int cl (1-?)
= int (1-?)=0,
Int cl (1-µ) =
int (1-µ)= 0,
Int cl (1-?) =int (1-?)= 0.
Now 1-?, 1-µ,
1-?'s are fuzzy generalized ?-nowhere dense set.
Example 3.6
Let X = {a,b,c}. The fuzzy sets
?, µ and ? are defined on X as follows:
? : X
0,1 defined as ?(a) = 0.7; ?(b) = 0.6.
µ : X 0,1 defined as µ(a) = 0.8; µ(b) =
0.6.
? : X 0,1 defined as ?(a) = 0.9; ?(b) =
0.7.
Then T = {0, ?, µ, ?,1} is fuzzy
topology on X. The fuzzy sets ?, µ, and ? are fuzzy open sets.
Now 1-? < ? ? cl (1-?) ? ?,
1-µ < ? ?
cl (1-µ) ? ?,
1-? < ? ?
cl (1-?) ? ?,
Now we define
the fuzzy sets ?, ? and ? on X as
follows:
? : X ? 0,1 defined as ?(a) =
0.7 ; ?(b) = 0.7.
? : X 0,1 defined as ?(a) = 0.8 ; ?(b) =
0.5.
? : X 0,1 defined as ?(a) = 0.9 ; ?(b) =
0.6.
The
fuzzy subsets ?, ? and ? are not fuzzy generalized
? – nowhere dense sets. Since int cl(?) = 1?0, int cl(?) =1? 0 and int cl(?) =1? 0. Therefore ?, ? and ? are not of
fuzzy generalized ?-
nowhere dense set.
Fuzzy
?-generalized Baire spaces, Fuzzy weakly generalized Baire space, Fuzzy
generalized ?-Baire space:
We introduce fuzzy ? –
generalized Baire space, fuzzy weakly generalized Baire space, fuzzy
generalized ?-Baire space with suitable examples.
Definition 4.1
A fuzzy topological space (X,T) is called fuzzy ?-generalized
Baire space if
, where
) 's are fuzzy ?-nowhere dense sets in (X,T).
In
example 3.1, The fuzzy sets 1-?, 1-µ, 1-? ,1-(??µ)'s are fuzzy ?-generalized nowhere
dense sets. Now ?-int(1-?) ?
(1-µ) ? (1-?) ? (1-(??µ))= ?-int(1-?) = 0. Therefore the fuzzy topological
space (X,T) is fuzzy ?-generalized Baire space.
Definition 4.2
A fuzzy topological space (X,T) is called fuzzy weakly generalized
Baire space if
, where
) 's are fuzzy nowhere dense sets in (X,T).
In
example 3.3, The fuzzy sets 1-?, 1-µ, 1-? ,1-(??µ),(1-(??µ),1-(µ??),1-(µ??)'s are fuzzy weakly generalized
nowhere dense sets. Now int(1-?) ?
(1-µ) ? (1-?) ? (1-(??µ)) ?
(1(??µ))? (1-(µ??))?(1-(µ??))= int(1-(??µ)) = 0. Therefore the fuzzy topological
space (X,T) is fuzzy weakly generalized Baire space.
Definition 4.3
A fuzzy topological space (X,T) is called fuzzy generalized ?-
Baire space if
, where
) 's are fuzzy nowhere dense sets in (X,T).
In
example 3.5, The fuzzy sets 1-?, 1-µ, 1-?'s are fuzzy generalized ?-
nowhere dense sets. Now ?-int(1-?) ?
(1-µ) ? (1-?)= int(1-µ)
= 0. Therefore the fuzzy topological space (X,T) is fuzzy generalized ?-Baire space.
Some
relations of fuzzy ?–generalized, fuzzy weakly generalized Baire space and
fuzzy generalized ?– Baire space:
Proposition 5.1:
A
fuzzy generalized ?-Baire space is also
a fuzzy ?-generalized Baire space.
Consider
the following example.
Let X = {a,b,c}.
The fuzzy sets ?, µ and ? are defined on X as follows:
? : X
0,1 defined as ?(a) = 0.9 ; ?(b) = 0.7; ?(c) = 0.6.
µ : X 0,1 defined as µ(a) = 0.8 ; µ(b) =
0.5; µ(c) = 0.4.
? : X 0,1 defined as ?(a) = 0.7 ; ?(b) =
0.6 ; ?(c) = 0.6.
Then
T = {0, ?, µ, ?, (µ??),(µ??),1} is fuzzy
topology on X.
1-? < ? ?
? is open ?cl (1-?) ? ?,
1-µ < ? ?
? is open ?cl (1-µ) ? ?,
1-? < ? ?
? is open ?cl (1-?) ? ?,
1-µ??
< ? ? ? is open ?cl
(1- µ??) ? ?,
1- µ??
< ? ? ? is open ?cl
(1- µ??) ? ?.
(1-?),(1-µ),(1-?),(1-
µ??),( 1-µ??)'s are fuzzy
generalized ?- closed set.
Int cl (1-?) =
int (1-?) = 0,
Int cl (1-µ) =
int (1-µ) = 0,
Int cl (1-?) =
int (1-?) = 0,
Int cl (1-µ??)
= int (1- µ??) = 0,
Int cl (1- µ??)
= int (1- µ??) = 0.
(1-?),(1-µ),(1-?),(1-
µ??),( 1-µ??)'s are
fuzzy generalized ?- nowhere dense
set.Now
Int (1-?) ?
(1-µ) ? (1-?)
? (1-µ??) ?
(1-µ??) = 0
Int (1-µ??)
= 0.
Therefore (X,T)
is a fuzzy generalized ?- Baire space. Now to say that it is fuzzy
?-generalized Baire space we have to show that µ is ?-open Int cl int (?) ? ?.
Int cl int (?) =
int cl (?) = int (1) = 1,
Int cl int (µ) =
int cl (µ) = int (1) = 1,
Int cl int (?) =
int cl (?) = int (1) = 1,
Int cl int (µ??)
= int cl (µ??) = int (1) = 1,
Int cl int (µ??)
= int cl (µ??) = int (1) = 1.
Thus generalized
?- Baire space is also a ?-generalized Baire space.
Proposition 5.2:
A
weakly generalized Baire space is also a generalized ?- Baire space.
Consider
a example.
Let X = {a,b,c}.
The fuzzy sets ?, µ and ? are defined on X as follows:
? : X
0,1 defined as ?(a) = 0.9 ; ?(b) = 0.9; ?(c) = 0.8.
µ
: X 0,1 defined as µ(a) = 0.8 ; µ(b) = 0.7;
µ(c) = 0.6.
?
: X 0,1 defined as ?(a) = 0.9
; ?(b) = 0.8 ; ?(c) = 0.6.
Then
T = {0, ?, µ, ?,1} is fuzzy topology on X.
Now1-? < ? ?
cl int (1-?) = cl (0) ? ?,
1-µ < ? ?
cl int (1-µ) = cl (0) ? ?,
1-? < ? ?
cl int (1-?) = cl (0) ? ?.
(1-?), (1-µ),
(1-?)'s are fuzzy weakly generalized closed set.
Int cl int (1-?)
= int (1-?) = 0,
Int cl int (1-µ)
= int (1-µ) = 0,
Int cl int (1-?)
= int (1-?) = 0.
(1-?), (1-µ),
(1-?)'s are fuzzy weakly generalized nowhere dense set.
Int ((1-?) ?
(1-µ) ? (1-?) = 0
Int (1-µ) = 0.
Therefore (X,T) is a fuzzy weakly
generalized Baire spaces. Now to say that it is generalized ?- Baire space we
have to show that cl (?) ?µ ? µ is open in X.
(1-?) < ? ?
µ is open ? cl (1-?) ? ?,
(1-µ) < ? ?
µ is open ? cl (1-µ) ? ?,
(1-?) < ? ?
µ is open ? cl (1-?) ? ?.
(1-?), (1-µ),
(1-?)'s are fuzzy generalized ?- closed set.
Int cl int (1-?)
= int (1-?) = 0,
Int cl int (1-µ)
= int (1-µ) = 0,
Int cl int (1-?)
= int (1-?) = 0.
(1-?), (1-µ),
(1-?)'s are fuzzy generalized ?- nowhere dense set.Now
Int ((1-?) ?
(1-µ) ? (1-?) = 0
Int (1-µ) = 0.
Therefore (X,T) is a fuzzy generalized ?-baire spaces.Thus weakly generalized
baire space is also a generalized ?- baire space.
Proposition 5.3:
A
weakly generalized Baire space is also a ?-generalized Baire space.
Consider
an example.
Let X = {a,b,c}.
The fuzzy sets ?, µ and ? are defined on X as follows:
?
: X 0,1 defined as ?(a) = 0.9
; ?(b) = 0.7; ?(c) = 0.6.
µ
: X 0,1 defined as µ(a) = 0.8
; µ(b) = 0.8; µ(c) = 0.5.
?
: X 0,1 defined as ?(a) = 0.8 ;
?(b) = 0.8 ; ?(c) = 0.8.
Then
T = {0, ?, µ, ?, ??µ, ??µ, ???, 1}
is fuzzy topology on X.
1-? < ??µ ?cl
int (1-?) = cl(0) = 0 ? ??µ,
1-µ < ??µ ?cl
int (1-µ) = cl(0) = 0 ? ??µ,
1-?
< ??µ ?cl
int (1-?) = cl(0) = 0 ? ??µ,
1-??µ
< ??µ ?cl
int (1-??µ) = cl(0) = 0 ? ??µ,
1-??µ
< ??µ ?cl
int (1-??µ) = cl(0) = 0 ? ??µ ,
1-???
< ??µ ?cl
int (1-???) = cl(0) = 0 ? ??µ.
(1-?),(1-µ),(1-?),(1-???),(1-??µ),(1-??µ)'s
are fuzzy weakly generalized closed set.
Int cl (1-?) =
int (1-?) = 0,
Int cl (1-µ) =
int (1-µ) = 0,
Int cl (1-?)
= int (1-?) = 0,
Int cl (1-??µ)
= int (1-??µ) = 0,
Int cl (1-??µ)
= int (1-??µ) = 0,
Int cl (1-???)
= int (1-???) = 0.
(1-?),(1-µ),(1-?),(1-???),(1-??µ),(1-??µ)'s
are fuzzy weakly generalized nowhere dense set.
Int (1-?) ? (1-µ) ? (1-?) ?
(1-??µ) ?
(1-??µ) ? (1-???)= 0
Int (1-??µ) = 0. (X,T) is a fuzzy weakly generalized Baire space. Now
to say that it is ?-generalized Baire space we have to show that it µ ?- is open and cl (?) ? µ.
1-? < ??µ
? ??µ is ?-open ?
cl (1-?) ? ??µ,
1-µ < ??µ
? ??µ is ?-open ?
cl (1-µ) ? ??µ,
1-?
< ??µ ? ??µ
is ?-open ? cl (1-?) ? ??µ,
1-??µ<
??µ ? ??µ
is ?-open ? cl (1-??µ) ? ??µ,
1-??µ
< ??µ ? ??µ
is ?-open ? cl (1-??µ) ? ??µ,
1-???
< ??µ ? ??µ
is ?-open ? cl (1-???) ? ??µ.
(1-?),(1-µ),(1-?),(1-???),(1-??µ),(1-??µ)'s
are fuzzy generalized closed set.
Int cl (1-?) =
int (1-?) = 0,
Int cl (1-µ) =
int (1-µ) = 0,
Int cl (1-?)
= int (1-?) = 0,
Int cl (1-??µ)
= int (1-??µ) = 0,
Int cl (1-??µ)
= int (1-??µ) = 0,
Int cl (1-???)
= int (1-???) = 0.
(1-?),(1-µ),(1-?),(1-???),(1-??µ),(1-??µ)'s
are fuzzy ?-generalized nowhere dense set.
Int (1-?) ? (1-µ) ? (1-?) ?
(1-??µ) ?
(1-??µ) ? (1-???)= 0
Int (1-??µ) = 0. (X,T) is a fuzzy ?-generalized
Baire space. Thus weakly generalized
Baire space is also ?-generalized Baire space.
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