Download Between semi-closed and GS

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El-Maghrabi & Nasef / JTUSCI 2: 78-87 (2009)
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Between semi‐closed and GS‐closed sets A.I. El‐Maghrabi 1 & A.A.Nasef 2 1
Department of Mathematics, Faculty of Education, 2 Department of Physics and Engineering , Faculty of
Engineering, Kafr El-Sheikh University, Kafr El-Sheikh, Egypt.
Received 21 April 2008; revised 20 October 2008; accepted 26 October 2008
Abstract
In this paper, we introduce the concept of strongly generalized semi- closed (= g ∗ s − closed) sets, strongly
generalized semi-open (= g ∗ s − open) sets and strongly semi-T1/2-spaces (= st. semi-T1/2) which are
stronger forms of gs-closed sets, gs-open sets and semi-T1/2 spaces respectively. Furthermore, we study
some of their properties.
(2000) Math. Subject Classification. 54C05.
Keywords and Phrases.
g ∗ s − closed sets;
semi- T p - spaces.
-------------------------------------------E-mail address:[email protected]
g ∗ s − open sets; st. semi-T1/2; semi- Tb
and
El-Maghrabi & Nasef / JTUSCI 2: 78-87 (2009)
79
Definition 2.1.
1. Introduction
In 1963, Levine [10] introduced the concept of
a semi-open set. The initiation of the study of
generalized closed sets was done by Aull [2] in
A subset A of a topological space (X,τ) is said
to be :
(1) a semi- open set [10] if A ⊆ cl (int(A)) and
a semi-closed set if int (cl ( A) ) ⊆ A ,
1968 as he considered sets whose closure belongs
to every open superset. The notion of generalized
semi-closed sets was introduced by Arya and Nour
(2) a preopen set [13] if A⊆ int (cl ( A) ) and a
preclosed set if cl (int ( A)) ⊆ A ,
[1]. In 1987, Bhattacharyya and Lahiri defined and
studied the concept of semi- generalized closed sets
(3) an
α − open
set
[14]
if
via the notion of a semi-closed set. In 1994, Maki,
A ⊆ int (cl (int ( A))) and an α − closed set if
Devi and Balachandran [12] introduced the class of
cl (int (cl ( A))) ⊆ A ,
α − generalized closed sets.
(4) a regular set [16] if int (cl ( A)) = A and a
As a continuation of this work, we introduce
and study in Section 3, a new class of sets namely
g ∗ s − closed sets which is properly placed in
between the class of semi-closed sets and the class
of gs-closed sets due to Arya and Nour [1]. In
Section 4, the class of g ∗ s − open sets introduced
and investigated. We also introduce in Section 5,
an application of topological spaces under the title
of strongly semi-T1/2 spaces. All definitions of the
several concepts used throughout the sequel are
explicitly stated in the following section.
regular closed set if cl (int ( A)) = A ,
(5) a Q-set [9] if int ( cl ( A)) = cl ( int ( A)) .
The intersection of all semi-closed (resp.
preclosed, α-closed) sets containing a subset A of
(X, τ ) is called semi-closure [6] (resp. preclosure,
α-closure) of A and is denoted by scl(A) (resp.
pcl(A), clα (A) ). The semi- interior of A is the
largest semi-open set contained in A and denoted
by s-int(A).
Lemma 2.1.
Let A be a subset of a topological space X. Then:
2.
(1) s − cl ( A) = A U int(cl ( A)).
Preliminaries
Throughout this paper (X,τ) , (Y,σ) and ( Z, η )
represent non-empty topological spaces on which
no
separation
axioms
are
assumed,
(2) s − cl ( X − A) = X − ( s − int( A)).
(3) s − int( X − A) = X − scl ( A).
unless
otherwise mentioned. For a subset A of a space
Definition 2.2.
(X,τ), cl(A) and int (A) represent the closure of A
A subset A of a space (X,τ) is called :
with respect to τ and the interior of A with respect
(1)
a generalized closed (= g-closed) set [2,11]
to τ respectively. (X, τ ) will be replaced by X if
if cl(A) ⊆ U whenever A ⊆ U and U is
there is no chance of confusion.
open in (X,τ),
Let us recall the following definitions which we
shall require later.
Between semi‐closed and GS‐closed sets (2)
a generalized open (= g-open) set [11] if
X / A is g-closed ,
El-Maghrabi & Nasef / JTUSCI 2: 78-87 (2009)
(3)
(3)
a semi-generalized closed (= sg-closed) set
[4] if scl(A) ⊆U whenever A⊆U and U is
(4)
(5)
open in (X,τ),
a generalized semi-open (= gs-open) set [1]
an
a
T p − space [17] if every strongly
g-closed set in X is closed,
(6)
if X / A is gs-closed ,
(6)
a semi- T1/2 space [4] if every sg-closed set
in X is semi-closed,
a generalized semi-closed (= gs-closed) set
[1] if scl(A) ⊆U whenever A⊆U and U is
(5)
a Td-space [8] if every gs-closed set in X is
g-closed,
semi-open in (X,τ),
(4)
80
α − generalized closed (= αg − closed)
set [12] if clα ( A) ⊆ U whenever A⊆U and
a Tb − space [8] if every gs-closed set in
X is closed.
3.
Strongly generalized semi-closed sets
We start this section with the following basic
definition .
U is open in (X,τ).
Definition 3.1.
Definition 2.3 [17].
Let A be a subset of a topological space. Then
A
The definition of g*s-closed set in B is called
is called a strongly generalized closed
(=
a strongly
∗
generalized semi-closed (= g s −
g ∗ − closed) set of B if cl B ( A) ⊆ G whenever
closed) set if scl(B) ⊆ U whenever B ⊆ U and U
A ⊆ G and G is g-open in B.
is g-open in (X,τ).
Definition 2.4.
Remark 3.1.
Recall that a mapping
f : ( X ,τ ) → ( Y , σ ) is
(1)
irresolute [7] if f
−1
in
X
is
The converse is not true as may be seen
(U ) is semi-open in
from Example 3.1.
(X,τ), for every semi-open U of (Y, σ ),
(2)
set
g ∗ s − closed.
called:
(1)
Every semi-closed
pre-semi-closed [7] if for each semi-closed
(2)
Every g ∗ s − closed set in X is gs-closed.
set B of ( X , τ ), f( B ) is semi-closed in
(Y, σ ),
(3)
Example 3.1.
(U ) is g-closed in
Let X={p,q,r} with τ={φ,{p},{p,r},X}. Then
(X,τ), for every g-closed set U of ( Y, σ ).
a subset B={p,q} is g ∗ s − closed but not semi-
Gc-irresolute [3] if f
−1
closed.
Definition 2.5.
A topological space X is called :
(1)
(2)
Example 3.2. If X ={p,q,r} with τ={φ,{p},X}, then
a T1/2 –space [11] if every g-closed set in
a subset B = {p,q} is gs-closed but not
X is closed,
g ∗ s − closed.
an α-space [14] if every α-closed set in
X is closed,
Between semi‐closed and GS‐closed sets From the above definition and examples ,we
have the following diagram.
El-Maghrabi & Nasef / JTUSCI 2: 78-87 (2009)
semi-closed set → g ∗ s − closed set → g s-closed.
81
Corollary 3.1.
Let B be open and g ∗ s − closed sets in X. Then B
is semi-closed.
Remark 3.2 .
(1)
Every g ∗ s − closed set in X is g-closed and
αg − closed in X . However the converse is
not true as shown from Example 3.3.
(2)
The union of two g ∗ s − closed sets need
∗
not be strongly g s − closed .
Lemma 3.3.
Let H ⊆ B ⊆ X . If H is g-open and B is clopen in
X. Then H is g-open in B .
Proof.
Let F be a closed subset of B such that F ⊆ X .
Since B is closed in X, then F is closed in X . By
Example 3.3.
hypothesis, then F ⊆ int( H ) , but B is g-open in X,
Let X={a,b,c,d} and τ = {X , φ , {a}, {a, b}, {a, c, d }} .
hence
Then the sets {b} and {c} are g ∗ s − closed but
F = F I B ⊆ int B ( H ) .Therefore, H is g-open in B.
int B ( H ) = B I int( H )
holds.
Hence,
their union {b,c} is not g ∗ s − closed. Further,
a subset {b,c} of X is g-closed and αg − closed .
Lemma 3.4.
If V ⊆ B ⊆ X ,V is g ∗ s − closed in B and B is
Lemma 3.1.
clopen, then V g ∗ s − closed in X.
Let B be a g ∗ s − closed set in X .Then scl(B)\B
Proof.
does not contain any non-empty g-closed set.
Let H be a g-open set in X and V ⊆ H .
Proof.
Then V ⊆ H I B and H I B is g-open in X.
Assume that V is a g-closed subset of scl(B)\B.
Hence by using Lemma 3.3, H I B is g-open in B.
This implies that V ⊆ scl ( B) and V⊆X\B. Since
X\V is a g-open set , B is g ∗ s − closed and
Since V g ∗ s − closed in B, then scl B (V ) ⊆ H I B .
Since
B
is
closed
in
X,
then
scl ( B) ⊆ X \ V .Therefore,
scl ( B ) = B [ 6, Theorem 1.4] . Hence, by using
V ⊆ scl ( B ) I ( X \ scl ( B )) = φ .Hence scl(B)\B does
[ 15, Theorem 2.4], we have scl (V ) ⊆ H .
not contain any non-empty g-closed set.
shows that V is g ∗ s − closed in X.
This
Lemma 3.2.
If B is g-open and g ∗ s − closed sets in X, then B
is semi-closed.
Let V ⊆ B ⊆ X . If V g ∗ s − closed in X and B is
open, then V g ∗ s − closed in B.
Proof.
Since B is g-open and
Lemma 3.5.
g ∗ s − closed, then
scl ( B) ⊆ B , but B ⊆ scl ( B ) .Therefore scl ( B ) = B .
Hence , B is semi-closed.
Between semi‐closed and GS‐closed sets Proof.
If H is a g-open set in B such that V ⊆ H and B is
g-open in X, then by [ 11, Theorem 4.6]
H is
El-Maghrabi & Nasef / JTUSCI 2: 78-87 (2009)
g-open in X. Since, V is g ∗ s − closed in X, then
scl (V ) ⊆ H .
Hence
by
scl B (V ) = B I scl ( B ) ⊆ H .
[ 15, Theorem
Therefore
V
3.5]
82
Theorem 3.3.
If B is a subset of a space X, the following are
equivalent:
is
(1) B is clopen,
g ∗ s − closed in B.
(2) B is open, a Q-set and g ∗ s − closed.
Proof.
Theorem 3.1.
(1)⇒(2). Since B is clopen, then B is both open
For a space X, if V ⊆ B ⊆ X and B is clopen in X,
and a Q-set. Let H be a g-open set in X and B ⊆ H .
then the following are equivalent:
Then
∗
B U int(cl ( B )) ⊆ H
and so scl ( B) ⊆ H .
(1) V is g s − closed in B,
Hence, B is g ∗ s − closed in X.
(2) V is g ∗ s − closed in X.
(2)⇒(1).
Hence,
by
Theorem
3.2,
B
is
Proof.
regular-open. Since every regular-open set is open,
(1)⇒(2).Let V be g ∗ s − closed in B. Then by
then B is open. Also, B is a Q-set, then B is closed.
Lemma 3.4, V is g ∗ s − closed in X.
(2)⇒(1). If V is g ∗ s − closed in X, then by
Lemma 3.5, V is g ∗ s − closed in B.
Therefore B is clopen.
4.
Strongly generalized semi- open sets
The aim of this section is to introduce the
concept of a strongly generalized semi-open set
and study some of their properties.
Theorem 3.2.
Let B be a subset of a space X, the following are
Definition 4.1.
equivalent:
(1) B is regular- open,
∗
(2) B is open and g s − closed .
Proof.
A subset B of a topological space X is called a
strongly generalized semi- open (= g ∗ s − open ) set
if X\B is g ∗ s − closed .
(1)⇒(2). Let H be a g-open set in X containing B
and every regular- open set is open, then
Theorem 4.1. A subset B of a space X is
B U int(cl ( B )) ⊆ B ⊆ H . Hence, scl ( B) ⊆ H and
g ∗ s − open if and only if F ⊆ s − int( B) whenever
therefore B is g ∗ s − closed.
F is g-closed and F⊆B .
(2)⇒(1). Since B is open and g ∗ s − closed, then
Proof. Suppose that B is g ∗ s − open in X, F is
scl ( B) ⊆ B and so B U int(cl ( B)) ⊆ B , but B is
g-closed and F ⊆ B. Then X\F is g-open and
open, we have int(cl ( B )) ⊆B . Since every open
X\B⊆X\F. Since, X\B is
set is preopen, then B ⊆ int(cl ( B)) . Therefore,
scl(X\B) ⊆X\F. But, scl(X\B) =X\ s-int(B)⊆X\F.
B = int(cl ( B )) and hence B is regular-open.
Hence F ⊆ s-int(B).
Between semi‐closed and GS‐closed sets g ∗ s − closed, then
El-Maghrabi & Nasef / JTUSCI 2: 78-87 (2009)
83
Conversely, Suppose that F ⊆ s-int(B) whenever
Theorem 4.2.
F⊆B and F is g-closed. If H is a g-open set in
For each x∈X, then either {x} is
X containing X\B, then X\H is a g-closed set
X\{x} is g ∗ s − closed.
contained
in
B.
Hence
by
hypothesis,
X\H ⊆ s-int(B), then by taking the complements,
we have, scl(X\B) ⊆ H. Therefore X\B is
∗
∗
g s − closed in X and hence B is g s − open in X.
g-closed or
Proof.
If {x} is not g-closed, then the only g-open set
containing X\{x} is X, hence, scl(X\ {x})⊆ X is
contained
in
X
and
therefore,
X\{x}
is
g ∗ s − closed.
Remark 4.1.
∗
The intersection of two g s − open sets need not
5.
Strongly semi − T 1 spaces
2
to be g ∗ s − open.
In
this
semi − T 1
Example 4.1.
If
section,
we
spaces and
introduce
strongly
discuss some of their
2
X = {1,2,3,4} with τ = {X , φ , {1}, {1,2}, {1,3,4}} ,
then the sets {1,3,4} and {1,2,4} are g ∗ s − open
∗
sets but their intersection {1,4} is not g s − open.
properties.
Definition 5.1.
A topological space X is said to be :
(1) strongly semi − T 1 (= st. semi − T 1 ) if
2
Corollary 4.1.
2
every gs-closed set in X is g ∗ s − closed,
If B is g ∗ s − open in X, then H=X, whenever H is
(2) semi − T p if every
g-open and s − int( B ) U ( X \ B ) ⊆ H .
g ∗ s − closed set in
X is closed,
Proof.
Assume
that
H
is
g-open
(3) semi − Tb
and
s-int(B)) ∪ (X\B) ⊆ H.
if every gs-closed set in
X is semi-closed.
Hence X\H⊆scl(X\B) ∩B = scl(X\B)\(X\B).
Since, X\H is g-closed and X\B is g ∗ s − closed,
then by Lemma 3.1, X\H=∅ and hence, H=X.
Remark 5.1.
semi − T p
and
st.
semi − T 1
spaces
are
2
independent as may be seen from Example 5.1.
Lemma 4.1.
∗
∗
Example 5.1.
If B is g s − closed, then scl(B)\ B is g s − open.
If
Proof .
τ 2 = {X , φ , {u , v}} , then ( X ,τ 1 ) is semi − T p but
∗
Suppose that B is g s − closed. Then by Lemma
X={u,
Between semi‐closed and GS‐closed sets w}
with
τ 1 = {X , φ , {u}} and
not st. semi − T 1 , since a subset {u, v} is gs-closed
3.1, scl(B)\B does not contain any non-empty
g-closed set. Therefore, scl(B)\B is g ∗ s − open.
v,
2
but not
g ∗ s − closed. Also,
( X ,τ 2 )
is st.
El-Maghrabi & Nasef / JTUSCI 2: 78-87 (2009)
semi − T 1 but not semi − T p , where a subset
84
(3) Every semi − T p space is T p .
2
{u, w} is g ∗ s − closed but not closed.
The converses of the above corollary need not be
true as may be seen from the following examples.
Theorem 5.1.
For a space ( X ,τ ) , the following statements hold :
(1) Every
T0 − space
semi − Tb ,
(resp.
semi − T 1 , α − space ) is st. semi − T 1 ,
2
2
Example 5.3 .
A space ( X ,τ 2 ) in Example 5.2, is a semi − T 1
2
space but it is not semi − Tb .
(2) Every semi − Tb space is semi − T p .
The converses of the above theorem need not be
true as may be seen from the following examples.
Example 5.4 .
If X= {p, q, r} with τ = {X , φ , {p}} , then ( X ,τ ) is
a T p - space but it is not semi − T p , since a subset
Example 5.2 .
Let X={p, q, r} with the following topologies:
{p, q} is g ∗ s − closed but it is not closed.
(1) τ 1 is the indiscrete topology,
(2)
τ 2 ={φ,{p, q},X},
(3)
τ 3 ={φ,{p},{q, r},X}.
Remark 5.2.
(1) st. semi − T 1 and semi-T1\2 spaces are
2
Then ( X ,τ 1 ) is st. semi − T 1 space but it is not
independent,
(2) st.
semi − T 1
and
Td-spaces
are
2
independent.
2
a T0 − space, since X is the only open set contains
any two distinct points from X.
Further ( X ,τ 2 ) is a st. semi − T 1 space but
2
Example 5.5.
(1) A space (X,τ 3) in Example 5.2, is
a semi-T1\2 space but it is not st.
it is not semi − Tb ( resp. T 1 − space , α − space ).
2
semi − T 1 , since a subset {q} is gs-closed
2
Furthermore, ( X ,τ 3 ) is a semi − T p space but
but it is not g ∗ s − closed. Also, a space
it is not semi − Tb , since a subset {p, q} is
(X,τ2) in Example 5.2, is
gs-closed but not semi-closed.
semi − T 1 space but it is not semi-T1\2.
a st.
2
(2) A space (X,τ) in Example 5.4, is
Corollary 5.1.
a Td-space but it is not st. semi − T 1 ,
For a space ( X ,τ ) , the following are hold :
2
(1) Every Tb − space is semi − Tb ,
where a subset {p,q} is gs-closed but it is
(2) Every semi − Tb space is semi − T 1 ,
not g ∗ s − closed.
2
Between semi‐closed and GS‐closed sets El-Maghrabi & Nasef / JTUSCI 2: 78-87 (2009)
(3) A space (X,τ) where X= {p, q, r} and
τ = {X , φ , {p}, {p, r}}
is
a
85
Remark 5.3.
We can summarize
st.
the following diagram by
using [11,17] and the above results.
semi − T 1 space but it is not a Td-space.
2
T p - space
←
Td-space
semi − T p
semi − T1 space
T0 − spacest.
Tb − space
2
T1 − space
←
semi − T 1 space
semi − Tb
2
2
Theorem 5.2.
If X is a st. semi − T 1 space, then the following
2
Theorem 5.3.
The image of a g ∗ s − closed set is g ∗ s − closed
statements are hold:
(1) Every singleton in X is closed or
g ∗ s − open in X,
(2) Every singleton in X is closed or gs-open.
(3)
under gc-irresolute and pre-semi-closed mappings.
Proof. If f(B)⊆H, where H is g-open in Y, then
B ⊆ f-
−1
(H) and scl ( B ) ⊆ f
−1
(H ) .
Hence, f(scl(B)) ⊆ H and f(scl(B)) is a semi-closed
set in Y. Since,
Proof.
(1) Suppose that {x} is not closed, for
some x ∈ X . Then the only open set
scl ( f ( B )) ⊆ scl ( f ( scl ( B ))) = f ( scl ( B )) ⊆ H , then
f(B) is g ∗ s − closed in Y.
containing X\{x} is X , hence X\{x} is gsclosed. Since X is a st. semi − T 1 , then
2
Theorem 5.4.
X\{x} is g ∗ s − closed.
If B is a g ∗ s − closed ( resp. g ∗ s − open ) subset
Therefore, {x} is g ∗ s − open in X.
of Y, ƒ : (X,τ)→(Y,σ) is a bijection irresolute and
(2) Assume that {x} is not closed, for every
x ∈ X . Then X is the only open set
containing X\{x} ,
hence X\{x} is gs-closed and therefore,
{x} is gs-open in X.
Between semi‐closed and GS‐closed sets closed mappings, then f(resp. g ∗ s − open) in X.
−1
(B) is
g ∗ s − closed
El-Maghrabi & Nasef / JTUSCI 2: 78-87 (2009)
86
Proof.
of their properties. Furthermore, we discuss the
Assume that B is a g ∗ s − closed subset of Y and
conditions which are added to these concepts in
f-
−1
(B) ⊆ H, where H is g-open in X. Then we
need
to
show
that
scl ( f
−1
( B)) ⊆ H or
scl ( f −1 ( B)) I X \ H = φ .Now,
order to coincide with the concept of semi-colsed
[5],
regular
[16]
and
clopen
sets.
Furthermore, we define some spaces on these
concepts
f ( scl ( f −1 ( B))) I X \ H ) ⊆ scl ( B ) \ B ,
–open
such
as:
strongly
semi- T 1
2
then by
(= st. semi- T 1 ), semi − Tb and semi − T p spaces
2
Lemma 3.1,
We have scl ( f
f-
−1
and we give the relation between these spaces and
−1
( B )) I ( X \ H ) = φ . Therefore,
(B) is g ∗ s − closed in X.
f-
−1
(B) is g ∗ s − open
in X.
Conclusion
During the last few years the study of
generalized closed sets has found considerable
interest among general topologists. One reason is
these objects are natural generalizations of closed
sets. More importantly, generalized closed sets
suggest some new separation axioms which have
been found to be very useful in the study of certain
objects of digital topology.
The initiation of the study of generalized closed
sets was done by Aull [2] in 1968 as he considered
sets whose closure belongs to every open superset.
The concept of strongly generalized closed set was
introduced by Sundaram and Pushpalatha in [17].
The aim of this paper is to introduce the
concepts of strongly generalized semi- closed
(= g ∗ s − closed)
sets,
investigate some of their characterizations.
References
By taking complements, we can show that, if B is
g ∗ s − open in Y, then
other spaces which are defined above. Finally, we
strongly
generalized
semi-open (= g ∗ s − open) sets and we study some
Between semi‐closed and GS‐closed sets [1] S.P. Arya and T. Nour, Characterizations of
s-normal spaces, Indian J. Pure Appl. Math.,
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