Download on fuzzy almost p-spaces

IJISET - International Journal of Innovative Science, Engineering & Technology, Vol. 2 Issue 4, April 2015.
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ISSN 2348 – 7968
ON FUZZY ALMOST P-SPACES
G.Thangaraj, C.Anbazhagan
Department
Department
of Mathematics, Thiruvalluvar
of
University, Vellore - 632 115, Tamilnadu, India.
Mathematics, Jawahar Science College, Neyveli – 607 803, Tamilnadu, India.
ABSTRACT
In
this paper we
conditions
discuss several
under which
fuzzy
characterizations
topological spaces
investigated. Some results concerning
the context
of
images
KEY WORDS : Fuzzy
and
of fuzzy
functions
preimages
G-set, fuzzy
are
Fσ -set,
almost
P-spaces and
become fuzzy almost
that
the
P-spaces, are
preserve fuzzy almost P-spaces
in
obtained.
fuzzy dense set, fuzzy
nowhere dense set ,
fuzzy submaximal space, fuzzyD-Bairespace,fuzzy strongly irresolvable
space , somewhat
fuzzy
continuous function , somewhat fuzzy open
function.
1 .INTRODUCTION
In order to deal with uncertainties,
were
first
introduced
by
the concept
of
fuzzy sets
and fuzzy set operations
L.A.ZADEH [23]
in
his classical paper in
the year 1965,
describing fuzziness mathematically for the first time. This inspired mathematicians to fuzzify
mathematical structures. The concepts of fuzzy topology was defined by C.L.CHANG [3] in
the year 1968. The paper of Chang paved the way
the
numerous fuzzy topological
for the subsequent tremendous growth of
concepts. Since then much attention has been paid
to
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ISSN 2348 – 7968
generalize the basic concepts of general topology in fuzzy setting and thus a modern
theory of fuzzy topology has been developed.
A.K.MISHRA [8] introduced
additive spaces
the
concepts
of
P-spaces
generalization
of -
of SIKORSKI [10] and COHEN, L.W. and C. GOFFMAN [5]. The concept
of P-spacesin fuzzy setting was introduced by
G. BALASUBRAMANIAN in [13].
P-spaces in classical topology was introduced by
further by
as a
A .I. Veksler [21] and wasalso studied
R.Levy [7]. Chang IL Kim [4] studied
spaces.The concept
Almost
several characterizations
of almost P-spaces in fuzzy setting
was
introduced by
almost
the
P-
authors
in [16]. In this paper we discuss several characterizations of fuzzy almost P-spaces and
the conditions
under which
fuzzy
topological spaces
investigated.Some results concerning
the context
of
images
and
functions
that
become fuzzy almost
preservefuzzy
almost
P-spaces, are
P-spaces
in
preimages are obtained.
2. PRELIMINARIES
Now we
In
introduce some basic notions
this work by (X,T) or simply
due
and results used
by X, we will denote a
in the sequel.
fuzzy
topological space
to CHANG (1968).
Definition 2.1 : Let
and μ be any two fuzzy sets in a fuzzy topological
space ( X,T
). Then we define :
(i).  μ : X  [0,1] as follows : (  μ) ( ) = Max { ( ), μ ( ) }.
(ii).  μ : X  [0,1]
(iii).
μ as follows : (  μ) ( ) = Min { ( ), μ ( ) }.
⇔ μ ( )
= 1
( ).
390 IJISET - International Journal of Innovative Science, Engineering & Technology, Vol. 2 Issue 4, April 2015.
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More
∨
) and
,
∧
generally , for a family {
/ ∈I } of fuzzy sets in ( X,T),
, are defined respectively as a
fuzzy
topological
space
in (X,T). We define the interior and the closure of
(i). int (λ)
˅ { μ / μ
(ii). cl (λ)
˄ { μ / λ
Lemma 2.1 [1] : For
a
(ii). 1
respectively
as follows :
μ, 1 μ ∈ T }.
of a
fuzzy topological space X,
int ( 1 – ).
A fuzzy set
in
a
dense
if there exists no fuzzy closed set
cl( )
1.
Definition 2.4 [14] : A fuzzy set
if
there exists
μ ˂ cl ( ). That is,
in
if
μ in (X,T) such that
a fuzzy topological
is,
space (X,T) is called fuzzy
fuzzy open set
μ in
(X,T) such that
in a fuzzy topological space (X,T) is called a
⋁ λ where 1  i T
λ
Definition 2.6 [2] : A fuzzy set
λ
fuzzy G-set in (X,T)
⋀ λ where
if
< μ < 1. That
0.
int cl ( )
(X,T)
fuzzy topological space (X,T) is called fuzzy
no non - zero
Definition 2.5 [2] : A fuzzy set λ
Fσ -set in
be any fuzzy set
λ , μ ∈ T }
fuzzy set
cl ( )
Definition 2.3 [14] :
dense
λ
and cl ( 1 ,
(i). 1 int (
fuzzy
∈ X }and
,
∈ X }.
Definition 2.2 : Let ( X,T) be
nowhere
λ
Definition 2.7 [1]: A fuzzy set
λ
for i.
in a fuzzy topological space (X,T) is called a
in a
i
T
for i.
fuzzy topological space (X,T) is called
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(i).a fuzzy regular openset in (X,T)ifint cl ( λ (ii).a fuzzy regular closed set in (X,T) if cl int ( λ λ,
setin (X,T) if cl int ( λ ,
(ii).a fuzzy semi-open
(ii).a fuzzy semi- closed set in (X,T) if int cl ( λ Lemma 2.2 [1] : For
ISSN 2348 – 7968
λ ,
a
of {λ } of
family
( X,T ),  cl λ  cl (λ . In case
λ.
fuzzy sets of a fuzzy topological
space
set, cl λ  cl( λ )). Also
 int
is a finite
λ  int (λ in (X,T).
Definition 2.9 [14] : A fuzzy
set
⋁
in
if
sets
other fuzzy set
(X,T). Any
a
fuzzy
topological space ( X,T )
λ ,where where
fuzzy first category set
in
λ
λ
in (X,T)
is
′
are fuzzy
said to be of
is called a
nowhere
dense
fuzzy second
category.
Definition 2.10 [14]: A fuzzy
if
⋁
λ
1 ,where
which is not of
fuzzy
topological space (X,T)
′
are fuzzy
is
called
first category, is said to be of fuzzy
–1
() is
the fuzzy
Also the image of
defined
category
second category.
topological spaces. Let f
function from the fuzzy topological space ( X ,T) to the fuzzy topological space
(Y,S). Let  be a fuzzy set in (Y,S). The inverse
f
first
nowhere dense sets in (X,T). A topological space
Definition 2.11 [3 ] : Let (X ,T) and (Y,S ) be any two fuzzy
be a
fuzzy
image of  under f written
–1
set in (X ,T ) defined by f () ( ) =  ( f ( ))
for all
as
 X.
 in (X,T) under f written as f (  ) is the fuzzy set in (Y,S)
by
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ISSN 2348 – 7968
 f
∈f y
f (  ) (y) =
y isnon
f : (X ,T ) →(Y, S)
of
(X ,T)
and
;
for each y  Y.
0
Lemma 2.3[3] :Let
.
be
a
mapping. For
fuzzy
sets
λ and μ
(Y, S) respectively, the following statements hold :
(μ) ≤ μ ;
(1)
ff
(2)
f
(3)
f (1 – λ)  1 – f (λ);
(4)
f
(5)
If
f
is
one - to - one , then
(6)
If
f
is
onto ,
(7)
If
f is

f(λ)
λ;
1–f
(1 – μ)
(μ) ;
ff
then
f
(μ)
f (λ)
λ ;
μ;
one - to - one and onto, then
LEMMA 2.4 [ 1 ] : Let f : (X ,T ) →(Y, S)
be
f (1–λ)
a
1 – f( λ).
mapping and{
} be a family
of fuzzy sets of Y. Then
(a) f
(⋃α λ )
f
(⋂ λ )
(b)
⋃α f
⋂ f
.
).
Lemma 2.5 [6] : Let f : (X ,T ) →(Y, S) be
a
mapping and { A }, jϵJ be a family
of fuzzy sets in X. Then
(a)
f (⋃ ϵ A )
(b)
f (⋂ ∈ A )
⋃ ∈ f A .
⋂ ∈ f A ).
3. FUZZY ALMOST P-SPACES
Almost P-spaces in classical topology was introduced
by A .I. Veksler [21] and was also
studied further by R.Levy [7]. Chang IL Kim [4 ] studied several characterizations ofalmost P393 IJISET - International Journal of Innovative Science, Engineering & Technology, Vol. 2 Issue 4, April 2015.
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ISSN 2348 – 7968
spaces.Motivated
bythese ideas,theconcept of
almost
P-spaces in
fuzzy
setting
,isintroduced in [ 16].
Definition 3.1 [16]: A
fuzzy topological space (X,T) is called a fuzzy almostP-space if
for every non- zero fuzzyG-set λ in (X,T), int (λ)
Proposition 3.1 : If
is
a
fuzzy Fσ -set
in
0 in (X,T).
a
fuzzyalmost
fuzzy
almost
P- space ( X, T), then
cl λ 1.
Proof:
Let
be
( 1 is
fuzzy
hence
a
fuzzyG-set in
G-set ( 1 , we
we
have
Remark 3.1:In
The
in a
(X,T) .Since
P- space
(X,T).
Then,
(X,T) is a fuzzy almostP-space, for the
have int ( 1 0. This implies
that
1 cl λ 0
and
cl λ 1.
view
fuzzy G-set in
Fσ -set
a fuzzy
of
the
a fuzzy almost
following
result : “ If
proposition 3.1 , we have the following
P- space ( X, T),
propositions give
the
then
cl 1
conditions
λ for a
is
a
1 .”
fuzzy topological space
to be a fuzzy almost P-space.
Proposition 3.2 : If eachnon-zero fuzzy G-set
is a fuzzy
regular closed
set in a fuzzy
topological space (X,T), then ( X, T) is a fuzzy almost P- space.
Proof: Let
be a
non-zero
fuzzy
G-set
in ( X,T ) such that
cl int ( )
int ( 0.Assume the contrary. Then int ( 0, will imply that cl int ( )
hence we will have
0, a contradiction to
beinga
non-zero
fuzzy
.We claim that
cl 0 G-set
Hence we must haveint ( 0,for a fuzzy G-set in (X,T)and therefore( X, T) is
0and
in ( X,T ).
a
fuzzy
almost P - space.
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Proposition 3.3 : If eachnon-zero fuzzy G-set
fuzzy
semi-open set in a
topological space (X,T), then ( X, T) is a fuzzy almost P - space.
Proof: Let
be a
non-zero
G-set
fuzzy
that int ( 0. Assume the contrary.
cl 0 G-set
is a fuzzy
0 and hence we will have
in ( X,T ). Hence
therefore ( X, T) is a
in ( X,T ) such that
cl[ int ( ) .We claim
Then int ( 0, will
imply
that cl int ( )
0, a contradiction to
beinga
non-zero
we must have int ( 0, for a fuzzy G-set in
fuzzy
(X,T) and
fuzzy almost P - space.
The following proposition
shows that
a fuzzy
first category set, is not a fuzzy
dense
set in a fuzzy almost P-space.
Proposition 3.4 : If
is a
fuzzyfirst category set in
a
fuzzy almost P – space (X,T), then
cl ( ) 1, in (X,T).
Proof : Let
fuzzy
be a fuzzy
nowhere
dense
⋁ cl λ and
hence
λ
a
cl ⋁ cl λ first category set in
sets
we
in
(X,T). Then,λ
(X,T).Now
have
cl (
….(1). Now ⋁ cl λ is
⋁
λ ,where
cl λ , implies that
λ
⋁
a fuzzy
λ
λ ′ sare
⋁
λ
cl ⋁ cl λ . Thencl (
Fσ -set in (X,T). Since ( X, T) is
fuzzy almost P-space, by proposition 3.1 , cl ⋁ cl λ 1. ….(2). Now we claim that is
not a fuzzy
, then
dense set in (X,T).Assume the
cl (
) 1,
implies
contrary. Suppose that is
from (1) ,
that
1
afuzzy
cl ⋁ cl λ dense
set
. That
is,
cl ⋁ cl λ 1, a contradiction to (2). Hence we must have cl ( ) 1, in (X,T).
Proposition 3.5 :If eachnon-zero fuzzy first category
set is a fuzzy dense set
in a fuzzy topological space (X,T), then ( X, T) is not a fuzzy almost P -space.
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Proof :Let λ be a fuzzy first
λ
⋁
fuzzy open set
T).Now
fuzzy
nowhere dense
in (X,T). Let μ ⋀ 1 1
μ ⋀ 1 1
Then int μ
Hence,
λ ′ sare
λ ,where
cl λi
cl λi 1 [⋁ cl λ int 1
) and
set in (X,T) such that cl( ) 1.Then
category
sets
. Thenμ is
a fuzzy
1 [⋁ λ hence int μ
1
for the fuzzyG -set μin (X, T),int μ
cl( )
1 cl λ
in (X,T). Now
1
1
is a
G -set in (X,
is,μ 1
. That
0. That is, int μ
1
0. Therefore( X, T) is not a
.
0.
fuzzy almost
P -space.
Proposition 3.6 : If
is
a
fuzzy residual set in
a
fuzzy almost P - space (X,T), then
int
( ) 0, in (X,T).
Proof : Let be a fuzzy
in (X,T). Then, ( 1 γ is
residual set
cl ( 1 γ)
set in (X,T) and hence by proposition 3.4,
( )
cl 1
γ Proposition 3.7 : If
is
a
Proof : Let be a fuzzy
a
fuzzy almost P - space (X,T), then there
μ
in (X,T).
such that
a
in (X,T). Then, ( 1 γ is
residual set
λ
⋁
G -set in (X, T) . Since (X, T) is
Let
int
(
cl λi 1 [⋁ cl λ fuzzy residual set in
μ in (X, T) such that μ λ ,where
a
fuzzyfirst category
λ ′ sare fuzzy
nowhere dense
is a fuzzy open set in(X,T). Let μ ⋀ 1 1
sets in (X,T).Now [ 1 cl λ
μ ⋀ 1 1
1 in (X,T). Therefore 1 int
fuzzy residual set in
set in (X,T). Let λ 1 γ . Then
(X,T).
fuzzyfirst category
1.Therefore int ( ) 0, in (X,T).
exists a fuzzy G -set μ in (X, T)
is a fuzzy
a
a
μ
a
fuzzy
)
1 [⋁ λ almost
. That
fuzzy almost P - space (X,T), then there
. Thenμ
P-space, int ( μ )
whereδ ∈ T.
,
1
cl λi
is, μ exists
0 ,in
Now
. Thus, if
is
a fuzzy G -set
in(X,T).
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Proposition 3.8 : If μis
then
fuzzy residual set in
μ is not a fuzzy nowhere
Proof : Letμbe a fuzzy
3.6,
a
int (μ) 0,
a
fuzzy almost P - space (X,T),
dense set in (X,T).
residual set in a
in (X,T).We claim
cl(μ 0 and int (μ) int cl(μ ,will
fuzzy almost P – space(X,T).Then, by proposition
that
int cl (μ 0. Assume
0,a
imply that int (μ)
a fuzzy nowhere
dense set in (X,T).
Remark 3.1: If
is a fuzzy semi-open
set
in
the
contrary. Thenint
contradiction. Henceμ is not
a fuzzy topological space (X,T), then
cl( ) is a fuzzy regular closed set in (X,T).
For, if
is a
cl(cl[int
semi-open set in (X,T), then λ fuzzy
cl int λ
(λ)])
cl λ … . . 2 .From (1) and
cl [int (λ)] and hence
cl intcl λ ……(1).Alsowe
(2), cl int cl λ
cl λ
havecl intcl λ
and hence
cl (λ)
cl cl λ
cl( ) is a fuzzy
regular
closed set in (X,T).
Proposition 3.9 : If (
)’s
space(X,T), then cl ( ⋁
Proof: Let (
regular
)’s
be
open sets
in
a fuzzy
G-set in
λ fuzzy
fuzzy
regular
are
regular
closed
fuzzy
open sets
in
a fuzzy almost P -
sets
in
(X,T). Then, (1
Proposition 3.10 : If (
)’sarefuzzy semi-open sets
λ
)’sare fuzzy
open sets are fuzzy open setsin
in
(X,T) and hence
0 and thereforecl ( ⋁
in
λ a fuzzy
⋀ 1
(X,T). Since (X, T) is a fuzzy almost P-space, int ⋀ 1
int (1 ⋁
sets
1, in (X,T).
(X,T). Then we have
( ⋁ cl λ closed
(X,T). Since thefuzzy regular
)’s
topological space ,(1
are
λ is
λ 0 in
1, in (X,T).
a fuzzy almostP – space(X,T), then cl
1, in (X,T).
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Proof: Let (
)’s
are
)’s
fuzzy
⋁ cl λ 4. FUZZY
be
fuzzy
regular
semi-open
closed
sets
sets
in (X,T). Then , by remark 3.1, cl (
in(X,T)
and hence,by
proposition
3.8, cl (
1, in (X,T).
ALMOST P-SPACESand
OTHER
FUZZY
TOPOLOGICAL
SPACES
Definition 4.1 [13] : A
fuzzy
topological
space ( X,T)
if countable intersection of fuzzy open sets in (X,T)
is
called
a
fuzzy
P-space
is fuzzy open. That is , every
non-
zero fuzzy G-set in (X,T), is fuzzy open in (X,T).
Proposition 4.1 : Ifa fuzzy topological
T) is a
fuzzy
the fuzzy topological
T) is a
fuzzy
sets
proposition
⋁
1
λ
a fuzzy
Definition
fuzzy almost P -space, then (X,
3.1 ,
λ
fuzzy almost P –space.Suppose that(X,
λ
cl λ , implies that ⋁
cl ⋁ cl λ cl ⋁ cl λ . This implies
1 ..…(2).
that
cl( 1 cl ⋁ cl λ 1, a
λ ′ sare fuzzy nowhere
1,where
Fσ -set in (X,T). Since ( X, T) is
cl ⋁ cl λ . That is,
have ⋁
space (X, T) be a
first category space. Then⋁
in (X,T).Now
⋁ cl λ is
a
second category space.
Proof : Let
dense
space (X, T) is
⋁ cl λ … . 1 .Also
λ
a
fuzzy almost P-space, by
From (1),
we
havecl (
cl ⋁ cl λ and hence we have
contradiction
to (2). Hence
we
must
λ 1, in (X,T) and therefore (X, T) is a fuzzy second category space.
4.2 [20] : A fuzzy topological
space (X,T) is said to be a
fuzzy strongly
irresolvable space if cl int (λ) = 1 , for each fuzzy dense set λ in ( X,T ).
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Proposition 4.2 : Ifeach non-zero fuzzy
fuzzy strongly irresolvable
Proof : Let λbe a
Gδ -set
is
a
fuzzy dense
set
in
a
space, then (X, T) is a fuzzy almost P –space.
non-zero fuzzy Gδ -set
in (X,T). Then, by hypothesis, λ
is a
fuzzy
dense set in (X,T).Since ( X, T) is a fuzzy strongly irresolvable space, we have
cl (int ( ) ) 1. Then, we have int ( )
0, will imply cl (int ( ) ) cl
0,[ Otherwise int ( )
(0) 0, a contradiction ]. Hence , for every non- zero fuzzyG-set λ in
(λ)
0 in (X,T). Therefore ( X, T) is
a
(X,T), we have int
fuzzy almost P - space.
Definition 4.3 [15]: A fuzzy topological space (X,T) is called a fuzzy resolvable space
if there
exists
a
(X,T) is called
fuzzy
a
dense
set
 in (X,T) such that cl ( 1 ) = 1. Otherwise
fuzzy irresolvable space.
Theorem 4.1 [ 11] :If λ is a fuzzy dense and fuzzy Gδ -set in a fuzzy topological
space (X,T), then (1 – λ )
is a fuzzy first category set in (X,T).
Proposition 4.3 : Ifeachnon-zero fuzzy dense set
is a fuzzyGδ -set
in a fuzzy almost P -space,
then(X, T) is a fuzzy irresolvable space.
Proof : Let λbe a non-zero fuzzy dense set in(X,T). Then, by hypothesis , λ
set and hence λ
is
a
fuzzy
first
proposition 3.4,
we
have
is a
cl ( 1
category
set
fuzzy Gδ -
in (X,T). Then , by theorem 4.1, 1 – λ
in (X,T).Since(X, T)
is
a
cl ( 1 ) 1, in (X,T). Thus , for each
fuzzy almost
fuzzy dense set λ
) 1, in (X,T). Hence (X, T) is not a
therefore (X, T) is a
Definition
fuzzydense andfuzzyGδ –set
is a
fuzzy
P-space, by
in (X,T) ,
resolvable space
and
fuzzyirresolvable space.
4.4 [ 2] : A
fuzzy
space if for each fuzzy set
topological
space ( X,T)
in ( X,T) such that cl (
is
)
called a
fuzzy submaximal
1 , then λ ∈ T in(X,T).
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Proposition 4.4: Ifeach non-zero fuzzy
fuzzy submaximal
Proof : Let λbe a
Gδ -set
a
fuzzy dense
set
in
a
space , then ( X,T ) is a fuzzy almost P -space.
non-zero fuzzyGδ -set in (X,T) such that cl( )
a fuzzy submaximal space, the fuzzy dense set
and hence
is
1.Since(X,T) is
in ( X,T), is a fuzzy open set in ( X,T)
is a fuzzysemi-open set in (X ,T). Thus, each non-zero fuzzy G-set
is
a fuzzy semi-open set in ( X,T ).Then , by proposition 3.3,( X, T) is a fuzzy almost
P -space.
Definition4.5 [12] : A fuzzy
topological space (X, T) is
if every fuzzy first category set in (X,T), is
a
called a
fuzzy
fuzzy D- Baire
nowhere
The following propositiongivesacondition for a fuzzy almost P -space
space
dense set in (X,T).
to be
a
fuzzy D-Baire
μ in
a
fuzzy
almost
fuzzy
almost
space.
Proposition 4.5 : If int( μ)
P -space, then
(X, T)
is
a
Proof : Let λ be a fuzzy
P-space, by
proposition
0 , for each
first
(X,T). Hence
D-Baire
category
cl ( ) 1,
3.4,
Then int [cl ( )] int [ μ]
in
fuzzy
fuzzy closed set
space.
set
in
in (X,T). Since (X, T) is
(X,T). Let
0 ( by hypothesis). Thus
each fuzzy first category
λ is
cl ( )
a
μ, where
fuzzy
set in (X,T), is a
a
nowhere
fuzzy
1
dense
∈ T.
set
nowhere dense
set in (X,T). Therefore (X, T) is a fuzzy D-Baire space.
Definition 4.6[18]: A fuzzy
topologicalspace ( X,T) is
eachfuzzy dense and fuzzy G-set
Theorem
4.2 [ 18] : If each
(X,T) is
a
fuzzy
almost
called
in ( X,T ), cl int ( )
fuzzyG-set
is
fuzzy
a fuzzyGID-space if
for
1,in ( X,T).
densein
afuzzy
GID
space , then
P -space .
400 IJISET - International Journal of Innovative Science, Engineering & Technology, Vol. 2 Issue 4, April 2015.
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ISSN 2348 – 7968
Definition 4. 6 [9] : A
hyperconnected
fuzzy
if every non- null
topological
space
fuzzy open subset
That is , a fuzzy topological space ( X,T)
is fuzzy
of
X
is
said
to be fuzzy
X
is fuzzy dense in
hyperconnected if cl (μi
X.
1, for all
μi ∈ T.
Proposition 4.7 : If
a fuzzy almost P space (X,T)
then( X,T) is a fuzzy
Proof : Let
fuzzy dense
fuzzy almost P – space, for the
we have
we have
fuzzy G-set
1. Hence
for
a fuzzy
ALMOST
in ( X,T ).Since (X, T) is
in (X,T ), we have int (λ)
the fuzzy
dense
open set
and fuzzy
a
0 in (X,T).
int (λ) in (X,T),
G-set
in ( X,T ),
P-SPACESandFUNCTIONS.
Theorem 5.1 [22] : Letf : ( X,T) → (Y,S)
topological
fuzzy set β
fuzzy hyperconnected space,
1, in ( X,T). Therefore ( X,T) is a fuzzy GID space.
cl int ( )
5 . FUZZY
fuzzy
G-set
and fuzzy
a fuzzy hyperconnected,for
cl (int (λ) )
a
GID space.
be a
Also, since (X, T) is
is
space (X,T)
in (Y,S), f
1
into
( cl (β) )
cl f
be
a fuzzy
1
a
fuzzy
topological
open
space
function
from
a
(Y,S). Then, for every
(β).
Theorem 5.2 [19] : Letf : ( X,T) → (Y,S)
be
topological
topological space (Y,S). Then, for every fuzzy
space
( X,T)
set δ in (Y,S), int f
1
Definition 5.1 [14]: A
(X,T)
into
another
(δ)
into
f
1
function
fuzzy
a fuzzy
a
fuzzy open
function from
a
fuzzy
( int (δ)).
f : ( X,T) → ( Y,S)
topological
space
from
( Y, S)
a
fuzzy
is called
topological space
somewhat
fuzzy
401 IJISET - International Journal of Innovative Science, Engineering & Technology, Vol. 2 Issue 4, April 2015.
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continuous
open
set
∈S
if
δ in ( X ,T)
f ()
and
such that
δ
X,T)
into another fuzzy topological
∈T
and
0
and
topological
(X,T)
is
be
a
space ( Y,S)
fuzzy
fuzzy
topological space (
is called somewhat
fuzzy
function from
the
open set
fuzzy open if
η in ( Y,S)
such that
fuzzy topological space (X ,T)
space (Y,S).Under what conditions
fuzzy
almost
propositions
another fuzzy
on “ f ” may
P- space, then (Y,S)
is
a
to the
we assert that if
fuzzy almost P- space? .The
establish the desired conditions.
Proposition 5.1 : Ifa function
into
exist a
f 1(  ) .
δ
a
there
f().
η
fuzzy
following
0 and
0 implies that there exists
Let f
a
0 implies that
f : ( X,T) → ( Y,S) from a
Definition 5.2 [14] : A function

ISSN 2348 – 7968
1
f : ( X,T) → (Y,S)
topological space (Y,S) is
from
a
fuzzy
topological space (X,T)
fuzzy continuous function
and
ifλ is
a
fuzzy G-set in (Y,S), then f 1 (λ) is a fuzzy G-set in (X,T).
Proof:Let
be a
open
in (Y,S). Now f
sets
function
f 1 f
is
fuzzy
fuzzy
⋀∞
G-set in (Y,S ).Then,
1
continuous
’s are fuzzy open sets
f
1
and
⋀∞
1
)) ⋀∞
λi ′ s are
1
fuzzy
in (X,T). Hencef 1 λ)
), where λi ′ sare
1
f
1
open
is
a
fuzzy
) ] . Since
sets
fuzzy
in
the
(Y,S),
G- set
in
(X,T).
Proposition 5.2 : Ifafunction
onto
another fuzzy
f : ( X,T) → (Y,S)
from a
fuzzy topological
space ( X,T)
topological space (Y,S) is fuzzy continuous, and fuzzy open function
and if (X,T) is a fuzzy almost P- space , then (Y,S) is a fuzzy almost P- space.
402 IJISET - International Journal of Innovative Science, Engineering & Technology, Vol. 2 Issue 4, April 2015.
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ISSN 2348 – 7968
be a fuzzy
Proof:Let
by
proposition 5.1, f
1
space, int [f
(Y,S),for the
[f
1
(
)]
(
1
f f
)is
f
is
a
fuzzy open
1
in (Y,S), by theorem 5.2, int f
is a
continuousfunction,
a fuzzy G-set in (X,T). Since (X,T) is
0, in (X,T). Since
fuzzy set
( )] . Then, μ
f ( μ) 1
G-set in (Y,S ). Since f is a fuzzy
fuzzy
open
set
( int ( ))]. Sincethe function f
is onto , f f
1
fuzzy almost P
function from
1
f
( )
in (X,T). Now
a
μ
f
1
( int ( ))]
( int ( )).Let μ
( int ( ))
-set
a
fuzzy open
in (Y,S). Therefore(Y,S) is
set in (Y,S). Hence we haveint ( )
onto
somewhat
a
another
) f
1
be
⋀
1
(X,T). Thenf
1
(
]
a
topological
space
(Y,S)
if ( X,T) is a
λi )
1
(
⋀
0,for the fuzzyG
a
is
fuzzy
a
topological
space
fuzzy continuous
and
fuzzy almost P-space , then ( Y,S)
is
1
f f
have
f (μ) 1 f
1
in
fuzzy
a
fuzzy
open
λi .Since f is
topological space (Y,S). Then,
sets
in
(Y,S).Now
a fuzzy continuous
in (Y,S), f
1
f
1
(
function from ( X,T)
λi ′ s are fuzzy open sets in
(λi )’s are fuzzy open
sets
) is
in (X,T). Since(X,T) is a fuzzy almost P-space,int
0.Then there
f (μ) fuzzy G-set
λi ′ sare
), where
1
onto (Y,S) and
[f
fuzzy open set
almost P-space.
Proof : Let
⋀∞
fuzzy
fuzzy open function and
fuzzy
implies that
a fuzzy almost P- space.
Proposition 5.3 : If the function f : (X,T) → (Y,S) from
(X,T)
int
int ( ) and hence f ( μ)
int ( ). Since f is a fuzzy open function from ( X,T) onto (Y,S),for the
μ in (X,T), f ( μ) is
( X,T)onto
a fuzzy
exists a
G-set
fuzzy open set μ in (X,T) such that μ
( )]. Since the function f is onto, by lemma 2.3,f [ f
. Since f is a somewhat
fuzzy open set μ in (X,T),there exists
1
λ ]
f
1
( ). Then,
. Hence we
fuzzy openfunction from ( X,T) onto (Y,S),for the
a
fuzzy
open set
η in ( Y,S)
such that
403 IJISET - International Journal of Innovative Science, Engineering & Technology, Vol. 2 Issue 4, April 2015.
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0
int ( ) and
f (μ) . Thus we have η
η
0,for the fuzzyG -set
in (Y,S). Therefore (Y,S) is
a
Proposition 5.4 : If the function f : (X,T) → (Y,S) from
(X,T)
onto
another
continuous, one-to-one
then (X,T)
and
topological
fuzzy open function and
be
), where
1
λi ′ sare
1
)
1 – f ( λ) and hence 1 – f ( λ)
]
1
⋁∞
)
1
1
G-set inthe
fuzzy
⋀∞
).
fuzzy
Since f
is
( μ fuzzy G-set
,there
fuzzy
topological
is
a
space
somewhat fuzzy
fuzzy almost P-space ,
topological
1
)
space (X,T). Then
in (X,T) . Then
1
1
and onto, by lemma 2.3, f ( 1 – λ
⋁∞
]
1 f
1
)
⋀∞
i 1 f λi ).Since f is a
⋁∞
fuzzy
1 1
f
)
open function
sets λi ′ s in (X,T), we have f λi ′ s are
is a
is a
fuzzy
fuzzy G-set
almost
in ( Y,S). Then, f ( λ)
P- space,
int [ f ( λ) ] 0.
fuzzy open set μ in (Y,S) such that μ f ( λ). This implies thatf
( since f is one - to - one , f
1
f (λ)
λ ). Hence we have,f
1
( μ 1
.
a somewhat fuzzy continuous function , for the fuzzy open set μ in (Y,S)
exist a
f 1 (μ)
1
open
in( Y,S). Since ( Y,S)
exists a
f 1 f ( λ)
Since f is
fuzzy
sets in ( Y,S) and hence ⋀∞
i 1 f λi ),
open
Then , there
a
if (Y,S) is a
one - to - one
f ( λ)
from ( X,T) onto (Y,S), for the fuzzy
is a
have
fuzzy almost P-space.
(Y,S)
open sets
f ⋁∞
⋀∞
i 1 f λi ). Then , we have
fuzzy
space
.Hence we
is a fuzzy almost P- space.
Proof : Let
⋀∞
fuzzy
ISSN 2348 – 7968
f (μ) and henceη
fuzzy
open
set δ
in ( X ,T)
such that
δ
and hence int ( ) 0 in ( X,T).Therefore X, T is a
0 and
δ
f 1 (μ). Thenδ
fuzzy almost P-space.
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