International Journal of Scientific & Engineering Research, Volume 3, Issue 6, June-2012 1

ISSN 2229-5518

sg-Interior and sg-Closure in Topological spaces

S.Sekar and K.Mariappa

Abstract: In this paper, we introduce sg-interior, sg-closure and some of its basic properties.

Keywords: sg-open; sg-closed; sg-int(A); sg-cl(A); sg-Hausdorff space. AMS Subject Classification: 54C10, 54C08, 54C05, 54E55.

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1. INTRODUCTION AND PRELIMINARIES

evine [6] introduced generalized closed sets in topology as a generalization of closed sets. This concept was found to be useful and many results in general topology were improved. Many researchers like Arya et al [2], Balachandran et al [3], Bhattarcharya et al [4], Arockiarani et al [1], Gnanambal [5] Malghan [7], Nagaveni [8] and Palaniappan et al [9] have worked on generalized closed sets. In this paper, the notion of sg-interior is defined and some of its basic properties are investigated. Also we introduce the idea of sg- closure in topological spaces using the notions of sg-closed sets and
The complements of the above mentioned closed sets are their respective open sets.

Defintion 1.3: Let X be a topological space and let x X. A subset N of X is said to be sg-neighbourhood of x if there exists a sg-open set G such that x G N.

2. SG–CLOSURE AND INTERIOR IN TOPOLOGICAL SPACE.

obtain some related results.

Definition 2.1: Let A be a subset of X. A point x

A is said to be

Throughout the paper, X and Y denote the topological spaces
sg-interior point of A is A is a sg-neighbourhood of x. The set of all

X , and Y ,

respectively and on which no separation axioms
sg-interior points of A is called the sg-interior of A and is denoted by
are assumed unless otherwise explicitly stated.

Definition 1.1 A subset A of a space X is called

1) A preopen set if A int(cl(A)) and a preclosed if cl(int(A)) A
2) A regular open set if A = int(cl(A)) and regular closed set if A =
cl(int(A))

3) A semi open set if A cl(int(A)) and semi closed set if int(cl(A))
sg-int(A).

Theorem 2.1: If A be a subset of X. Then sg-int(A) = { G : G is a sg-open, G A}.

Proof: Let A be a subset of X.

x sg-int(A) x is a sg-interior point of A.

A is a sg-nbhd of point x.
A
The intersection of all preclosed subsets of X containing A
is called pre-closure of A and is denoted by pcl(A)

Definition1.2: A subset A of a space X is called

1) g-closed set[6] if if cl(A) U whenever A U and U is open in
X

2) semi generalized closed set [4] if scl(A) U whenever A U
and U is semi open in X.

3) generalized preclosed set [ 7] if clint(A) U whenever A U
and U is open in X.

there exists sg-open set G such that x G A.

x {G:G is a sg-open, G A} Hence sg-int(A) = {G : G is a sg-open, G A}.

Theorem 2.2: Let A and B be subsets of X. Then

(i) sg-int(X) = X and sg-int( ) = (ii) sg-int(A) A.

(iii) If B is any sg-open set contained in A, then B sg - int(A). (iv) If A B, then sg-int(A) sg-int(B).
(v) sg-int(sg-int(A)) = sg-int(A).

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Proof: (i) Since X and are sg open sets, by Theorem
sg-int(X) = { G : G is a sg-open, G X}

sg-int(A) sg-int(B) sg-int(A B).

Theorem 2.5: If A and B are subsets of X, then sg-int(A B) = sg- int(A) sg-int(B).

= X all sg open sets
= X.

Proof: We know that A B A and A B B. We have sg- int(A B) sg-int(A) and sg-int(A B) sg-int(B).

(ie) int(X) = X. Since is the only sg- open set contained in , sg-int( ) =
This implies that sg-int(A B) sg-int(A) sg-int(B)
-----(1)

(ii) Let x sg-int(A) x is a interior point of A.

A is a nbhd of x.

Again let x sg-int(A) sg-int(B). Then x sg-int(A) and x sg-int(B). Hence x is a sg-int point of each of sets A and B. It follows that A and B is sg-nbhds of x, so that their intersection A B is also a sg-nbhds of x. Hence x sg-int(A B). Thus
x sg-int(A) sg-int(A) implies that x sg-int(A B).

Thus, x sg

int( A)

A .

x A .

Therefore sg-int(A) sg-int(B) sg-int(A B) ------(2)
From (1) and (2),

Hence sg-int(A) A.

(iii) Let B be any sg-open sets such that B A. Let x

B . Since

We get sg-int(A B)=sg-int(A) sg-int(B).

Theorem 2.6: If A is a subset of X, then int(A) sg-int(A).

B is a sg-open set contained in A. x is a sg-interior point of A. (ie) x sg-int(A).

Hence B sg-int(A).

(iv) Let A and B be subsets of X such that A B. Let x sg- int(A). Then x is a sg-interior point of A and so A is a sg-nbhd

of x. Since B A, B is also sg-nbhd of x. x sg-int(B). Thus
we have shown that x sg-int(A) x sg-int(B).

Theorem 2.3: If a subset A of space X is sg-open, then sg-int(A)

=A.

Proof: Let A be sg-open subset of X. We know that sg-int(A) A. Also, A is sg-open set contained in A. From Theorem

(iii) A sg-int(A). Hence sg-int(A) = A.
The converse of the above theorem need not be true, as seen from the
following example.

Example 2.1: Let X = {a,b,c} with topology

={X, , {b},{c},{a,b},{b,c}}. Then sg-O(X) = {X,
,{a},{b},{c},{a,b},{b,c}}. sg-int({a,c}) ={a} {c} { } =
{a,c}. But {a,c} is not sg-open set in X.

Theorem 2.4: If A and B are subsets of X, then sg-int(A) sg- int(B) sg-int(A B).

Proof. We know that A A B and B A B. We have Theorem 2.2

(iv) sg-int(A) sg-int(A B), sg-int(B) sg-int(A B).
This implies that

Proof: Let A be a subset of X.

Let x int(A) x {G : G is open, G A}.

there exists an open set G such that x G A. there exist a sg-open set G such that x G A,
as every open set is a sg-open set in X .

x {G : G is sg- open, G A}. x sg-int(A).
Thus x int(A) x sg-int(A). Hence int(A) sg-int(A).

Remark.2.1: Containment relation in the above theorem may be proper as seen from the following example.

Example 2.2: Let X ={a,b,c} with topology ={X, ,
{b},{c},{b,c}}. Then sg-O(X)={X,
,{b},{c},{a,b},{a,c},{b,c}}.

Let A = {a,b}. Now sg-int(A) = {a,b} and int(A) = {b}. It follows
that int(A) sg-int(A) and int(A) sg-int(A).

Theorem 2.7: If A is a subset of X, then g-int(A) sg-int(A), where g-int(A) is given by g-int(A) = {G : G is g-open, G A}. Proof: Let A be a subset of X.

Let x int(A) x {G : G is g-open, G A}.

there exists a g-open set G such that x G A

there exists a sg-open set G such that x G A, as every
g- open set is a sg-open set in X

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x {G : G is sg-open, G A}.

x sg-int(A). Hence g-int(A) sg-int(A).

Remark 2.2: Containment relation in the above theorem may be proper as seen from the following example.

Example 2.3: Let X ={a,b,c} with topology ={X, ,

{b},{c},{a,c}}. Then sg-o(X) = { X, , {a},{c},{a,b},{a,c},{b,c}}.

& g – open (X) = { X, ,{a},{c},{a,c}}. Let A = {b,c}, sg-int(A)

= {b,c} & g-int(A) = {c}. It follows g-int(A) sg-int(A) and g- int(A) sg-int(A) .

Definition 2.2: Let A be a subset of a space X. We define the sg-

closure of A to be the intersection of all sg-closed sets containing A.
In symbols, sg-cl(A) = {F : A F sgc(X)}.

Theorem 2.8: If A and B are subsets of a space X. Then

(i) sg-cl(X) = X and sg-cl( ) = (ii) A sg-cl(A).

(iii) If B is any sg-closed set containing A, then sg-cl(A) B. (iv) If A B then sg-cl(A) sg-cl(B).

Proof: (i) By the definition of sg-closure, X is the only sg-closed set containing X. Therefore sg-cl(X) = Intersection of all the sg-closed sets containing X = {X} = X. That is sg-cl(X) = X. By the definition of sg-closure, sg-cl( ) = Intersection of all the sg-clsed sets containing = { } = . That is sg-cl( ) = .

(ii) By the definition of sg-closure of A, it is obvious that A sg- cl(A).

(iii) Let B be any sg-closed set containing A. Since sg-cl(A) is the intersection of all sg-closed sets containing A, sg-cl(A) is contained in every sg-closed set containing A. Hence in particular sg-cl(A)
B.

(iv) Let A and B be subsets of X such that A B. By the definition sg-cl(B) = { F: B F sg-c(X)}. If B F sg-c(X), then sg- cl(B) F. Since A B, A B F sg-c(X), we have sg-cl(A)

F. There fore sg-cl(A) {F : B F sg-c(X)} = sg-cl(B). (i.e) sg-cl(A) sg-cl(A).

Theorem 2.9: If A X is sg-closed, then sg-cl(A) = A.

Proof: Let A be sg-closed subset of X. We know that A sg- cl(A). Also A A and A is sg-closed. By theorem (iii) sg-cl(A) A. Hence sg-cl(A) = A.

Remarks 2.3: The converse of the above theorem need not be true as seen from the following example.

Example 2.4: Let X ={a,b,c} with topology ={X, ,
{b},{c},{a,b},{b,c}}. Then sg-C(X)={X,
,{a},{c},{a,b},{b,c},{a,c}}. sg-cl({b}) ={b}. But {b} is not sg- closed set in X.

Theorem 2.10: If A and B are subsets of a space X, then sg-cl(A B) sg-cl(A) sg-cl(B).

Proof: Let A and B be subsets of X. Clearly A B A and A B B.

By theorem sg-cl(A B) sg-cl(A) and sg-cl(A B) sg-

cl(B).
Hence sg-cl(A B) sg-cl(A) sg-cl(B).

Theorem 2.11: If A and B are subsets of a space X then sg-cl(A B)= sg-cl(A) sg-cl(B).

Proof: Let A and B be subsets of X. Clearly A A B and B A B. We have sg-cl(A) sg-cl(B) sg-cl( A B)

----(1) Now to prove
sg-cl( A B) sg-cl(A) sg- cl(B).

Let x sg-cl(A B) and suppose x sg-cl(A) sg- cl(B). Then there exists sg-closed sets A1 and B1 with A A1, B B1 and x
A1 B1. We have A B A1 B1 and A1 B1 is sg-closed set

by theorem such that x A1 B1. Thus x sg-cl(A B) which is a contradiction to x sg-cl(A B). Hence sg-cl(A B)
sg-cl(A) sg-cl(B)

----(2) From (1) and (2), we have
sg-cl(A B)= sg-cl(A) sg-cl(B).

Theorem 2.12: For an x X, x sg-cl(A) if and only if V A

for every sg-closed sets V containing x.

Proof: Let x X and x sg-cl(A). To prove V A for every sg-open set V containing x. Prove the result by contradiction.

Suppose there exists a sg-open set V containing x such that

V A = . Then A X-V and X-V is sg-closed. We have sg-

cl(A) X - V. This shows that x sg-cl(A), which is a contradiction.
Hence V A for every sg-open set V containing x.

Conversly, let V A for every sg-open set V containing x. To prove x sg-cl(A). We prove the result by contradiction.
Suppose x sg-cl(A). Then x X – F and S – F is sg-open. Also
(X–F) A = , which is a contradiction. Hence x sg-cl(A).

Theorem 2.13: If A is a subset of a space X, then sg-cl(A) cl(A).

Proof: Let A be a subset of a space S. By the definition of closure, cl(A) = {F: A F C(X)}.

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If A F C(X)}, Then A F sg-C(X), because every closed set is sg-closed. That is sg-cl(A) F. There fore sg-cl(A) {F X

: F C(X)} = cl(A). Hence sg-cl(A) cl(A).

Remark 2.4: Containment relation in the above theorem may be proper as seen from the following example.

Example 2.5: Let X ={a,b,c} with topology ={X, ,

{b},{c},{a,c}}. Then sg-cl(X) = {X, , {a},{b},{c},{a,b},{b,c}} and g – cl (X) = { X, , {b},{a,b},{b,c}}. Let A = {b,c}, sg-cl(A) = {b,c} and g-cl(A) = {b}. It follows g-cl(A) sg-cl(A)
and g-cl(A) sg-cl(A) .

Theorem 2.14 : If A is a subset of X, then sg-cl(A) g-cl(A), where g-cl(A) is given by g-cl(A) = {F X : A F and f is a g-closed set in X}.

Proof: Let A be a subset of X. By definition of g-cl(A) =

{F X : A F and f is a g-closed set in X}. If A F and F is

g-closed subset of x, then A F sg-cl(X), because every g closed is sg-closed subset in X. That is sg-cl(A) F.
Therefore sg-cl(A) {F X : A F and f is a g-closed set in

X} = g-cl(A).
Hence sg-cl(A) g-cl(A).

Corrolory2.1: Let A be any subset of X. Then

(i) sg-int(A))c = sg-cl(Ac) (ii) sg-int(A) = (sg-cl(Ac)) (iii) sg-cl(A) = (sg-cl(Ac))

Proof: Let x sg-int(A))c. Then x sg-int(A). That is every sg- open set U containing x is such that U A. That is every sg-open
set U containing x is such that U Ac . By theorem

Defintion 3.1: A topological space x is said to be g-Hausdorff if whenever x and y are distinct points of X there are disjoint g-open sets U and V with x U and y V.

It is obvious that every Hausdorff space is g-Hausdorff space. The following example shows that the converse is not true.

Example 3.1: Let X = {a,b,c} and ={X, , {a}}. It is clear that X is not Hausdorff Space. Since {a}, {b} and {c} are all g- open, it follows that H is sg-Hausdorff Space.

Theorem3.1: Let X be a topological space and Y be Hausdorff. If f: X Y is injective and g-continous, then x is g-Hausdorff.

Proof: Let x and y be any two distinct points of X. Then f(x) and f(y) are distinct points of Y, because f is injective. Since Y is Hausdorff, there are disjoint open sets U and V in Y containing f(x) and f(y) respectively. Since f is g-continous and U V = , we

have f-1(U) and f-1(V) are disjoint g-open sets in X such that x f-1(U) and y f-1(V). Hence X is g-Hausdorff space.

Defintion3.2: A topological space X is said to be sg-Hausdorff Space if whenever x and y are distinct points of X there are disjoint sg-open sets U and V with x U and y V.

It is obvious that every g-Hausdorff space is a sg-Hausdorff space. The following example shows that the converse is not true.

Example 3.1: Let X = {a,b,c} and ={X, , {a}}. Since {a},
{b} and {c} are all sg-open, it implies that X is sg-Hausdorff space. Since {a}, {b} and {c} are not g-open in X , it follows that „a‟ and „c‟ can not be separated by any two disjoint g-open sets in X. Hence X is not g-Hausdorff Space.

Theorem3.2: Let X be a topological space Y be Hausdorff space. If f: X Y is injective and sg-continuous, then X is sg-Hausdorff Space.

Proof: Let x and y be any two distinct points of X. Then f(x) and f(y) are distinct points of Y, because f is injective. Since Y is Hausdorff, there are disjoint open sets U and V in Y containing f(x) and f(y) respectively. Since f is sg-continous and U V= , we

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x sg-int(A))c and there fore sg-int(A))c sg-cl(Ac).

have f (U) and f (V) are disjoint sg-open sets in X such that x

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Conversely, let x sg-cl(Ac).

Then by theorem, every sg-open set U containing x is such that U Ac . That is every sg-open set U containing x is such that U

A. This implies by definition of sg-interior of A, x sg-int(A). That is x sg-int(A))c and sg-cl(Ac) ( sg-int(A))c. Thus sg- int(A))c = sg-cl(Ac)
(ii) Follows by taking complements in (i).
(ii) Follows by replacing A by Ac in (i).
f (U) and y f (V). Hence X is sg-Hausdorff space.

Theorem3.3: Let X be a topological space Y be sg-Hausdorff Space. If f: X Y is injective and sg-irresolute, then X is sg- Hausdorff space.

Proof: Let x and y be any two distinct points of X. Then f(x) and

f(y) are distinct points of Y, because f is injective. Since Y is sg- Hausdorff, there are disjoint sg- open sets U and V in Y containing f(x) and f(y) respectively. Since f is sg-irresolute and U V = ,

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3. PRESERVATION THEOREMS CONCERNING G-

we have f (U) and f (V) are disjoint sg-open sets in X such that

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HAUSDORFF AND SG-HAUSDORFF SPACES

In this section we investigate preservation theorems concerning sg- Hausdorff spaces.
x f (U) and y f (V). Hence X is sg-Hausdorff space.

4. CONCLUSION

From the definitions of g-Hausdorff space and sg-Hausdorff space, we have result.

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X is a Hausdorff Space X is a g- Hausdorff Space X is a sg- Hausdorff Space.

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