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JTUSCI

Available online at www.jtusci.info ISSN: 1658-3655

El-Maghrabi & Nasef / JTUSCI2: 78-87 (2009)

■ of Taibah University

I for Science Journal

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; g*s - open sets; st. semi-Ty2; semi- Tb and semi- Tp - spaces.

E-mail address:aelmaghrabi@yahoo.com

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 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 [1]. In 1987, Bhattacharyya and Lahiri defined and studied the concept of semi- generalized closed sets via the notion of a semi-closed set. In 1994, Maki, Devi and Balachandran [12] introduced the class of a - generalized closed sets.

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-Ti/2 spaces. All definitions of the several concepts used throughout the sequel are explicitly stated in the following section.

2. Preliminaries

Throughout this paper (X,t) , (Y,c) and ( Z, rj)

represent non-empty topological spaces on which no separation axioms are assumed, unless otherwise mentioned. For a subset A of a space (X,t), cl(A) and int (A) represent the closure of A with respect to T and the interior of A with respect to t respectively. (X, t ) will be replaced by X if there is no chance of confusion.

Let us recall the following definitions which we shall require later.

Definition 2.1.

A subset A of a topological space (X,t) is said to be :

(1) a semi- open set [10] if A c cl(int(A)) and a semi-closed set if int(cl(A)) c A ,

(2) a preopen set [13] if Acint(cl(A)) and a preclosed set if cl (int (A)) c A,

(3) an a- open set [14] if A c int (cl (int (A))) and an a- closed set if cl (int (cl (A))) c A ,

(4) a regular set [16] if int(cl (A))=A and a 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, a-closed) sets containing a subset A of (X, t ) is called semi-closure [6] (resp. preclosure, a-closure) of A and is denoted by scl(A) (resp. pcl(A), cla (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:

(1) s - cl (A) = A U int(cl (A)).

(2) s - cl(X - A) = X - (s - int(A)).

(3) s - int(X - A) = X - scl(A).

Definition 2.2.

A subset A of a space (X,t) is called :

(1) a generalized closed (= g-closed) set [2,11] if cl(A) c U whenever A c U and U is open in (X,t),

(2) a generalized open (= g-open) set [11] if X / A is g-closed ,

(3) a semi-generalized closed (= sg-closed) set [4] if scl(A) cU whenever AcU and U is semi-open in (X,t),

(4) a generalized semi-closed (= gs-closed) set [1] if scl(A) cU whenever AcU and U is open in (X,t),

(5) a generalized semi-open (= gs-open) set [1] if X / A is gs-closed ,

(6) an a - generalized closed (= ag - closed)

set [12] if cla (A) c U whenever AcU and U is open in (X,t).

(3) a Td-space [8] if every gs-closed set in X is g-closed,

(4) a semi- T1/2 space [4] if every sg-closed set in X is semi-closed,

(5) a Tp - space [17] if every strongly

g-closed set in X is closed,

(6) 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 .

Definition 2.3 [17].

Let A be a subset of a topological space. Then A is called a strongly generalized closed (=

g *- closed) set of B if clB (A) c G whenever

A c G and G is g-open in B.

Definition 3.1.

The definition of g*s-closed set in B is called

a strongly generalized semi-closed (= g * s -

closed) set if scl(B) c U whenever B c U and U is g-open in (X,t).

Definition 2.4.

Recall that a mapping f:(X ,t) ^ (Y,a) is

called:

(1) irresolute [7] if f _1(U) is semi-open in (X,t), for every semi-open U of (Y, a),

(2) pre-semi-closed [7] if for each semi-closed set B of ( X , t ), f( B ) is semi-closed in (Y, a),

(3) Gc-irresolute [3] if f _1(U) is g-closed in (X,t), for every g-closed set U of ( Y, a ).

Definition 2.5.

A topological space X is called :

(1) a T12 -space [11] if every g-closed set in X is closed,

(2) an a-space [14] if every a-closed set in X is closed,

Remark 3.1.

(1) Every semi-closed set in X is

g *s - closed.

The converse is not true as may be seen from Example 3.1.

(2) Every g *s - closed set in X is gs-closed. Example 3.1.

Let X={p,q,r} with T={f{p},{p,r},X}. Then a subset B={p,q} is g *s - closed but not semi-closed.

Example 3.2. If X ={p,q,r} with T={^,{p},X}, then a subset B = {p,q} is gs-closed but not

g *s - closed.

From the above definition and examples ,we have the following diagram.

semi-closed set ^ g *s - closed set ^ g s-closed.

Remark 3.2 .

(1) Every g *s - closed set in X is g-closed and ag - 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 .

Example 3.3.

Let X={a,b,c,d} and t = {x{a},{a,b},{a,c,d}}.

Then the sets {b} and {c} are g*s - closed but

their union {b,c} is not g*s - closed. Further, a subset {b,c} of X is g-closed and ag - closed .

Lemma 3.1.

Let B be a g*s - closed set in X .Then scl(B)\B does not contain any non-empty g-closed set. Proof.

Assume that V is a g-closed subset of scl(B)\B. This implies that V c scl(B) and VcX\B. Since

X\V is a g-open set , B is g*s - closed and scl(B) cX \ V .Therefore, V c scl(B) I (X \ scl(B)).Hence scl(B)\B does not contain any non-empty g-closed set.

Lemma 3.2.

If B is g-open and g *s - closed sets in X, then B

is semi-closed.

Proof.

Since B is g-open and g *s - closed, then scl(B) c B, but B c scl(B) .Therefore scl(B) = B . Hence , B is semi-closed.

Corollary 3.1.

Let B be open and g *s - closed sets in X. Then B is semi-closed.

Lemma 3.3.

Let H c B c 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 c X . Since B is closed in X, then F is closed in X . By hypothesis, then F c int(H), but B is g-open in X,

hence int B (H) = BI int(H) holds. Hence, F = FIBcintB (H) .Therefore, H is g-open in B.

Lemma 3.4.

If V c B c X ,V is g *s - closed in B and B is

clopen, then V g * s - closed in X. Proof.

Let H be a g-open set in X and V c H . Then V c HIB and HIB is g-open in X. Hence by using Lemma 3.3, HIB is g-open in B.

Since V g*s - closed in B, thensclB (V) cHIB . Since B is closed in X, then scl(B) = B [ 6, Theorem 1.4] . Hence, by using

[ 15, Theorem 2.4], we have scl(V) c H . This shows that V is g*s - closed in X.

Lemma 3.5.

Let V c B c X . If V g *s - closed in X and B is

open, then V g*s - closed in B. Proof.

If H is a g-open set in B such that V c H and B is g-open in X, then by [ 11, Theorem 4.6] H is

g-open in X. Since, V is g *s - closed in X, then scl(V) cH . Hence by [ 15, Theorem 3.5] sclB (V) = BI scl(B) c H . Therefore V is g * s - closed in B.

Theorem 3.1.

For a space X, if V c B c X and B is clopen in X, then the following are equivalent:

(1) V is g*s - closed in B,

(2) V is g *s - closed in X. Proof.

(1)^(2).Let V be g*s - closed in B. Then by 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.

Theorem 3.2.

Let B be a subset of a space X, the following are equivalent:

(1) B is regular- open,

(2) B is open and g *s - closed . Proof.

(1)^(2). Let H be a g-open set in X containing B and every regular- open set is open, then B U int(cl(B)) c B c H . Hence, scl(B) c H and

therefore B is g *s - closed.

(2)^(1). Since B is open and g*s - closed, then scl(B) c B and so B U int(cl(B)) c B, but B is open, we have int(cl(B)) cB. Since every open set is preopen, then B c int(cl (B)). Therefore, B = int(cl (B)) and hence B is regular-open.

Theorem 3.3.

If B is a subset of a space X, the following are equivalent:

(1) B is clopen,

(2) B is open, a Q-set and g *s - closed. Proof.

(1)^(2). Since B is clopen, then B is both open and a Q-set. Let H be a g-open set in X and B c H . Then B U int(cl(B)) c H and so scl(B) c H .

Hence, B is g *s - closed in X.

(2)^(1). Hence, by Theorem 3.2, B is regular-open. Since every regular-open set is open, then B is open. Also, B is a Q-set, then B is closed. 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.

Definition 4.1.

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 .

Theorem 4.1. A subset B of a space X is g *s - open if and only if F c s - int(B) whenever F is g-closed and FcB .

Proof. Suppose that B is g *s - open in X, F is g-closed and F c B. Then X\F is g-open and X\BcX\F. Since, X\B is g *s - closed, then scl(X\B) cX\F. But, scl(X\B) =X\ s-int(B)cX\F. Hence F c s-int(B).

Conversely, Suppose that F c s-int(B) whenever FcB and F is g-closed. If H is a g-open set in X containing X\B, then X\H is a g-closed set contained in B. Hence by hypothesis, X\H c s-int(B), then by taking the complements, we have, scl(X\B) c H. Therefore X\B is

g * s - closed in X and hence B is g *s - open in X. Remark 4.1.

The intersection of two g * s - open sets need not to be g *s - open.

Example 4.1.

If X ={1,2,3,4} with t ={x{l}, {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.

Corollary 4.1.

If B is g *s - open in X, then H=X, whenever H is

g-open and s - int(B) U (X \ B) c H .

Proof.

Assume that H is g-open and

s-int(B)) u (X\B) c H.

Hence X\Hcscl(X\B) nB = 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=0 and hence, H=X.

Lemma 4.1.

If B is g *s - closed, then scl(B)\ B is g *s - open. Proof .

Suppose that B is g *s - closed. Then by Lemma 3.1, scl(B)\B does not contain any non-empty g-closed set. Therefore, scl(B)\B is g *s - open.

Theorem 4.2.

For each xeX, then either {x} is g-closed or

X\{x} is g*s - closed.

Proof.

If {x} is not g-closed, then the only g-open set containing X\{x} is X, hence, scl(X\ {x})c X is contained in X and therefore, X\{x} is g *s - closed.

5. Strongly semi - T1 spaces

In this section, we introduce strongly

semi - T1 spaces and discuss some of their 2

properties. Definition 5.1.

A topological space X is said to be :

(1) strongly semi - T1 (= st. semi - T1 ) if

every gs-closed set in X is g *s - closed,

(2) semi - Tp if every g *s - closed set in

X is closed, ( 3 ) semi - Tb if every gs-closed set in X is semi-closed.

Remark 5.1.

semi - Tp and st. semi - T1 spaces are

independent as may be seen from Example 5.1. Example 5.1.

If X={u, v, w} with t1 ={x {u}}and T2 ={x{u, v}}, then (X,T1)is semi - Tp but

not st. semi - T1 , since a subset {u, v} is gs-closed

but not g *s - closed. Also, (X,t2) is st.

semi - Tj but not semi - Tp, where a subset 2

{u, w} is g*s - closed but not closed. Theorem 5.1.

For a space (X,t) , the following statements hold :

(1) Every T0 - space (resp. semi - Tb ,

semi - Ti , a - space ) is st. semi - T^ , 2 2

(2) Every semi - Tb space is semi - Tp.

The converses of the above theorem need not be true as may be seen from the following examples.

Example 5.2 .

Let X={p, q, r} with the following topologies:

(1) t1 is the indiscrete topology,

(2) T2={^,{p, q},X},

(3) T3 ={^,{p},{q, r},X}.

Then (X) is st. semi - T1 space but it is not

a T0 - space, since X is the only open set contains any two distinct points from X.

Further (X, t2 ) is a st. semi - T1 space but

it is not semi - Tb ( resp. T1 - space , a - space ).

Furthermore, (X,t3) is a semi - Tp space but it is not semi - Tb , since a subset {p, q} is gs-closed but not semi-closed.

Corollary 5.1.

For a space (X,t) , the following are hold :

(1) Every Tb - space is semi - Tb ,

(2) Every semi - Tb space is semi - T1 ,

(3) Every semi - Tp space is Tp .

The converses of the above corollary need not be true as may be seen from the following examples.

Example 5.3 .

A space (X,T2)in Example 5.2, is a semi - Tx

space but it is not semi - Tb . Example 5.4 .

If X= {p, q, r} with т ={X ,ф, {p}}, then (X ,т) is a Tp - space but it is not semi - Tp, since a subset

{p, q} is g*s - closed but it is not closed.

Remark 5.2.

(1) st. semi - T1 and semi-Tn2 spaces are

independent,

(2) st. semi - T1 and Td-spaces are

independent.

Example 5.5.

(1) A space (Х,т 3) in Example 5.2, is

a semi-T1\2 space but it is not st.

semi - T1 , since a subset {q} is gs-closed 2

but it is not g *s - closed. Also, a space (Х,т2) in Example 5.2, is a st.

semi - T1 space but it is not semi-T1\2. 2

(2) A space (Х,т) in Example 5.4, is

a Td-space but it is not st. semi - T1 ,

where a subset {p,q} is gs-closed but it is not g *s - closed.

(3) A space (X,t) where X= {p, q, r} and Remark 5.3.

t ={X{p},{p,r}} is a st. We can summarize the following diagram by

^ u . .. • . T using [11,17] and the above results.

semi - Ti space but it is not a Td-space. & L 1

T - space <--► Td-space ^ Tb - space

+ __ t

T0 - spacest. -► semi - T1 space <——► semi - Tp

t / * 1

T1 - space m-y/--► semi - T1 space ^ semi - Tb

Theorem 5.2.

If X is a st. semi - T1 space, then the following

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) Proof.

(1) Suppose that {x} is not closed, for some x e X . Then the only open set containing X\{x} is X , hence X\{x} is gs-

closed. Since X is a st. semi - T1 , then

X\{x} is g*s - closed.

Therefore, {x} is g*s - open in X.

(2) Assume that {x} is not closed, for every x e 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.

Theorem 5.3.

The image of a g *s - closed set is g *s - closed under gc-irresolute and pre-semi-closed mappings.

Proof. If f(B)cH, where H is g-open in Y, then

B cf (H) and scl(B) c f _1(H).

Hence, fscl(B)) c H and f(scl(B)) is a semi-closed set in Y. Since,

scl (f (B)) c scl(f (scl(B))) = f (scl(B)) c H , then f(B) is g *s - closed in Y.

Theorem 5.4.

If B is a g *s - closed ( resp. g *s - open ) subset of Y, / : (X,t)^(Y,c) is a bijection irresolute and

closed mappings, then f (B) is g s - closed (resp. g *s - open) in X.

Proof.

Assume that B is a g *s - closed subset of Y and

f (B) c H, where H is g-open in X. Then we need to show that scl(f B)) c H or

scl(f (B)) nX \ H = 0 .Now, f (scl (f _1(B))) IX \ H) c scl (B)\ B, then by

Lemma 3.1,

We have scl(f _1(B)) I (X \ H) = 0. Therefore,

f (B) is g *s - closed in X.

By taking complements, we can show that, if B is

* -1 *

g *s - open in Y, then f (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, strongly generalized semi-open (= g * s - open) sets and we study some

of their properties. Furthermore, we discuss the conditions which are added to these concepts in order to coincide with the concept of semi-colsed [5], regular -open [16] and clopen sets. Furthermore, we define some spaces on these

concepts such as: strongly semi- T1

(= st. semi- T1), semi - Tb and semi - Tp spaces 2

and we give the relation between these spaces and other spaces which are defined above. Finally, we investigate some of their characterizations.

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