Gen. Math. Notes, Vol. 22, No. 2, June 2014, pp. 93-102
ISSN 2219-7184; Copyright © ICSRS Publication, 2014
www.i-csrs.org
Available free online at http://www.geman.in
Generalized Alpha Star Star Closed Sets in
Bitopological Spaces
Qays Hatem Imran
Al-Muthanna University, College of Education
Department of Mathematics, Al-Muthanna, Iraq
E-mail: alrubaye84@yahoo.com
(Received: 12-2-14 / Accepted: 20-3-14)
Abstract
The aim of this paper is to introduce the concepts of generalized alpha star
star closed sets, generalized alpha star star open sets and studies their basic
properties in bitopological spaces.
Keywords: Bitopological space, τ 1τ 2 - generalized alpha star closed sets,
τ 1τ 2 - generalized alpha star star closed sets, τ 1τ 2 - generalized alpha star star
open sets.
1
Introduction
Levine, [6] initiated the study of generalized closed sets in topological spaces in
1970. In 1963, J.C. Kelly, [2] defined: a set equipped with two topologies is called
a bitopological space, denoted by ( X ,τ 1 ,τ 2 ) where ( X ,τ 1 ) and ( X ,τ 2 ) are two
topological spaces. In 1986, T. Fukutake, [7] generalized this notion to
bitopological spaces and he defined a set A of a bitopological space X to be an ijgeneralized closed set (briefly ij-g-closed) if j - cl ( A) ⊆ U whenever A ⊆ U and
U is τ i - open in X, i, j = 1,2 and i ≠ j . Semi generalized closed sets and
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Qays Hatem Imran
generalized semi closed sets are extended to bitopological settings by F. H. Khedr
and H. S. Al-saadi, [1]. K. Chandrasekhara Rao and K. Kannan, [4, 5] introduced
the concepts of semi star generalized closed sets in bitopological spaces.
The aim of this communication is to introduce the concepts of τ 1τ 2 - generalized
alpha star star closed sets, τ 1τ 2 - generalized alpha star star open sets and study
their basic properties in bitopological spaces.
2
Preliminaries
Throughout this paper, spaces always mean a bitopological spaces, for a subset A
of X τ i - cl ( A) (resp. τ i - int ( A) , τ i - αcl ( A) ) denote the closure (resp. interior, α closure) of A with respect to the topology τ i , for i = 1, 2 .
Definition 2.1: A subset A of a bitopological space ( X ,τ 1 ,τ 2 ) is called
τ 1τ 2 - α - open [3] if A ⊆ τ 1 - int(τ 2 - cl (τ 1 - int( A))) .
τ 1τ 2 - α - closed [3] if X − A is τ 1τ 2 - α - open. Equivalently, a subset A
of a bitopological space ( X ,τ 1 ,τ 2 ) is called τ 1τ 2 - α - closed if
τ 2 - cl (τ 1 - int (τ 2 - cl ( A))) ⊆ A .
(iii)
τ 1τ 2 - generalized closed (briefly τ 1τ 2 - g - closed) [7] if τ 2 - cl ( A) ⊆ U
whenever A ⊆ U and U is τ 1 - open in X.
(iv)
τ 1τ 2 - generalized open (briefly τ 1τ 2 - g - open) [7] if X − A is
τ 1τ 2 - g - closed.
τ 1τ 2 - alpha generalized closed (briefly τ 1τ 2 - αg - closed) [3] if
(v)
τ 2 - α cl ( A ) ⊆ U whenever A ⊆ U and U is τ 1 - open in X .
(vi)
τ 1τ 2 - alpha generalized open (briefly τ 1τ 2 - αg - open) [3] if X − A is
τ 1τ 2 - αg - closed.
(vii) τ 1τ 2 - generalized alpha closed (briefly τ 1τ 2 - gα - closed) [3] if
τ 2 - α cl ( A ) ⊆ U whenever A ⊆ U and U is τ 1 - α - open in X.
(viii) τ 1τ 2 - generalized alpha open (briefly τ 1τ 2 - gα - open) [3] if X − A is
τ 1τ 2 - gα - closed.
(i)
(ii)
Definition 2.2: A subset A of a bitopological space ( X ,τ 1 ,τ 2 ) is called
τ 1τ 2 - generalized alpha star closed (briefly τ 1τ 2 - gα * - closed) if τ 2 - cl ( A) ⊆ U
whenever A ⊆ U and U is τ 1 - α - open in X. The family of all τ 1τ 2 - gα * - closed
sets of X is denoted by τ 1τ 2 - gα *C ( X ) .
Generalized Alpha Star Star Closed Sets in…
95
Example 2.3: Let X = {a, b, c} , τ 1 = {φ , X ,{a},{b, c}} , τ 2 = {φ , X ,{a},{b},{a, b}} .
Then {a,b} is τ 1τ 2 - gα * - closed and {a} is not τ 1τ 2 - gα * - closed.
Definition 2.4: A subset A of a bitopological space ( X ,τ 1 ,τ 2 ) is called
τ 1τ 2 - generalized alpha star open (briefly τ 1τ 2 - gα * - open) if and only if
X − A is τ 1τ 2 - gα * - closed. The family of all τ 1τ 2 - gα * - open sets of X is
denoted by τ 1τ 2 - gα *O( X ) .
3
Generalized Alpha Star Star Closed Sets
In this section we define and study the concept of τ 1τ 2 - generalized alpha star star
closed sets in bitopological spaces.
Definition 3.1: A subset A of a bitopological space ( X ,τ 1 ,τ 2 ) is called
τ 1τ 2 - generalized alpha star star closed (briefly τ 1τ 2 - gα ** - closed) if
τ 2 - cl ( A) ⊆ U whenever A ⊆ U and U is τ 1 - gα * - open in X. The family of all
τ 1τ 2 - gα ** - closed sets of X is denoted by τ 1τ 2 - gα **C ( X ) .
Example 3.2: Let X = {a, b, c, d } , τ 1 = {φ , X ,{a, b}} , τ 2 = {φ , X ,{b},{c},{b, c}} .
Then ϕ, X,{c},{d}, {a,c},{a,d},{b,c},{b,d},{c,d},{a,b,c},{a,c,d},{a,b,d},{b,c,d}
are τ 1τ 2 - gα ** - closed sets.
Now, the characterization of τ 1τ 2 - gα ** - closed sets by using different types of
generalization of closed sets and τ 1 - gα * - open sets are established in the
following theorem.
Theorem 3.3: Let ( X ,τ 1 ,τ 2 ) be a bitopological space and A ⊆ X . Then the
following are true.
(i)
(ii)
(iii)
If A is τ 2 - closed, then A is τ 1τ 2 - gα ** - closed.
If A is τ 1 - gα * - open and τ 1τ 2 - gα ** - closed, then A is τ 2 - closed.
If A is τ 1τ 2 - gα ** - closed, then A is τ 1τ 2 - g - closed.
(iv)
If A is τ 1τ 2 - gα ** - closed, then A is τ 1τ 2 - αg - closed.
Proof:
(i) It is obvious that every τ 2 - closed set is τ 1τ 2 - gα ** - closed.
(ii) Suppose that A is τ 1 - gα * - open and τ 1τ 2 - gα ** - closed. Then A ⊆ A implies
that τ 2 - cl ( A) ⊆ A . Obviously, A ⊆ τ 2 - cl ( A) . Therefore, A is τ 2 - closed.
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Qays Hatem Imran
(iii) Suppose that A is τ 1τ 2 - gα ** - closed. Let A ⊆ U and U is τ 1 - open in X. Since
every τ 1 - open set is τ 1 - gα * - open in X, we have U is τ 1 - gα * - open in X. Then,
τ 2 - cl ( A) ⊆ U since A is τ 1τ 2 - gα ** - closed. Consequently, A is τ 1τ 2 - g - closed.
(iv) Suppose that A is τ 1τ 2 - gα ** - closed. Let A ⊆ U and U is τ 1 - open in X. Since
every τ 1 - open set is τ 1 - gα * - open in X, we have U is τ 1 - gα * - open in X.
Then, τ 2 - cl ( A) ⊆ U since A is τ 1τ 2 - gα ** - closed. Since τ 2 - αcl ( A) ⊆ τ 2 - cl ( A) ,
we have τ 2 - αcl ( A) ⊆ U . Consequently, A is τ 1τ 2 - αg - closed.
In the following examples it is proved that the converses of the assertions of the
above theorem are not true in general.
Example 3.4: In example (3.2), {c} is τ 1τ 2 - gα ** - closed but not τ 2 - closed. Also
{a, d} is τ 2 - closed, τ 1τ 2 - gα ** - closed but not τ 1 - gα * - open.
Example 3.5: Let X = {a, b, c} , τ 1 = {φ , X ,{b, c}} , τ 2 = {φ , X ,{a}} . Then {b} is
τ 1τ 2 - g - closed but not τ 1τ 2 - gα ** - closed in X.
Example 3.6: In example (3.2), {a} is τ 1τ 2 - αg - closed but not τ 1τ 2 - gα ** - closed
in X.
Remark 3.7: τ 1τ 2 - gα - closed sets and τ 1τ 2 - gα ** - closed sets are independent in
general. The following example supports our claim. In Example (3.2), {a} is
τ 1τ 2 - gα - closed but not τ 1τ 2 - gα ** - closed in X. Also {a,b,c} is τ 1τ 2 - gα ** closed but not τ 1τ 2 - gα - closed in X.
Theorem 3.8: If A is τ 1τ 2 - gα ** - closed, τ 1 - gα * - open in X and F is τ 2 - closed
in X then A ∩ F is τ 2 - closed in X.
Proof: Since A is τ 1τ 2 - gα ** - closed, τ 1 - gα * - open in X, we have A is τ 2 - closed
in X [by theorem (3.3) (ii)]. Since F is τ 2 - closed in X, A ∩ F is τ 2 - closed in X.
Remark 3.9:
(i) τ 1τ 2 - gα * - closed and τ 1τ 2 - α - closed sets are independent in general.
(ii) τ 1τ 2 - gα ** - closed and τ 1τ 2 - gα * - closed sets are independent in general.
Example 3.10: In example (3.2), {c} is τ 1τ 2 - gα ** - closed but not τ 1τ 2 - gα * closed in X.
Generalized Alpha Star Star Closed Sets in…
97
Theorem 3.11: If A is τ 1τ 2 - gα ** - closed in X and A ⊆ B ⊆ τ 2 - cl ( A) , then B is
τ 1τ 2 - gα ** - closed.
Proof: Suppose that A is τ 1τ 2 - gα ** - closed in X and A ⊆ B ⊆ τ 2 - cl ( A) . Let
B ⊆ U and U is τ 1 - gα * - open in X. Since A ⊆ B and B ⊆ U , we have A ⊆ U .
Hence τ 2 - cl ( A) ⊆ U (Since A is τ 1τ 2 - gα ** - closed). Since B ⊆ τ 2 - cl ( A) , we
have τ 2 - cl ( B) ⊆ τ 2 - cl ( A) ⊆ U . Therefore, B is τ 1τ 2 - gα ** - closed.
Theorem 3.12: If A and B are τ 1τ 2 - gα ** - closed sets then so is A ∪ B .
Proof: Suppose that A and B are τ 1τ 2 - gα ** - closed sets. Let U be τ 1 - gα * - open
in X and A ∪ B ⊆ U . Since A ∪ B ⊆ U , we have A ⊆ U and B ⊆ U . Since U is
τ 1 - gα * - open in X and A and B are τ 1τ 2 - gα ** - closed sets, we have
τ 2 - cl ( A) ⊆ U and τ 2 - cl ( B) ⊆ U . Therefore, τ 2 - cl ( A ∪ B) ⊆ τ 2 - cl ( A) ∪
τ 2 - cl ( B) ⊆ U . Hence A ∪ B is τ 1τ 2 - gα ** - closed.
Remark 3.13: The following diagram shows the relations among the different
types of weakly closed sets that were studied in this section:
τ 1τ 2 - α - closed
τ 1τ 2 - gα - closed
τ 1τ 2 - αg - closed
τ 2 - closed
τ 1τ 2 - gα ** - closed
τ 1τ 2 - g - closed
τ 1 - gα * - open
+
τ 1τ 2 - gα * - closed
Theorem 3.14: The arbitrary union of τ 1τ 2 - gα ** - closed sets Ai , i ∈ I in a
bitopological space ( X ,τ 1 ,τ 2 ) is τ 1τ 2 - gα ** - closed if the family { Ai , i ∈ I } is
locally finite in ( X ,τ 1 ) .
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Qays Hatem Imran
Proof: Let { Ai , i ∈ I } be locally finite in X and Ai is τ 1τ 2 - gα ** - closed in X for
each i ∈ I . Let ∪ Ai ⊆ U and U is τ 1 - gα * - open in X. Then Ai ⊆ U and U is
τ 1 - gα * - open in X for each i. Since Ai is τ 1τ 2 - gα ** - closed in X for each i ∈ I ,
we have τ 2 - cl ( Ai ) ⊆ U . Consequently, ∪ [τ 2 - cl ( Ai )] ⊆ U . Since the family
{ Ai , i ∈ I } is locally finite in X, τ 2 - cl [ ∪ ( Ai )] = ∪ [τ 2 - cl ( Ai )] ⊆ U . Therefore,
∪ Ai is τ 1τ 2 - gα ** - closed in X.
Remark 3.15: The intersection of any two τ 1τ 2 - gα ** - closed sets is not
necessary τ 1τ 2 - gα ** - closed set as in the following example.
Example 3.16: In example (3.2), A = {a, c} , B = {a, d } are τ 1τ 2 - gα ** - closed but
A ∩ B = {a} is not τ 1τ 2 - gα ** - closed in X.
Theorem 3.17: If a set A is τ 1τ 2 - gα ** - closed in X , then τ 2 - cl ( A) − A contains
no nonempty τ 1 - gα * - closed set.
Proof: Suppose that A is τ 1τ 2 - gα ** - closed in X. Let F be τ 1 - gα * - closed and
F ⊆ τ 2 - cl ( A) − A . Since F is τ 1 - gα * - closed, we have F c is τ 1 - gα * - open.
Since F ⊆ τ 2 - cl ( A) − A , we have F ⊆ τ 2 - cl ( A) and F ⊆ A c . Hence A ⊆ F c .
Consequently τ 2 - cl ( A) ⊆ F c {Since A is τ 1τ 2 - gα ** - closed in X}. Therefore,
F ⊆ [τ 2 - cl ( A)]c . Hence F ⊆ [τ 2 - cl ( A)]c ∩ τ 2 - cl ( A) = φ . Hence τ 2 - cl ( A) − A
contains no nonempty τ 1 - gα * - closed set.
Corollary 3.18: Let A be τ 1τ 2 - gα ** - closed. Then A is τ 2 - closed if and only if
τ 2 - cl ( A) − A is τ 1 - gα * - closed.
Proof: Suppose that A is τ 1τ 2 - gα ** - closed and τ 2 - closed. Since A is τ 2 - closed,
we have τ 2 - cl ( A) = A . Therefore, τ 2 - cl ( A) − A = φ which is τ 1 - gα * - closed.
Conversely, suppose that A is τ 1τ 2 - gα ** - closed and τ 2 - cl ( A) − A is τ 1 - gα * closed. Since A is τ 1τ 2 - gα ** - closed, we have τ 2 - cl ( A) − A contains no
nonempty τ 1 - gα * - closed set {by Theorem 3.17}. Since τ 2 - cl ( A) − A is itself
τ 1 - gα * - closed, we have τ 2 - cl ( A) − A = φ . Therefore, τ 2 - cl ( A) = A implies
that A is τ 2 - closed.
A ⊆ B ⊆ τ 2 - cl ( A) then
τ 2 - cl ( B) − B contains no nonempty τ 1 - gα - closed set.
Theorem 3.19: If A is τ 1τ 2 - gα ** - closed and
*
Generalized Alpha Star Star Closed Sets in…
99
Proof: Let A be τ 1τ 2 - gα ** - closed and A ⊆ B ⊆ τ 2 - cl ( A) . Then B is
τ 1τ 2 - gα ** - closed {by theorem (3.11)}. Therefore, τ 2 - cl ( B) − B contains no
nonempty τ 1 - gα * - closed set {by theorem (3.17)}.
Theorem 3.20: For each x ∈ X , the singleton {x} is either τ 1 - gα * - closed or its
complement {x}c is τ 1τ 2 - gα ** - closed in ( X ,τ 1 ,τ 2 ) .
Proof: Let x ∈ X . Suppose that {x} is not τ 1 - gα * - closed. Then {x}c is not
τ 1 - gα * - open. Consequently, X itself is the only τ 1 - gα * - open set
containing X − {x} . Therefore, τ 2 - cl ( X − {x}) ⊆ X which implies that X − {x}
is τ 1τ 2 - gα ** - closed in ( X ,τ 1 ,τ 2 ) .
4
Generalized Alpha Star Star Open Sets
We begin this section with a relatively new definition.
Definition 4.1: A subset A of a bitopological space ( X ,τ 1 ,τ 2 ) is called
τ 1τ 2 - generalized alpha star star open (briefly τ 1τ 2 - gα ** - open) if and only if
X − A is τ 1τ 2 - gα ** - closed. The family of all τ 1τ 2 - gα ** - open sets of X is
denoted by τ 1τ 2 - gα **O ( X ) .
Example 4.2: In example (3.2), ϕ, X, {a}, {b}, {c}, {d}, {a, b}, {a, c}, {a, d},
{b, c}, {b, d}, {a, b, c}, {a, b, d} are τ 1τ 2 - gα ** - open sets in X.
The following theorem will give an equivalent definition of τ 1τ 2 - gα ** - open sets.
Theorem 4.3: A set A is τ 1τ 2 - gα ** - open if and only if F ⊆ τ 2 - int ( A) whenever
F is τ 1 - gα * - closed and F ⊆ A .
Proof: Suppose that A is τ 1τ 2 - gα ** - open. Then A c is τ 1τ 2 - gα ** - closed. Suppose
that F is τ 1 - gα * - closed and F ⊆ A . Then F c is τ 1 - gα * - open and A c ⊆ F c .
Therefore, τ 2 - cl ( A c ) ⊆ F c (since A c is τ 1τ 2 - gα ** - closed). Since τ 2 - cl ( A c ) =
[τ 2 - int ( A)]c , we have [τ 2 - int ( A)]c ⊆ F c . Hence F ⊆ τ 2 - int ( A) .
Conversely, suppose that F ⊆ τ 2 - int ( A) whenever F is τ 1 - gα * - closed and
F ⊆ A . Then A c ⊆ F c and F c is τ 1 - gα * - open. Take U = F c .
Since F ⊆ τ 2 - int ( A) , we have [τ 2 - int ( A)]c ⊆ F c = U . Since τ 2 - cl ( A c ) =
[τ 2 - int ( A)]c , we have τ 2 - cl ( A c ) ⊆ U . Therefore, A c is τ 1τ 2 - gα ** - closed.
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Thus, A is τ 1τ 2 - gα ** - open.
Remark 4.4: Every τ 1 - open set is τ 1τ 2 - gα ** - open but the converse is not true
in general as can be seen from the following example.
Example 4.5: In example (3.2), {a, c} is τ 1τ 2 - gα ** - open in X but not τ 1 - open
in X.
Remark 4.6: τ 1τ 2 - gα ** - open and τ 1τ 2 - gα * - open sets are in general,
independent as can be seen from the following two examples.
Example 4.7: Let X = {a, b, c} , τ 1 = {φ , X , {a},{b, c}} , τ 2 = {φ , X ,{b},{a, c}} .
Then {c} is τ 1τ 2 - gα * - open in X but not τ 1τ 2 - gα ** - open in X.
Example 4.8: In example (3.2), {d} is τ 1τ 2 - gα ** - open in X but not
τ 1τ 2 - gα * - open in X.
Remark 4.9: The union of any two τ 1τ 2 - gα ** - open sets is not necessary
τ 1τ 2 - gα ** - open set as in the following example.
Example 4.10: In example (3.2), A = {b, c} , B = {b, d } are τ 1τ 2 - gα ** - open sets
but A ∪ B = {b, c, d } is not τ 1τ 2 - gα ** - open in X.
Theorem 4.11: If A and B are τ 1τ 2 - gα ** - open sets then so is A ∩ B .
Proof: Suppose that A and B are τ 1τ 2 - gα ** - open sets. Let F be τ 1 - gα * - closed
in X and F ⊆ A ∩ B . Since F ⊆ A ∩ B , we have F ⊆ A and F ⊆ B . Since A and
B are τ 1τ 2 - gα ** - open sets. Then F ⊆ τ 2 - int ( A) and F ⊆ τ 2 - int ( B) . Therefore,
F ⊆ τ 2 - int ( A) ∩ τ 2 - int( B) ⊆ τ 2 - int( A ∩ B) . Hence A ∩ B is τ 1τ 2 - gα ** - open.
Theorem 4.12: The arbitrary intersection of τ 1τ 2 - gα ** - open sets Ai , i ∈ I in a
bitopological space ( X ,τ 1 ,τ 2 ) is τ 1τ 2 - gα ** - open if the family { Ai , i ∈ I } is
c
locally finite in ( X ,τ 1 ) .
Proof: Let { Ai , i ∈ I } be locally finite in ( X ,τ 1 ) and Ai is τ 1τ 2 - gα ** - open in X
c
for each i ∈ I . Then Ai
c
is τ 1τ 2 - gα ** - closed in X for each i ∈ I . Then by
theorem (3.14), we have ∪ ( Ai ) is τ 1τ 2 - gα ** - closed in X. Consequently, let
c
(∩ Ai ) c is τ 1τ 2 - gα ** - closed in X. Therefore, ∩ Ai is τ 1τ 2 - gα ** - open in X.
Generalized Alpha Star Star Closed Sets in…
101
Theorem 4.13: If A is τ 1τ 2 - gα ** - open in X and τ 2 - int ( A) ⊆ B ⊆ A , then B is
τ 1τ 2 - gα ** - open.
Proof: Suppose that A is τ 1τ 2 - gα ** - open in X and τ 2 - int ( A) ⊆ B ⊆ A . Let F is
τ 1 - gα * - closed and F ⊆ B . Since F ⊆ B and B ⊆ A , we have F ⊆ A . Since A
is τ 1τ 2 - gα ** - open, we have F ⊆ τ 2 - int ( A) . Since τ 2 - int ( A) ⊆ B , we have
F ⊆ τ 2 - int ( A) ⊆ τ 2 - int ( B) . Hence B is τ 1τ 2 - gα ** - open in X.
Theorem 4.14: If a set A is τ 1τ 2 - gα ** - closed in X, then τ 2 - cl ( A) − A is
τ 1τ 2 - gα ** - open set.
Proof: Suppose that A is τ 1τ 2 - gα ** - closed in X. Let F be τ 1 - gα * - closed and
F ⊆ τ 2 - cl ( A) − A . Since A is τ 1τ 2 - gα ** - closed in X, we have τ 2 - cl ( A) − A
contains no nonempty τ 1 - gα * - closed set. Since F ⊆ τ 2 - cl ( A) − A , we have
F = φ ⊆ τ 2 - int[τ 2 - cl ( A) − A] . Therefore, τ 2 - cl ( A) − A is τ 1τ 2 - gα ** - open.
Theorem 4.15: If a set A is τ 1τ 2 - gα ** - open in a bitopological space ( X ,τ 1 ,τ 2 ) ,
then G = X whenever G is τ 1 - gα * - open and τ 2 - int ( A) ∪ A c ⊆ G .
Proof: Suppose that A is τ 1τ 2 - gα ** - open in a bitopological space ( X ,τ 1 ,τ 2 ) and
G is τ 1 - gα * - open and τ 2 - int ( A) ∪ A c ⊆ G . Then G c ⊆ [τ 2 - int ( A) ∪ A c ]c =
τ 2 - cl ( A c ) − A c . Since G is τ 1 - gα * - open, we have G c is τ 1 - gα * - closed.
Since A is τ 1τ 2 - gα ** - open, we have A c is τ 1τ 2 - gα ** - closed. Therefore,
τ 2 - cl ( A c ) − A c contains no nonempty τ 1 - gα * - closed set in X {by theorem
(3.17)}. Consequently, G c = φ . Hence G = X.
Remark 4.16: The converse of the above theorem is not true in general as can
seen from the following example.
Example 4.17: In example (3.2), if we take A = {c,d}, then τ 2 - int ( A) ∪ A c ⊆ X ,
X is τ 1 - gα * - open, but A is not τ 1τ 2 - gα ** - open.
Lemma 4.18: The intersection of τ 1τ 2 - gα ** - open set and τ 2 - open set is always
τ 1τ 2 - gα ** - open.
Proof: Suppose that A is τ 1τ 2 - gα ** - open and B is τ 2 - open. Since B is
τ 2 - open, we have B c is τ 2 - closed. Then B c is τ 1τ 2 - gα ** - closed {by theorem
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Qays Hatem Imran
(3.3) (i)}. Hence, B is τ 1τ 2 - gα ** - open. Hence A ∩ B is τ 1τ 2 - gα ** - open {by
theorem (4.11)}.
Remark 4.19: The following diagram shows the relations among the different
types of weakly open sets that were studied in this section:
τ 1τ 2 - α - open
τ 1τ 2 - gα - open
τ 1τ 2 - αg - open
τ 2 - open
τ 1τ 2 - gα ** - open
τ 1τ 2 - g - open
τ 1 - gα * - closed
+
τ 1τ 2 - gα * - open
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