Some properties of gi closed sets in topological space.docx

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Vol.4, No.9, 2014
Some properties of g*i- Closed Sets in Topological Space
Dr. Haji M. Hasan
University of Duhok, Kurdistan-Region, Iraq
E-mail: dr.hajihasan@uod.ac
Dllveen Abdall
University of Mousl , Mousl, Iraq
E-mail: dllveen_abd@yahoo.com
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Abstract
In this paper we introduce a new class of sets called generalized *i-closed sets in
topological spaces (briefly g*i-closed set). Also we study some of its basic properties and
investigate the relations between the associated topology.
Keywords: g-closed, i-open, gi-closed.
1-Introduction
In 1970 Levine [6], first considered the concept of generalized closed (briefly, g-closed)
sets were defined and investigated. Arya and Nour [1], defined generalized semi open sets
[briefly, gs-open] using semi open sets. Maki Devi and Balachandram [2, 3]. On generalized
α-closed maps and semi-generalized homeorphisems. Dontchev and Maki, in 1999 [4, 5],
introduced the concept of (δ-generalized, θ-generalized) respectively. Mohammed and
Askander [7], in 2011, introduced the concept of i-open sets. Mohammed and Jardo [8], in
2012, introduced the concept of generalized i-closed sets.
We introduced a new class of sets called g*i-closed sets and study some properties.
2- Preliminaries
Throughout this paper (X, τ) or simply X represent nonempty topological spaces on which
no separation axioms are assumed, unless otherwise mentioned. For a subset A of (X, τ), cl(A)
and int(A), represent the closure of A and the interior of A respectively. A subset A of a space
(X, τ) is called semi-open [2] (resp; α-open[11], b- open[9]), if A  cl(int(A)); (resp, A 
int(cl(int(A))), A  cl(int(A))  int(cl(A)) ). The family of all semi-open (resp; α- open, b-open)
sets of (X, τ) denoted by SO(X) (resp; αO(X), BO(X)). The complement of a semi-open
(resp; α-open, b-open) set is said to be semi-closed (resp; α-closed, b-closed). The semi
closure, α-closure, b-closure of A are similarly defined and are denoted by Cls(A), Clα(A),
Clb(A). And a subset A of (X, τ) is called( δ-open, θ-open) set [10], if A=clδ(A) where clδ(A)
= {x X : int(cl(U))∩ A ≠ Ø,U τ and x U}, A=clθ(A) where clθ(A) = {x X :

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int(cl(U))∩ A ≠ Ø, U τ and x U} respectively, and the family of all ( δ-open, θ-open) sets
of (X, τ) denoted by δO(X), θO(X) respectively, and the complement of (δ-open, θ-open) sets
is called ( δ-closed, θ-closed) sets the family of all (δ-closed, θ-closed) sets of (X, τ) is
denoted by δC(X), θC(X) respectively. And a subset A of (X, τ) is called an i-open set [7], if A
cl(A∩G), if there exists an open set G whenever (G≠X,Ø), and the complement of i-open
sets is called i-closed sets. The family of all i-closed sets of (X, τ) is denoted by IC(X), and the
family of all i-open sets of (X, τ) is denoted by IO(X). If A is a subset of a space (X, τ), then
the i-closure of A, denoted by cli(A) is the smallest an i-closed set containing A. The i-interior
of A denoted by inti(A) is the larges an i- open set contained in A.
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Some definitions used throughout this paper.
Definition 2.1:
For any subset A of topological spaces (X, τ) we have
1- Generalized closed (briefly g-closed) [6], if cl(A) U whenever A U and U is open
in X, the complement of g-closed is called g-open.
2- Generalized α –closed (briefly gα-closed) [2], if clα(A) U whenever A U and U
is α-open in(X, τ), the complement of gα -closed is called gα-open.
3- α -Generalized closed (briefly αg-closed) [2], if clα(A) U whenever A U and U is
open in(X, τ), the complement of αg -closed is called αg-open.
4- Generalized b-closed (briefly gb-closed) [9], if clb(A) U whenever A U and U is
open in (X, τ), the complement of gb -closed is called gb-open.
5- Generalized i-closed (briefly gi-closed) [8], if cli(A) U whenever A U and U is
open in (X, τ), the complement of gi -closed is called gi-open.
6- Generalized semi -closed (briefly gs-closed) [1], if cls(A) U whenever A U and
U is open in (X, τ), the complement of gs -closed is called gs-open.
7- Generalized θ-closed (briefly gθ-closed) [5], if clθ(A) U whenever A U and U is
open in(X, τ), the complement of gθ -closed is called gθ-open.
8- Generalized δ-closed (briefly gδ-closed) [4], if clδ(A) U whenever A U and U is
open in(X, τ), the complement of gδ -closed is called gδ-open.
9- Generalized g*closed (briefly g*-closed) [11], if cl(A) U whenever A U and U
is g-open in (X, τ), the complement of g* -closed is called g*-open.
Theorem 2.3[6]: Every open set is g-open set.
Theorem 2.4[6]: Every closed set is g-closed set.
Theorem 2.5[3]: Every semi-closed set is gs-closed set.
3- properties of g*i-closed sets in topological spaces
In this section, we introduce a new class of closed set called g*i-closed set and study
some of their properties.

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Definition 3.1: A subset A of topological spaces (X, τ) is called a g*i-closed set if cli(A) U
whenever A U, U is g-open in (X ,τ), and (G≠X,Ø), the set of all family g*i-closed denoted
by g*i C(X).
Theorem 3.2: Every closed set in a space X is g*i-closed, but converse need not be true in
general.
Proof: Let A be a closed set in (X ,τ) such that A U , where U is g-open .Since A is closed
,that is cl(A) =A , since cli (A) cl(A) = A , and A U therefore cli (A) U . Hence A is
g*i-closed set in (X, τ).
The converse of the above theorem is not true in general as shown from the following
example.
Example 3.3: Consider the topological spaces X= {a, b, c} with the topology
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τ = {Ø, {a}, {a, b}, X}
C(X) = {Ø, {b, c}, {c}, X}
GC(X) = {Ø, {c}, {b, c}, {a, c}, X}
GO(X) = {Ø, {a}, {b}, {a, b}, X}
IO(X) = {Ø, {a}, {b}, {a, c}, {a, b}, {a, c}, {b, c}, X}
IC(X) = {Ø, {a}, {b}, {c}, {a, c}, {b, c}, X}
g*iC(X) = {Ø, {a}, {b}, {c}, {a, c}, {b, c}, X}
Let A = {a, c}, here A is a g*i-closed set but not a closed set.
Theorem 3.4: Every i-closed in X is g*i-closed set
Proof: Let A be i-closed in X such that A U, where U is g-open. Since A is an i-closed set,
then cli(A) =A ,and A U, therefore cli(A) U . Hence A is g*i-closed set in X.
The converse of the above theorem is not true in general as shown in the following example.
Example 3.5: Let X = {a, b, c}, with the topology τ = {Ø, {b}, {a, b}, {b, c}, X}
C(X) = {Ø, {a}, {c}, {a, c}, X}
GC(X) = {Ø, {a}, {c}, {a, c}, X}
GO(X) = {Ø, {a, b}, {b, c}, {b}, X}
IO(X) = {Ø, {a}, {b}, {c}, {a, b}, {b, c}, X}
IC(X) = {Ø, {a}, {c}, {a, b}, {a, c}, {b, c}, X}

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g*iC(X) = {Ø, {a}, {b}, {c}, {a, b}, {a, c}, {b, c}, X}
Let A = {b}. So A is g*i-closed set but not i-closed in (X, τ)
Theorem 3.6: Every g*i-closed set in topological spaces (X, τ) is gi-closed set.
Proof: let A be a g*i-closed set in X such that A U, where U is open. Since every open set is
g-open by Theorem (2.3), and A is g*i-closed, cli(A) U . Hence A is gi-closed.
example.
Example 3.7: let X = {a, b, c}, τ = {Ø, {a}, X}
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C(X) = {Ø, {b, c}, X}.
IO(X) = {Ø, {a}, {a, b}, {a, c}, X}
IC(X) = {Ø, {b}, {c}, {b, c}, X}
GC(X) = {Ø, {b}, {c}, {a, b}, {a, c}, {b, c}, X}
GO(X) = {Ø, {a}, {c}, {b}, {a, b}, {a, c}, X}
g*iC(X) = {Ø, {b}, {c}, {b, c}, X}
giC(X) = {Ø, {b}, {c}, {a, c}, {b, c}, {a, b}, X}
Let A = {a, b}, then A is gi-closed set but not g*i-closed set.
Theorem 3.8: Every semi-closed set in topological spaces (X, τ) is g*i-closed set.
Proof: let A be semi-closed set in (X, τ) , such that A U, where U is g-open. Since A is
semi-closed and by Theorem (2.2), then cli(A) cls(A) U
, therefore cli(A) U and U is g-open . Hence A is g*i-closed set.
example.
Example 3.9: let X = {a, b, c} and τ = {Ø, {c}, {a, c}, {b, c}, X}, then
C(X) = {Ø, {a}, {b}, {a, b}, X}, and
IO(X) = {Ø, {a}, {b}, {c}, {a, c}, {b, c}, X}
IC(X) = {Ø, {a}, {b}, {a, b}, {a, c}, {b, c}, X}
SO(X) = {Ø, {c}, {a, c}, {b, c}, X}
SC(X) = {Ø, {a}, {b}, {a, b}, X}
GC(X) = {Ø, {a}, {b}, {a, b}, X}

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GO(X) = {Ø, {c}, {a, c}, {b, c}, X}
Let A ={c}. Then A is g*i-closed set but not semi-closed set.
Theorem 3.10: Every g*-closed set in topological spaces (X, τ) is g*i-closed set.
Proof: let A be a g*-closed set in (X, τ) such that A U, where U is g-open .Since A is g*-
closed and by Theorem (2.2), then cli(A) cl(A) U. Hence A is a g*i-closed set in (X, τ).
example.
Example 3.11: Let X = {a, b, c}, τ = {Ø, {b}, {a, b}, {b, c}, X}
C(X) = {Ø, {a}, {c}, {a, c}, X}
IO(X) = {Ø, {a}, {b}, {c}, {a, b}, {b, c}, X}
IC(X) = {Ø, {a}, {c}, {a, b}, {a, c}, {b, c}, X}
GC(X) = {Ø, {a}, {c}, {a, c}, X}
GO(X) = {Ø, {b}, {a, b}, {b, c}, X}
g*C(X) = {Ø, {a}, {c}, {a, c}, X}
Let A = {b}, so A is a g*i-closed set but not g*-closed set of (X, τ)
Corollary 3.12: Every g-closed set in (X, ) is g*i-closed.
Proof: By Theorem (2.4), and Theorem (3.2).
Corollary 3.13: Every gs-closed set in (X, ) is g*i-closed.
Proof: By Theorem (2.5), and by Theorem (3.8).
Theorem 3.14: Every δg-closed set in topological spaces(X, τ) is g*i-closed set.
Proof: Let A be a δg-closed set in (X, τ) such that A U where U is g-open. Since A is δg-closed
and by Theorem (2.2), then cli(A) clδ(A) U , so we get cli(A) U . Hence A is
g*i-closed set.
example.
Example 3.15: let X = {a, b, c}, τ = {Ø, {a}, {b}, {a, b}, X}

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C(X) = {Ø, {c}, {a, c}, {b, c}, X}
IO(X) = {Ø, {a}, {b}, {a, b}, {a, c}, {b, c}, X}
IC(X) = {Ø, {a}, {b}, {c}, {a, c}, {b, c}, X}
GC(X) = {Ø, {c}, {a, c}, {b, c}, X}
GO(X) = {Ø, {a}, {b}, {a, b}, X}
δO(X) = {Ø, {a}, {b}, {a, b}, X}
δ C(X) = {Ø, {c}, {a, c}, {b, c}, X}
δ GC(X) = {Ø, {c}, {b, c}, {a, c}, X}
g*iC(X) = {Ø, {a}, {b}, {c}, {a, c}, {b, c}, X}
Let A = {a}, so A is g*i-closed set in (X, τ), but not δg-closed set.
Theorem 3.16: Every θg-closed set in topological spaces (X, τ) is g*i-closed set.
Proof: Let A be θg-closed set in (X, τ), such that A U where U is g-open. Since A is θg-closed
and by Theorem (2.2), then cli(A) clθ(A) U, so we have cli(A) U. Hence A is a
g*i-closed set.
example.
Example 3.17: Let X = {a, b, c}, and τ = {Ø, {b}, {a, b}, {b, c}, X}, then C(X) = {Ø, {a}, {c},
{a, c}, X}, θO(X) = {Ø, X}, and θC(X) = {Ø, X}
IO(X) = {Ø, {a}, {b}, {c}, {a, b}, {b, c}, X}
IC(X) = {Ø, {a}, {c}, {a, b}, {a, c}, {b, c}, X}
GC(X) = {Ø, {a}, {c}, {a, c}, X}
GO(X) = {Ø, {b}, {a, b}, {b, c}, X}
g*iC(X ) = {Ø, {a}, {b} ,{c}, {a, b}, {a, c}, {b, c}, X}
θGC(X) = {Ø, {a, c}, X}
Let A = {a, b}. So A is g*i-closed set in (X, τ) but not θg-closed set.
Remark 3.18: By the above results we have the following diagram.

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Closed
g*-closed
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g-closed
gs-closed
Semi-closed
i- g*i-closed
closed
gi-closed
g-closed
Theorem 3.19: A set A is g*i-closed if and only if
δg-closed
cli(A) A contains no non-empty g-closed set .
Proof : Necessity let F be a g-closed set in (X, τ) such that F cli(A) A then cli(A) X F .
This implies F X cli(A), so F (X cli(A)) ∩ (cli(A) A) (X cli(A)) ∩ cli(A)=Ø .
Therefore F = Ø.
Sufficiency: Assume that cli(A) A contains no non –empty g-closed set. And let A U, U is
g-open. Suppose that cli(A) is not contained in U, cli(A) ∩ U is a non-empty g-closed set of
cli(A) A which is a contradiction. Therefore cli(A) U , Hence A is g*i-closed .
Theorem 3.20: A g*i-closed set A is an i-closed set if and only if cli(A) A is an i-closed set.
Proof: If A is an i-closed set, then cli(A) A=Ø. Conversely, suppose cli(A) A is an i-closed set
in X. Since A is g*i-closed. Then cli(A) A contain no non-empty g-closed set in X. Then
cli(A) A =Ø . Hence A is an i-closed set.
Theorem 3.21: If A and B are two g*i-closed, then A∩B is g*i-closed.
Proof: Let A and B be two g*i-closed sets in X. And let A ∩ B U,
U is g-open set in X. Since A is g*i-closed, then cli(A) U , whenever
A U, and U is g-open in X. Since B is g*i-closed, then cli(B) U whenever B U, and U
is g-open in X. Now cli(A)∩ cli(B) U, therefore A∩B is g*i-closed.

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Corollary 3.22: The intersection of g*i-closed set and closed set is g*i-closed set .
Proof: By Theorem (3.2) and Theorem (3.21) we get the result.
Note. If A and B are g*i-closed then their union need not be g*i-closed as shown in the
following example.
Example 3.23: let X = {a, b, c}, Ʈ = {Ø, {a}, {b}, {a, b}, X}
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g*iC(X) = {Ø, {a}, {b}, {c}, {a, c}, {b, c}, X}
Let A = {a} and B = {b} are g*i-closed but A  B= {a, b} is not g*i-closed.
Theorem 3.24: If A is both g-open and g*i-closed set of X, then A is an i-closed set.
Proof: Since A is g-open and g*i-closed in X, then cli(A) U , U is g-open.
Since A U and A is g-open then cli(A) A. but always A cli(A), therefore A = cli(A).
Hence A is an i-closed set.
Theorem 3.25: For x X, the set X {x} is g*i-closed or g-open.
Proof: Suppose X {x} is not g-open. Then X is the only g-open set containing X {x}. This
implies cli(X {x}) is g*i-closed. Then X {x} is g*i-closed.
Theorem 3.26: If A is g*i-closed and A B cli (A), then B is g*i-closed.
Proof: Let U be g-open set of X such that B U. Then A U. Since A is g*i-closed. Then
cli(A) U, now cli(B) cli (cli(A)) = cli(A) U. Therefore B is g*i-closed.
Theorem 3.27: Let A Y X, and suppose that A is g*i-closed in X, then A is g*i-closed
relative to Y
Proof: Given that A Y X and A is g*i-closed in X. To show that A is g*i-closed relatives
Y. Let A Y ∩ U, where U is g-open in X. Since A is g*i-closed A U, implies cli(A) U.
It follows that Y ∩ cli(A) Y ∩ U. Thus A is g*i-closed relative to Y.
Proposition 3.28: If a set X is finite and a topology on X is T1 – space, then gi-closed = g*i- closed
Proof: Let X be a finite set and T1 – space, let A gi-closed. If A = Ø, then A g*i – closed.
If A ≠ Ø, then let A X and for each x A. {x} is closed therefore A = x A {x}, then A is
closed. By Theorem (3.2), A g*i-closed. Hence gi-closed g*i-closed but by Theorem
(3.6), g*i-closed gi-closed therefore gi-closed = g*i-closed.
Corollary 3.29: If a topological on X is discrete topology, then
gi-closed = g*i-closed.
Proof: Obvious

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Proposition 3.30: For a subset A of a topological space (X, ) the following statements are
true.
1- If A is gi-closed then cl (A) g*i-closed.
2- If A is g*i-closed then gi-int (A) g*i-closed.
3- If A is g*i-closed then gi-cl (A) g*i-closed.
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Proof: Obvious.
Definition 3.31: A subset A of a space X is called g*i-open if X A is g*i-closed. The family
of all g*i-open subset of a topological space (X, ) is denoted by g*iO(X, or g*iO(X).
All of the following results are true by using complement.
Proposition 3.32: The following statements are true:
1- Every open is g*i-open.
2- Every g*i-open is gi-open.
3- Every i-open is g*i-open.
4- Every -open is g*i-open.
Proof: By using the complement of the definition of g*i-closed.
Proposition 3.33: Let A be subset of a topological space (X, ). If A is g*i-open, then for each
x A there exists g*i-open set be such that x B A.
Proof: Let A be g*i-open set in a topological space (X, ) then for each x A, put A = B is g*i-open
containing x such that x B A.
4. Some properties of g*i-open and g*i-closed sets in a topological space
Definition 4.1: Let (X, ) be a topological space and x X. A subset N of X is said to be g*i-neighborhood
of x if there exists g*i-open set Y in X such that x Y N.
Definition 4.2: Let A be subset of a topological space (X, ), a point x X is called g*i-interior
point of A, if there exist g*i-open set U such that x U A. The set of all g*i-interior
points of A is called g*i-interior of A and is denoted by g*i-int(A).
Proposition 4.3: For any subsets A and B of a space X, the following statements hold:
1. g*i-int(Ø) = Ø and g*i-int(X) = X.
2. g*i-int (A) is the union of all g*i-open sets which are contained in A.
3. g*i-int(A) is g*i-open set in X.
4. g*i-int (A) A.
5. If A B, then g*i-int(A) g*i-int (B).
6. If A B = Ø, then g*i-int(A) g*i-int(B) = Ø.
7. g*i-int (( g*i-int(A)) = g*i-int(A)
8. A is g*i-open if and only if A = g*i-int (A).

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9. g*i-int(A) g*i-int(B) g*i-int(A B ).
10. g*i-int(A B) g*i-int(A) g*i-int(B).
Proof : The prove of (1), (2), (3), (4), (5), (6), (7) and (8) is Obvious, only to prove (9) and
(10)
Proof (9): Let A and B be subset of X, since A A B and B A B. Then by Proposition
(4.3) (5), we have g*i-int(A) g*i-int (A B) and g*i-int(B) g*i-int(A B). Hence g*i-int(
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A) g*i-int(B) g*i-int(A B)
Proof (10): Since A B A and A B B then by Proposition (4.3) (5), we have g*i-int(A
B) g*i-int(A) and g*i-int(A B) g*i-int(B). Hence g*i-int(A B) g*i-int(A) g*i-int(
B).
In general the equalities of (9) and (10) and the converse of (5) does not hold, as shown in
the following example.
Example 4.4: From Example (3.7)
g*iO(X) = { ,X, {a}, {a, b}, {a, c}}
Let A = {a} and B = {b}, then g*i-int({a}) = {a} and g*i-int({b})= Ø.
g*i-int({a}) g*i-int({b}) = {a} Ø = {a}
g*i-int ({a} {b}) = g*i-int ({a , b }) = { a , b}
g*i-int (A B ) g*i-int (A) g*i-int (B)
g*iO(X) = { , X {a}, {a, b}, {b}, {a, c},{b, c}}
Let A = {a, c} , B = {b, c}
g*i-int (A B ) = g*i-int ({a, c} {b, c}) = g*i-int{c} = Ø.
g*i-int (A) g*i-int (B) = {a, c} {b, c} = {c}.
g*i-int (A) g*i-int (B) g*i-int (A B ).
g*iO(X) = { , X, {a}, {a, b}, {a, c}}
Let A = {b, c},and B = {a, c}
g*i-int (A) = and g*i-int (B) = {a, c}
g*i-int(A) g*i-int (B), but A B.

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Proposition 4.7: For any subset A of X, g*i-int (A) gi-int (A)
Proof: Let A be a subset of a space X and let x g*i-int(A), then x {G : G is g*i-open, G
A }. Then there exists a g*i-open set G such that x G A. Since every g*i-open set is gi-open
then there exists a gi-open set G such that x G A, this implies that x gi-int(A).
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Hence g*i-int(A) gi-int(A).
The converse of the above proposition is not true in general as shown from the following
example.
g*iO(X)= { , X, {a}, {a, b} ,{a, c}}, let A={b}
g*i-int{b}=Ø, gi-int{b}={b}, this implies gi-int(A) g*i-int(A).
Definition 4.9: Let A be a subset of a space X. A point x X is called to be g*i-limit point of
A if for each g*i-open set U containing x, U (A {x}) ≠ Ø
The set of all g*i-limit points of A is called the g*i-derived set of A and is denoted by g*i-
D(A).
Proposition 4.10: Let A and B be subsets of a space X, then we have the following
properties:
1- g*i-D (Ø) = Ø.
2- If x g*i-D(A), then x g*i-D(A {x}).
3- If A B, Then g*i-D(A) g*i-D(B).
4- g*i-D(A) g*i-D(B) g*i-D(A B).
5- g*i-D(A B) = g*i-D(A) g*i-D(B).
6- g*i-D(g*i-D(A)) A g*i-D(A).
7- g*i-D(A g*i-D(A) A g*i-D(A).
Proof: We only prove (6) and (7) since the other part can be proved obviously.
6- If x g*i-D(g*i-D(A)) A, then x g*i-D(g*i-D(A) and x A ,and U is g*i-open set
containing x. Then U (g*i-D(A) {x} ≠ Ø . Let y U (g*i-D(A) {x}. Since y U and
y g*i-D(A). U (A {y}) ≠ Ø, Let z U (A {y}), Then z ≠ x for z A, and x A ,
U (A {x}) ≠ Ø, Therefore x g*i-D(A).
7- Let x g*i-D(A g*i-D(A)). If x A the result is obvious. Let x A, and
x g*i-D(A g*i-D(A)) A then for any g*i-open set U containing x,
U (A g*i-D(A)) {x} ≠ Ø. It following similarly from (6). Thus U (g*i-D(A))
{x} ≠ Ø. Then ( U A) {x} ≠ Ø. Hence x g*i-D(A). Therefore g*i-D( A g*i-
D(A)) A g*i-D(A).

Vol.4, No.9, 2014
The converse of above proposition (3) and (4) is not true in general as shown the following
example.
74
Example 4.11: From example (3.7)
g*iO(X) = { , X, {a}, {a, b}, {a, c}}
Let A = {b, c} and B = {a, c}
g*i-D(A) = Ø, g*i-D(B) = {b, c} , then g*i-D(A) g*i-D(B), but A B
g*iO(X) = { , X, {a}, {a, b}, {b}, {a, c}, {b, c}
Let A = {a} and B = {b}
g*i-D(A) = Ø and g*i-D(B) = Ø, then g*i-D(A B) = {c}
g*i-D(A) g*i-D(B) g*i-D(A B), but g*i-D(A B) g*i-D(A) g*i-D(B).
Proposition 4.13: If X a topological space and A is subset of X, then gi-D(A) g*i-D(A).
Proof: Let x g*i-D(A). This implies that there exists g*i-open set U containing x such that U
(A {x})= Ø, U is g*i-open. Since every g*i-open is gi-open. Then U is gi-open set
containing x and U (A {x}) = Ø, then x g*i-D(A). Hence gi-D(A) g*i-D(A)
The converse of above proposition is not true in general as shown the following example .
Example 4.14: Let X = {a, b, c} , = {Ø, {c}, X }
Let A = {c}, gi-D(A) = Ø and g*i-D(A) = {a, b}
gi-D(A) g*i-D(A) but g*i-D(A) gi-D(A) .
Definition 4.15: For any subset A in space X, the g*i-closure of A, denoted by g*i-cl(A), and
defined by the intersection of all g*i-closed sets containing A.
Proposition 4.16: Let X be a topological space . If A and B are subsets of space X, then
1. The g*i-closure of A is the intersection of all g*i-closed sets containing A
2. g*i-cl(X) =X and g*i-cl( ) = .
3. A g*i-cl(A).
4. g*i-cl(A) is g*i-closed set in X.
5. if g*i-cl(A) g*i-cl(B) = Ø , then A B = Ø.
6. If B is any g*i-closed set containing A. Then g*i-cl(A) B.
7. If A B then g*i-cl(A) g*i-cl(B).
8. g*i-cl(g*i-cl(A) = g*i-cl(A).
9. A is g*i-closed if and only if g*i-Cl(A)=A.

Vol.4, No.9, 2014
75
Proof: It is obvious
Proposition 4.17: If A and B are subset of a spare X then
1- g*i-cl(A) g*i-cl(B) g*i-cl(A B).
2- g*i-cl (A B ) g*i-cl(A) g*i-cl(B).
Proof: Let A and B be subsets of space X
1- Since A A B and B A B. Then by Proposition (4.16)(7), the g*i-cl(A) g*i-cl(
A B ) and g*i-cl(B) g*i-cl(A B). Hence g*i-cl(A) g*i-cl(B) g*i-cl(A B).
2- Since A B A and A B B. Then by Proposition (4.16)(7). The g*i-cl(A B)
g*i-cl(A) and g*i-cl (A B) g*i-cl(B). Hence g*i-cl(A B) g*i-cl(A) g*i-cl(
B).
The converse of the above Proposition is not true in general as shown the following
example.
g*iO(X) = { , X, {a}, {a, b}, {b}, {a, c}, {b, c}}
g*I = { , {a}, {b}, {c}, {a, c}, {b, c}, X }
g*i-cl({a}) = {a}, g*i-cl({b}) = {b}
g*i-cl( A B) = g*i-cl({a, b}) =X
g*i-cl(A) g*i-cl(B) = {a} {b} = {a, b}
g*i-cl(A B) = X {a, b}
1- Let A = {a, b}, and B = {c}, then A B = {a, b} {c} = , therefore g*i-cl(A B) =
, but g*i-cl(A)=X and g*i-cl (B) = {c}.Therefore g*i-cl(A) g*i-cl (B) = X {c} =
{c}, but g*i-cl(A B ) = , implies that g*i-cl(A) g*i-cl (B) g*i-cl(A B).
Proposition 4.19: If A is subset of a space X. Then gi-cl(A) g*i-cl(A)
Proof: Let A be a subset of a space X. By Definition of g*i-closed.
g*i-cl(A) = { F : A F is g*i-closed}, since A F g*i-closed . Then by Theorem (3.6),
A F gi-closed and by Proposition (4.16),(7), gi-cl(A) F, therefore gi-cl(A) {F : A
F g*i-closed } = g*i-cl(A). Hence gi-cl(A) g*i-cl(A).
Proposition 4.20: A subset A of a topological space is g*i-closed if and only if it contains the
set of all g*i-limit points.
Proof : Assume that A is g*i-closed we will prove that A it contains the set of its g*i-limit
points. And assume that if possible that x is g*i-limit points of A which be longs to X A.

Vol.4, No.9, 2014
Then X A is g*i-open set containing the g*i-limit point of A. Therefore by definition of g*i-limit
points A X A ≠ Ø which is contradiction.
Conversely, assume that A contains that set of its g*i-limit points. For each x X A, there
exists g*i-open set U containing x such that A U = Ø. That is x U X A, by Proposition
( 3.33), X A is g*i-open set. Hence A is g*i-closed.
Proposition4.21: Let A be subset of space X, then g*i- cl(A) = A g*i- D(A)
Proof: Since A g*i-Cl(A) and g*i-D(A) g*i-Cl(A), then (A g*i-D(A) g*i-cl(A). To
prove that g*i-cl(A) A g*i-D(A), but g*i-Cl(A) is the smallest g*i-closed containing A, so
we prove that A g*i-D(A) is g*i-closed, let x A g*i-D(A). This implies that x A and
x g*i-D(A). Since x g*i-D(A), there exists g*i-open set Gx of x which contains no point
of A other than x but x A. So Gx Contains no point of A. Then Gx agin Gxis an g*i-open
set of each of its points but as Gx does not contain any point of A no point of Gx can be
g*i-limit points of A, this implies that Gx X g*i-D(A), hence x Gx Cl(A) A
X g*i-D(A) X (A g*i-D(A)). Therefore A g*i-D(A) is g*i-closed. Hence g*i-cl(A)
76
g*i-D(A). Thus g*i-cl(A)=A g*i-D(A) .
Proposition 4.22: Let A be subset of a topological space (X, .And for any x X , then
x g*i-Cl(A) if and only if A U ≠ for every g*i-open set U containing x.
Proof : Let x X and x g*i-cl(A).We will prove A U for every g*i-open set U
containing x, we will proved by contradiction, suppose that there exists g*i-open set U
containing x such that A Then A X U and XA is g*i-closed. g*i-cl(A) ,
then x g*i-cl(A). which is contradiction. Hence A
Conversely, let A For every g*i-open set U containing x, we will proved by
contradiction. Suppose x g*i-cl(A). Then there exists a g*i-closed set F containing A such
that x F, hence x X F and X F is g*i-open set, then A X F = which is contradiction.
Proposition 4.23: For any subset A of a topological space X, the following statements are
true:
1- X g*i-cl(A) = g*i-int (X A).
2- X g*i-int(A) = g*i-cl(X A ).
3- g*i-cl(A) = X g*i-int(X A).
4- g*i-int(A) = X g*i-cl(X A).
Proof:
1- For any x X, then x X g*i-cl(A) implies that x g*i-cl(A), then there exists g*i-open
set G containing x such that A G = Ø, then x G X A. Thus x g*i-int(XA).
conversely, by reversing the above steps, we can prove this part.
2- Let x X g*i-int(A), then x g*i-int(A), so for any g*i-open set B containing x

Vol.4, No.9, 2014
B A. This implies that every g*i-open set B containing x ,B XA . This means x
g*i-cl(X A). Hence (Xg*i-int(A)) g*i-cl(XA). Conversely, by reversing the above
steps, we can prove this part
Proposition 4.24: Let A be subset of a space X. If A is both g*i-open and g*i-closed, then A=
g*i-int(g*i-cl(A))
Proof: If A is both g*i-open and g*i-closed, then g*i-int(g*i-cl(A))= g*i-int(A)=A.
77
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[4] J. Dontchev and M. Ganster , On δ- generalized closed sets and T 3/4 –spaces,Mem.Fac
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