Download A STUDY ON FUZZY LOCALLY δ- CLOSED SETS

Asian Journal of Current Engineering and Maths 2 : 4 July – August (2013) 244 - 247
Contents lists available at www.innovativejournal.in
ASIAN JOURNAL OF CURRENT ENGINEERING AND MATHS
Journal homepage: http://www.innovativejournal.in/index.php/ajcem
A STUDY ON FUZZY LOCALLY - CLOSED SETS
D. Amsaveni, Dr. B. Amudhambigai, R. Dhivya
Department of Mathematics, Sri Sarada College for Women, Salem-16, Tamil Nadu, India.
ARTICLE INFO
ABSTRACT
Corresponding Author
Dr. B. Amudhambigai
Department of Mathematics, Sri
Sarada College for Women, Salem16, Tamil Nadu, India.
In this paper, the interrelations of fuzzy locally -regular closed sets and fuzzy
locally -closed sets are studied with suitable counter examples. Also, the
interrelations of fuzzy locally -regular continuous functions with other types
of fuzzy locally -continuous functions are established with the required
counter examples. Finally the properties and the characterizations of fuzzy
locally -compact spaces and fuzzy locally -compact* spaces are discussed.
Key Words: fuzzy locally -closed
sets, fuzzy locally  pre-closed sets,
fuzzy locally  regular-closed sets.
©2013, AJCEM, All Right Reserved.
1.INTRODUCTION AND PRELIMINARIES
The concept of fuzzy set was introduced by zadeh
[13] in his classical paper. The concept of fuzzy topological
spaces was introduced and developed by Chang [5].Fuzzy
sets have application in many fields such as information
theory [11] and control [12]. The first step of locally
closedness was done by Bourbaki [4]. Ganster and Reily
used locally closed sets in [7] to define LC-continuity and
LC-irresoluteness. The notions of fuzzy -closure derived
from regular closed sets and fuzzy -closure of fuzzy sets in
a fuzzy topological spaces were introduced by Ganguly and
Saha [6] and Mukherjee and Sinha [9], respectively. Fuzzy
-compactness in fuzzy topological spaces was discussed
by Hanafy [8]. The concepts of fuzzy -continuity and fuzzy
-compactness are studied by Seok Jong Lee and Sang Min
Yin [10]. The concepts of r-fuzzy - -locally closed sets
and fuzzy
- -locally continuous functions were
introduced by B.Amudhambigai, M.K. Uma, and E. Roja [1].
In this paper, the interrelations of fuzzy locally -regular
closed sets and fuzzy locally -closed sets are studied with
suitable counter examples. Also, the interrelations of fuzzy
locally -regular continuous functions with other types of
fuzzy locally -continuous functions are established with
the required counter examples. Finally the properties and
the characterizations of fuzzy locally -compact spaces and
fuzzy locally -compact* spaces are discussed.
2. PRELIMINARIES
Definition 2.1 A fuzzy set  in a fuzzy topological space ( X,
T ) is said to be
a)
fuzzy pre-closed if cl ( int (  ) )  . [3]
b)
fuzzy regular-closed if  = cl ( int (  ) ). [2]
Definition 2.3 [6] A fuzzy point X is said to be a fuzzy cluster point of a fuzzy set A if and only if every regular
open q-neighbourhood U of
in q-coincident with A. The
set of all fuzzy -cluster points of A is called the fuzzy closure of A, and denoted by cl ( A ).
Definition 2.4 [10] For a fuzzy subset A in a fuzzy
topological space X, the fuzzy -interior is defined as
follows : int ( A ) = 1 – cl ( 1 – A ).
Definition 2.5 [8] A collection { Ui | i  I } of fuzzy -open
sets in a fuzzy topological space ( X, T ) is called a fuzzy open cover of a fuzzy set A if A 
holds.
Definition 2.6 [8]
A fuzzy topological space ( X, T ) is said to be a
fuzzy -compact space if every fuzzy -open cover of ( X, T
) has a finite subcover. A fuzzy subset A of a fuzzy
topological space ( X, T ) is said to be fuzzy -compact in X
provided for every collection { Ui / i  I } of fuzzy -open
sets of X such that A 
, there exists a finite
subset I0 of I such that A 
.
3. ON FUZZY LOCALLY -REGULAR CLOSED SETS
Definition 3.1 Let ( X, T ) be a fuzzy topological space. Any
X
  I is called a fuzzy locally -closed set ( briefly, fl-cls )
if  =   , where  is fuzzy -open and  is a fuzzy closed
set.
Definition 3.2 Let ( X, T ) be a fuzzy topological space. Any
X
  I is called a fuzzy locally  pre-closed set (briefly, fl
pre-cls) if  =   , where  is fuzzy -open and  is a fuzzy
pre-closed set.
Definition 3.3 Let ( X, T ) be a fuzzy topological space. Any
X
  I is called a fuzzy locally  regular-closed set (briefly,
fl-reg cls) if  =   , where  is fuzzy -open and  is a
fuzzy regular closed set.
Proposition 3.1 Every fuzzy locally -regular closed set is
fuzzy locally -closed.
Remark 3.1
The converse of the above Proposition 3.1 need
not be true.
Example 3.1 Every fuzzy locally -closed set need not be
fuzzy locally -regular closed.
244
Amudhambigai et.al/A Study on Fuzzy Locally - Closed Sets
X
Let X = { a, b } and 1, 2, 3  I be defined as, 1( a ) = 0.4,
1( b ) = 0.3; 2( a ) = 0.5,
2( b ) = 0.7; 3( a ) = 0.45, 3( b
) = 0.65. Define the fuzzy topology T as T = { 0, 1, 1, 2, 3 }.
Clearly, (X,T) is a fuzzy topological space. For any fuzzy open set 1 and fuzzy closed set 1 - 3, 1  ( 1 - 3 ) = (1 - 3)
is fuzzy locally -closed. But,  is not fuzzy locally regular closed.
Proposition 3.2 Every fuzzy locally -closed set is fuzzy
locally  pre-closed.
Proposition 3.3 Every fuzzy locally -regular closed set is
fuzzy locally  pre-closed.
Remark 3.2 The converse of the above Proposition 3.2
and Proposition 3.3 need not be true.
Example 3.2 Every fuzzy locally  pre-closed set need not
be fuzzy locally - closed and fuzzy locally -regular closed.
of all fuzzy -open sets containing  That is, f - int () = {
µ : µ is fuzzy -open and µ ≤  }.
Remark 4.1 Let ( X, T ) be any fuzzy topological space and
let  be any fuzzy set in ( X, T ). Then,
(a) f  - cl (  ) = 1− f  - int (  )
(b) f  - int (  ) = 1− f  - cl (  ).
Proposition 4.1 Let ( X, T ) and ( Y, S ) be any two fuzzy
topological spaces and then for any function f : ( X, T )  (
Y, S ), the following statements are equivalent:
(a) f is fuzzy locally -continuous.
X
(b) For every   I , f ( fl -cl (  ) )  cl ( f (  ) )
Y
-1
Y
-1
(c) For every   I , f ( cl (  ) )  fl -cl f -1 (  ) )
(d) For every   I , f ( int (  ) )  fl -int f -1 (  ) )
Proposition 4.2 Every fuzzy locally -regular continuous
function is fuzzy locally -continuous.
Let X = { a, b }, and 1, 2, 3  I be defined as,
Remark 4.2 The converse of the above Proposition 4.1
need not be true.
) = 0.5, 2( b ) = 0.7; 3( a ) = 0.4, 3( b ) = 0.6. Let   IX be
Example 4.1 Every fuzzy locally -continuous function
need not be fuzzy locally -regular continuous.
X
1( a ) = 0.4,
1( b ) = 0.3;
 2( a
defined as  ( a ) = 0.6 and  ( b ) = 0.4. Then,  is fuzzy preclosed. Thus, 1   =  is fuzzy locally  pre-closed. But,  is
not fuzzy locally -regular closed and hence not fuzzy
locally -closed. Since it is not an intersection of any fuzzy
-open set with fuzzy regular closed set.
Remark 3.3From the above discussions the following
implications hold.
Let X = { a, b }. Define the fuzzy topology T as T = { 0, 1, 1,
X
2, 3 }, where 1, 2, 3  I be defined as follows, 1( a ) =
0.4, 1( b ) = 0.3; 2( a ) = 0.5,
2( b ) = 0.7; 3( a ) =
0.45, 3( b ) = 0.65. Define S = { 0, 1,  } and f : ( X, T )  ( Y,
S ) as f ( a ) = b ; f ( b ) = a.
-1
f (1 – ) = ( 0.55, 0.35 ) = 1 - 3 is fuzzy locally -closed
but not fuzzy locally -regular closed. Therefore, every
fuzzy locally -continuous function need not be fuzzy
locally -regular continuous function.
Proposition 4.3 Every fuzzy locally -continuous function
is fuzzy locally  pre-continuous function.
4. ON FUZZY LOCALLY -REGULAR CONTINUOUS
FUNCTIONS
Definition 4.1 Let ( X, T ) and ( Y, S ) be any two fuzzy
topological spaces. A function f : ( X, T )  ( Y, S ) is said to
be a fuzzy locally -regular continuous function (briefly, flY
-1
X
rcf) if for each fuzzy closed set   I , f (  ) I is fuzzy
locally -regular closed.
Definition 4.2 Let ( X, T ) and ( Y, S ) be any two fuzzy
topological spaces. A function f : ( X, T )  ( Y, S ) is said to
be a fuzzy locally -continuous function (briefly, fl-cf)if for
Y
-1
X
each fuzzy closed set   I , f () I is fuzzy locally closed sets.
Definition 4.3 Let ( X, T ) and ( Y, S ) be any two fuzzy
topological spaces. A function
f : ( X, T )  ( Y, S ) is
said to be a fuzzy locally  pre-continuous function (briefly,
Y
-1
X
fl p-cf) if for each fuzzy closed set   I , f (  ) I is
fuzzy locally  pre-closed.
Definition 4.4 A fuzzy topological space and let  be any
fuzzy set in ( X, T ). Then fuzzy -closure of  denoted by fcl (  ) is defined as the intersection of all fuzzy -closed
sets containing . That is, f - cl (  ) = ∧ { µ : µ is a fuzzy closed set and µ ≥  }.
Definition 4.5 Let ( X, T ) be any fuzzy topological space
and let  be any fuzzy set in ( X, T ). Then fuzzy -interior
of , denoted by f - int (  ) is defined us the intersection
Proposition 4.4 Every fuzzy locally -regular continuous
function is fuzzy locally  pre-continuous function.
Remark 4.3 The converse of the above Proposition 4.2
and Proposition 4.3 need not be true.
Example 4.2 Every fuzzy locally  pre-continuous function
need not be fuzzy locally -regular continuous and locally
-continuous.
Let X = { a, b }. Define the fuzzy topology T as T = { 0, 1, 1,
2, 3 }, where 1, 2, 3  IX be defined as follows : 1( a ) =
0.4, 1( b ) = 0.3; 2( a ) = 0.5 2( b ) = 0.7; 3( a ) = 0.4,
Y
3( b ) = 0.6. Define S as S = {0, 1, }. Let   I be defined as
( a ) = 0.6 and ( b ) = 0.4.
-
Define f : ( X, T )  ( Y, S ) as f( a ) = b and f( b ) = a. Then, f
1
(1 – ) = (1 - 3) is fuzzy locally  pre-closed. Therefore, f is
-1
fuzzy locally  pre-continuous. But, f (1 – ) is not fuzzy
locally -regular closed set and hence not fuzzy locally
-closed set.
Therefore, every fuzzy locally  pre-continuous function
need not be fuzzy locally -regular continuous function and
fuzzy locally -continuous function.
Remark 4.4 From the above discussions the following
implications hold.
245
Amudhambigai et.al/A Study on Fuzzy Locally - Closed Sets
Proposition 5.4 Let A and B be fuzzy subsets of a fuzzy
topological space ( X, T ) such that A is fuzzy locally compact in X and B is fuzzy locally -closed in ( X,T ). Then
A  B is fuzzy locally -compact in X.
5. FUZZY LOCALLY -COMPACT SPACES AND FUZZY
LOCALLY -COMPACT* SPACES
Definition 5.1 Let ( X, T ) be a fuzzy topological space. The
X
collection {
I :
is fuzzy locally -open, i
} is
called the fuzzy locally -open cover of ( X, T ) if 
=
iI
1.
Definition 5.2 Any fuzzy topological space ( X, T ) is called
fuzzy locally -compact if every fuzzy locally -open cover
( X, T ) has a finite subcover.
Definition 5.3 A collection { Ui | i  I } of fuzzy locally open sets in a fuzzy topological space ( X, T ) is called a
fuzzy locally -open cover of a fuzzy set A if A 
holds.
Definition 5.4 A fuzzy topological space ( X, T ) is said to
be a fuzzy locally -compact space if every fuzzy locally open cover of ( X, T ) has a finite subcover. A fuzzy subset A
of a fuzzy topological space ( X, T ) is said to be fuzzy
locally -compact in X provided for every collection { Ui / i
 I } of fuzzy locally -open sets of X with A 
, there exists a finite subset I0 of I such that A 
.
Definition 5.5 A fuzzy topological space X is said to be
fuzzy locally -compact* at a fuzzy point x if there is a
fuzzy locally -open subset U and a fuzzy set F which is
fuzzy locally -compact in X such that x  F  U. If X is
fuzzy locally -compact* at each of its fuzzy point, X is said
to be a fuzzy locally -compact* space.
Definition 5.6 A fuzzy subset A of a fuzzy topological space
X is said to be fuzzy locally -compact* in X provided for
each fuzzy point x in A, there is a fuzzy locally -open
subset u and a fuzzy subset F which is fuzzy locally compact in X such that x  F  U.
Definition 5.7 Let ( X, T ) and ( Y, S ) be any two fuzzy
topological spaces. Any function f : ( X, T ) → ( Y, S ) is
-1
X
called a fuzzy locally  - irresolute function if f (  ) I
is fuzzy locally -open for every fuzzy locally -open set 
Y
I .
Remark : 5.1 Every fuzzy locally -open set is fuzzy
locally open.
Proposition 5.1 Every fuzzy locally compact space is locally
-compact.
Proposition 5.2 ( X, T ) is fuzzy locally -compact if and
only if every family of fuzzy locally -closed subsets of ( X, T
) which has the finite intersection property has a nonempty
intersection.
Corollary 5.1 A fuzzy topological space ( X, T ) is fuzzy
locally compact if and only if every family of fuzzy T-closed
subsets of (X, T ) with the finite intersection property has a
non empty intersection.
Proposition 5.3 Let F be a fuzzy locally -closed subset of a
fuzzy locally -compact space ( X, T ), then F is also fuzzy
locally -compact in X.
Proposition 5.5 Let ( X,T ) and ( Y, S ) be any two fuzzy
topological space. Let f : ( X, T )  ( Y, S ) be a fuzzy locally
-irresolute and surjective function. If ( X, T ) is a fuzzy
locally -compact space, then ( Y, S ) is also a fuzzy locally
-compact space.
Proposition 5.6 Let f : ( X, T )  ( Y, S ) be a fuzzy locally continuous. If a fuzzy subset A is fuzzy locally -compact in
( X, T ), then the image f ( A ) is fuzzy locally -compact in
( Y, S ).
Proposition 5.7 Let f : ( X, T )  ( Y, S ) be a fuzzy locally irresolute, fuzzy locally -open and injective mapping. If a
fuzzy subset B of Y is fuzzy locally -compact in ( Y, S ),
-1
then the image f ( B ) is fuzzy locally -compact in ( X, T ).
Proposition 5.8 Let X be a fuzzy locally -compact* space
and A a fuzzy subset of ( X, T ). If A is fuzzy locally -closed
in X, then A is fuzzy locally -compact* in X.
Proposition 5.9 Let a fuzzy topological space ( X, T ) be
fuzzy locally -compact* and A be a fuzzy open subset of X.
Then A is fuzzy locally -compact* in X.
Proposition 5.10 Let ( X, T ) and ( Y, S ) be two fuzzy
topological spaces and f : ( X, T )  ( Y, S ) be a fuzzy locally
-irresolute, fuzzy locally -open and surjective function. If
(X, T ) is fuzzy locally -compact*, then ( Y, S ) is also fuzzy
locally -compact*.
Proposition 5.11 Let (X, T) and (Y, S) be fuzzy topological
spaces and f : (X, T)  (Y, S) be a fuzzy locally -irresolute,
fuzzy locally -open and injective function. If ( Y, S ) is fuzzy
locally -compact*, then ( X, T ) is also fuzzy locally compact*.
ACKNOWLEDGEMENT: The authors express their sincere
thanks to the referees for their valuable comments
regarding the improvement of the paper.
REFERENCES
1. Amudhambigai.B, Uma.M.K, And Roja.E :r-Fuzzy - Locally Closed Sets And Fuzzy - -Locally Continuous
Functions, Int. J. Of Mathematical Sciences And
Applications, 1, Sep 2011.
2. Azad.k.k : on fuzzy semicontinuity, Fuzzy almost
continuity and fuzzy weakly continuity, J.math. Appl.,
82 (1981), 14-32
3. Bin Shahna.A.S : On Fuzzy Strong Semi Continuity And
fuzzy Pre Continuity, Fuzzy Sets And Systems 44
(1991), 303-308.
4. Bourbaki.N: General Topology, Part I. Addison-Wesley,
Reading, Mass 1996.
5. Chang.C.L: Fuzzy Topological Spaces, J.Math. Anal.
Appl.,24 (1968),182-190.
6. Ganguly.S And Saha.S :A Note On -Continuity Connected Sets In Fuzzy Set Theory, Simon Stevin 62
(1998), 127-141
7. Ganster.M And Reily.I.L : Locally Closed Sets And LcContinuous functions Int. J.Math and Math.Sci 12
(1989), 417-424.
8. Hanafy.I.M : Fuzzy -Compactness In Fuzzy
Topological Spaces Math. Japonica 51(2000), 229-234.
246
Amudhambigai et.al/A Study on Fuzzy Locally - Closed Sets
9.
Mukherjee.M.N And Sinha.S.P: On Some new fuzzy
Continuous Functions Between Fuzzy Topological
Spaces, Fuzzy Sets And Systems 34 (1990), 245-254.
10. Seok Jong Lee And Sang Min Yun., Fuzzy  -Topology
And Compactness, Commun. Korean Math. Soc. 27
(2012), 357-368.
11. Smets. P : The Degree Of Belief In A Fuzzy event
Inform. Sci., 25 (1981), 1-19.
12. Sugeno.M : An Introductory Survey Of Fuzzy Control,
Inform, Sci., 36 (1985), 59-83.
13. Zadeh. L. A: Fuzzy Sets, Information And Control, 8
(1965), 338-353.
247