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International Journal of Mathematical Archive-4(8), 2013, 67-73
Available online through www.ijma.info ISSN 2229 – 5046
NEW NOTION OF CLOSED SETS IN FUZZY TOPOLOGY
1
M. Jeyaraman, 2S. Vijayalakshmi* and 3O. Ravi
1
Department of Mathematics, H. H. The Rajah’s College, Pudukottai,
Pudukottai District, Tamil Nadu, India
2
Department of Mathematics, St. Michael College of Engg & Tech, Kalaiyarkoil,
Sivaganga 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.
 -closed set, f g -open set, f ω -closed set.
Key words and Phrases: Fuzzy Topological space, fg-closed set, f g
1. INTRODUCTION
The study of fuzzy sets was initiated with the famous paper of Zadeh [21] and thereafter Chang [7] paved the way for
subsequent tremendous growth of the numerous fuzzy topological concepts. The theory of fuzzy topological spaces was
developed by several authors by considering the basic concepts of general topology. The extensions of functions in
fuzzy settings can be carried out using the concepts of quasi coincidences and q-neighborhoods by Pu and Liu [13].
Generalized fuzzy closed sets and regular generalized fuzzy closed sets were defined in [15, 13].
 -closed (briefly f g -closed) sets in fuzzy topological
In this paper, we introduce a new class of sets namely fuzzy g
spaces. This class lies between the class of fuzzy closed sets and the class of fg-closed sets. This class also lies between
the class of closed sets and the class of f 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 subset A of a fuzzy space (X, τ), cl(A), int(A) and Ac denote the closure
of A, the interior of A and the complement of A respectively.
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 [3] (briefly fs-open) if A cl(int(A));
(ii) Fuzzy preopen set [6] (briefly fp-open) if A int(cl(A));
(iii) Fuzzy α -open set [6] (briefly f α -open) if A int(cl(int(A)));
(iv) Fuzzy β-open set [20] (briefly f β -open) if A cl(int(cl(A)));
(v) Fuzzy regular open set [3] if A = int(cl(A)).
Corresponding author: 2S. Vijayalakshmi*
Department of Mathematics, St. Michael College of Engg & Tech, Kalaiyarkoil,
Sivaganga District, Tamil Nadu, India
2
International Journal of Mathematical Archive- 4(8), August – 2013
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M. Jeyaraman, S. Vijayalakshmi* and O. Ravi/ New Notion Of Closed Sets In Fuzzy Topology / IJMA- 4(8), August-2013.
The complements of the above mentioned fuzzy open sets are called their respective closed sets.
The semi closure (resp. α -closure, semi-pre-closure) of a 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 semipreclosed) set. For any fuzzy subset A of an arbitrarily chosen fuzzy topological space, the semi-interior (resp. α interior, preinterior) of A, denoted by sint(A) (resp. α int(A), pint(A)) is defined to be the union of all fuzzy semiopen (resp. fuzzy α -open, fuzzy preopen) sets of (X, τ) contained in A.
Definition 2.2: A subset A of a space (X,τ) is called:
(i) a fuzzy generalized closed (briefly fg-closed) set [4] if cl(A) U whenever A U and U is fuzzy open in (X, τ).
The complement of fg-closed set is called fg-open set;
(ii) a fuzzy semi-generalized closed (briefly fsg-closed) set [5] 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 [17] 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 [15] if α cl(A)
U and U is fuzzy
U whenever A
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 [17] 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 f ω -closed set [19] 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;
 -closed (resp. f ω -closed, fg-closed, fgs-closed, fgsp-closed, f α g-closed, fsgRemark 2.3: The collection of all f g
 C(X) (resp. F ω C(X), F G C(X), F GS C(X), F GSP
closed, f α -closed, fuzzy semi-closed) sets is denoted by F G
C(X), F αg C(X), F SG C(X), F α C(X), F S C(X)).
 -open (resp. f ω -open, fg-open, fgs-open, fgsp-open, f α g-open, fsg-open, f α -open, fuzzy
The collection of all f g
 O(X) (resp. FωO(X), F G O(X), F GS O(X), F GSP O(X), FαGO(X), F SG O(X),
semi-open) sets is denoted by F G
F α O(X), F S O(X)).
We denote the power set of X by P(X).
3. FUZZY
g -CLOSED SETS
We introduce the following definition.
 -closed set (briefly f g -closed set) if cl(A)
Definition 3.1: A subset A of X is called a fuzzy g
and U is fsg-open in (X, τ).
U whenever A
U
 -closed.
Proposition 3.2: Every fuzzy closed set is f g
 -closed.
A = cl(A). Hence A is f g
Proof: If A is any fuzzy closed set in (X,τ) and G is any fsg-open set such that G
The converse of Proposition 3.2 need not be true as seen from the following example.
Example 3.3: Let X = {a, b} and
α
:X
[ 0,1] be defined by 𝛼𝛼 (a) = 𝛼𝛼 (b) =
5
10
and 𝛽𝛽(a) = 𝛽𝛽(b) =
is a fuzzy topological space with τ = {0x, 𝛼𝛼, 𝛽𝛽, 1x}. In (X, τ), the fuzzy subset 𝜆𝜆 = (
but not fuzzy closed in (X, τ).
3
20
,
3
20
6
10
. Then (X, τ)
 -closed set in (X, τ)
) is f g
 -closed set is fgsp-closed.
Proposition 3.4: Every f g
 -closed subset of (X, τ) and G is any fuzzy open set such that G ≥ A, every fuzzy open set is fsgProof: If A is a f g
open, we have G
cl(A)
spcl(A). Hence A is fgsp-closed in (X,τ).
© 2013, IJMA. All Rights Reserved
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M. Jeyaraman, S. Vijayalakshmi* and O. Ravi/ New Notion Of Closed Sets In Fuzzy Topology / IJMA- 4(8), August-2013.
The converse of Proposition 3.4 need not be true as seen from the following example.
Example 3.5: Let X = {a, b} and
α:X
[ 0,1] be defined by 𝛼𝛼 (a) = 𝛼𝛼 (b) =
space with τ = {0x, 𝛼𝛼, 1x}. In (X, τ), the fuzzy subset 𝜆𝜆 = (
 -closed set is f ω -closed.
Proposition 3.6: Every f g
4
10
,
4
10
5
10
. Then (X, τ) is a fuzzy topological
 -closed set in (X, τ).
) is fgsp-closed set but not f g
Proof: Suppose that A G and G is fuzzy semi-open in (X, τ). Since every fuzzy semi-open set is fsg-open and A is f
g -closed, therefore cl(A) G. Hence A is f ω -closed in (X, τ).
The converse of Proposition 3.6 need not be true as seen from the following example.
Example 3.7: Let X = {a, b} and
α:X
[ 0,1] be defined by 𝛼𝛼 (a) = 𝛼𝛼 (b) =
space with τ = {0x, 𝛼𝛼, 1x}. In (X, τ), the fuzzy subset 𝜆𝜆 = (
 -closed set is fg-closed.
Proposition 3.8: Every f g
4
,
10
4
10
5
10
. Then (X, τ) is a fuzzy topological
 -closed set in (X, τ).
) is f ω -closed but not f g
 -closed subset of (X, τ) and G is any fuzzy open set such that G
Proof: If A is a f g
fsg-open, we have G
A, since every fuzzy open set is
cl(A). Hence A is fg-closed in (X, τ).
The converse of Proposition 3.8 need not be true as seen from the following example.
Example 3.9: Let X = {a, b} and
α:X
[ 0,1] be defined by 𝛼𝛼 (a) = 𝛼𝛼 (b) =
space with τ = {0x, 𝛼𝛼, 1x}. In (X, τ), the fuzzy subset 𝜆𝜆 = (
 -closed set is f α g-closed.
Proposition 3.10: Every f g
4
,
10
4
10
5
10
. Then (X, τ) is a fuzzy topological
 -closed set in (X, τ).
) is fg-closed but not f g
 -closed subset of (X, τ) and G is any fuzzy open set such that G
Proof: If A is a f g
fsg-open, we have G
cl(A)
A, since every fuzzy open set is
α cl(A). Hence A is f α g-closed in (X, τ).
The converse of Proposition 3.10 need not be true as seen from the following example.
Example 3.11: Let X = {a, b} and
α :X
[ 0,1] be defined by 𝛼𝛼 (a) = 𝛼𝛼 (b) =
space with τ = {0x, 𝛼𝛼, 1x}. In (X, τ), the fuzzy subset 𝜆𝜆 = (
 -closed set is fgs-closed.
Proposition 3.12: Every f g
4
10
,
4
10
5
10
. Then (X, τ) is a fuzzy topological
 -closed set in (X, τ).
) is f α g-closed set but not f g
 -closed subset of (X, τ) and G is any fuzzy open set such that G
Proof: If A is a f g
fsg-open, we have G
cl(A)
A, since every fuzzy open set is
scl(A). Hence A is fgs-closed set in (X, τ).
The converse of Proposition 3.12 need not be true as seen from the following example.
Example 3.13: Let X = {a, b} and
α :X
[ 0,1] be defined by 𝛼𝛼 (a) = 𝛼𝛼 (b) =
space with τ = {0x, 𝛼𝛼, 1x}. In (X, τ), the fuzzy subset 𝜆𝜆 = (
4
10
,
4
10
5
10
. Then (X, τ) is a fuzzy topological
 -closed set in (X, τ).
) is fgs-closed set but not f g
α -closed set (briefly f gα -closed set) if
Definition 3.14: A subset A of X is called a fuzzy g
α cl(A)
U whenever A
α -closed set is called f gα -open set.
U and U is fsg-open in (X, τ). The complement of f g
 -closed set is f gα -closed.
Proposition 3.15: Every f g
 -closed subset of (X, τ) and G is any fsg-open set such that G
Proof: If A is a f g
A, then G
cl(A)
α cl(A).
α -closed in (X,τ).
Hence A is f g
© 2013, IJMA. All Rights Reserved
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M. Jeyaraman, S. Vijayalakshmi* and O. Ravi/ New Notion Of Closed Sets In Fuzzy Topology / IJMA- 4(8), August-2013.
The converse of Proposition 3.15 need not be true as seen from the following example.
Example 3.16: Let X = {a, b} and
α:
X
[ 0,1] be defined by 𝛼𝛼 (a) = 1, 𝛼𝛼 (b) =
topological space with τ = {0x, 𝛼𝛼,1x}. In (X, τ), the fuzzy subset 𝜆𝜆 = (0,
closed in (X, τ).
5
10
. Then (X, τ) is a fuzzy
α -closed set in (X, τ) but not f g ) is f g
α -closed.
Proposition 3.17: Every f𝛼𝛼-closed set is f g
A, then G
Proof: If G is any fsg open set such that G
cl(A)
α cl(A).
α -closed.
Hence f g
The converse of Proposition 3.17 need not be true as seen from the following example
Example 3.18: Let X = {a, b} and
α:
X
[ 0,1] be defined by 𝛼𝛼 (a) = 1, 𝛼𝛼 (b) =
topological space with τ = {0x, 𝛼𝛼, 1x}. In (X, τ), the fuzzy subset 𝜆𝜆 = (0,
(X, τ).
5
10
. Then (X, τ) is a fuzzy
α -closed but not f𝛼𝛼 -closed set in
) is f g
Definition 3.19: A subset A of X is called fuzzy ψ -closed set (briefly fψ -closed set) if scl(A)
U whenever A
U
and U is fsg-open in (X, τ). The complement of fψ -closed set is called fψ -open set.
 -closed set is fψ -closed.
Proposition 3.20: Every f g
 -closed subset of (X, τ) and G is any fsg-open set such that G
Proof: If A is a f g
A, then G
A is fψ -closed in (X, τ).
cl(A)
scl(A). Hence
The converse of Proposition 3.20 need not be true as seen from the following example.
[ 0,1] be defined by 𝛼𝛼 (a) = 1, 𝛼𝛼 (b) = . Then (X, τ) is a fuzzy
Example 3.21: Let X = {a, b} and α : X
5
 -closed set in
topological space with τ = {0x, 𝛼𝛼, 1x}. In (X, τ), the fuzzy subset 𝜆𝜆 = (0, ) is fψ -closed set but not f g
10
(X, τ).
Proposition 3.22: Every fψ -closed set is fsg-closed.
Proof: Suppose that A G and G is fuzzy semi-open in (X, τ). Since every fuzzy semi-open set is fsg-open and A is f
ψ -closed, therefore scl(A) ≤ G. Hence A is fsg-closed in (X, τ).
The converse of Proposition 3.22 need not be true as seen from the following example.
α :X
5
. Then (X, τ) is a fuzzy topological
Proof: If A is fs-closed set in (X, τ) and G is any fuzzy semi-open set such that G
set is fsg-open, we have G A = scl(A). Hence A is fψ -closed in (X, τ).
A, since every fuzzy semi-open
Example 3.22: Let X = {a, b} and
[ 0,1] be defined by 𝛼𝛼 (a) = 𝛼𝛼(b) =
space with τ = {0x, 𝛼𝛼, 1x}. In (X, τ), the fuzzy subset 𝛽𝛽= (
Proposition 3.23: Every fs-closed set is fψ -closed.
4
10
,
4
10
10
) is fsg-closed set but not fψ -closed set in (X, τ).
The converse of Proposition 3.23 need not be true as seen from the following example.
[ 0,1] be defined by 𝛼𝛼(a) = 1, 𝛼𝛼(b) = . Then (X, τ) is a fuzzy topological
Example 3.24: Let X = {a, b} and α : X
5
space with τ = {0x, 𝛼𝛼, 1x}. In (X, τ), the fuzzy subset 𝜆𝜆 = (0, ) is fψ -closed set but not fs-closed set in (X, τ).
10
α cl(A) U
open in (X, τ). The complement of f αgs -closed set is called f αgs -open set;
Definition 3.25: A subset A of X is called a f αgs -closed set if
whenever A
U and U is fuzzy semi-
Proposition 3.26: Every f ω -closed set is f αgs -closed.
© 2013, IJMA. All Rights Reserved
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M. Jeyaraman, S. Vijayalakshmi* and O. Ravi/ New Notion Of Closed Sets In Fuzzy Topology / IJMA- 4(8), August-2013.
Proof: If A is a f ω -closed subset of (X, τ) and G is any fuzzy semi-open set such that G
cl(A). Hence A is f αgs -closed in (X, τ).
A, then G ≥ cl(A)
α
The converse of Proposition 3.26 need not be true as seen from the following example.
[ 0,1] be defined by 𝛼𝛼 (a) = 1,
(b) = . Then (X, τ) is a fuzzy
Example 3.27: Let X = {a, b} and α : X
5
topological space with τ = {0x, 𝛼𝛼, 1x}. In (X, τ), the fuzzy subset β = (0, ) is f αgs -closed but not f ω -closed set in
10
(X, τ).
 -closed set is f αgs -closed.
Proposition 3.28: Every f g
 -closed subset of (X, τ) and G is any fuzzy semi-open set such that G
Proof: If A is a f g
open set is fsg-open, we have G
cl(A)
A, since every fuzzy semi-
α cl(A). Hence A is f αgs -closed in (X, τ).
The converse of Proposition 3.28 need not be true as seen from the following example.
Example 3.29: Let X = {a, b} and
α:X
[ 0,1] be defined by 𝛼𝛼 (a) =𝛼𝛼 (b) =
5
. Then (X, τ) is a fuzzy topological
10
 -closed set in (X, τ).
space with τ = {0x, 𝛼𝛼, 1x}. In (X, τ), the fuzzy subset β = ( , ) is f αgs -closed but not f g
4
4
10
10
Proposition 3.30: Every f αgs -closed set is f α g-closed.
G and G is fuzzy semi-open such that G
Proof: Suppose that A
we have G cl(A) 𝛼𝛼cl(A). Hence A is f α g-closed in (X, τ).
A. Since every fuzzy semi-open set is fsg-open,
The converse of Proposition 3.30 need not be true as seen from the following example.
Example 3.31: Let X = {a, b} and
α:X
[ 0,1] be defined by 𝛼𝛼 (a) =𝛼𝛼 (b) =
5
. Then (X, τ) is a fuzzy topological
10
space with τ = {0x, 𝛼𝛼, 1x}. In (X, τ), the fuzzy subset β = ( , ) is f α g -closed set but not f αgs -closed set in (X, τ).
4
4
10
10
Proposition 3.32: Every f ω -closed set is fg-closed.
Proof: Suppose that A G and G is fuzzy open set such that G
we have G cl(A). Hence A is fg-closed in (X, τ).
A. Since every fuzzy open set is fuzzy semi open set,
The converse of Proposition 3.32 need not be true as seen from the following example.
Example 3.33: Let X = {a, b} and
α:X
[ 0,1] be defined by 𝛼𝛼 (a) =𝛼𝛼 (b) =
5
. Then (X, τ) is a fuzzy topological
10
space with τ = {0x, 𝛼𝛼, 1x}. In (X, τ), the fuzzy subset β = ( , ) is fg-closed but not f ω -closed set in (X, τ).
4
4
10
10
 -closed sets are independent of f α -closed sets and fuzzy semiRemark 3.34: The following examples show that f g
closed sets.
Example 3.35: Let X = {a, b} and
α:X
[ 0,1] be defined by 𝛼𝛼 (a) = 𝛼𝛼 (b) =
5
6
10
and β (a) =β (b) = . Then (X, τ)
10
 -closed but it is neither
is a fuzzy topological space with τ = {0x, 𝛼𝛼, β , 1x}. In (X, τ), the fuzzy subset λ = ( , ) is f g
f α -closed nor fuzzy semi-closed in (X, τ).
9
9
20
20
Example 3.36: Let X = {a, b} and α : X
[ 0,1] be defined by 𝛼𝛼 (a) = 1,
(b) = . Then (X, τ) is a fuzzy
5
topological space with τ = {0x, 𝛼𝛼, 1x}. In (X, τ) the fuzzy subset β = (0, ) is f α -closed as well as fuzzy semi-closed
 -closed in (X, τ).
in (X, τ) but it is not f g
Remark 3.37: we obtain the following diagram, where A → B (resp. A
conversely (resp. A and B are independent of each other).
© 2013, IJMA. All Rights Reserved
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B) represents A implies B but not
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M. Jeyaraman, S. Vijayalakshmi* and O. Ravi/ New Notion Of Closed Sets In Fuzzy Topology / IJMA- 4(8), August-2013.
f α -closed
α -closed
fg
f αgs -closed
f-closed
 -closed
fg
f ω -closed
fψ -closed
fsg-closed
f-semi-closed
f α g-closed
fg-closed
fgs-closed
None of the above implications is reversible as shown in the remaining examples and in the related papers [17, 20, 21].
 -CLOSED SETS
4. PROPERTIES OF f g
 -closed sets.
In this section, we discuss some basic properties of f g
 -closed sets in (X, τ), then A
Theorem 4.1: If A and B are f g
Proof: If A
B
G and G is fsg-open, then A
cl(B) and hence G
cl(A)
cl(B) = cl(A B). Thus A
 -closed in (X, τ) and A
Theorem 4.2: If A is f g
Proof: Let B
Since B
G and B
B
B is
 -closed, G
G. Since A and B are f g
cl(A) and G
 -closed set in (X, τ).
fg
 -closed in (X, τ).
cl(A), then B is f g
U where U is fsg-open in (X, τ). Since A
cl(A), cl(B)
 -closed in (X, τ).
B is f g
B, A
 -closed in (X, τ), cl(A)
U. Since A is f g
U.
 -closed in (X, τ).
cl(A) ≤ U. Therefore B is f g
 -closed in (X, τ), then A is fuzzy closed in (X, τ).
Proposition 4.3: If A is a fsg-open and f g
 -closed, cl(A)
Proof: Since A is fsg-open and f g
A and hence A is fuzzy closed in (X, τ).
 -closed set of a fuzzy topological space (X, τ)
Theorem 4.4: Let A be a f g
(i)
 -closed set.
If A is fuzzy regular open, then scl(A) is f g
 -closed set.
(ii) If A is fuzzy regular closed, then sint(A) is also f g
Proof:
 -closed in
(i) Since A is fuzzy regular open in X, A = int(cl(A)). Then scl(A) = A ∨ int(cl(A)) = A. Thus, scl(A) is f g
(X, τ).
 (ii) Since A is fuzzy regular closed in X, A = cl(int(A)). Then sint(A) = A cl(int(A)) = A. Thus, sint(A) is f g
closed in (X, τ).
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Source of support: Nil, Conflict of interest: None Declared
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