 Open Access
 Total Downloads : 14
 Authors : M. Jeyaraman, J. Rajalakshmi, R. Muthuraj
 Paper ID : IJERTCONV3IS33005
 Volume & Issue : RACMS – 2015 (Volume 3 – Issue 33)
 Published (First Online): 24042018
 ISSN (Online) : 22780181
 Publisher Name : IJERT
 License: This work is licensed under a Creative Commons Attribution 4.0 International License
gmClosed and gmOpen Maps in Fuzzy Topological Spaces
Special Issue – 2015
International Journal of Engineering Research & Technology (IJERT)
ISSN: 22780181
RACMS2014 Conference Proceedings
g Closed and g Open Maps in Fuzzy Topological Spaces
1M. Jeyaraman,
1Department of Mathematics, Raja Duraisingam Govt.Arts College,
Sivagangai, Sivagangai District, Tamil Nadu.
2J. Rajalakshmi,
2Department of Mathematics,
St. Michael College of Engg & Tech, Kalaiyarkovil, Sivagangai District, Tamil Nadu.
3R. Muthuraj, Department of Mathematics, H.H.The Rajahs College,
Pudukkottai, Pudukkottai District, Tamil Nadu.
Abstract: A fuzzy set A in a fuzzy topological space (X, ) is said to be fuzzy g closed set if cl(A) U whenever A U
and U is fgsopen in (X, ). In this paper, we introduce fuzzy g closed map from a fuzzy topological space X to a
fuzzy topological space Y as the map which maps every fuzzy closed set to a fuzzy g closed, and notice that the
composition of two fuzzy g closed maps need not be fuzzy g closed map. We also obtain some properties of fuzzy g closed maps.
Key words : Fuzzy Topological space, fuzzy g closed map, fuzzy g *closed map and fuzzy g open map, fuzzy g * open map.

INTRODUCTION
In the classical paper [19] of 1965, L.A.Zadeh generalized the usual notion of a set by introducing the important and useful notion of fuzzy sets. Subsequently many researchers have worked on various basic concepts from general topology using fuzzy sets and developed the theory of fuzzy topological spaces. The notion of fuzzy sets naturally plays a very significant role in the study of fuzzy topology introduced by C. L. Chang [5].
Fuzzy continuous functions is one of the main topics in fuzzy topology. Various authors introduce various types of fuzzy continuity. The decomposition of fuzzy continuity is one of the many problems in fuzzy topology. Tong [16] obtained a decomposition of fuzzy continuity by introducing two weak notions of fuzzy continuity namely, fuzzy strong semicontinuity and fuzzy precontinuity. Rajamani [9] obtained a decomposition of fuzzy continuity.
In this section, we introduce fuzzy g closed maps,
fuzzy g open maps, fuzzy g *closed maps and
fuzzy g *open maps in fuzzy topological spaces and obtain certain characterizations of these maps.

PRELIMINARIES
Throughout this paper, (X, ), (Y, ) and (Z, ) (or X, Y and Z) represent fuzzy topological spaces on which no separation axioms are assumed unless otherwise mentioned. For any fuzzy subset A of a space (X, ), the closure of A, the interior of A and the complement of A are denoted by cl(A), int(A) and Ac respectively.
We recall the following definitions which are useful in the sequel.
Definition 2.1
A fuzzy subset A of a space (X, ) is called: fuzzy semiopen set [1] if A cl(int(A))
The complement of fuzzy semiopen set is fuzzy semiclosed.
The fuzzy semiclosure [18] of a fuzzy subset A of X, denoted by scl(A), is defined to be the intersection of all fuzzy semiclosed sets of (X, ) containing A. It is known that scl(A) is a fuzzy semiclosed set. For any fuzzy subset A of an arbitrarily chosen fuzzy topological space, the fuzzy semiinterior [18] of A, denoted by sint(A), is defined to be the union of all fuzzy semiopen sets of (X, ) contained in A.
Definition 2.2
A fuzzy subset A of a space (X, ) is called:

a fuzzy generalized closed (briefly fgclosed) set
[2] if cl(A) U whenever A U and U is fuzzy open in (X, ). The complement of fgclosed set is called fgopen set; 
a fuzzy closed set (= f closed set) [13] if
cl(A) U whenever A U and U is fuzzy semi
Special Issue – 2015
open in (X, ). The complement of fuzzy – closed set is called fuzzy open set;

a fuzzy semigeneralized closed (briefly fsgclosed) set [3] if scl(A) U whenever A U and U is fuzzy semiopen in (X, ). The complement of fsgclosed set is called fsgopen set;

a generalized fuzzy semiclosed (briefly fgs closed) set [11] if scl(A) U whenever A U and U is fuzzy open in (X, ). The complement of fgs closed set is called fgsopen set;

a fuzzy g closed set [7] if cl(A) U whenever
A U and U is fgsopen in (X, ). The complement of fuzzy g closed set is called
fuzzy g open set;

a fuzzy g*sclosed set [7] if scl(A) U whenever A U and U is fgsopen in (X, ). The complement of fg*sclosed set is called fg*sopen set.
The collection of all fuzzy g closed sets is
denoted by G C(X).
Remark 2.3[7]
Every fuzzy closed set is fuzzy g closed set but not conversely.
Example 2.4
Let X = {a, b} with = {0x, A, 1x} where A is fuzzy set in X defined by A(a)=1,A(b)=0 .Then (X, ) is a fuzzy topological space. Clearly B defined by
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ISSN: 22780181
RACMS2014 Conference Proceedings


FUZZY g INTERIOR AND FUZZY g CLOSURE
Definition 3.1

For any fuzzy subset A of X, fuzzy g int(A) is defined as the union of all fuzzy g open sets contained in A.
i.e., f g int(A) = {G : G A and G is
fuzzy g open}.

For every fuzzy subset A of X, we define the fuzzy g closure of A to be the intersection of all fuzzy g closed sets containing A.
In symbols, f g cl(A) = {F : A F
F G C(X)}.
Definition 3.2
Let (X, ) be a fuzzy topological space. Let G be a fuzzy subset of X. Then G is called an fuzzy g – neighborhood of A (briefly, f g nbhd of A) iff there
exists an f g open set U of X such that A< U < G.
Definition 3.3
A fuzzy topological space (X, ) is called a

T f space if every f closed set in it is fuzzy closed.

T fgspace if every f g closed set in it is fuzzy
closed.
Example 3.4
Let X = {a, b} with = {0x, , 1x} where is fuzzy set
B(a)=0.5,B(b)=1 is fuzzy g
closed set.
Lemma 2.5[5]
closed set but not fuzzy
in X defined by (a)=0.4, (b)=0.5 .Then (X, ) is a fuzzy topological space. Clearly (X, ) is a T f space.
Example 3.5
Let f : (X, ) (Y, ) be a fuzzy function. For fuzzy sets A and B of X and Y respectively, the following statements hold:

ff1(B) B;

f1f(A) A;

f(A) (f(A)); (iv) f1(B) = (f1(B));

if f is injective, then f1(f(A)) = A;

if f is surjective, then f f1(B) = B;

if f is bijective, then f(A) = (f(A)).
Definition 2.6
A fuzzy function f : (X, ) (Y, ) is called

fuzzy closed [5] if the image of every fuzzy closed set of X is a fuzzy closed in Y.

fuzzy open [5] if the image of every fuzzy open set of X is a fuzzy open in Y.

fuzzy continuous [5] if the inverse image of every fuzzy open set in (Y, ) is a fuzzy open set in (X, ).

fuzzy continuous [13] if the inverse image of
every fuzzy closed Set in (Y, ) is a f closed set in (X,
).
Let X = {a, b} with = {0x, , 1x} where is fuzzy set in X defined by (a)=0.5, (b)=0.5 .Then (X, ) is a fuzzy topologicalspace. Clearly (X, ) is a T fg space.
Definition 3.6
A fuzzy map f : (X, ) (Y, ) is called

fuzzy g continuous [8] if the inverse image of every fuzzy closed set in (Y, ) is f g closed in (X, ).

fuzzy g irresolute if the inverse image of every f g closed set in (Y, ) is f g closed in (X, ).

strongly fuzzy g continuous if the inverse image of every f g open set in (Y, ) is fuzzy open in (X, ).

fuzzy gsirresolute if f1(V) is fgsopen in (X, ) for every fgsopen subset V in (Y, ).
Proposition 3.7
If A is fuzzy g open, then fuzzy g int(A) = A. But
the converse is not true.
Proposition 3.8
If A is fuzzy g closed , then fuzzy g cl(A) = A. But the converse is not true.
Special Issue – 2015
International Journal of Engineering Research & Technology (IJERT)
ISSN: 22780181
RACMS2014 Conference Proceedings
Proposition 3.9
For any two fuzzy subsets A and B of (X, ), the following hold:

If A B, then f g cl(A) f g cl(B).

f g cl(A B) f g cl(A) f g cl(B).
Definition 3.10
A fuzzy map f : (X, ) (Y, ) is called:

fgclosed if f(V) is fgclosed in (Y, ) for every fuzzy closed set V of (X, ).

fsgclosed if f(V) is fsgclosed in (Y, ) for every fuzzy closed set V of (X, ).

fgsclosed if f(V) is fgsclosed in (Y, ) for every fuzzy closed set V of (X, ).

fg*sclosed if f(V) is fg*sclosed in (Y, ) for every fuzzy closed set V of (X, ).



FUZZY g CLOSED MAPS We introduce the following definition:
Definition 4.1
A fuzzy map f : (X, ) (Y, ) is said to be fuzzy g closed if the image of every fuzzy closed set in (X, ) is f g closed in (Y, ).
Example 4.2
Let X = Y = {a, b} with = {0x , , 1x } where
(a)= 0.5, (b)= 0 and = {0x , , 1x } where (a) =1,
(b)= 0. Then (X, ) and (Y, ) are fuzzy topological spaces. Let f : (X, ) (Y, ) be the identity fuzzy map. Clearly f is a fuzzy g closed map.
Proposition 4.3
A fuzzy map f : (X, ) (Y, ) is fuzzy g –
closed if and only if fuzzy g cl(f(A)) f(cl(A)) for every fuzzy subset A of (X, ).
Proof
Let A be a fuzzy closed set in (X, ). Then by hypothesis f g cl(f(A)) f(cl(A)) = f(A) and so f g – cl(f(A)) = f(A). Therefore f(A) is f g closed in (Y, ).
Theorem 4.5
A fuzzy map f : (X, ) (Y, ) is fuzzy g –
closed if and only if for each fuzzy subset S of (Y, ) and each fuzzy open set U containing f1(S) there is a fuzzy g –
open set V of (Y, ) such that S V and f1(V) U.
Proof
Suppose f is fuzzy g closed. Let S be a fuzzy
subset of Y and U be a fuzzy open set of (X, ) such that f 1(S) U. Then V = (f(Uc))c is a fuzzy g open set
containing S such that f1(V) U.
For the converse, let F be a fuzzy closed set of (X,
). Then f1((f(F))c) Fc and Fc is fuzzy open. By assumption, there exists a fuzzy g open set V in (Y, )
such that (f(F))c V and f1(V) Fc and so F (f1(V))c. Hence Vc f(F) f((f1(V))c) Vc which implies f(F) = Vc. Since Vc is fuzzy g closed, f(F) is f g closed and therefore f is f g closed.
Proposition 4.6
If f : (X, ) (Y, ) is fuzzy gsirresolute fuzzy g closed and A is a fuzzy g closed subset of (X, ), then f(A) is fuzzy g closed in (Y, ).
Proof
Let U be a fgsopen set in (Y, ) such that f(A)
U. Since f is fgsirresolute, f1(U) is a fgsopen set containing A. Hence cl(A) f1(U) as A is fuzzy g closed in (X, ). Since f is fuzzy g closed, f(cl(A)) is an
fuzzy g closed set contained in the fgsopen set U, which implies that cl(f(cl(A))) U and hence cl(f(A)) U.
Proof
Suppose that f is fuzzy g
closed and A is a
Therefore, f(A) is a fuzzy g closed set in (Y, ).
fuzzy subset of X. Then cl(A) is fuzzy closed in X and so f(cl(A)) is fuzzy g closed in (Y, ). We have f(A) f(cl(A)) and by Propositions 3.8 and 3.9, f g cl(f(A))
f g cl(f(cl(A))) = f(cl(A)).
Conversely, let A be any fuzzy closed set in (X,
). Then A = cl(A) and so f(A) = f(cl(A)) f g cl(f(A)), by hypothesis. We have f(A) f g cl(f(A)) by Proposition 3.8. Therefore f(A) = f g cl(f(A)). i.e., f(A) is f g closed by Proposition 3.8 and hence f is f g – closed.
Proposition 4.4
Let f : (X, ) (Y, ) be a fuzzy map such that f g cl(f(A)) f(cl(A)) for any fuzzy closed subset A of

Then the image f(A) is f g closed in (Y, ).
The following example shows that the composition of two fuzzy g closed maps need not be fuzzy g closed.
Example 4.7
Let X = Y = Z = {a, b} with = {0x , , 1x } where
(a)= 0.4, (b)= 0 , = {0x , , 1x } where (a) =1, (b)= 0 and = {0x , , , 1x } where (a) = 0.5, (b) = 0. Then (X, ), (Y, ) and (Z, ) are fuzzy topological spaces. Let f : (X, ) (Y, ) be the identity map and g : (Y, ) (Z, ) be the identity fuzzy map. Clearly both f and g are fuzzy g closed maps but their composition g f : (X, )
(Z, ) is not a fuzzy g closed map.
Corollary 4.8
Let f : (X, ) (Y, ) be fuzzy g closed and g
: (Y, ) (Z, ) be fuzzy g closed and fuzzy gs
irresolute, then their composition g f : (X, ) (Z, ) is fuzzy g closed.
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ISSN: 22780181
RACMS2014 Conference Proceedings
Proof
Let A be a fuzzy closed set of (X, ). Then by
Definition 4.14
A fuzzy map f : (X, ) (Y, ) is said to be a
hypothesis f(A) is a fuzzy g closed set in (Y, ). Since g is both fuzzy g closed and fgsfuzzy irresolute by Proposition 4.6, g(f(A)) = (g f) (A) is fuzzy g closed in (Z, ) and therefore g f is fuzzy g closed.
Proposition 4.9
Let f : (X, ) (Y, ), g : (Y, ) (Z, ) be fuzzy g closed maps and (Y, ) be a T fg space.
Then their composition g f : (X, ) (Z, ) is fuzzy g closed.
Proof
Let A be a fuzzy closed set of (X, ). Then by assumption f(A) is fuzzy g closed in (Y, ). Since (Y, ) is a T fgspace, f(A) is fuzzy closed in (Y, ) and again
fuzzy g open map if the image f(A) is fuzzy g open in (Y, ) for each fuzzy open set A in (X, ).
Proposition 4.15
For any fuzzy bijection f : (X, ) (Y, ), the following statements are equivalent:

f1: (Y, ) (X, ) is fuzzy g continuous.

f is fuzzy g open map.

f is fuzzy g closed map.
Proof

(ii). Let U be an fuzzy open set of (X, ). By assumption, (f1)1(U) = f(U) is fuzzy g open in (Y, ) and so f is fuzzy g open.

(iii). Let F be a fuzzy closed set of (X, ). Then Fc is fuzzy open set in (X, ). By assumption, f(Fc) is
by assumption g(f(A)) is fuzzy g closed in (Z, ). i.e., (g
f) (A) is fuzzy g closed in (Z, ) and so g f is fuzzy g closed.
fuzzy g open in (Y, ). That is f(Fc) = (f(F))c is
fuzzy g open in (Y, ) and therefore f(F) is fuzzy g – closed in (Y, ). Hence f is fuzzy g closed.
Proposition 4.10
If f : (X, ) (Y, ) is fuzzy g
closed, g : (Y,

(i). Let F be a fuzzy closed set of (X, ). By assumption, f(F) is fuzzy g closed in (Y, ). But f(F)


) (Z, ) is fuzzy g closed (resp. fgclosed, fg*s
closed, fsgclosed and fgsclosed) and (Y, ) is a T fg
space, thn their composition g f : (X, ) (Z, ) is fuzzy g closed (resp. fgclosed, fg*sclosed, fsgclosed
and fgsclosed).
Proof
Similar to Proposition 4.9.
Proposition 4.11
Let f : (X, ) (Y, ) be a fuzzy closed map and g : (Y, ) (Z, ) be an fuzzy g closed map, then their composition g f : (X, ) (Z, ) is fuzzy g closed.
Proof
Similar to Proposition 4.9.
= (f1)1(F) and therefore f1 is fuzzy g continuous.
Theorem 4.16
A fuzzy map f : (X, ) (Y, ) is f g open if and only if for any fuzzy subset S of (Y, ) and for any fuzzy closed set F containing f1(S), there exists an
f g closed set K of (Y, ) containing S such that f1(K)
F.
Proof
Similar to Theorem 4.5.
Corollary 4.17
A fuzzy map f : (X, ) (Y, ) is fuzzy g open
if and only if f1( g cl(B)) cl(f1(B)) for each fuzzy subset B of (Y, ).
Remark 4.12
If f : (X, ) (Y, ) is a fuzzy g
closed and g
Proof
Suppose that f is fuzzy g open. Then for any
: (Y, ) (Z, ) is fuzzy closed, then their composition need not be a fuzzy g closed map as seen from the
following example.
Example 4.13
Let X = Y = Z = {a, b} with = {0x , , 1x } where
(a)= 0.4, (b)= 0 , = {0x , , 1x } where (a) =1, (b)= 0 and = {0x , , , 1x } where (a) = 0.5, (b) = 0. Then (X, ), (Y, ) and (Z, ) are fuzzy topological spaces. Let f : (X, ) (Y, ) be the identity fuzzy map and g : (Y,
) (Z, ) be the identity fuzzy map. Clearly both f is a fuzzy g closed map and g is fuzzy closed map but their
fuzzy subset B of Y, f1(B) cl(f1(B)). By Theorem 4.16, there exists a fuzzy g closed set K of (Y, ) such that B K and f1(K) cl(f1(B)). Therefore, f1( g –
cl(B)) (f1(K)) cl(f1(B)), since K is a fuzzy g – closed set in (Y, ).
Conversely, let S be any fuzzy subset of (Y, )
and F be any fuzzy closed set containing f1(S). Put K =
g cl(S). Then K is a fuzzy g closed set and S K. By assumption, f1(K) =
f1( g cl(S)) cl(f1(S)) F and therefore by Theorem
composition
g f : (X, ) (Z, ) is not an fuzzy g closed map.

, f is fuzzy g open.
Finally in this section, we define another new class
Analogous to a fuzzy g closed map, we define a fuzzy g open map as follows:
of fuzzy maps called f g *closed maps which are stronger than f g closed maps.
Special Issue – 2015
Definition 4.18
A fuzzy map f : (X, ) (Y, ) is said to be f g *closed if the image f(A) is fuzzy g closed in (Y,
) for every fuzzy g closed set A in (X, ).
For example the fuzzy map f in Example 4.2 is an f g *closed map.
Remark 4.19
Since every fuzzy closed set is a fuzzy g closed set we have f g *closed map is a f g closed map. The converse is not true in general as seen from the following example.
Example 4.20
Let X = Y = {a, b} with = {0x , , 1x } where
(a) =1, (b)= 0 and = {0x , , , 1x} where (a) =0.5,
(b)= 0 . Then (X, ) and (Y, ) are fuzzy topological spaces. Let f : (X, ) (Y, ) be the identity fuzzy map. Then f is a f g closed but it is not f g *closed map.
Proposition 4.21
A fuzzy map f : (X, ) (Y, ) is f g *open if and only if f g cl(f(A)) f( g cl(A)) for every fuzzy subset A of (X, ).
Proof
Similar to Proposition 4.3.
Analogous to f g *closed map we can also define f g *open map.
Proposition 4.22
For any fuzzy bijection f : (X, ) (Y, ), the following statements are equivalent:

f1 : (Y, ) (X, ) is fuzzy g irresolute.

f is f g *open map.

f is f g *closed map.
Proof
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Similar to Proposition 4.15.
Proposition 4.23
If f : (X, ) (Y, ) is fgsirresolute and f g –
closed, then it is a f g *closed map.
Proof
The Proof follows from Proposition 4.6.