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 Total Downloads : 163
 Authors : M. A. M. Talukder, D. M. Ali
 Paper ID : IJERTV2IS90163
 Volume & Issue : Volume 02, Issue 09 (September 2013)
 Published (First Online): 11092013
 ISSN (Online) : 22780181
 Publisher Name : IJERT
 License: This work is licensed under a Creative Commons Attribution 4.0 International License
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 .

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 .

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.

Definition (10) : Let X be a nonempty 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

Definition (10)
: Let X be a nonempty 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}.

Definition ( 9 )
0 or .
: A fuzzy set is empty iff its grade of membership is identically zero . It is denoted by

Definition ( 9 ) : A fuzzy set is whole iff its grade of membership is identically one in X . It is
denoted by 1 or X .

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

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

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

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

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

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


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 .

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)

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)) .

DeMorgans laws (10)
sets in X, then
: DeMorgans 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 .

Definition ( 3 )
: Let X be a nonempty 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

ui t for each iJ , then ui t
i

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 tclosed iff its complements is topen. In the sequel, when no confusion is likely to arise, we shall call a topen ( tclosed ) fuzzy set simply an open ( closed ) fuzzy set .


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 sopen fuzzy set is topen .

Definition ( 9 )
: Let X , t be an fts and A X. Then the collection t A = { uA : 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 .

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 .

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 .

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 .

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 .

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 .

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 .

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 .


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 .


Characterizations of fuzzy compactness .
Now we obtain some tangible properties of fuzzy compact spaces .

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 .

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 .

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
.

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 .

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

(x) > and
i
i
k

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 .


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)

(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



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


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

(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 .


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

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 =

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


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 .
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