fuzzy weakly semi continuous functions

8/6/2019 Fuzzy Weakly Semi Continuous Functions

http://slidepdf.com/reader/full/fuzzy-weakly-semi-continuous-functions 1/7

On fuzzy weakly semi-cont inuous funct ions

S . D a n g a, A . B e h e r a b '* , S . N a n d a c

a Depar tm en t o f M a thema t i c s , G overnmen t Co ll ege , Rourke la , 769 004, I nd ia

b Depa r tmen t o f M a themat i c s , Reg iona l E ng ineer ing Co l lege, Rourke la , 769 008, I nd ia

c Depar tm en t o f M a themat i c s , I nd ian I n s t i tu t e o f Techno logy , Kharagpur , 721302 , I nd ia

Received February 1993; revised March 1994

Abstrac t

This paper is devoted to the intro duction and s tudy of fuzzy weakly semi-continuous functions between fuztopological spaces. Some properties of these functions are characterized in term s of quasi coincidence, quas i neighbo

hoo ds, 0-neighborhoods etc. Fu rthe rm ore , som e relations connecting fuzzy weakly semi-continuous functions and fuzretractions have been established.

Key words." Fu zzy weakly semi-continuous function; Qu asi coincidence; Quasi-neighborhood; 0-neighborhood; Sem

open set ; Semi-closed set ; Regular open set ; Fuz zy retract

I . I n t r o d u c t i o n

T h e c o n c e p t o f f u z z y s e t w a s i n t r o d u c e d b y

Z a d e h i n h i s c l as s ic p a p e r [ 1 6 ]. A z a d [ 2 ] h a s i n t r o -

d u c e d t h e c o n c e p t s o f f u z z y s e m i - o p e n a n d s e m i -

c l o s e d s e t s . S i n c e t h e n , m a n y a u t h o r s i n c l u d i n g

A z a d h a v e u s e d t h e s e c o n c e p t s t o d e f i n e a n d s t u d y

f u z z y s e m i - c o n t i n u o u s , s e m i - o p e n a n d s e m i - c lo s e d

m a p p i n g s b e t w e e n f u z z y t o p o l o g i c a l s p a c e s (f ts ).

N o t e w o r t h y a m o n g t h e m a re M u k h e r je e a n d S i n h a

[ 7 - 9 ] , E 1 - M o n s e f a n d G h a n i m [ 4 ] a n d G h o s e [ 5 ] .

T h e c o n c e p t o f a l m o s t c o n t i n u i t y , s t r o n g c o n t i n u i t y

a n d p r i n c i p a l s u p e r - c o n n e c t e d n e s s f o r f t s h a v e

b een , re sp ec t i v e ly , d i scu ssed b y N an d a [1 0 1 2] . In

t h i s n o t e w e i n t r o d u c e f u z z y w e a k l y s e m i - c o n t i n u -

o u s f u n c t i o n s w h i c h a r e n a t u r a l g e n e r a l i z a t i o n o f

* Corresponding author.

f u z z y s e m i - c o n t i n u o u s f u n c t io n s . W e h a v e b r o u g h

o u t c h a r a c t e r i z a t i o n o f s u c h f u n c t i o n s a n d a l s

h a v e i n v e s t i g a t e d s o m e o f t h e i r p r o p e r t i e s i n S e c

i o n 4 b y u s i n g t h e n o t i o n s o f q u a s i- c o i n c i d e n c

( q - c o i n c i d e n c e ) , q u a s i - n e i g h b o r h o o d ( q - n b d ) , a n

0 - c l u s t e r p o i n t s a s i n t r o d u c e d b y M i n g a n d M i n

[ 1 3 ] a n d K o t z e [ 6 ] . T h i s c o n c e p t o f q - c o i n c i d e n c

i s fo u n d t o b e we l l su i t ed t o t h e fu zzy s i t u a t i o n s . I

S e c t i o n 5 , w e h a v e i n v e s t i g a t e d s o m e r e l a t i o n s c o n

n e c t i n g f u z z y w e a k l y s e m i - c o n t i n u o u s f u n c t i o n

a n d f u z z y re t r a c t io n s .

2 . P r e l i m i n a r i e s

T h e d e f i n i ti o n s a n d r e s u l t s n o t e x p l a i n e d i n th

p a p e r a r e a lr e a d y s t a n d a r d i z e d b y n o w a n d c a n b

fo u n d i n [1 -3 , 1 0, 1 4 -1 6 ] . Bu t fo r th e sak e o

co m p le t en ess , we reca l l so m e o f t h ese d e f i n i t i o n

Published in Fuzzy Sets and Systems, 1994 Vol 67, Iss 2, P 239-245 email [email protected]



a n d r e s u lt s . S o fa r a s t h e n o t a t i o n s a r e c o n c e r n e d ,

we sh a l l w r i t e A , A ° , A , A i an d A ' t o m e an , r e sp ec -

t iv e l y , t h e f u z z y c l o s u r e , f u z z y i n t e r i o r , f u z z y s e m i -

c l o s ur e , fu z z y s e m i - i n te r i o r a n d f u z z y c o m p l e m e n t

o f a f u z z y s e t A . T h e w o r d s ' fu z z y ' , ' n e i g h b o r h o o d '

a n d ' f u z z y t o p o l o g i c a l s p a c e ' w i ll b e a b b r e v i a t e d a s

T , ' n b d ' a n d ' f ts ', r e s p e c t iv e l y . F S O ( X ) a n d I x will ,

r e s p e c t i v e ly , m e a n t h e s e t o f f u z z y s e m i - o p e n s e t s

a n d t h e s e t o f a l l f u z z y s e t s o n a n o n e m p t y s e t X .

De f in i t i o n 2 .1 [ 2 ] . Le t A b e a f u zzy se t in an f t s

( X , T ) . T h e n A i s c a l le d

( a) f - s e m i - o p e n i f t h e r e i s a n f - o p e n s e t B i n ( X , T )

suc h th a t B ~< A ~</3 ,

( b ) f - sem i - c lo s ed i f t h e r e i s an f - c lo sed se t B in

(X, T ) suc h th a t B ° ~< A ~< B,

( c) f - r e g u la r o p en i f 4 ° = A ,

( d ) f - r e g u la r c lo s ed i f ,~o = A .

D e f i n i t i o n 2 .2 [ 3 ] . A m a p p i n g f : ( X , T 1 ~ ( Y, T 2)

i s sa id to b e( a) f u z z y c o n t i n u o u s ( f - co n t i n u o u s) if f - I ( B ) e T 1

for all B e 7"2,

( b ) f u zz y o p e n ( f - o p en ) i f f ( A) e T2 f o r a l l A e T1 ,

( c) f u zzy c lo sed ( f - c lo sed ) i f f ( A ' ) i s f u zzy c lo sed f o r

a l l A e T1 .

D e f i n i t i o n 2 .3 [ 4 , 2 ] . A m a p p i n g f : X ~ Y f r o m a n

f t s X to an f t s Y i s sa id to b e

( a ) f - s e m i - c o n t i n u o u s i f f - 1 ( B ) e F S O ( X ) f o r e a c h

f - o p en se t B in K

( b ) f - sem i - o p en ( sem i - c lo sed ) i f f ( A) is f - sem i - o p en

( sem i - c lo sed ) i n Y f o r each f - o p e n ( f - c lo sed ) se t

A i n X ,

( c ) f - a l m o s t o p e n [ 4 ] i f f ( A) i s f - o p en in Y f o r each

f - r e g u l a r o p e n s e t A i n X ,

( d) f - ir r e s o lu t e [ 7 ] i f f - l ( B ) e F S O ( X ) f o r e a c h

B e F S O ( Y ) .

D e f i n i t i o n 2 .4 [ 1 3 ] . ( a ) A f u zzy p o in t x ~ in X i s

a f u zzy se t d e f in ed a s f o l lo ws :

x~(y) = o t h e r w i s e ,

w he re 0 < ~ ~< 1; c~ is calle d i ts va lu e a nd x is i ts

s u p p o r t .

( b) A f u z z y p o i n t x ~ i s sa i d t o b e c o n t a i n e d i n

fuzz y se t A o r x~ bel ong s to A, i .e . x~ e A i f~ ~< A ( x

T h e o r e m 2 .5 [ 1 3 ] . Le t A e I x . T hen A is the un i

o f a l l i t s fu zz y po in ts , i .e ., A = ~/~ ~A x~.

T h e o r e m 2 .6 [ 3 ] . L e t f b e a f u n c t i o n f r o m a n t

X to a n t i s Y . Th en th e f o l lo win g s ta t em en t s h o ld

(1 ) I r A <~ B , t h e n f ( A ) < ~ f ( B ) f o r a ll A , B e I x .

(2 ) I f C <~ D , t h e n f - l ( C ) <~ - l ( D ) f o r a l l C , D e I(3 ) F o r a n y A e l x , A < ~ f - 1 ( f ( A ) ) .

(4 ) F o r an y B e I r , f ( f - l ( B ) ) < ~ B .

(5 ) F o r a n y B e f f , f I (B ' ) = ( f - l ( B ) ) ' .

(6 ) F o r a n y A e I X , f ( A ') ~ ( f ( A ) ) ' .

T h e o r e m 2 .7 [ 5 ] . A ma p p in g f : X - ~ Y f ro m a n t

X to a n t i s Y i s f - semi - co n t in u o u s i f f fo r a n y f - p o in t

in X and a ny f -op en se t B in Y w i th f (x~) e B , the

ex i s t s A e FS O ( X ) su ch th a t x~ e A a n d f ( A ) <~ B

Def in i t i o n 2 .8 [ 1 3 ] . An f - se t A is sa id to b e q u as

c o i n c i d e n t ( q - c o i n c i d e n t ) w i t h a n f - se t B , if t h e

e x is t s a t le a s t o n e p o i n t x e X s u c h th

A ( x ) + B ( x ) > 1 . I t i s d en o ted b y A q B . A ¢

m e a n s t h a t A a n d B a r e n o t q - c o i n c i d e n t .

F o r t w o f u z z y s e ts A a n d B , A < ~ B

A (x ) <% B (x ) f o r e a c h x e X . N o t e t h a t A ~< B

Ara B ' .

Def in i t i o n 2 .9 [ 5 ] . ( 1) A n f - se t A i s ca l l ed an f -

n b d ( f - sem i - q - n b d ) o f an f - p o in t , x ~ in an f t s ( X ,

i f f t h e r e ex i s t s an f - o p en se t ( f - sem i - o p e n se t) B

X suc h tha t x~ q B ~< A.

( 2) An f - p o in t x ~ in a n f t s ( X , T ) i s ca l l ed

f - sem i - c lu s t e r p o in t o f an f - se t A i f f ev e r y f- sem i -

n b d o f x~ i s q - c o i n c i d e n t w i t h A . T h e s e t o f

f - sem i - c lu s t e r p o in t s o f an f - se t A i s ca l l ed f -sem

c l o s u r e o f A a n d d e n o t e d b y .4 . I t is p r o v e d i n [

th a t , 4 = / ~ { V : V ~> A , w h e r e V is f - sem i - c lo sed

X} . A i s f - sem i - c lo sed i f f A = A .

(3 ) T h e f - s e m i - i n t e r io r o f a n f - se t A , d e n o t e d

A i , is de fin ed as A i --- V {U : U ~< A, U 6 F S O ( X )

T h e o r e m 2. 10 [ 4 ] . F o r a n y f - s e t A in a n t i s ( X , TA ° <~ A i <~ A <~ A <~ ,4.

T h e o r e m 2 .1 1 [ 7 ] . I f f : X ~ Y i s f - semi - co n t in u o

a n d a lmo s t f - o p en th en f i s f - i r re so lu t e .



D e f i n i t i o n 2 .1 2 [ 8 ] . An f t s ( X , T ) is f u zzy r eg u la r i f f

f o r e a c h f - p o i n t x ~ i n X a n d e a c h f - o p e n q - n b d U o f

x ~, t h e r e e x i s ts f - o p e n q - n b d I / o f x ~ s u c h t h a t

17~< U .

D e f i n i t i o n 2 .1 3 [ 8 ] . ( 1) An f - se t A in an f t s ( X , T ) i s

s a i d t o b e a n f - 0 - n b d ( f -s e m i - 0 - n bd ) o f a n f - p o i n t x ~

i f f t h e r e e x i s t s a n f - c lo s e d (f - s em i - c l o se d ) q - n b d U o f

x~ suc h th a t U ~A ' , i .e ., U ~< A.

( 2 ) An f - p o in t x , i s sa id to b e an f - sem i - 0 - c lu s t e r

p o i n t o f a n f - s e t A i f f f o r e v e r y s e m i - o p e n q - n b d

U o f x ~ , U is q - co in c id en t w i th A . Th e se t o f a l l

f - sem i - 0 - c lu s t e r p o in t s o f an f - se t A is ca l l ed f -sem i -

0 - c l o s u r e o f A a n d d e n o t e d b y [ A ] a . A n f - s et A i s

f - s e m i - 0 -c l o s e d i f f A = [ A ] a . T h e c o m p l e m e n t o f a n

f - sem i - 0 - c lo sed se t i s ca l l ed an f - sem i - 0 - o p en se t .

T h e o r e m 2 .1 4 [ 1 5 ] . L e t f : X ~ Y b e a n y f u n c t i o n

a n d x~ b e a n y f - p o in t i n X , t h en

(1 ) f o r A c X a n d x , q A , w e h av e f ( x , ) q f ( A ) ,

(2 ) f o r B c Y a n d f ( x , ) q B , w e h av e x , q f - l ( B ) .

3 . F u z z y w e a k l y s e m i - c o n t i n u o u s f u n c t i o n s

D e f i n i t i o n 3 . 1 . A m a p p i n g f : X ~ Y f r o m a n f l s

X t o a n f ts Y i s c a l l e d f u z z y w e a k l y s e m i - c o n t i n u -

o u s ( f wsc ) if f f o r an y f - p o in t x~ in X an d an y f - o p en

s e t B i n Y c o n t a i n i n g f ( x , ) , t h e r e e xi s ts a n f - s em i -

o p e n s e t A c o n t a i n i n g x ~ s u c h t h a t f ( A ) <~ B .

N o t e 3 . 2 . I t i s c l e a r f r o m T h e o r e m s 2 . 1 0 a n d 2 . 7

t h a t e v e r y f - s e m i - c o n t i n u o u s f u n c t i o n i s f w s c

w h e r e a s t h e c o n v e r s e is n o t t r u e a s c a n b e s e e n f r o m

t h e e x a m p l e g i v e n b e lo w .

E x a m p l e 3 . 3. L e t A , B a n d C b e t h e f u z z y s e ts o f

I = [ 0 , 1 ] d e f in ed a s f o l lo ws :

A ( x ) = ~0 if 0 ~< x ~< 1/2,

( 2 x - 1 i f 1 / 2 ~ < x ~ < 1,

B ( x ) =

C(x ) =

1 i f O ~ < x ~ < 1 / 4,

- - 4 x + 2 i f 1/4 <. x <~1/2,

0 i f 1 /2 ~ <x ~ < 1 ,

0 i f O~< x ~< 1 /4,

( 4 x - 1 )/3 i f 1 / 4 ~ < x ~ < 1.

C o n s id e r t h e f u zzy to p o lo g ie s T1 = {0, C , 1} an

T 2 = { 0 , 1 , A , B , A v B } o n I a n d th e m a p p in

f : ( I , T 1 ) ~ ( I , T z ) d e f i n e d b y f ( x ) = x / 2 fo r a l l x e

Th en we see th a t f i s f wsc ; b u t f i s n o t an f - sem

c o n t i n u o u s f u n c ti o n s i n c e f - ~ ( B ) = A ' w h i c h is n o

an f - sem i - o p en se t i n ( I , T~).

T h e o r e m 3.4. I f Y i s a n f - reg u la r sp a ce , th en a m a p

p in g f : X ~ Y i s fws c i f f f i s f - semi - co n t in u o u s .

P r o o f . T h e n e c e s s a r y p a r t f o l lo w s f r o m N o t e 3 .

W e p r o v e o n l y t h e s u ff ic i en t p a r t . L e t f b e f w sc a n

Y b e a n f - r e g u l a r s p a ce . L e t x ~ b e a n y f - p o i n t o

X a n d B b e a n y f -o p e n s e t i n Y c o n t a i n i n g f ( x , )

Sin ce Y i s f - r eg u la r ( cf . De f in i t i o n 2 .1 2) t h e r e e x i s

a n f -o p e n - q -n b d C o f f ( x ~ ) = y , ( w h er e y = f ( x )

su ch th a t ( 7 ~< B . S in ce f i s f wsc an d C i s an f - o p en

q - n b d o f f ( x , ) , t h e r e ex i st s A 6 F S O ( X ) w i t h x~ e

s u c h t h a t f ( A ) <~ C. B y T h e o r em 2 .1 0, C ~< C an

so f ( A ) <~ C <~ C ~ B. T h u s f i s f - s e m i - c o n t i n u o u

b y T h e o r e m 2.7 a n d t h is c o m p l e t e s th e p r o o f . [

I n t h e f o l l ow i n g t h e o r e m s w e g i v e s o m e c h a r a c

t e r i z a t i o n o f f w s c f u n c ti o n s .

T h e o r e m 3.5. A ma p p in g f : ( X , T1 ) ~ ( Y , T2 ) f ro m

a n t i s X to a n t i s Y is fw sc i f f f o r ea ch f - o p en se t B

Y , f ' ( B ) ~ ( f - l ( ~ ) ) i .

P r o o f . L e t f b e f w s c a n d B ~ 7 " 2 . L e t x , b e a

f - p o i n t i n f - I (B ) . T h u s f ( x , ) ~ B . f i s fwsc impl ie

t h a t t h e r e e x is ts a n A e F S O ( X ) s u c h t h a t x , ~

a n d f ( A ) <~ B . B y T h e o r e m 2 . 6 ( 2 ) a n d ( 3 ) w e h a v

A ~ < f - l ( / ~ ) . H e n c e h i ~ ( f - l ( j ~ ) ) i a n d s i n c e A

f - sem i-op en , A ~< ( f - 1 (/~)) i. So f - ~ B) <<,A <

i f - 1 ~ ) ) i .

C o n v e r s e l y l e t x~ b e a n f - p o i n t in X a n d B b

a n y f - o p e n s e t i n Y s u c h t h a t f ( x ~ ) ~ B . B y h y p o

thesis , f - l ( B ) <~ ( f - l ( / ~ ) ) i = A ( sa y ). H e n c

x ~ ~ f - 1 B ) ~< A , wh ich im p l i e s t h a t A i s an f - sem

o p e n s e t i n X c o n t a i n i n g x ~ . S o A = i f - 1 ( / ~ ) ) i <

f - 1(/~) , i .e . , f ( A ) ~ B ( cf . T h e o r em 2 .6 (4 )) . H en c

f i s f w s c a n d t h i s p r o v e s t h e r e s u lt . [ ]

T h e o r e m 3.6. A m a p p in g f : X ~ Y f ro m a n t i s X

a n f t s Y i s fw sc i f f o r ea ch f - o p en se t B in

f - ' ( /~ ) e F S O ( X ) .

P r o o f . S t r a i g h t f o r w a r d . [ ]



T h e f o l l o w i n g e x a m p l e s h o w s t h a t t h e c o m p o s i -

t i o n o f t w o f w s c f u n c t i o n s n e e d n o t b e f w sc .

E x a m p l e 3 . 7. L e t I = [ 0 , 1 ] a n d E t b e t h e E u c l i d -

e a n s u b s p a c e t o p o l o g y o n I . L e t /~ , b e t h e f u z z y

t o p o l o g y o n I i n d u c e d b y t h e u s u a l to p o l o g y o n I .

L e t X = Y = Z = I w i t h t h e f u z z y t o p o l o g y / ~ ,.

C o n s i d e r t h e f u n c t i o n s f : X - + Y a n d g : Y ~ Z d e -

f i n e d b y

4 . F u zzy w ea k l y s em i - co nt i nuo u s f unc t i o ns i n t erm

o f q - co i nc i dence , q - ne i g hbo rho o ds a nd 0 - c l us t

po i n t s

I n t h i s s e c t i o n w e c h a r a c t e r i z e f u z z y w e a k

s e m i - c o n t i n u o u s f u n c t i o n s a n d i n v e s t i g a t e s o m e

t h e i r p r o p e r t i e s b y u s i n g t h e n o t i o n s o f q u a s i - c o i

c i d e n ce , q u a s i - n e i g h b o r h o o d s a n d 0 - c lu s t e r p o i n

a s i n t r o d u c e d b y M i n g a n d M i n g [ 1 3 ].

x , 0 ~ < x ~ < 1 / 2 ,

f ( x ) = O, 1 / 2 < x ~ < 1 ,

0 , 0 ~ < x < 1 /2 ,

9 ( x ) = 1 , 1 / 2 ~ < x ~ < 1 .

I t is e a s y t o c h e c k t h a t f a n d 9 a r e f u z z y s e m i -

c o n t i n u o u s a n d h e n c e f w s c ( cf. N o t e 3 .2 ); b u t 9 o f i s

no t f w sc .

U n d e r r e a s o n a b l e c o n d i t i o n s , t h e fo l l o w i n g r e -

s u l t s h o w s t h a t 9 o f is f w sc .

T h e o r e m 3.8. I f f : X ~ Y i s f - i r r e s o l u t e a n d

9 : Y ~ Z i s f ws c , t h en 9 ° f : X ~ Z i s f ws c .

P r o o f . L e t x , b e a n y f - p o i n t i n X a n d C b e a n y

f - o p e n s e t i n Z c o n t a i n i n g ( ( g o f ) ( x , ) ) = 9 ( f ( x , ) ) .

S inc e g i s f w sc t he r e e x i s t s a n f - ope n se t B in

Y c o n t a i n i n g f ( x , ) s u c h t h a t g(B) <~ C. A lso s inc e

f i s f uz z y i r r e so lu t e a nd B i s f - op e n i n Y , i t

f o l l o w s t h a t f - l ( B ) i s f - s e m i - o p e n in X . L e t

A = f - l ( B ) . N o w (9 o f ) (A ) = 9 ( f (A ) ) <~ 9 (B ) <~ C .

S o 9 o f i s f w s c a n d t h i s c o m p l e t e s t h e

p r o o f . [ ]

C o ro l l a ry 3.9. I f f : X ~ Y i s f - s e m i - c o n t i n u o u s a n d

f -a l m o s t o p en (o r f - o p en ) a n d 9 : Y ~ Z i s f ws c , t h en

g o f i s f ws c .

P r o o f . T h e p r o o f f o l lo w s fr o m T h e o r e m s 2 .1 1 a n d

3.8. []

T h e o r e m 4.1. A m a p p i n g f : X ~ Y i s f w s c i f f co r r e

p o n d i n g t o ea ch f -o p e n q -n b d B o f y , i n Y , t h e r e ex i

a n f - s e m i - o p e n s e m i - q - nb d A o f x , in X s u c h t h

f ( A ) <~ B , w h e r e f ( x , ) = ( f ( x ) ) , = y , .

P r o o f . L e t f b e fw s c a n d B b e a n f- o p e n q - n b d o f y

w h e r e f ( x ) = y in Y. So, B ( y ) + ~ > 1 . W e c

c h o o s e a p o s i t i v e r e a l n u m b e r ? s u c h t h

B ( y ) > ? > 1 - ~ . H e n c e B is a n f - o p e n n b d o f y ~

Y . S inc e f i s f w sc , t he r e e x i s t s a n f - se m i - op e n sA c o n t a i n i n g x ~ s u c h t h a t f ( A ) ~ </~ . N o w A (x ) >

i m pl ie s A ( x ) > l - ~ , i.e., A ( x ) + ~ > l . T h

x ~ q A . S o A i s a n f - s e m i - o p e n s e m i - q - n b d o f x= .

C o n v e r s e l y , le t th e c o n d i t i o n o f t h e t h e o r e

ho ld , i . e . , l e t x , be a n f - po in t i n X a nd B be

f - o p e n s e t i n Y c o n t a i n i n g y , = ( f ( x ) ) , . S

x~ ~ f - 1 ( B ) = C ( s a y ) . H e nc e C ( x ) > - ~ . W e c

c h o o s e a p o s i t i v e i n t e g e r 7 s u c h t h a t C(x )> ~ 1

P u t a , = 1 + ( i / n ) - C ( x ) , f o r a n y p o s i t iv e i n t e g

n > ~ 7 . C l e a r l y 0 < a , ~ < 1 f o r a ll n ~ > 7 . N o

B ( y ) + ~ n = B ( y ) + 1 + ( l / n ) - - C ( x ) = 1 + ( I /

> 1 ( s ince C ( x ) = f - l ( B ) ( x ) = B ( f ( x ) ) = B( y

H e n c e Y , , q B , i .e ., B is a n f - o p e n q - n b d o f y , , f o r

n > i 7 . S o b y h y p o t h e s i s t h e r e e x i s t s a n f - s e m i - o p

q - n b d A n o f x ~ , s u c h t h a t f ( A , ) ~</~, for a l l n / >

N o w A = V , >--~ A n i s f -s e m i - o p e n i n X . I t r e m a i

t o s h o w t h a t x , E A . W e h a v e A n ( x ) + C~n > 1 fo r

n ~> 7- Th is im plie s A ,(x ) > 1 - ctn = C (x) - ( i /

fo r al l n ~> 7. T h us A ( x ) > C ( x ) - ( l /n ) fo r a l l n ~>

S i n c e x~ E C, A (x ) >~ C (x ) >1 or. So A i s a n f - se m

o p e n s e t i n X s u c h t h a t f ( A ) = f ( V , > , ~ A n ) =

V , ~ r f (A n ) <~ B . H e n c e f i s f ws c a n d t hi s co m p l e t

t h e p r o o f . [ ]

R e m a r k 3 . 1 0 . F u z z y w e a k l y s e m i - c o n t i n u i t y o f

a f u n c t i o n i s b o t h a l o c a l a n d g l o b a l p r o p e r t y .

L e m m a 4 .2 [ 8 ] . F o r a n y t w o f - s e t s A a n d B i n

A <~ B i f f f or each x~ in X , x~ e A impl ies x~ ~ B .



T h e o r e m 4 .3 . I f f : X ~ Y i s an fwsc func t ion f rom an

fts X to an f ts Y , then fo r each f-open set B in Y ,

f - ' ( B ) < . f - l ( B ) .

Pro o f . An f - se t i s t h e u n io n o f a ll o f i t s f -p o in t s

[ 1 3 ] . S u p p o s e t h a t t h e r e i s a n f - p o i n t x , E f - I ( B )

b u t x , 6 f - l ( B ) . Since f ( x , ) ¢ B t h e re ex i s t s an f -

o p en se t C i n Y w i t h f ( x , ) ~ C s u c h t h a t C ~ / ~ . T h u s

C ~ B an d b y P ro p o s i t i o n 1 .4 o f [1 ] C ¢ 1B. S i n c e f i s

f w sc , t h e r e e x i s ts a n A E F S O ( X ) w i t h x , ~ A s u c hth a t f (A) ~< t~ . H e n c e f ( A ) C l n , s in ce f( A ) ~< C ~< C.

B u t o n t h e o t h e r h a n d , s in c e e a c h f - se m i - o p e n s e t i s

f - se m i q - n b d o f e a c h o f i ts f - p o i n t , x , ~ j T - I ( B ) a n d

A i s an f - semi q -n b d o f x ~ . By Def in i t i o n 2 .9 (2 ) ,

A q f - 1 (B) . By Th eo r em 2 .1 4 (1 ), f ( A ) q f ( f - i (B)),

a n d h e n c e b y T h e o r e m 2 . 6 ( 4 ) , f ( A ) q B w h i c h i s

a c o n t r a d i c t i o n . T h i s c o m p l e t e s t h e p r o o f o f t h e

t h e o r e m . [ ]

Th eo rem 4 .4 . Let f : X ~ Y be an f -open and fwsc

mapping. Then f( A ) <~ (A ), fo r each f-open set A inX .

P r o o f . L e t A b e a n f- o p e n s e t i n X a n d l e t f ( A ) = B .

S in ce f i s f -o p en , we see t h a t B i s an f -o p en se t i n

Y . H e n c e b y T h e o r e m 2 . 6 ( 3 ) , A < ~ f - l ( f ( A ) ) =f - t ( B ) . S i n c e f i s f w s c, w e h av e f ro m T h e o r e m 4 .3 ,

f - a ( B ) <<, - I ( B ) . Th us ,4 ~< f- ~ (B ), i . e . , f ( ,4 ) ~< B =

f ( A ) an d t h i s p ro v es t h e re su l t. [ ]

Th eo rem 4 .5 . A func t ion f : X ~ Y i s fwsc i f f o r each

f-open set B in Y , x~ q f - 1 ( B ) implies x~ q f - 1 ( ~ ) i f o r

each f-point x~ in X .

Pro o f . Le t f b e fwsc . Le t x~ b e an f - p o in t i n X a n d

B b e an y f -o p en se t i n Y su ch t h a t x ~ q f - 1 B). Th en

f ( x ~ ) q B . S i n c e f i s f w s c b y T h e o r e m 4 .1 , t h e r e e x i s ts

an f - s emi -o p e n se t A i n X su ch t h a t x ~ q A a n d

f ( A ) <~ B. By Th eo rem 2 .6 (2 ) an d (3 ) we h av e

A ~ < f - t ( /~ ) . S i nc e A e F S O ( X ) , A < < , f - I ( B ) ~. S oX~ q f - 1(/~)

Co n v e rse ly , l e t t h e co n d i t i o n g iv en i n t h e s t a t e -

m e n t h o l d . L e t x ~ b e a n f - p o i n t i n X a n d B b e a nf - o p e n q - n b d o f f ( x ~ ) s u c h t h a t x ~ q f - l ( B ) . B y

h y p o th es i s , x ~ q f - 1 /~) i. Pu t A = f - 1 /~ )i . He n ce

A ~ FSO (X) . x ~ q A imp l i e s t h a t A i s an f - sem i -o p en

q -n b d o f x ~ . Al so f ( f - , ( ~ ) i ) < f ( f - l ( ~ ) ) , i.e.,

f ( A ) <% B. Th u s f i s fwsc an d t h i s co mp le t e s t

p ro o f i [ ]

T h e o r e m 4 . 6 . I f f : X ~ Y i s fwsc then for each

po in t x , in X and each fuz zy O-nbd B o f f (x~ ) , f - I (

is an f-sem i q-nbd of x , .

P r o o f . L e t f : X ~ Y b e a n f w s c f u n c t i o n a n d x ,

a n f - p o i n t in X . L e t /~ b e a n f - 0 - n b d o f f ( x , ) .

t h e r e is a n f - o p e n q - n b d C o f f ( x , ) s u c h t h a t ( ~ ¢1i .e ., t~ ~< B. Since C i s an f-op en q -nb d o ff (x , ) ,

T h e o r e m 4 .1 , t h e r e is a n f - s e m i - o p e n q - n b d A o f

such th a t f( A ) ~< C an d t h us f( A ) ~< (~ ~< C ~< B.

A ~< - 1 B) . H en ce f - x B) i s an f - semi q -n b d o f

an d t h i s co m p le t e s t h e p ro o f . [ ]

Theorem 4.7. I f f : X ~ Y is a function such that f

each f point x~ in X and each f-sem i O -nbd B o f f (

in Y , f - l (B) is f -sem i q-nbd of x~ in X, then f is fws

P r o o f . L e t x ~ b e a n f - p o i n t i n X a n d B b e af - o p e n q - n b d o f f ( x ~ ) . W e n o t e t h a t B i s a n f -s e m

o p e n q - n b d o f f ( x ~ ) . S o / ~ i s a n f - s em i 0 - n b d o f f ( x

By h y p o th es i s , f - 1 /~ ) i s an f - semi q -n b d o f x ~.

t h e re ex i s t s an f - semi -o p en se t A i n X su ch t h

x~ q A ~< f - 1 /~), i.e., f( A ) <~ B. He n ce f i s fwsc a

th i s co mp le t e s t h e p ro o f .

Theorem 4.8. I f f : X ~ Y i s fwsc then

(i) f ( ~ <~ [ . f (A)]# for each f - se t A in X ,

(ii) f [ f - l ( B ) i ~ ] <% [ B ] a fo r e a ch f - s e t B in Y .

Pro o f . ( i) Le t x ~ ~ ,4 an d S b e an f -c lo sed q -n b d

f (x~) . T h e n t h e r e ex i st s a n f - o p e n q - n b d V o f f ( x

suc h th a t V ~< S. Since f i s fwsc , by Th eo re m

t h e r e e x i st s a n f - s e m i - o p en q - n b d U o f x~ s u c h t h

f ( U ) <% ~'. Since x~ E A, by D efin i t ion 2 .9(2) , x~ i s

f - semi -c lu s t e r p o in t o f A. Hen ce U q A a n d a l

f ( U ) q f ( A ) . Sin ce f (U)<< , V , V q f ( A ) . Ag a in s i n

S i s f -c losed , we hav e 17~< I7~< S. Th eref o

S q f ( A ) . B y D e f i n i t i o n 2 . 1 3 ( 2 ) , f ( x ~ ) e [ f ( A ) ] g , i

x , E - 1 [ f (A )]g . Th e re fo re , A -< - 1.~ f [ f (A )]~ ; wh i

imp l ies f ( A ) ~< [-f(A)]~, p ro vin g ( i) .( ii ) L e t B b e an f - se t in Y an d x ~ b e an f -p o in t

X s u ch t h a t x~ e f - l ( B ) i - . L e t V b e a n y f -o p

q -n b d o f f (x ~) . By Th eo rem 4 .1 , t h e re ex i s t s f - sem

o p e n q - n b d U o f x ~ s u c h t h a t f (U)<~ ~ ' . Sin



y - 1 B )i ~< f - 1 B ), w e h a v e x ~ ~ f - 1 B ) i ~ . B y D e f -

i n i t i o n 2 . 13 ( 2 ), U q f - 1 B ), i . e . , f ( U ) q B~ T hu s 17q B ,

w h i c h i m p l i e s f ( x ~ ) e [ B ] ¢ . S o f [ f - l ( B ) i ~ ] <<,

[ B ] ~ , p r o v i n g ( ii) . [ ]

d i s j o i n t s e m i q - n b d s , i.e ., if x , a n d y p a r e d i s t i n

f u z z y p o i n t s i n X , t h e n t h e r e e x i st f u z z y s e m i q - n b

V 1 a n d V 2 s u c h t h a t x ~ q V 1 , y a q V z a n

V 1 A V 2 = 0 .

5. Fu zzy weakly semi-continuous functions

and fuzzy retracts

Theorem 5.6 . I f Y i s a n f - q u a s i U r y s o h n s p a c e a

f : X ~ Y i s a n f w s c i n j ec t io n , t h en X i s a f u z z y q u a

s e m i - H a u s d o r f f s p ac e .

Definition 5 .1 . L e t X b e a n f ts a n d A c X . T h e n t h e

s u b s p a c e ( c ri sp ) A o f X is c a ll e d a f u z z y r e t r a c t o f

X i f t h e r e e x i s ts a f u z z y c o n t i n u o u s f u n c t i o n

r : X ~ A s u c h t h a t r (a) = a f o r a l l a e A . I n t h i s c a s e

r i s c a l l e d a f u z z y r e t r a c t i o n . A is c a l l e d a n f - o p e n ,

s e m i - c o n t i n u o u s r e t r a c t o f X i f r is f - o p e n a n d

f - s e m i - c o n t i n u o u s ; s i m i l a r l y A is c a l l e d f w s c r e t r a c t

o f X i f r i s fw s c .

Proof . L e t x ~ a n d y a b e t w o d i s t i n c t f u z z y p o i n t s X . f b e i n g i n j e c ti v e , f ( x ~ ) a n d f ( y a ) a r e d i s t i n

f u z z y p o i n t s i n Y . S i n c e Y i s f u z z y q u a s i U r y s o h

t h e r e e x i s t s f - o p e n s e t s V a a n d V 2 i n Y s u c h t h

f ( x ~ ) q V a , f ( y a ) q V z an d 171 /x 1 7 2 = 0 , i .

f - l ( 1 ? l ) i A f - l (1 ? 2 ) i = 0 . B y T h e o r e m 3 .5 ,

x ~ q f - l ( V ~ ) ~ < f - l ( f f l ) i - ~ < f - 1 1 7 1) i.

Theorem 5.2 . I f A i s a n f - o p e n , f - s e m i - c o n t i n u o u s

r e t r a c t o f th e f t s X t h e n f o r e v e r y f t s Y , a n y f w s c

f u n c t i o n g : A ~ Y c a n b e e x t e n d e d t o a n f w s c f u n c -

t ion o f X in to Y .

P r o o f . L e t Y b e a n a r b i t r a r y f t s a n d g : A ~ Y b e a n

f w s c f u n c t i o n . B y C o r o l l a r y 3 .9 , g o r : X ~ Y i s f w s c

a n d g o r ( a ) = g ( r ( a ) ) = g (a ) f o r a l l a ~ A , w h e r e

r : X ~ A is a n f - o p e n , f - s e m i - c o n t i n u o u s r e t r a c t i o n .

H e n c e g o r is a n f w s c e x t e n s i o n o f g to X a n d t h i s

c o m p l e t e s t h e p r o o f . [ ]

Theorem 5 . 3 . I f A i s a n f - o p e n , f - s e m i - c o n t i n u o u s

r e t r a c t o f X a n d B i s a n f w s c r e t r a c t o f A t h en B i s a n

f w s c r e t ra c t o f X .

S i m i l a r l y

y p q f - l ( V 2 ) <~ l ( v 2 ) i ~ f - 1 ( 1 7 2 ) i .

S o , f - 1 ( i ? I )i a n d f - 1 1 72 )i a r e d i s j o i n t f u z z y s e mq - n b d s o f x ~ a n d y p , r e s p e c ti v e l y . S o X

f u z z y q u a s i s e m i - H a u s d o r f f a n d t h is p r o v e s t

r e s u l t . [ ]

Acknowledgement

I t i s a p l e a s u r e t o t h a n k t h e r e f e r e e s f o r t h e

c o m m e n t s w h i c h r e s u lt ed i n an i m p r o v e d p r e s en t

t i o n o f t h e p a p e r .

Proof . L e t r : X ~ A b e a n f - o p e n a n d f - s e m i - c o n -

t i n u o u s m a p p i n g s u c h th a t r (a) = a f o r a l l a e A .

L e t s : A ~ B b e a n f w s c r e t r a c t i o n o f A s u c h t h a t

s (b) = b f o r a l l b ~ B . B y C o r o l l a r y 3 .9 , s o r : X ~ B

i s f w s c a n d s o r(b) = b f o r a l l b e B . H e n c e B i s f w s c

r e t r a c t o f X a n d t h i s p r o v e s t h e r e s u l t.

Definition 5 . 4. A n f t s X i s c a l l e d a n f - q u a s i

U r y s o h n s p a c e i f f o r a n y t w o d i s ti n c t f u z z y p o i n t s

x ~ a n d yp , t h e r e e x i s t f - o p e n s e t s U 1 a n d U 2 inX s u c h t h a t x ~ q U 1 , y a q U 2 a n d /.71 ^ L72 = 0 .

Definition 5 . 5 . A n f t s X i s s a i d t o b e f u z z y q u a s i

s e m i - H a u s d o r f f i f d i s ti n c t f u z z y p o i n t s i n X h a v e

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