Certain Aspects of Fuzzy Alpha Compactness

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. *

       • subshading )

         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

      subshading , say { :

      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

subshading of { vi

} . Thus B , sB is compact .

REFERENCES

1. K. K. Azad , On Fuzzy semi continuity , Fuzzy almost continuity and Fuzzy weakly continuity , J. Math. Anal. Appl. 82(1) (1981) , 14 32 .

2. S. S. Benchalli & G. P. Siddapur , On the Level Spaces of Fuzzy Topological Spaces * , B. Math. Appl. Vol 1. Issue 2 (2009 ) , 57 65 .

3. C. L. Chang , Fuzzy Topological Spaces , J. Math. Anal. Appl. , 24(1968) , 182 190 .

4. H. Foster , David , Fuzzy Topological Groups , J. Math. Anal. Appl. , 67(1979) , 549 564 .

5. T. E. Gantner, R. C. Steinlage and R. H. Warren , Compactness in Fuzzy Topological Spaces , J. Math. Anal. Appl. , 62(1978) , 547 562 .

6. A. J.Klein , Closure in Fuzzy Topology , Rocky Mount . J. Math. Anal. Appl. 11(1981) , 553

   560 .

7. S. Lipschutz , Theory and problems of general topology , Schaums outline series , McGraw-Hill book publication company , Singapore , 1965 .

1. . S. R. Malghan, and S. S. Benchalli , On Fuzzy Topological Spaces , Glasnik Mathematicki , 16(36) (1981) , 313- 325 .

2. . Pu Pao Ming and Liu Ying – Ming, Fuzzy topology. I. Neighborhood Structure of a fuzzy point and Moore Smith Convergence ; J. Math. Anal. Appl. 76 (1980) , 571 599 .

3. . L. A. Zadeh , Fuzzy Sets , Information and Control , 8(1965) , 338 353 .