Download SOME STRONGER FORMS OF FUZZY CONTINOUS MAPPINGS 1

SOOCHOW JOURNAL OF MATHEMATICS
Volume 22, No. 1, pp. 17-32, January 1996
SOME STRONGER FORMS OF FUZZY CONTINOUS MAPPINGS
BY
SUNDER LAL AND PUSHPENDRA SINGH
Abstract.
In this paper some stronger forms of fuzzy continuous
mappings have been introduced and some of their preservation properties are discussed.
1. Introduction
In this paper we introduce and study some stronger forms of fuzzy continuous mappings in fuzzy topological spaces. These are all generalizations of
mappings that already exist in general topology. In topology we come across
with a large number of dierent types of sets. Using these sets we may dene a lot of mappings. It may not be pleasant task to nd adjectives for all
these mappings. It would be still more dicult to nd adjectives which would
reect the true nature of the mappings as also its relationship with other similar mappings. It is, therefore, proposed in 8] a more suggestive terminology.
The names suggested under this scheeme may appear to be little crude for
established names like continuous mappings, but are fairly appropriate for
the names not so established and also for the mappings which are yet to be
dened.
Let and be set adjectives (or their abbreviations) associated with the
topologies of X and Y . Then we say that a mapping f : X ! Y is I ( ) if
the inverse image f ;1(U ) is an set in X for each set U in Y . For example,
f is I (open, open) or I (o o) (in short), if f ;1(U ) is an open set in X for each
open set U in Y . When X and Y are fuzzy topological spaces, then we say
Received October 26, 1993.
17
18
SUNDER LAL AND PUSHPENDRA SINGH
that f : X ! Y is fuzzy I ( ) if f ;1(U ) is fuzzy set of X whenever U is a
fuzzy set of Y . Thus f : X ! Y is fuzzy I (o o) if f ;1(U ) is fuzzy open set of
X whenever U is fuzzy open set of Y . Here we study fuzzy I ( ) mappings
for some combinations of and where 2 f open(o), regular open (ro),
clopen (clo), -open (po), -open (do) g and 2 f semi open (so), open (o) g.
We study their interrelationships as also some of their preservation properties.
2. Preliminaries
Throughout this paper X and Y mean fuzzy topological spacs. I denotes
the closed unit interval. The denitions of fuzzy sets, fuzzy topological spaces
and other concepts about functions can be found in 3, 12, 13].
Let U be a fuzzy set of X . U is said to be fuzzy semiopen set of X if
there exists a fuzzy open set V such that V U cl V . Complement of a
fuzzy semi open set is called fuzzy semi cloed. FSO(X ) denotes the family
of all fuzzy semi open sets of X . S -int U (semi interior of U ) is dened as the
suprimum of all fuzzy semi open sets contained in U and S -cl U (semi closure
of U ) as the inmum of all fuzzy semi closed sets containing U .
A fuzzy set U of a fuzzy space X is called a fuzzy regular open set of
X if int cl U = U , and a fuzzy regular closed set of X if cl int U = U .
A fuzzy set U of a fuzzy space X is called a fuzzy -open set of X if
it is a nite union of fuzzy regular open sets of X . Complement of a fuzzy
-open set is called fuzzy -closed set.
A fuzzy space X is said to be fuzzy S-closed (resp: fuzzy strongly Sclosed), if every fuzzy semiopen cover of X has a nite subfamily the closures
(resp: semi closures) of whose members cover X .
A fuzzy space X is said to be (i) fuzzy compact 3] if every fuzzy open
cover of X has a nite subcover, (ii) fuzzy nearly compact 5] if every fuzzy
regular open cover of X has a nite subcover, (iii) fuzzy slightly compact,
if every fuzzy clopen cover of X has a nite subcover, (iv) fuzzy almost
compact 5] if every fuzzy open cover of X has a nite subfamily the closures
SOME STRONGER FORMS OF FUZZY CONTINOUS MAPPINGS
19
of whose members cover X , (v) fuzzy semi compact if every fuzzy semi
open cover of X has a nite subcover.
A fuzzy space X is said to be (a) fuzzy semi T0 , if every fuzzy set U of
X can be written in the form U = Wi2I Vj2J Uij , where Uij are fuzzy semi
open or fuzzy semi closed sets (b) fuzzy semi T1 , if every fuzzy set U of X
W
can be written in the form U = i2I Ui , where Ui are fuzzy semi closed sets
(c) fuzzy semi T2 , if every fuzzy set U of X can be written in the form
i
U=
_^
i2I j 2Ji
Uij =
_ ^ S-cl U
i2I j 2Ji
ij where Uij are fuzzy semi open sets (d) fuzzy S-regular, if every fuzzy open
W
set U can be written in the form U = i2I Ui , where Ui are fuzzy semi open
sets with S -cl Ui U (e) fuzzy S-normal, if for any fuzzy closed set K
and fuzzy open set U such that K U , there exists a fuzzy set V such that
K S -int V S -cl V U (f) fuzzy rT0, if every fuzzy set U of X can be
W V
written in the form U = i2I j2J Uij , where Uij are fuzzy regular open or
fuzzy regular closed sets of X (g) fuzzy rT1 , if every fuzzy set U of X can
W
be written in the form U = i2I Ui , where Ui are fuzzy regular closed sets of
X (h) fuzzy quasi-normal, if for any fuzzy -closed set K and fuzzy
-open set U such that K U , there exists a fuzzy set V such that K int V cl V U (i) fuzzy ultra-normal, if for any fuzzy closed set A and
fuzzy open set B such that A B , there exists a fuzzy clopen set U such that
A U B (j) Completely ultra-normal, if for any fuzzy set K and U
such that cl K U and K int U there exists a fuzzy clopen set V such that
K V U.
i
3. Fuzzy I (o so) Mappings
Denition 3.1. A mapping f : X ! Y from a fuzzy space X to fuzzy
space Y is said to be fuzzy I (o so) if f ;1 (U ) is fuzzy open set of X for every
fuzzy semi open set U of Y .
Remark 3.2. I (o so) mappings have been studied in topological spaces
20
SUNDER LAL AND PUSHPENDRA SINGH
under the name S-continuous mappings by Singal and Yadav 9]. Here we
generalize this concept to the fuzzy setting.
Theorem 3.3. For f : X ! Y , where X and Y are fuzzy spaces, the
following statements are equivalent:
(a) f is fuzzy I (o so)
(b) for every fuzzy semi closed set V of Y , f ;1(V ) is fuzzy closed set of X (c) f (clU ) S -cl (f (U )), for every fuzzy set U of X (d) cl f ;1(V ) f ;1 (S -cl V ), for every fuzzy set V of Y ,
(e) f ;1 (S -int V ) int f ;1 (V ), for every fuzzy set V of Y .
Chang 3] dened a fuzzy continuous mapping, which is fuzzy I (o o)
in our terminology. Azad 1] studied fuzzy semi continuous mappings
which are fuzzy I (so o) in our notations. Fuzzy irresolute mappings of
Mukherjee and Sinha 7] are fuzzy I (so so) mappings. Among these mappings
the following implications hold.
Fuzzy I (o o)
&
Fuzzy I (so so)
%
Fuzzy I (so o)
The following example shows that a mapping may be fuzzy I (o o) as well
as fuzzy I (so so) but may fail to be fuzzy I (o so).
Example 3.4. Let U1 , U2, V1 and V2 be fuzzy sets of I dened as follows:
for each x 2 I
8> x
0 x 21 <3
U1(x) = > 2 ; 2x 12 x 34 :
3
( 20x 0 x4 x1 1
2
U2(x) =
1
1 2 x 1
x
V1(x) = 2 0 x 1
V2(x) = x 0 x 1:
SOME STRONGER FORMS OF FUZZY CONTINOUS MAPPINGS
21
Consider the fuzzy topologies 1 = f0 V1 V2 1g and 2 = f0 U1 U2 1g on I
and the mapping f : (I 1 ) ! (I 2 ) dened by setting f (x) = x2 , for each
x 2 I . It is easy to see that f ;1(0) = 0, f ;1(1) = 1, f ;1(U1 ) = V1, and
f ;1(U2) = V2 . Thus f is fuzzy I (o o). cl U1 = U10 and hence U10 is fuzzy semi
open in (I 2 ). From f ;1 (U10 ) = V10 ; cl V1 , it follows that the inverse image of
fuzzy semi open set U10 in (I 2 ) is V10 which is fuzzy semiopen but not fuzzy
open in (I 1 ). Hence mapping is not fuzzy I (o so).
It is also easy to see that the inverse image of every fuzzy semi open set in
(I 2 ) is fuzzy semi open in (I 1 ). Hence the mapping is also fuzzy I (so so).
Theorem 3.5. A fuzzy I (o so) image of a fuzzy almost compact space
is fuzzy strongly S-closed.
Proof. Let f : X ! Y be a fuzzy I (o so) surjection and let X be fuzzy
almost compact. Let fUi gi2I be a fuzzy semi open cover of Y . Since f is fuzzy
I (o so). It follows that ff ;1(Ui )gi2I is a fuzzy open cover of X . Since the
space is fuzzy almost compact, there exists a nite subject K of I such that
_ cl f ;1(U ) = 1
i
i2K
X:
Since f is surjective, we have
f(
_ cl f ;1(U )) = _ f (cl f ;1(U )) = 1
i
i2K
i2K
i
Y:
Using (c) of Theorem 3.3, we get
1Y =
_ f (cl f ;1(U )) _ S-cl f (f ;1(U )) = _ S-cl U :
i2K
i
i2K
i
i2K
i
Corollary 3.6. A fuzzy I (o so) image of a fuzzy almost compact space
is fuzzy S-closed.
Theorem 3.7. Fuzzy I (o so) image of a fuzzy compact space is fuzzy
semi compact.
The proof is similar to that of Theorem 3.5.
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SUNDER LAL AND PUSHPENDRA SINGH
Corollary 3.8. Fuzzy I (o so) image of fuzzy compact space is fuzzy
compact.
! Y , be a fuzzy I (o so) injection.
fuzzy semi P then X is fuzzy P , where P 2 fT0 T1 T2 g 4].
Theorem 3.9. Let f : X
If Y is
Proof. Suppose Y is fuzzy semi T0. Let A be a fuzzy set of X . Since
Y is fuzzy semi T0 , f (A) can be written in the form f (A) = Wi2I Vj2J Uij ,
where Uij are fuzzy semi open or fuzzy semi closed sets of Y . Since f is fuzzy
I (o so) and injective, we have
i
A = f ;1(f (A)) = f ;1(
_^U
i2I j 2Ji
ij ) =
_ ^ f ;1(U
i2I j 2Ji
ij )
where f ;1(Uij ) are fuzzy open or fuzzy closed sets of X . Hence X is fuzzy T0 .
Similar arguments work for P = T1 and P = T2 .
Theorem 3.10. Let f : X
!Y
be a fuzzy I (o so) and fuzzy closed
injection. Then
(a) X is fuzzy regular 4] if Y is fuzzy S-regular.
(b) X is fuzzy normal 4] if Y is fuzzy S-normal.
Proof. (a) Let A be a fuzzy open set of X . Then f (A0) is fuzzy closed set
W
of Y . Y being fuzzy S-regular, we have f (A0 )0 = i2I Ui , where Ui are fuzzy
open sets of Y with S -cl Ui f (A0 )0 . Since f is fuzzy I (o so) and injective,
W
W
we have A = f ;1(f (A0 )0 ) = f ;1 ( i2I Ui ) = i2I f ;1 (Ui ), where f ;1(Ui ) are
fuzzy open sets of X with cl f ;1 (Ui ) f ;1(S -cl Ui ) U . Thus X is fuzzy
regular.
(b) Let Y be fuzzy S-normal. Let K and U be, respectively fuzzy closed
and fuzzy open sets of X , such that K U . Thus f (K ) f (U 0 )0 . Y being
fuzzy S-normal, there exists a fuzzy set V of Y such that f (K ) S -int V
S-cl V f (U 0)0. Since f is fuzzy I (o so) and injective, we have
K = f ;1(f (K )) f ;1(S -int V ) int f ;1(V ) cl f ;1(V )
f ;1(S-cl V ) f ;1(f (U 0)0) = U:
SOME STRONGER FORMS OF FUZZY CONTINOUS MAPPINGS
23
Thus X is fuzzy normal.
4. Fuzzy I (ro so) Mappings
Denition 4.1. A mapping f : X ! Y from a fuzzy space X to fuzzy
space Y is said to be fuzzy I (ro so) if for every fuzzy semi open (fuzzy semi
closed) set U of Y , f ;1(U ) is fuzzy regular open (fuzzy regular closed) set of
X.
Remark 4.2. Fuzzy I (ro so) mappig is a generalization of the concept of
completely S-continuous mapping in topological spaces introduced by Singal
and Yadav 11].
Obviously every fuzzy I (ro so) mapping is fuzzy I (o so) as well as I (ro o)
but the converse is not true, as is shown by the following examples.
Example 4.3. Let U and V be fuzzy sets of I dened as follows: for
each X 2 I
( 0
if x < 13
1 if x 13
( 0 if x < 1
2
V (x) =
1 if x 12 .
U (x) =
Consider fuzzy topologies 1 = f0 1g _ fW : U W 1g and 2 =
f0 U 1g on I and the mapping f : (I 1) ! (I 2) dened by f (x) = x for
each x 2 I . It is easy to see that the inverse image of every fuzzy semi open
set in (I 2 ) is fuzzy open in (I 1 ) and hence f is fuzzy I (o so).
Since U V cl U = 1, therefore V is fuzzy semi open in (I 2 ). Because
int cl f ;1(V ) = 1 6= f ;1 (V ), f is not fuzzy I (ro so).
Example 4.4. Let X = fx y zg, and U1 , U2 and U3 be fuzzy sets of X
dened as follows:
U1(x) = 0
U2(x) = 0:2
U3(x) = 0:3
U1(y) = 0:2
U2(y) = 0:5
U3(y) = 0:5
U1 (z) = 0:4
U2 (z) = 0:4
U3 (z) = 0:6:
24
SUNDER LAL AND PUSHPENDRA SINGH
Consider fuzzy topologies 1 = f0 U1 U2 1g and 2 = f0 U2 1g on X
and identity mapping f : (X 1 ) ! (X 2 ). It is clear that f ;1 (U2 ) is fuzzy
regular open in (I 1 ), where U2 is fuzzy open set of (I 2 ) and hence mapping
is fuzzy I (ro o). Since inverse image of fuzzy semi open set U3 in (I 2 )
is fuzzy semi open which is not fuzzy regular open in (I 1 ), because
int cl f ;1(U3 ) = U2 6= f ;1 (U3 ). Hence mapping is not fuzzy I (ro so).
Remark 4.5. In general topology I (ro o) mappings have been studied
under the name completely continuous 2].
Theorem 4.6. A fuzzy I (ro so) image of a fuzzy nearly compact space
is fuzzy semi compact.
Proof. Let f : X ! Y be a fuzzy I (ro so) mapping from a fuzzy nearly
compact space X on to a fuzzy space Y . Suppose fUi gi2I be a fuzzy semi
open cover of Y . Since f is fuzzy I (ro so), therefore ff ;1 (Ui )gi2I is a fuzzy
regular open cover of X . Since the space X is fuzzy nearly compact therefore,
W
there exists a nite subset K of I such that i2K f ;1 (Ui ) = 1X .
By surjectivity of f , we have
f(
_ f ;1(U )) = _ f (f ;1(U )) = _ U = 1
i2K
i
i2K
i
i2K
i
Y:
Thus Y is fuzzy semi compact.
Theorem 4.7. Let f : X ! Y be a fuzzy I (ro so) injection from a fuzzy
space X to fuzzy space Y . If Y is fuzzy semi T0 , then X is fuzzy rT0 .
Proof. Let A be a fuzzy set of X . Then f (A) being a fuzzy set of a fuzzy
W V
semi T0 space Y , can be written in the form f (A) = i2I j2J Uij , where Uij
are fuzzy semi open or fuzzy semi closed sets of Y . Since f is fuzzy I (ro so)
injection we have
A = f ;1(f (A)) = f ;1(
_^U
i2I j 2Ji
ij ) =
_ ^ f ;1(U
i2I j 2Ji
i
ij )
SOME STRONGER FORMS OF FUZZY CONTINOUS MAPPINGS
25
where f ;1 (Uij ) are fuzzy regular open or fuzzy regular closed sets of X . Thus
X is fuzzy rT0.
Using similar argument one can prove.
Theorem 4.8. Let f : X
!Y
fuzzy semi T1 , then X is fuzzy rT1 .
be a fuzzy I (ro so) injection. If Y is
5. Fuzzy I (clo so) Mappings
Denition 5.1. A mapping f : X ! Y from a fuzzy space X to fuzzy
space Y is said to be fuzzy I (clo so) if the inverse image of every fuzzy semi
open (or fuzzy semi closed) set of Y is fuzzy clopen set of X .
Remark 5.2. Fuzzy I (clo o) mappings are a generalization of totally
S-continuous mappings 11].
Obviously every fuzzy I (clo so) mapping is fuzzy I (ro so) as well as fuzzy
I (clo o) mapping, but the converse need not be true.
Example 5.3. Let U and V be fuzzy sets of I dened as follows: for
each x 2 I
( 1 ; x 0 x 1 2
U (x) =
1 x 1
x
2
( x 0 x 1
2
V (x ) =
1
1 ; x 2 x 1.
Consider fuzzy topologies 1 = fW W 0 : 0 W < V g _ fV g and 2 =
f0 U 1g and the mapping f : (I 1) ! (I 2) dened by f (x) = 1 ; x, for
each x 2 I . It is clear that the mapping is fuzzy I (ro so). Since U is
fuzzy open and hence fuzzy semi open in (I 2 ). f ;1(U ) = V is fuzzy regular
open in (I 1 ) which is not fuzzy clopen. Thus the mapping is not fuzzy
I (clo so).
Example 5.4. Suppose that U1 and U2 are fuzzy sets of X as described
26
SUNDER LAL AND PUSHPENDRA SINGH
in Example 4.4. Consider fuzzy topologies 1 = f0 U1 U10 U2 1g and 2 =
f0 U1 1g. Let f : (X 1) ! (X 2) be the identity mapping. Clearly, inverse
image of every fuzzy open set in (X 2 ) is fuzzy clopen in (X 1 ). Hence the
mapping is fuzzy I (clo o). If we take inverse image of fuzzy semi open set U2
in (X 2 ), then it is fuzzy open which is not fuzzy clopen in (X 1 ). Hence
the mapping is not fuzzy I (clo so).
Remark 5.5. In topology I (clo o) mappings have been studied under
the name totally continuous 6].
Theorem 5.6. A fuzzy I (clo so) image of a fuzzy slightly compact space
is fuzzy semi compact.
Proof. Let f : X ! Y be a fuzzy I (clo so) mapping from a fuzzy slightly
compact space X on to fuzzy space Y . Let fUi gi2I be a fuzzy semi open cover
of Y . Since f is fuzzy I (clo so), ff ;1 (Ui )gi2I is a fuzzy clopen cover of X .
Since the space is fuzzy slightly compact, there exists a nite subset K of I
W
such that i2K f ;1(Ui ) = 1X . Since f is surjective, we have
f(
_ f ;1(U ) = _ f (f ;1(U )) = _ U = 1
i2K
i
Thus Y is fuzzy semi compact.
i2K
i
i2K
i
Y:
6. Fuzzy I (clo o), Fuzzy I (ro o), Fuzzy I (po o), Fuzzy I (do o) and
Fuzzy Clopen Mappings
Denition 6.1. A mapping f : X ! Y from a fuzzy space X to a fuzzy
space Y is said to be
(a) fuzzy I (do o), if f ;1 (U ) is fuzzy -open set of X for each fuzzy open set
U of Y (b) fuzzy I (po o), if f ;1(U ) is fuzzy -open set of X for each fuzzy open set
U of Y (c) fuzzy I (ro o), if f ;1(U ) is fuzzy regular open set of X for each fuzzy open
set U of Y SOME STRONGER FORMS OF FUZZY CONTINOUS MAPPINGS
27
(d) fuzzy I (clo o), if f ;1 (U ) is fuzzy clopen set of X for each fuzzy open set
U of Y .
Among these various types of mappings the following implications hold.
Fuzzy I (clo o) )Fuzzy I (ro o) ) Fuzzy I (po o) ) Fuzzy I (do o)
)Fuzzy I (o o):
However, none of these implications is reversible as is shown by the following examples.
Example 6.2. Let U1 and U2 be fuzzy sets of I dened as follows: for
every x 2 I ,
( x 0 x 1 ,
2
U1 (x) =
1
1 ; x 2 x 1
( 2x
0 x 12 ,
U2 (x) =
2(1 ; x) 21 x 1.
Consider fuzzy topology = f0 U1 U2 1g on I and the mapping f :
(I ) ! (I ) dened by f (x) = x, for every x 2 I . We see that inverse
image of every fuzzy open set is fuzzy open and hence f is fuzzy I (o o).
Inverse image of fuzzy open set U2 is not fuzzy -open. Hence f is not fuzzy
I (do o).
Example 6.3. Let us consider fuzzy set fab of R dened as follows: for
every x 2 R,
fab = 1
= 0
if a < x < b
otherwise:
Consider fuzzy topology 1 on R generated by fuzzy sets ffab : a b 2
R a < bg and its relative topology 2 on the interval ;1 1]. Then mapping
f : (R 1 ) ! (;1 1] 2 ) dened by f (x) = sin x is fuzzy I (do o), since 1 is
fuzzy semi regular space, i.e: int cl fab = fab in (R 1 ). It is clear that fuzzy
28
SUNDER LAL AND PUSHPENDRA SINGH
set g dened by setting
g(x) = 1 ; 21 < x < 12 = 0
otherwise
is fuzzy open in (;1 1] 2 ). We have
sin;1 (g(x)) = g(sin x)
i.e.
; 12 < sin x < 12 = 1
if
= 0
otherwise
sin;1 (g(x)) = 1 if x 2 (n ; =6 n + 6 )
= 0 otherwise
=
_1 nf
n=;1
ab (x) : x
;1 n 1
o
2 (n ; 6 n + 6 ) which being an innite union of fuzzy regular open set is not fuzzy -open.
The mapping is not fuzzy I (po o).
Example 6.4. Let U1, U2 and U3 be fuzzy sets of I dened as follows:
for every x 2 I ,
( 0
0 x 21 ,
1
8> 21x ; 1 2 0x x1 1 ,
4
<
1
U2(x) = > ;4x + 2 4 x 21 ,
: 0
1 x 1,
2
8> 1
0
x 41 ,
<
U3(x) = > ;4x + 2 14 x 21 ,
: 2x ; 1 12 x 1.
Consider fuzzy topologies 1 = f0 U1 U2 U1 _ U2 1g and 2 = f0 U3 1g on
I and the mapping f (I 1) ! (I 2 ) dened by f (x) = x, for each x 2 I .
f ;1(U3) = U1 _ U2 , which is fuzzy -open but not fuzzy regular open in
(I 1 ). Thus mapping is fuzzy I (po o) which is not fuzzy I (ro o).
U1(x) =
SOME STRONGER FORMS OF FUZZY CONTINOUS MAPPINGS
29
Example 6.5. Let us take fuzzy sets U1 and U2 of I which are dened
in Example 6.4. Consider fuzzy topologies 1 = f0 U1 U2 U1 _ U2 1g and
2 = f0 U1 1g on I and the mapping f : (I 1 ) ! (I 2 ) dened by f (x) = x,
for every x 2 I . It is clear that inverse image of fuzzy open set U1 in (I 2 ) is
fuzzy regular open in (I 1 ) which is not fuzzy clopen. Hence the mapping is
fuzzy I (ro o) but not fuzzy I (clo o).
Denition 6.6. A mapping f : X ! Y from a fuzzy space X to fuzzy
space Y is said to be fuzzy clopen mapping if f (U ) is fuzzy clopen set of
Y for each fuzzy clopen set of X .
Clearly, every map which is both fuzzy open as well as fuzzy closed is a
fuzzy clopen map, but the converse is not true.
Example 6.7. Let U1 and U2 be fuzzy sets of I dened by setting
( x 0 x 1 ,
2
U1 (x) =
1
1 ; x 2 x 1
U2 (x) = x 0 x 1:
Consider fuzzy topologies 1 = f0 U1 U10 U2 1g and 2 = f0 U1 U10 1g on
I and the mapping f : (I 1 ) ! (I 2 ) dened as f (x) = x, for every x 2 I .
We see that image of every fuzzy clopen set in (I 1 ) is fuzzy clopen in (I 2 ),
the mapping is fuzzy clopen. Image of fuzzy open set U2 in (I 1 ) is not fuzzy
open in (I 2 ) and image of fuzzy closed set U20 in (I 1 ) is not fuzzy closed
in (I 2 ). Thus the mapping is neither fuzzy open nor fuzzy closed.
Theorem 6.8. Let f : X ! Y be a fuzzy I (o o) fuzzy clopen mapping
from fuzzy space X on to fuzzy space Y . If Y is fuzzy ultra-normal or fuzzy
completely ultra-normal then so is Y .
Proof. Let X be a fuzzy ultra-normal and let A and B be, respectively
fuzzy closed and fuzzy open sets of Y such that A B . Since f is fuzzy
I (o o), f ;1(A) and f ;1(B ) are respectively fuzzy closed and fuzzy open sets
of X such that f ;1(A) f ;1(B ). Since X is fuzzy ultra-normal there exists
30
SUNDER LAL AND PUSHPENDRA SINGH
a fuzzy clopen set U such that f ;1 (A) U f ;1 (B ). By surjectivety of f ,
we have
A = f (f ;1(A)) f (U ) f (f ;1(B )) = B:
Since f is fuzzy clopen mapping f (U ) is fuzzy clopen set of Y . Thus Y is
fuzzy ultra-normal.
Next, let Y be fuzzy completely ultra-normal, and let A and B be fuzzy
sets of Y such that clA B and A int B . Then f ;1 (A) and f ;1 (B ) are
fuzzy sets of X such that f ;1 (clA) f ;1 (B ) and f ;1 (A) f ;1(int B ). Since
f is fuzzy I (o o), f ;1(clA) is a fuzzy closed set and f ;1(int B ) is a fuzzy
open set. Now f ;1(clA) is a fuzzy closed set such that f ;1(A) f ;1 (clA).
Therefore cl(f ;1(A)) f ;1(clA) f ;1(B ). Similarly f ;1 (A) f ;1(
int B ) int f ;1(B ). Since X is fuzzy completely ultra-normal, there exists
a fuzzy clopen set V such that f ;1(A) V f ;1(B ). By surjectivity of f ,
we have A = f (f ;1(A)) f (V ) f (f ;1(B )) = B , with clA = f (f ;1(clA)) f (f ;1(B )) = B , and A = f (f ;1(A)) f (f ;1(int B )) =int B . Also f (V ) is
fuzzy clopen set, since f is a fuzzy clopen mapping. Thus Y is fuzzy completely
ultra-normal.
Theorem 6.9. If f : X
!Y
is a fuzzy open fuzzy closed and fuzzy
I (po o) surjection of a fuzzy space X to a fuzzy space Y . If X is fuzzy quasinormal, then Y is fuzzy normal.
Proof. Let A be a fuzzy closed and B be a fuzzy open set of Y such that
A B . Since f is fuzzy I (po o), f ;1(A) is fuzzy -closed and f ;1(B ) is fuzzy
-open set of X such that f ;1(A) f ;1(B ). By fuzzy quasi-normality of X ,
there exists a fuzzy open set V such that f ;1(A) V clV f ;1(B ). Since
f is fuzzy open and surjective, we have
A = f (f ;1(A)) f (V ) f (clV ) f (f ;1(B )) = B:
Also, f is fuzzy closed we have
A f (V ) clf (V ) B
SOME STRONGER FORMS OF FUZZY CONTINOUS MAPPINGS
31
where f (V ) is fuzzy open. Thus Y is fuzzy normal.
Theorem 6.10. If X is a fuzzy quasi-normal space and f : X ! Y is
fuzzy I (po o) surjection which takes fuzzy open sets of X to fuzzy clopen sets
of Y , then Y is fuzzy ultra-normal.
Proof. Let A and B be, respectively fuzzy closed and fuzzy open sets
of Y such that A B . Then f ;1(A) and f ;1(B ) are respectively fuzzy closed and fuzzy -open sets of X such that f ;1 (A) f ;1(B ), since f is fuzzy
I (po o). By fuzzy quasi-normality of X , there exists a fuzzy open set V such
that f ;1 (A) V clV f ;1 (B ).
By surjectivity of f , we have
A = f (f ;1(A)) f (V ) f (clV ) f (f ;1(B )) = B:
We have A f (V ) B , where f (V ) is fuzzy clopen set of Y . Thus Y is
fuzzy ultra-normal.
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32
SUNDER LAL AND PUSHPENDRA SINGH
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Department of Mathematics, Institute of Basic Science, Agra University, Khandari, Agra
282002, India.