# Certain Aspects of Fuzzy Alpha Compactness

DOI : 10.17577/IJERTV2IS90163

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#### Certain Aspects of Fuzzy Alpha Compactness

M. A. M. Talukder 1 and D. M. Ali 2

1 Department of Mathematics , Khulna University of Engineering & Technology , Khulna 9203 , Bangladesh .

2 Department of Mathematics , University of Rajshahi , Rajshahi 6205 , Bangladesh .

1 Corresponding author :

1 Presently on leave from : Department of Mathematics , Khulna University of Engineering & Technology , Khulna 9203 , Bangladesh .

ABSTRACT

In this paper , we study several aspects of fuzzy compactness due to T. E. Gantner et al. [5] in fuzzy topological spaces and also obtain its several other properties .

Keywords : Fuzzy topological spaces , compactness .

1. INTRODUCTION

The concept of fuzzy sets and fuzzy set operations was first introduced by L. A. Zadeh in his classical papers [10] in the year 1965 describing fuzziness mathematically first time . Compactness occupies a very important place in fuzzy topological spaces . The purpose of this paper is to study the concept due to T. E. Gantner et al. in more detail and to obtain several other features .

2. PRELIMINARIES

We briefly touch upon the terminological concepts and some definitions , which are needed in the sequel

. The following are essential in our study and can be found in the paper referred to.

1. Definition (10) : Let X be a non-empty set and I is the closed unit interval [0, 1]. A fuzzy set in X is a

function u : X I which assigns to every element x X. u(x) denotes a degree or the grade of

membership of x. The set of all fuzzy sets in X is denoted by fuzzy subset of X .

I X . A member of I X

may also be a called

2. Definition (10)

: Let X be a non-empty set and A X . Then the characteristic function 1A (x) : X

{0 , 1} defined by 1 (x) = 1 if

x A

0

0

A if

x A

Thus we can consider any subset of a set X as a fuzzy set whose range is {0 , 1}.

3. Definition ( 9 )

0 or .

: A fuzzy set is empty iff its grade of membership is identically zero . It is denoted by

4. Definition ( 9 ) : A fuzzy set is whole iff its grade of membership is identically one in X . It is

denoted by 1 or X .

5. Definition ( 3 ) : Let u and v be two fuzzy sets in X. Then we define

1. u = v iff u(x) = v(x) for all x X

2. u v iff u(x) v(x) for all x X

3. = u v iff (x) = (u v) (x) = max [ u(x) , v(x) ] for all x X

4. = u v iff (x) = (u v) (x) = min [ u(x) , v(x) ] for all x X

5. = uc iff (x) = 1 u(x) for all x X .

6. Definition ( 3 )

intersection ui

: In general , if { ui

are defined by

: i J } is family of fuzzy sets in X , then union ui

and

ui (x) = sup { ui (x) : i J and x X }

ui (x) = inf { ui (x) : i J and x X } , where J is an index set .

7. Definition ( 3 ) : Let f : X Y be a mapping and u be a fuzzy set in X. Then the image of u , written

f (u) , is a fuzzy set in Y whose membership function is given by

f(u) (y) =

sup{u (x) : x f 1 ( y)} if

f 1 ( y)

.

1

0

if f

( y)

8. Definition ( 3 ) : Let f : X Y be a mapping and v be a fuzzy set in Y. Then the inverse of v, written

f 1 (v) , is a fuzzy set in X whose membership function is given by ( f 1 (v)) (x) = v(f(x)) .

9. De-Morgans laws (10)

sets in X, then

: De-Morgans Laws valid for fuzzy sets in X i.e. if u and v are any fuzzy

(i) 1 (u v) = (1 u) (1 v)

(ii) 1 (u v) = (1 u) (1 v)

For any fuzzy set in u in X , u (1 u) need not be zero and u (1 u)need not be one .

10. Definition ( 3 )

: Let X be a non-empty set and t I X

i.e. t is a collection of fuzzy set in X. Then t is

called a fuzzy topology on X if

(i) 0 , 1 t

1. ui t for each iJ , then ui t

i

2. u , v t , then u v t

The pair X , t is called a fuzzy topological space and in short, fts. Every member of t is called a t- open fuzzy set. A fuzzy set is t-closed iff its complements is t-open. In the sequel, when no confusion is likely to arise, we shall call a t-open ( t-closed ) fuzzy set simply an open ( closed ) fuzzy set .

11. Definition ( 3 )

: Let X , t and Y , s be two fuzzy topological spaces. A mapping f : X , t

Y , s is called an fuzzy continuous iff the inverse of each s-open fuzzy set is t-open .

12. Definition ( 9 )

: Let X , t be an fts and A X. Then the collection t A = { u|A : ut } = { u A :

ut } is fuzzy topology on A, called the subspace fuzzy topology on A and the pair A , tA is referred to as a fuzzy subspace of X , t .

13. Definition ( 8 )

: An fts X , t is said to be fuzzy Hausdorfff iff for all x , y X , x y, there

exist u , v t such that u(x) = 1 , v(y) = 1 and u 1 v .

14. Definition ( 8 )

: An fts X , t is said to be fuzzy regular iff for each x X and u t c

with u(x) =

0 , there exist v , w t such that u(x) = 1 , u w and v 1 w .

15. Definition ( 4 )

: Let A , tA

and B , sB

be fuzzy subspaces of ftss X , t

and Y , s

respectively and f is a mapping from X , t to Y , s , then we say that f is a mapping from A , tA

to B , sB if f(A) B .

16. Definition ( 4 )

: Let A , tA

and B , sB

be fuzzy subspaces of ftss X , t

and Y , s

A

A

respectively. Then a mapping f : A , tA B , sB is relatively fuzzy continuous iff for each v sB ,

the intersection

f 1 (v) A t .

17. Definition (1)

: Let I X

and

I Y . Then ( ) is a fuzzy set in X Y for which ( )

( x , y ) = min { (x) , (y) } , for every ( x , y ) X Y .

18. Definition : Let X , t be an fts and I . A collection M of fuzzy sets is called an shading ( res. * shading ) of X if for each x X there exists a u M such that u(x) > ( res. u(x) ) . A subcollection of an shading ( res. * shading ) of X which is also an shading ( res. * shading ) is called an subshading ( res. * subshading ) of X .

19. Definition ( 5 )

: An fts X , t is said to be compact ( res. * compact ) if each shading

( res. *

• shading ) of X by open fuzzy sets has a finite subshading ( res. *

where I .

20. Definition ( 6 )

: Let X , t be an fts and 0 < 1 , then the family t

= { (u) : u t } of all

subsets of X of the form (u) = { x X : u(x) > } is called level sets , forms a topology on X and is called the level topology on X and the pair X , t is called level topological space .

3. Characterizations of fuzzy compactness .

Now we obtain some tangible properties of fuzzy compact spaces .

1. Theorem : Let 0 < 1. An fts X , t

is compact iff for every family { Fi } of level

n

closed subsets of X , Fi = implies { Fi } contains a finite ubfamily { Fik } ( k Jn ) with Fik = .

i J k 1

Proof : Let X , t be compact . Suppose M = { Fi

: i J } be a family level closed subsets of

X with F = . Then , since for each

F , there exists a t c such that F = ( ) , we have M =

i

i J

i i i i

c

c

c

{ ( ) : i J }. Then by De Morgans law X = c

= F

= F . Then the family H = {

c :

i i

i J

i i

i J

i J }is an open shading of X , t . To see this , let x X . Since M is a family of level closed

subsets of X , there is an

F M such that x F . But F = ( ) , for some t c

. Since X , t

i0 i0 i0 i0 i0

is compact , there exist

c H ( k J ) such that X = (

c ) = Fc . Then by De

Morgans law =

X c = F

ik n

c

= Fc .

i J

ik ik

i J

i J

ik

ik

i J

Conversely , suppose Fi = and M = { i

i J

: iJ } be an open shading of X , t . Then by De

c

c

c

Morgans law =

X c =

F

= F . Then the family H = { ( ) : i J } is a level open

i

i J

i i

i J

subsets of X , where

F = ( c ) . For let x X . Then there exists a M ( k J ) such that

i i ik n

k

k

i (x) > . Hence X , t is compact .

2. Theorem : Let 0 < 1 . Let X , t be an fts and X , t be a level topological space . Let

f : X , t X , t be continuous and onto . If X , t is compact , then X , t is compact

topological space . Proof : Let M = { Ui

: i J } be an open cover of X , t . Then , since for each Ui

, there exists a

gi t such that Ui = (

gi ) , we have M = { (

gi ) : i J } . Then the family W = {

gi : i J } is an

shading of X , t . Since f is continuous , then

f 1

( M ) = {

f 1 ( U ) : U

t

} is an open

i i

i i

shading of X , t . To see this , let x X . Since M is an open cover of X , t , then there is an

i i i

i i i

U M such that x U . But U = (

0 0 0

g ) for some

i

i

0

g t . Therefore x (

i

i

0

g ) which implies

i

i

0

that

g (x) > . Since f is continuous and onto , then U ( f(x) ) > which implies that

i i

i i

0 0

f 1 (U ) (x)

i

i

0

> . By compactness of X , t , W has a finite subshading , say { gik } ( k Jn ) such that

f 1 (

g ) (x) > or

i

i

k

g ( f(x) ) > for some x X . Thus { U } or { (

i i

i i

k k

gi ) } ( k Jn ) forms a

k

k

finite subcover of M . Hence X , t is compact topological space .

3. Theorem : Let X , t and Y , s be two fuzzy topological spaces and let f : X , t Y , s

be a continuous surjection . Let A be an compact subset of X , t . Then f(A) is also compact of Y , s .

Proof : Assume that f(X) = Y . Let { ui

: ui s } be an open shading of f(A) . Since f is continuous

, then

f 1 (

ui ) t . For if x A , then f(x) f(A) as A is compact subset of X , t . Thus we

see that

f 1 ( u ) (x) > and so

f 1 ( u ) is an open shading of A . Since A is compact , then {

i

i

i

i

i

i

f 1 ( u ) } has a finite subshading , say {

f 1 ( u ) } ( k J ) . Now if y f(A) , then y = f(x) for

i n

i n

k

k

k

some x A . Then there exists ui { ui } such that

f 1 ( u ) (x) > which implies that u ( f(x) ) >

i i

i i

k k

k k

k k

or ui (y) > . Thus { ui } has a finite subshading { ui } ( k Jn ) . Hence f(A) is compact

.

4. Definition : The mappings x : X Y X such that x ( x , y ) = x for all ( x , y ) X Y and

y : X Y Y such that y ( x , y ) = y for all ( x , y ) X Y are called projection mappings or

simply projection of X Y on X and Y respectively .

5. Theorem : Let X , t

and Y , s

be two fuzzy topological spaces . Then the product space

X Y , t s is compact iff X , t and Y , s are compact , where 0 < 1 .

Proof : First suppose that X Y , , where = {

gi hi :

gi t and hi s } is compact . Now

we can define a fuzzy projection mappings

x : X Y ,

X , t such that x ( x , y ) = x for

all ( x , y ) X Y and y

: X Y ,

Y , s such that y ( x , y ) = y for all ( x , y ) X Y

which we know are continuous . Hence X , t

and Y , s

are continuous images of X Y ,

which are therefore compact when X Y , is given to be compact .

Conversely , let X , t and Y , s be compact . Let = {

gi hi :

gi t and hi s } , where gi

and hi

are open fuzzy sets and {

gi : i J } is an shading of X , t and { hi

: i J } is an

shading of Y , s . That is

gi (x) > for all x X , hi (y) > for all y Y . We see that (

gi hi ) (

x , y ) = min {

gi (x) ,

hi (y) } > . As X , t and Y , s are compact , there exist

g t such

i

i

k

that

1. (x) > and

i

i

k

2. s such that

i

i

k

h (y) > respectively . Hence we have = {

i

i

k

gi hi :

gi t

k k

k k

and hi s } has a finite subshading , say {

gi hi } ( k Jn ) . Thus X Y , is compact .

6. Theorem : Let X , t be a fuzzy Hausdorff space and A be an compact ( 0 < 1 ) subset of

X , t . Suppose x Ac , then there exist u , v t such that u(x) = 1 , A

v 1 ( 0 , 1] and u 1 v .

Proof : Let y A . Since x A ( x Ac

) , then clearly x y . As X , t is fuzzy Hausdorff , then

there exist u y

, v y t such that u y (x) = 1 , v y (y) = 1 and u y

1 v y

. Let us take

I1 such that

v y (y) > > 0 . Thus we see that { vy : y A } is an shading of A . Since A is compact in

k

k

u

u

X , t , so it has a finite subshading , say { vy :

yk A } ( k Jn ) . Now, let v = vy

v

y

y

2

..

1

1

y

y

v

n

and u = u

y

y

1

y u y

. Thus we see that v and u are open fuzzy sets , as they are the

2

2

n

n

union and finite intersection of open fuzzy sets respectively i.e. v , u t . Moreover , A

y

y

and u(x) = 1 , as u (x) = 1 for each k .

k

v 1 ( 0 , 1]

Finally , we claim that u 1 v . As u y

1 v y

implies that u 1 v y

. Since u (x) 1 v (x)

y y

y y

k k

for each k , then u 1 v . If not , then there exists x X such that u y (x) 1 v y (x) . We have

u y (x)

1. (x) for all k . Then for some k , u (x) 1 v (x) . But this is a contradiction as u 1

y y y y

y y y y

k k k k

y

y

• v for all k . Henc u 1 v .

k

7. Theorem : Let X , t be a fuzzy Hausdorff space and A , B be disjoint compact ( 0 < 1 )

subsets of X , t . Then there exist u , v t such that A

u1 ( 0 , 1 ] , B

v1 ( 0 , 1 ] and u 1 v.

Proof : Let y A . Then yB , as A and B are disjoint . Since B is compact , then by theorem

( 3.6 ) , there exist u , v t such that u (y) = 1 , B v1 ( 0 , 1 ] and u 1 v . Let us take

y y y y y y

I1 such that vy (y) > > 0 . As u y (y) = 1 , then we observe that{ u y

: y A } is an shading of A .

k

k

Since A is compact in X , t , so it has a finite subshading , say { uy :

yk A } ( k Jn ) .

y

y

Furthermore , since B is compact , so B has a finite subshading , say { v :

k

yk B } ( k Jn )

as B v1 ( 0 , 1 ] for each k . Now , let u = u u u and v = v v .. v .

yk y1 y2 yn y1 y2 yn

Thus we see that A

u 1 ( 0 , 1] and B

v 1 ( 0 , 1] . Hence u and v are open fuzzy sets , as they are

the union and finite intersection of open fuzzy sets respectively i.e. u , v t .

y

y

Finally , we have to that u 1 v . First we observe that u

k

• v for each k and it is clearly shows that u 1 v .

1 v for each k , implies that u 1

y y

y y

k k

8. Theorem : Let X , t be an fts and A X .

( i ) If 0 < 1 and if A is compact , then A is closed in X . ( ii ) If 0 < 1 and if A is * compact , then A is closed in X .

y

y

Proof : ( i ) : Let x Ac

. We have to show that , there exist u t such that u(x) = 1 and u

Ap ,

where

Ap is the characteristic function of

Ac . Indeed , for each y A , there exist u

, vy t such that

u y (x) = 1 , vy (y) = 1 and u y

1 vy

. Let us take

I1 such that v y (y) > > 0 . Thus we see that

u

u

{ vy : y A } is an shading of A . Since A is compact in X , t

, so it has a finite

v

v

y

y

k

yk A } ( k Jn ) . Now , let u = u

y u y

. Thus we see that

y

y

k

k

1

1

2

2

n

n

u y (x) =1 and u 1 vy for each k and it is clear that u 1 v . For , each z A , there exists a k

such that

1. (z) > 0 and so u(z) = 0 . Hence u A

p

p

y

y

k

. Therefore ,

Ac is open in X . Thus A is

closed in X .

( ii ) The proof is similar .

9. Theorem :Let X , t be a fuzzy regular space and A be an compact subset of X , t .

Suppose x A and u t c

v1 ( 0 , 1 ] and v 1 w .

with u(x) = 0 . Then there exist v , w t such that v(x) = 1 , u w , A

Proof : Suppose x A and u t c

we have u(x) = 0 . As X , t is fuzzy regular , then there exist vx ,

wx t such that vx (x) = 1 , ux

wx

and

vx 1 wx

. Let us take

I1 such that vx (x) > > 0 .

Thus we observe that { vx : x A } is an open shading of A . Since A is compact in X , t ,

x

x

then it has a finite subshading , say { v :

k

xk A } ( k Jn ) . Let v = vx vx .. vx

and

1 2 n

1 2 n

x x x

x x x

w = w w .. w . Thus we see that v and w are open fuzzy sets , as they are the union and

1 2 n

finite intersection of open fuzzy sets respectively i.e. v , w t . Furthermore , u w , A

and v(x) = 1.

v1 ( 0 , 1 ]

x

x

Finally , we have to show that v 1 w . As v

k

k and hence it is clear that v 1 w .

1 w for each k implies that v

x x

x x

k k

1 w for each

10. Theorem : Let X , t

be a fuzzy regular space and A , B be disjoint compact subsets of

X , t . For each x X and u t c

with u(x) = 0 , there exist v , w t such that A

v1 ( 0 , 1 ] , B

w1 ( 0 , 1 ] and v 1 w .

Proof : Suppose for each x X and u t c we have u(x) = 0 . Let x A . Then x B , as A and B are

disjoint . As B is compact , then by theorem ( 3.9 ) , there exist vx

, wx t such that vx (x) = 1 , B

w1 ( 0 , 1 ] and v 1 w . Let us take I such that v (x) > > 0 . As v (x) = 1 , then we

x x x 1 x x

see that { vx

: x A } is an open shading of A . Since A is compact in X , t , then it has a

x

x

finite subshading , say { v :

k

xk A } ( k Jn ) . Further more , as B is compact , so it has a

finite subshading , say { w : x B } ( k J ) , as B w1 ( 0 , 1 ] for each k . Let v = v v

xk k n x x1 x2

x

x

.. v

n

and w =

1. w .. w . Thus we see that A

x x x

x x x

1 2 n

v1 ( 0 , 1 ] and B

w1 ( 0 , 1

] . Hence v and w are open fuzzy sets , as they are the union and finite intersection of open fuzzy sets respectively i.e. v , w t .

x

x

Lastly , we have to show that v 1 w . First , we observe that v

k

1

w for each k implies that

x

x

k

x

x

v 1 w for each k and hence it is clear that v 1 w .

k

11. Theorem : Let A , t A and B , sB be fuzzy subspaces of ftss X , t and Y , s respectively

with A , t A is compact . Let f : A , t A

and onto . Then B , sB is compact .

B , sB be relatively fuzzy continuous , one one

Proof : Let { vi

: vi

sB } be an open shading of B , sB

for every i J . As f is fuzzy

continuous , then

f 1 ( v

) t . By definition of subspace fuzzy topology , there exist ui s such that

i

i

vi = ui

B . We see that for every x X ,

f 1 ( v

) (x) =

f 1 ( u

B ) (x) > and so {

f 1 ( u

i

i

i

i

i

i

B ) } is an open shading of A , t A , i J . Since A , t A is compact , then {

f 1 ( u

i

i

B ) } has a finite subshading , say {

f 1 ( u

i

i

k

B ) } ( k Jn ) . Now , if y Y , then y = f(x) for

k

k

some x X . Then there exists vi { vi

} such that

f 1 ( v

i

i

k

) (x) > implies that

f 1 ( u

i

i

k

B ) (x)

i

i

> . So ( u

k

B ) f(x) > or ( u

i

i

k

B ) (y) > . Hence we observe that { vi } ( k Jn ) is a finite

k

k

} . Thus B , sB is compact .

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