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International Journal of Mathematical Archive-4(8), 2013, 187-192
Available online through www.ijma.info ISSN 2229 – 5046
ANOTHER GENERALIZATION OF CLOSED SETS
IN FUZZY TOPOLOGICAL SPACES
1
M. Jeyaraman, 2J. Rajalakshmi* and 3O. Ravi
1
Department of Mathematics, H. H. The Rajah’s College, Pudukkottai,
Pudukkottai District, Tamil Nadu, India
2
Department of Mathematics, St. Michael College of Engg & Tech, Kalaiyarkovil,
Sivagangai District, Tamil Nadu, India
3
Department of Mathematics, P. M. Thevar College, Usilampatti,
Madurai District, Tamil Nadu, India
(Received on: 07-07-13; Revised & Accepted on: 29-07-13)
ABSTRACT
In this paper, we offer a new class of sets called fuzzy g ′′′ -closed sets in fuzzy topological spaces and we study some of
its basic properties. It turns out that this class lies between the class of fuzzy closed sets and the class of fuzzy g-closed
sets.
2010 Mathematics Subject Classification: 54C10, 54C08, 54C05.
Key words and Phrases: Fuzzy Topological space, fuzzy g-closed set, fuzzy g ′′′ -closed set, fuzzy g ′′′ -open set, fuzzy ω closed set.
1. INTRODUCTION
The idea of fuzzy sets and fuzzy set operations were first introduced by L. A. Zadeh in his classical paper [16] in the
year 1965. 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].
Different types of generalizations of fuzzy continuous functions were introduced and studied by various authors in the
recent development of fuzzy topology. The decomposition of fuzzy continuity is one of the many problems in fuzzy
topology. Tong [14] obtained a decomposition of fuzzy continuity by introducing two weak notions of fuzzy continuity
namely, fuzzy strong semi-continuity and fuzzy precontinuity. Rajamani [8] obtained a decomposition of fuzzy
continuity.
In this paper, we introduce a new class of sets namely fuzzy g ′′′ -closed sets in fuzzy topological spaces. This class lies
between the class of fuzzy closed sets and the class of fuzzy g-closed sets. This class also lies between the class of
fuzzy closed sets and the class of fuzzy ω -closed sets.
2. PRELIMINARIES
Throughout this paper (X, τ) and (Y, σ) (or X and Y) represent fuzzy topological spaces on which no separation axioms
are assumed unless otherwise mentioned. For a fuzzy subset A of a space (X, τ), cl(A), int(A) and Ac denote the closure
of A, the interior of A and the complement of A respectively.
Corresponding author: 2J. Rajalakshmi*
Department of Mathematics, St. Michael College of Engg & Tech, Kalaiyarkovil,
Sivagangai District, Tamil Nadu, India
2
International Journal of Mathematical Archive- 4(8), August – 2013
187
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M. Jeyaraman, J. Rajalakshmi* and O. Ravi/ Another Generalization Of Closed Sets In Fuzzy Topological Spaces / IJMA- 4(8),
August-2013.
We recall the following definitions which are useful in the sequel.
Definition 2.1: A subset A of a space (X,τ) is called:
(i) fuzzy semi-open set [1] if A ≤ cl(int(A));
(ii) fuzzy preopen set [4] if A ≤ int(cl(A));
(iii) fuzzy α -open set [4] if A ≤ int(cl(int(A)));
(iv) fuzzy β-open set [13] (= fuzzy semi-preopen [13] ) if A ≤ cl(int(cl(A)));
(v) fuzzy regular open set [1] if A = int(cl(A)).
The complements of the above mentioned fuzzy open sets are called their respective fuzzy closed sets.
The fuzzy semi-closure [15] (resp. fuzzy α -closure [7], fuzzy semi-preclosure [13]) of a fuzzy subset A of X, denoted
by scl(A) (resp. α cl(A), spcl(A)) is defined to be the intersection of all fuzzy semi-closed (resp. fuzzy α -closed, fuzzy
semi-preclosed) sets of (X, τ) containing A. It is known that scl(A) (resp. α cl(A), spcl(A)) is a fuzzy semi-closed
(resp. fuzzy α -closed, fuzzy semi-preclosed) set.
Definition 2.2: A fuzzy subset A of a space (X, τ) is called:
(i) a fuzzy generalized closed (briefly, fuzzy g-closed) set [2] if cl(A) ≤ U whenever A ≤ U and U is fuzzy open in
(X, τ). The complement of fuzzy g-closed set is called fuzzy g-open set;
(ii) a fuzzy semi-generalized closed (briefly fsg-closed) set [3] if scl(A) ≤ U whenever A ≤ U and U is fuzzy semiopen in (X, τ). The complement of fsg-closed set is called fsg-open set;
(iii) a fuzzy generalized semi-closed (briefly fgs-closed) set [10] if scl(A) ≤U whenever A ≤ U and U is fuzzy open in
(X, τ). The complement of fgs-closed set is called fgs-open set;
(iv) a fuzzy α -generalized closed (briefly f α g-closed) set [11] if α cl(A) ≤ U whenever A ≤ U and U is fuzzy open
in (X, τ). The complement of f α g-closed set is called f α g-open set;
(v) a fuzzy generalized semi-preclosed (briefly fgsp-closed) set [9] if spcl(A) ≤ U whenever A ≤ U and U is fuzzy
open in (X, τ). The complement of fgsp-closed set is called fgsp-open set;
(vi) a fuzzy ω -closed set [12] (f ω -closed [12]) if cl(A) ≤ U whenever A ≤ U and U is fuzzy semi-open in (X, τ).
The complement of f ω -closed set is called f ω -open set;
3. FUZZY
g ′′′ -CLOSED SETS
We introduce the following definition.
Definition 3.1: A subset A of X is called a fuzzy g ′′′ -closed set if cl(A) ≤ U whenever A ≤ U and U is fgs-open in
(X, τ).
Definition 3.2: A subset A of X is called a fuzzy αgs -closed set if
semi-open in (X, τ).
α cl(A) ≤ U
whenever A ≤ U and U is fuzzy
Definition 3.3: A subset A of X is called a fuzzy g*s-closed set if scl(A) ≤ U whenever A ≤ U and U is fgs-open in
(X, τ).
Definition 3.4: A subset A of X is called a fuzzy g α′′′ -closed set if
(X, τ).
α cl(A) ≤ U
whenever A ≤ U and U is fgs-open in
Proposition 3.5: Every fuzzy closed set is fuzzy g ′′′ -closed.
Proof: If A is any fuzzy closed set in (X, τ) and G is any fuzzy gs-open set such that A ≤ G, then cl(A) = A ≤ G.
Hence A is fuzzy g ′′′ -closed.
The converse of Proposition 3.5 need not be true as seen from the following example.
Example 3.6: Let X = {a, b} with τ = {0x, A, 1x} where A is fuzzy set in X defined by A(a)=1, A(b)=0. Then (X, τ) is
a fuzzy topological space. Clearly B defined by B (a) =0.5, B (b) =1 is fuzzy g ′′′ -closed set but not fuzzy closed.
Proposition 3.7: Every fuzzy g ′′′ -closed set is fuzzy g α′′′ -closed.
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M. Jeyaraman, J. Rajalakshmi* and O. Ravi/ Another Generalization Of Closed Sets In Fuzzy Topological Spaces / IJMA- 4(8),
August-2013.
Proof: If A is a fuzzy g ′′′ -closed subset of (X, τ) and G is any fgs-open set such that A ≤ G, then
G. Hence A is fuzzy g α′′′ -closed in (X, τ).
α cl(A) ≤ cl(A) ≤
The converse of Proposition 3.7 need not be true as seen from the following example.
Example 3.8: Let X = {a, b} with τ = {0x, 𝜆𝜆, 1x} where 𝜆𝜆 is fuzzy set in X defined by 𝜆𝜆(a)=0.6, 𝜆𝜆(b)=0.5. Then (X, τ) is
a fuzzy topological space. Clearly 𝜇𝜇 defined by 𝜇𝜇(a)=0.4, 𝜇𝜇(b)=0.4 is fuzzy g α′′′ -closed set but not fuzzy g ′′′ -closed set
in (X, τ).
Proposition 3.9: Every fuzzy g ′′′ -closed set is fuzzy g*s-closed.
Proof: If A is a fuzzy g ′′′ -closed subset of (X, τ) and G is any fgs-open set such that A ≤ G, then scl(A) ≤ cl(A) ≤ G.
Hence A is fg*s-closed in (X, τ).
The converse of Proposition 3.9 need not be true as seen from the following example.
Example 3.10: Let X = {a, b} with τ = {0x, 𝛼𝛼, 1x} where 𝛼𝛼 is fuzzy set in X defined by 𝛼𝛼(a)=0.4, 𝛼𝛼(b)=0.5. Then (X, τ)
is a fuzzy topological space. Clearly 𝛼𝛼 is fuzzy g*s-closed but not fuzzy g ′′′ -closed set in (X,τ).
Proposition 3.11: Every fuzzy g ′′′ -closed set is fuzzy ω -closed.
Proof: Suppose that A ≤ G and G is fuzzy semi-open in (X,τ). Since every fuzzy semi-open set is fgs-open and A is
fuzzy g ′′′ -closed, therefore cl(A) ≤ G. Hence A is fuzzy ω -closed in (X,τ).
The converse of Proposition 3.11 need not be true as seen from the following example.
Example 3.12: Let X = {a, b}. Consider the fuzzy topology τ as in Example 3.8, where 𝜇𝜇 is fuzzy set in X defined by
𝜇𝜇(a)=0.4, 𝜇𝜇(b)=0.4. Clearly 𝜇𝜇 is fuzzy ω -closed but not fuzzy g ′′′ -closed set in (X, τ).
Proposition 3.13: Every fuzzy g*s-closed set is fuzzy sg-closed.
Proof: Suppose that A ≤ G and G is fuzzy semi-open in (X, τ). Since every fuzzy semi-open set is fgs-open and A is
fuzzy g*s-closed, therefore scl(A) ≤ G. Hence A is fuzzy sg-closed in (X,τ).
The converse of Proposition 3.13 need not be true as seen from the following example.
Example 3.14: Let X = {a, b}. Consider the fuzzy topology τ as in Example 3.10, where 𝛽𝛽 is fuzzy set in X defined by
𝛽𝛽(a)=0.4, 𝛽𝛽(b)=0.4. Clearly 𝛽𝛽 is fuzzy sg-closed but not fuzzy g*s-closed set in (X, τ).
Proposition 3.15: Every fuzzy ω -closed set is fuzzy αgs -closed.
Proof: If A is a fuzzy ω -closed subset of (X,τ) and G is any fuzzy semi-open such that A ≤ G, then
G. Hence A is f αgs -closed in (X, τ).
α cl(A) ≤ cl(A) ≤
The converse of Proposition 3.15 need not be true as seen from the following example.
Example 3.16: Let X = {a, b}. Consider the fuzzy topology τ as in Example 3.6, where C is fuzzy set in X defined by
C(a)=0, C(b)=0.5. Clearly C is fuzzy αgs -closed but not fuzzy ω -closed set in (X, τ).
Proposition 3.17: Every fuzzy g ′′′ -closed set is fuzzy g-closed.
Proof: If A is a fuzzy g ′′′ -closed subset of (X, τ) and G is any fuzzy open set such that A ≤ G, since every fuzzy open
set is fgs-open, we have cl(A) ≤ G. Hence A is fuzzy g-closed in (X, τ).
The converse of Proposition 3.17 need not be true as seen from the following example.
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M. Jeyaraman, J. Rajalakshmi* and O. Ravi/ Another Generalization Of Closed Sets In Fuzzy Topological Spaces / IJMA- 4(8),
August-2013.
Example 3.18: Let X = {a, b}. Consider the fuzzy topology τ as in Example 3.10, where 𝛾𝛾 is fuzzy set in X defined by
𝛾𝛾(a)=0.5, 𝛾𝛾(b)=0.5. Clearly 𝛾𝛾 is fuzzy g-closed but not fuzzy g ′′′ -closed set in (X, τ).
Proposition 3.19: Every fuzzy g ′′′ -closed set is f αgs -closed.
Proof: If A is a fuzzy g ′′′ -closed subset of (X, τ) and G is any fuzzy semi-open set such that A ≤ G, since every fuzzy
semi-open set is fgs-open, we have
α cl(A) ≤ cl(A) ≤ G. Hence A is f αgs -closed in (X, τ).
The converse of Proposition 3.19 need not be true as seen from the following example.
Example 3.20: Let X = {a, b}. Consider the fuzzy topology τ as in Example 3.6, where C is fuzzy set in X defined by
C(a)=0, C(b)=0.5. Clearly C is fuzzy αgs -closed but not fuzzy g ′′′ -closed set in (X, τ).
Proposition 3.21: Every fuzzy g ′′′ -closed set is fuzzy α g-closed.
Proof: If A is a fuzzy g ′′′ -closed subset of (X, τ) and G is any fuzzy open set such that A ≤ G, since every fuzzy open
set is fgs-open, we have
α cl(A) ≤
cl(A) ≤ G. Hence A is fuzzy α g-closed in (X, τ).
The converse of Proposition 3.21 need not be true as seen from the following example.
Example 3.22: Let X = {a, b}. Consider the fuzzy topology τ as in Example 3.6, where C is fuzzy set in X defined by
C(a)=0, C(b)=0.5. Clearly C is fuzzy α g-closed but not fuzzy g ′′′ -closed set in (X, τ).
Proposition 3.23: Every fuzzy g ′′′ -closed set is fgs-closed.
Proof: If A is a fuzzy g ′′′ -closed subset of (X, τ) and G is any fuzzy open set such that A ≤ G, since every fuzzy open
set is fgs-open, we have scl(A) ≤ cl(A) ≤ G. Hence A is fuzzy gs-closed in (X, τ).
The converse of Proposition 3.23 need not be true as seen from the following example.
Example 3.24: Let X = {a, b}. Consider the fuzzy topology τ as in Example 3.10, where 𝛾𝛾 is fuzzy set in X defined by
𝛾𝛾(a)=0.5, 𝛾𝛾(b)=0.5. Clearly 𝛾𝛾 is fuzzy gs-closed but not fuzzy g ′′′ -closed set in (X, τ).
Proposition 3.25: Every fuzzy g ′′′ -closed set is fuzzy gsp-closed.
Proof: If A is a fuzzy g ′′′ -closed subset of (X, τ) and G is any fuzzy open set such that A ≤ G, every fuzzy open set is
fuzzy gs-open, we have spcl(A) ≤ cl(A) ≤ G. Hence A is fgsp-closed in (X, τ).
The converse of Proposition 3.25 need not be true as seen from the following example.
Example 3.26: Let X = {a, b}. Consider the fuzzy topology τ as in Example 3.10, where 𝛾𝛾 is fuzzy set in X defined by
𝛾𝛾(a)=0.5, 𝛾𝛾(b)=0.5. Clearly 𝛾𝛾 is fuzzy gsp-closed but not fuzzy g ′′′ -closed set in (X,τ).
Remark 3.27: The following examples show that fuzzy g ′′′ -closed sets are independent of fuzzy α -closed sets and
fuzzy semi-closed sets.
Example 3.28: Let X = {a, b}. Consider the fuzzy topology τ as in Example 3.6, where B is fuzzy set in X defined by
B(a)=0.5, B(b)=1. Clearly B is fuzzy g ′′′ -closed but it is neither fuzzy α -closed nor fuzzy semi-closed in (X, τ).
Example 3.29: Let X = {a, b}. Consider the fuzzy topology τ as in Example 3.6, where C is fuzzy set in X defined by
C(a)=0, C(b)=0.5. Clearly C is fuzzy α -closed as well as fuzzy semi-closed in (X, τ) but it is not fuzzy g ′′′ -closed in
(X, τ).
Example 3.30: Let X = {a, b}. Consider the fuzzy topology τ as in Example 3.10. Clearly 𝛼𝛼 is fuzzy g α′′′ -closed but not
fuzzy
α -closed set in (X, τ).
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M. Jeyaraman, J. Rajalakshmi* and O. Ravi/ Another Generalization Of Closed Sets In Fuzzy Topological Spaces / IJMA- 4(8),
August-2013.
Example 3.31: Let X = {a, b} with τ = {0x, A, 1x}, where A is fuzzy set in X defined by A(a)=0.5, A(b)=0.5. Then (X,τ)
is a fuzzy topological space. Clearly B defined by B(a)=0.4, B(b)=0.5 is fuzzy αgs -closed but not fuzzy g α′′′ -closed set
in (X, τ).
Example 3.32: Let X = {a, b} with τ = {0x, R, 1x}, where R is fuzzy set in X defined by R(a)=0.3, R(b)=0.6.Then (X, τ)
is a fuzzy topological space. Clearly S defined by S(a)=0.4, S(b)=0.6 is fuzzy α g-closed but not fuzzy αgs -closed set
in (X, τ).
Example 3.33: Let X = {a, b}. Consider the fuzzy topology τ as in Example 3.8, where 𝜇𝜇 is fuzzy set in X defined by
𝜇𝜇(a)=0.4, 𝜇𝜇(b)=0.4. Clearly 𝜇𝜇 is both fuzzy 𝛼𝛼-closed and fuzzy semi-closed but not fuzzy closed set in (X, τ).
Example 3.34: Let X = {a, b}. Consider the fuzzy topology τ as in Example 3.32, where T is fuzzy set in X defined by
T(a)=0.3, T(b)=0.4. Clearly T is fuzzy 𝛼𝛼g-closed but not fuzzy g-closed set in (X, τ).
Example 3.35: Let X = {a, b}. Consider the fuzzy topology τ as in Example 3.10, where 𝛾𝛾 is fuzzy set in X defined by
𝛾𝛾(a)=0.5, 𝛾𝛾(b)=0.5. Clearly 𝛾𝛾 is fuzzy g-closed but not fuzzy ω -closed set in (X, τ).
Example 3.36: Let X = {a, b}. Consider the fuzzy topology τ as in Example 3.10. Clearly 𝛼𝛼 is fuzzy gs-closed but not
fuzzy g-closed set in (X, τ).
Example 3.37: Let X = {a, b}. Consider the fuzzy topology τ as in Example 3.10, where 𝛾𝛾 is fuzzy set in X defined by
𝛾𝛾(a)=0.5, 𝛾𝛾(b)=0.5. Clearly 𝛾𝛾 is fuzzy g*s-closed but not fuzzy semi-closed set in (X, τ).
Example 3.38: Let X = {a, b}. Consider the fuzzy topology τ as in Example 3.32, where T is fuzzy set in X defined by
T(a)=0.3, T(b)=0.4. Clearly T is fuzzy sg-closed but not fuzzy ω -closed set in (X,τ).
Example 3.39: Let X = {a, b}. Consider the fuzzy topology τ as in Example 3.32, where U is fuzzy set in X defined by
U(a)=0.3, U(b)=0.7. Clearly U is fuzzy gs-closed but not fuzzy sg-closed set in (X, τ).
Remark 3.40: From the above discussions and known results, we obtain the following diagram, where A → B
represents A implies B but not conversely.
f α -closed
f g α′′′ -closed
f αgs -closed
f-closed
f g ′′′ -closed
f ω -closed
f-semi-closed
fg*s-closed
f sg-closed
f α g-closed
fg-closed
fgs-closed
4. PROPERTIES OF FUZZY g ′′′ -CLOSED SETS
In this section, we discuss some basic properties of fuzzy g ′′′ -closed sets.
Theorem 4.1: If A and B are fuzzy g ′′′ -closed sets in (X, τ), then A ∨ B is fuzzy g ′′′ -closed in (X, τ).
Proof: If A ∨ B ≤ G and G is fgs-open, then A ≤ G and B ≤ G. Since A and B are fuzzy g ′′′ -closed, cl(A) ≤ G and
cl(B) ≤ G and hence cl(A ∨ B) = cl(A) ∨ cl(B) ≤ G. Thus A ∨ B is fuzzy g ′′′ -closed set in (X, τ).
Theorem 4.2: If A is fuzzy g ′′′ -closed in (X, τ) and A ≤ B ≤ cl(A), then B is fuzzy g ′′′ -closed in (X, τ).
Proof: Let B ≤ G where G is fgs-open. Since A ≤ B, A ≤ G. Since A is fuzzy g ′′′ -closed, cl(A) ≤ G. Since B ≤ cl(A),
cl(B) ≤ cl(A) ≤ G. Therefore B is fuzzy g ′′′ -closed in X.
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M. Jeyaraman, J. Rajalakshmi* and O. Ravi/ Another Generalization Of Closed Sets In Fuzzy Topological Spaces / IJMA- 4(8),
August-2013.
Theorem 4.3: If A is a fgs-open and fuzzy g ′′′ -closed in (X, τ), then A is fuzzy closed in (X, τ).
Proof: Since A is fgs-open and fuzzy g ′′′ -closed, cl(A) ≤ A and hence A is fuzzy closed in (X, τ).
Theorem 4.4: Let A be a fuzzy g ′′′ -closed set of a topological space (X, τ). If A is fuzzy regular open, then scl(A) is
also fuzzy g ′′′ -closed sets.
Proof: Since A is fuzzy regular open in X, A = int(cl(A)). Then scl(A) = A ∨ int(cl(A)) = A. Thus, scl(A) is fuzzy g ′′′ closed in (X, τ).
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Source of support: Nil, Conflict of interest: None Declared
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