Download Strongly Alpha Generalized Closed Sets in Intuitionistic Fuzzy

Int. J. Pure Appl. Sci. Technol., 3(1) (2011), pp. 50-58
International Journal of Pure and Applied Sciences and Technology
ISSN 2229 - 6107
Available online at www.ijopaasat.in
Research Paper
Strongly Alpha Generalized Closed Sets in Intuitionistic
Fuzzy Topological Spaces
R. Santhi 1 and K. Sakthivel 2,*
1
2
Department of Mathematics, NGM College, Pollachi, Tamilnadu, India
Department of Mathematics, SVS College of Engineering, Coimbatore, Tamilnadu, India.
* Corresponding author, e-mail: ([email protected])
(Received: 14-12-2010; Accepted: 27-01-2011)
Abstract: This paper is devoted to the study of intuitionistic fuzzy topological spaces. In this
paper intuitionistic fuzzy strongly alpha generalized closed sets and intuitionistic fuzzy
strongly alpha generalized open sets are introduced. Some of its properties are discussed
with existing intuitionistic fuzzy closed and intuitionistic fuzzy open sets.
Key words: Intuitionistic fuzzy topology, Intuitionistic fuzzy alpha generalized closed sets,
Intuitionistic fuzzy strongly alpha generalized closed sets, Intuitionistic fuzzy strongly alpha
generalized open sets.
1. Introduction
Zadeh [11] introduced the concept of fuzzy sets and later Atanassov [1] generalized this idea
to the new class of intuitionistic fuzzy sets using the notion of fuzzy sets. On the other hand
Coker [2] introduced intuitionistic fuzzy topological spaces using the notion of intuitionistic
fuzzy sets. This approach provided a wide field for investigation in the area of intuitionistic
fuzzy topology. In intuitionistic fuzzy sets we consider membership and non-membership
values of elements of the topological space.
In this paper, we introduce intuitionistic fuzzy strongly alpha generalized closed sets and
intuitionistic fuzzy strongly alpha generalized open sets. We established their properties and
relationships with other classes of early defined forms.
Int. J. Pure Appl. Sci. Technol., 3(1) (2011), 50-58
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2. Preliminaries
Definition 2.1: [1] Let X be a non empty fixed set. An intuitionistic fuzzy set (IFS in short) A
in X is an object having the form
A = { 〈 x, µA(x), νA(x) 〉 / x ∈ X }
where the functions µA(x): X → [0,1] and νA(x): X → [0,1] denote the degree of membership
(namely µA(x)) and the degree of non-membership (namely νA(x)) of each element x ∈ X to
the set A, respectively, and 0 ≤ µA(x) + νA(x) ≤ 1 for each x ∈ X. Denote by IFS(X), the set of
all intuitionistic fuzzy sets in X.
Definition 2.2: [1] Let A and B be IFSs of the form A = { 〈 x, µA(x), νA(x) 〉 / x ∈ X } and B
= { 〈 x, µB(x), νB(x) 〉 / x ∈ X }. Then
(a) A ⊆ B if and only if µA(x) ≤ µB (x) and νA(x) ≥ νB(x) for all x ∈ X
(b) A = B if and only if A ⊆ B and B ⊆ A
(c) Ac = { 〈 x, νA(x), µA(x) 〉 / x ∈ X }
(d) A ∩ B = { 〈 x, µA(x) ∧ µB(x), νA(x) ∨ νB(x) 〉 / x ∈ X }
(e) A ∪ B = { 〈 x, µA(x) ∨ µB(x), νA(x) ∧ νB(x) 〉 / x ∈ X }
For the sake of simplicity, we shall use the notation A = 〈 x, µA, νA 〉 instead of
〈 x, µA(x), νA(x) 〉 / x ∈ X }.
A={
Also for the sake of simplicity, we shall use the notation A = 〈 x, (µA, µB ), (νA, νB) 〉 instead
of A = 〈 x, (A/µA, B/µB), (A/νA, B/νB) 〉 .
Definition 2.3: [2] An intuitionistic fuzzy topology (IFT in short) on X is a family τ of IFSs in X
satisfying the following axioms.
(i) 0~, 1~ ∈ τ
(ii) G1 ∩ G2 ∈ τ, for any G1, G2 ∈ τ
(iii) ∪ Gi ∈ τ for any family { Gi / i ∈ J } ⊆ τ.
In this case the pair (X, τ) is called an intuitionistic fuzzy topological space (IFTS in short) and
any IFS in τ is known as an intuitionistic fuzzy open set (IFOS in short) in X.
The complement Ac of an IFOS A in an IFTS (X, τ) is called an intuitionistic fuzzy closed set
(IFCS in short) in X.
Definition 2.4: [2] Let (X, τ) be an IFTS and A = 〈 x, µA, νA 〉 be an IFS in X. Then the
intuitionistic fuzzy interior and an intuitionistic fuzzy closure are defined by
int(A) = ∪ { G / G is an IFOS in X and G ⊆ A },
cl(A) = ∩ { K / K is an IFCS in X and A ⊆ K }.
Definition 2.5: [4] An IFS A of an IFTS (X, τ) is an
(i) intuitionistic fuzzy pre closed set (IFPCS in short) if cl(int(A)) ⊆ A,
(ii) intuitionistic fuzzy pre open set (IFPOS in short) if A ⊆ int(cl(A)).
Note that every IFOS in (X, τ) is an IFPOS in X.
Definition 2.6: [4] An IFS A of an IFTS (X, τ) is an intuitionistic fuzzy semi pre closed set
(IFSPCS for short) if there exists an IFPCS B such that int(B) ⊆ A ⊆ B.
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Definition 2.7: [4] An IFS A of an IFTS (X, τ) is an
(i) intuitionistic fuzzy α-open set (IFαOS in short) if A ⊆ int(cl(int(A))),
(ii) intuitionistic fuzzy α-closed set (IFαCS in short) if cl(int(cl(A)) ⊆ A.
Definition 2.8: [5] An IFS A of an IFTS (X, τ) is an
(i) intuitionistic fuzzy γ-open set (IFγOS in short) if A ⊆ int(cl(A)) ∪ cl(int(A)),
(ii) intuitionistic fuzzy γ-closed set (IFγCS in short) if cl(int(A)) ∩ int(cl(A)) ⊆ A.
Definition 2.9: [8] An IFS A of an IFTS (X, τ) is an
(i) intuitionistic fuzzy regular open set (IFROS in short) if A = int(cl(A)),
(ii) intuitionistic fuzzy regular closed set (IFRCS in short) if A = cl(int(A)).
Definition 2.10: [8] An IFS A of an IFTS (X, τ) is an intuitionistic fuzzy generalized closed set
(IFGCS in short) if cl(A) ⊆ U whenever A ⊆ U and U is an IFOS in X.
Note that every IFCS in (X, τ) is an IFGCS in X.
Definition 2.11: [8] An IFS A of an IFTS (X, τ) is an intuitionistic fuzzy generalized open set
(IFGOS in short) if Ac is an IFGCS in X.
Definition 2.12: [7] An IFS A of an IFTS (X, τ) is an intuitionistic fuzzy generalized semi closed
set (IFGSCS in short) if scl(A) ⊆ U whenever A ⊆ U and U is an IFOS in X.
Note that every IFCS in (X, τ) is an IFGSCS in X.
Definition 2.13: [7] An IFS A of an IFTS (X, τ) is an intuitionistic fuzzy generalized semi open set
(IFGSOS in short) if Ac is an IFGSCS in X.
Definition 2.14: [6] An IFS A of an IFTS (X, τ) is an intuitionistic fuzzy generalized pre closed set
(IFGPCS in short) if pcl(A) ⊆ U whenever A ⊆ U and U is an IFOS in X.
Definition 2.15: [6] An IFS A of an IFTS (X, τ) is an intuitionistic fuzzy generalized pre open set
(IFGPOS in short) if Ac is an IFGPCS in X.
Definition 2.16: [9] Let an IFS A of an IFTS (X, τ). Then alpha closure of A (αcl(A) in
short) is defined as αcl(A) = ∩{ K / K is an IFαCS in X and A ⊆ K }.
Definition 2.17: [9] Let an IFS A of an IFTS (X, τ). Then the alpha interior of A (αint(A)) in
short) is defined as αint(A) = ∪{ K / K is an IFαOS in X and K ⊆ A}.
Definition 2.18: [6] An IFS A of an IFTS (X, τ) is an intuitionistic fuzzy alpha generalized closed
set (IFαGCS in short) if αcl(A) ⊆ U whenever A ⊆ U and U is an IFOS in X.
Definition 2.19: [4] For any two IFSs A and B of X, (A q B) if and only if A ⊆ Bc.
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3. Intuitionistic Fuzzy Strongly Alpha Generalized Closed Sets
In this section we introduce intuitionistic fuzzy strongly alpha generalized closed sets
and studied some of its properties.
Definition 3.1: An IFS A in (X, τ) is said to be an intuitionistic fuzzy strongly alpha generalized
closed set (IFsαGCS in short) if αcl(A) ⊆ U whenever A ⊆ U and U is an IFGOS in (X, τ).
The family of all IFsαGCSs of an IFTS (X, τ) is denoted by IFsαGC(X).
Example 3.2: Let X = { a, b } and let τ = {0~, T, 1~} is an IFT on X, where T = 〈 x, (0.2, 0.3),
(0.8, 0.7) 〉. Then the IFS A = 〈 x, (0.8, 0.7), (0.1, 0) 〉 is an IFsαGCS in (X, τ).
Theorem 3.3: Every IFCS is an IFsαGCS but not conversely.
Proof: Let A ⊆ U and U is an IFGOS in (X, τ). Since αcl(A) ⊆ cl(A) and A is an IFCS,
αcl(A) ⊆ cl(A) = A ⊆ U. Therefore A is an IFsαGCS in X.
Example 3.4: Let X = { a, b } and let τ = {0~, T, 1~} where T = 〈 x, (0.4, 0.2), (0.6, 0.7) 〉.
Then the IFS A = 〈 x, (0.6, 0.7), (0.1, 0.1) 〉 is an IFsαGCS but not an IFCS in X.
Theorem 3.5: Every IFαCS is an IFsαGCS but not conversely.
Proof: Let A ⊆ U and U is an IFGOS in (X, τ). By hypothesis αcl(A) = A. Hence αcl(A) ⊆
U. Hence A is an IFsαGCS in X.
Example 3.6: Let X = { a, b } and let τ = {0~, T1, T2, 1~}, where T1 = 〈 x, (0.4, 0.2), (0.6,
0.7) 〉, T2 = 〈 x, (0.8, 0.8), (0.2, 0.2) 〉. Then the IFS A = 〈x, (0.6, 0.7), (0.4, 0.3) 〉 is an
IFsαGCS but not an IFαCS in X.
Theorem 3.7: Every IFRCS is an IFsαGCS but not conversely.
Proof: Let A is an IFRCS in (X, τ). By Definition A = cl(int(A)).This implies
cl(A) = cl(int(A)). Therefore cl(A) = A. That is A is an IFCS in X. By Theorem 3.3, A is an
IFsαGCS in X.
Example 3.8: Let X = { a, b } and let τ = {0~, T, 1~} is an IFT on X, where T = 〈 x, (0.2, 0.4),
(0.8, 0.6) 〉. Then the IFS A = 〈 x, (0.8, 0.6), (0.2, 0.3) 〉 is an IFsαGCS in (X, τ) but not an
IFRCS in X.
Theorem 3.9: Every IFsαGCS is an IFαGCS but not conversely.
Proof: Let A ⊆ U and U is an IFGOS in (X, τ). By hypothesis, αcl(A) ⊆ U whenever A⊆U
and U is an IFGOS in X. Since every IFOS is an IFGOS, αcl(A) ⊆ U whenever A⊆U and U
is an IFOS in X. Hence A is an IFαGCS in X.
Example 3.10: Let X = { a, b } and let τ = {0~, T, 1~} where T = 〈 x, (0.6, 0.7),
(0.4,
0.2) 〉. Then the IFS A = 〈 x, (0.3, 0.2), (0.5, 0.5) 〉 is an IFαGCS but not an IFsαGCS in X.
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Int. J. Pure Appl. Sci. Technol., 3(1) (2011), 50-58
The following diagram implications are true:
IFαCS
IFRCS
IFsα
αGCS
IFCS
IFαGCS
Remark 3.11: An IFP closedness is independent of an IFsαG closedness.
Example 3.12: Let X = { a, b } and T = 〈 x, (0, 0.9), (0.5, 0.1) 〉 and let τ = {0~, T, 1~} is an
IFT on X. the IFS A = 〈 x, (0, 0.3), (0.7, 0.7) 〉 is an IFPCS but not an IFsαGCS in X.
Example 3.13: Let X = { a, b } and let τ = {0~, T1, T2, 1~}, where T1 = 〈 x, (0.5, 0.2), (0.5,
0.7) 〉, T2 = 〈 x, (0.8, 0.8), (0.2, 0.2) 〉. Then the IFS A = 〈x, (0.5, 0.7), (0.5, 0.3) 〉 is an
IFsαGCS but not an IFPCS in X.
Remark 3.14: An IFSPCS is need not be an IFsαGCS in general.
Example 3.15: Let X = { a, b } and T = 〈 x, (0.1, 0.9), (0.6, 0.1) 〉 and let
τ
= {0~, T, 1~} is an IFT on X. the IFS A = 〈 x, (0, 0.4), (0.7, 0.6) 〉 is an IFSPCS but not an
IFsαGCS in X.
Remark 3.16: An IFγCS is need not be an IFsαGCS in general.
Example 3.17: Let X = { a, b } and T = 〈 x, (0.4, 0.6), (0.2, 0.2) 〉 and let
τ
= {0~, T, 1~} is an IFT on X. Then the IFS A = 〈 x, (0.4, 0.3), (0.6, 0.2) 〉 is an IFγCS but not
an IFsαGCS in X.
Remark 3.18: The intersection of any two IFsαGCSs is not an IFsαGCS in general as seen
in the following example.
Example 3.19: Let X = { a, b } be an IFTS and τ = { 0~,T, 1~ } is an IFT on X, where T = 〈
x, (0.5, 0), (0.1, 1) 〉. Then the IFSs A = 〈 x, (0.2,1), (0.7, 0) 〉, B = 〈 x, (0.6, 0), (0.3, 1) 〉 are
IFsαGCSs but A ∩ B is not an IFsαGCS in X.
Theorem 3.20: Let (X, τ) be an IFTS. Then for every A∈IFsαGC(X) and for every IFS B in
X, A ⊆ B ⊆ αcl(A) implies B∈IFsαGC(X).
Proof: Let B ⊆ U and U be an IFGOS. Since A ⊆ B, A ⊆ U. Also since A is an IFsαGCS,
αcl(A) ⊆ U. By hypothesis B ⊆ αcl(A). This implies αcl(B) ⊆ αcl(A) ⊆ U. Therefore αcl(B)
⊆ U. Hence B is an IFsαGCS of X.
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Theorem 3.21: If A is an IFGOS and an IFsαGCS in (X, τ), then A is an IFαCS in X.
Proof: Let A is an IFGOS in X. Since A ⊆ A, by hypothesis αcl(A) ⊆ A. But from the
Definition, A ⊆ αcl(A). Therefore αcl(A) = A. Hence A is an IFαCS of X.
Theorem 3.22: Let (X, τ) be an IFTS. Then A is an IFsαGCS if and only if (A q F) implies
(αcl (A) q F) for every IFGCS F of X.
Proof: Necessity: Let F be an IFGCS and (A q F). Then by Definition, A ⊆ Fc where Fc is
an IFGOS in X. Then αcl(A) ⊆ Fc, by hypothesis. Hence (αcl(A) q F).
Sufficiency: Let F is an IFGCS in X. By hypothesis, (A q F) implies (αcl(A) q F). This
implies αcl(A) ⊆ Fc whenever A ⊆ Fc and Fc is an IFGOS in X. Hence A is an IFsαGCS in
X.
4. Intuitionistic Fuzzy Strongly Alpha Generalized Open Sets
In this section we introduce intuitionistic fuzzy strongly alpha generalized open sets and
studied some of its properties.
Definition 4.1: An IFS A is said to be an intuitionistic fuzzy alpha generalized open set (IFsαGOS
in short) in (X, τ) if the complement Ac is an IFsαGCS in X.
The family of all IFsαGOSs of an IFTS (X, τ) is denoted by IFsαGO(X).
Theorem 4.2: For any IFTS (X, τ), we have the following:
• Every IFOS is an IFsαGOS,
• Every IFαOS is an IFsαGOS,
• Every IFROS is an IFsαGOS,
• Every IFsαGOS is an IFαGOS. But the converses are not true in general.
Proof: Straight forward
Example 4.3: Let X = { a, b } and let τ = {0~, T, 1~} is an IFT on X, where
T = 〈 x, (0.4, 0.2), (0.6, 0.7) 〉. Then the IFS A = 〈 x, (0.1, 0.1), (0.6, 0.7) 〉 is an IFsαGOS in
(X, τ), but not an IFOS in X.
Example 4.4: Let X = { a, b } and let τ = {0~, T1, T2, 1~} is an IFT on X, where T 1 = 〈 x,
(0.4, 0.2), (0.6, 0.7) 〉 and T2 = 〈 x, (0.8, 0.8), (0.2, 0.2) 〉. Then the IFS A = 〈x, (0.5, 0.6), (0.4,
0.4) 〉 is an IFsαGOS but not an IFαOS in X.
Example 4.5: Let X = { a, b } and let τ = {0~, T, 1~} is an IFT on X, where T = 〈 x, (0.2, 0.4),
(0.8, 0.6) 〉. Then the IFS A = 〈 x, (0.2, 0.3), (0.8, 0.6) 〉 is an IFsαGOS but not an IFROS in
X.
Example 4.6: Let X = { a, b } and let τ = {0~, T, 1~} where T = 〈 x, (0.6, 0.7), (0.4, 0.2) 〉.
Then the IFS A = 〈 x, (0.5, 0.5), (0.3, 0.2) 〉 is an IFαGOS but not an IFsαGOS in X.
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Int. J. Pure Appl. Sci. Technol., 3(1) (2011), 50-58
Remark 4.7: The union of any two IFsαGOSs is not an IFsαGOS in general.
Example 4.8: Let X = { a, b } be an IFTS and let T = 〈 x, (0.5, 0), (0.1, 1) 〉. Then τ = { 0~,T,
1~ }is an IFT on X and the IFSs A = 〈 x, (0.7,0), (0.2,1) 〉, B = 〈 x, (0.3, 1), (0.6, 0) 〉 are
IFsαGOSs but A ∪ B is not an IFsαGOS in X.
Theorem 4.9: An IFS A of an IFTS (X, τ) is an IFsαGOS if and only if F ⊆ αint(A)
whenever F is an IFGCS in X and F ⊆ A.
Proof: Necessity: Suppose A is an IFsαGOS in X. Let F be an IFGCS in X and
F ⊆ A. Then Fc is an IFGOS in X such that Ac ⊆ Fc. Since Ac is an IFsαGCS, we have
αcl(Ac ) ⊆ Fc. Hence (αint(A))c ⊆ Fc. Therefore F ⊆ αint(A).
Sufficiency: Let A be an IFS in X and let F ⊆ αint(A) whenever F is an IFGCS in X and F ⊆
A. Then Ac ⊆ Fc and Fc is an IFGOS. By hypothesis, (αint(A))c ⊆ Fc, which implies αcl(Ac)
⊆ Fc. Therefore Ac is an IFsαGCS of X. Hence A is an IFsαGOS in X.
Theorem 4.10: If A is an IFsαGOS in (X, τ), then A is an IFGSOS in X.
Proof: Let A be an IFsαGOS in X. This implies A is an IFαGOS in X. Since every IFαGOS
is an IFGSOS, A is an IFGSOS in X.
5. Applications of Intuitionistic Fuzzy Strongly Alpha Generalized Closed
Sets
Definition 5.1: An IFTS (X, τ) is said to be an intuitionistic fuzzy
space if every IFαGCS in X is an IFsαGCS in X.
sαT1/2
(in short IFsαT1/2)
Definition 5.2: An IFTS (X, τ) is said to be an intuitionistic fuzzy aT1/2 (in short IFaT1/2)
space if every IFGCS in X is an IFCS in X.
Theorem 5.3: if an IFTS X is an IFsαT1/2 space, then every IFαGOS is an IFsαGOS.
Proof: Let A be an IFαGOS in X. This implies Ac is an IFαGCS in X. Since X is an IFsαT1/2
space, Ac is an IFsαGCS in X. Hence A is an IFsαGOS in X.
Theorem 5.4: Let an IFsαT1/2 space (X, τ) be an IFTS. If A is an IFS of X then the following
properties are hold.
(i) A ∈ IFsαGO(X)
(ii) U ⊆ int(cl(int(A))) whenever U ⊆ A and U is IFCS in X
(iii) There exists IFOSs G and G1 such that G1 ⊆ U ⊆ int(cl(G)).
Proof: (i) ⇒ (ii): Let A∈IFsαGO(X). This implies A is an IFαGOS in X. Then Ac is an
IFαGCS in X. Therefore αcl(Ac) ⊆ V whenever Ac ⊆ V and V is an IFOS in X. That is
cl(int(cl(Ac))) ⊆ V. This implies Vc ⊆ int(cl(int(A))) whenever Vc ⊆ A and Vc is IFCS in X.
Replacing Vc by U, U ⊆ int(cl(int(A))) whenever U ⊆ A and U is IFCS in X.
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(ii) ⇒ (iii): Let U ⊆ int(cl(int(A))) whenever U ⊆ A and U is IFCS in X. Hence int(U) ⊆ U ⊆
int(cl(int(A))). Then there exist IFOSs G and G1 in X such that G1 ⊆ U ⊆ int(cl(G)) where G
= int(A) and G1 = int(U).
(iii) ⇒ (i): Suppose that there exists IFOSs G and G1 such that G1 ⊆ U ⊆ int(cl(G)). It is clear
that (int(cl(G)))c ⊆ Uc. That is (int(cl(int(A))))c ⊆ Uc. This implies cl(int(cl(Ac))) ⊆ Uc , Ac ⊆
Uc and Uc is IFOS in X. This implies αcl(Ac) ⊆ Uc . That is Ac is an IFαGCS in X. This
implies A is an IFαGOS in X. Since X an IFsαT1/2 space, A ∈ IFsαGO(X).
Theorem.5.5: Let (X, τ) be an IFaT1/2 space and IFsαT1/2 space. Then IFαO(X) = IFαC(X) if
and only if every IFS in IFaT1/2 space (X, τ) is an IFsαGCS in X.
Proof: Necessity: Suppose that IFαO(X) = IFαC(X). Let A ⊆ U and U is an IFGOS in X.
Since X is an IFaT1/2 space, U is an IFOS in X. This implies αcl(A) ⊆ αcl(U) and U is an
IFαOS in X. Since by hypothesis, U is an IFαCS in X, αcl(U) = U. This implies αcl(A) ⊆ U.
Therefore A is an IFαGCS of X. Since A is an IFsαT1/2 space, A is an IFsαGCS of X.
Sufficiency: Suppose that every IFS is an IFsαGCS in (X, τ). Let U be an IFGOS
in X. Since X is an IFaT1/2 space, U is an IFOS in X. This implies U∈IFαO(X) and by
hypothesis αcl(U) ⊆ U ⊆ αcl(U). Therefore U∈IFαC(X). Hence IFαO(X) ⊆ IFαC(X). Let
A∈IFαC(X) then Ac is an IFαOS in X. But IFαO(X) ⊆ IFαC(X). Therefore A∈IFαO(X).
Hence IFαC(X) ⊆ IFαO(X). Thus IFαO(X) = IFαC(X).
Theorem 5.6: Let (X, τ) be an IFTS and X is an IFaT1/2 space. Then
(i) Any union of IFsαGCSs is an IFsαGCS in X.
(ii) Any intersection of IFsαGOSs is an IFsαGOS in X.
Proof: (i): Let {Ai}i∈J is a collection of IFsαGCSs in an IFaT1/2 space (X, τ). By hypothesis,
every IFGCS is an IFCS in X. But the union of IFCS is an IFCS in X. Therefore {UAi}i∈J is
an IFCS in X. Since every IFCS is IFsαGCS, {UAi}i∈J is an IFsαGCS in X. Hence any union
of IFsαGCSs is an IFsαGCS in X.
(ii): Can be proved by taking complement in (i).
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