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.

- - - - - - - - - - - - - - - - - - - - - - -

# 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.

# 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).

### 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).

International Journal of Scientific & Engineering Research, Volume 3, Issue 6, June-2012 2

ISSN 2229-5518

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

x

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).

=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.

### 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

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

ISSN 2229-5518

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).

### 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.

### 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).

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

International Journal of Scientific & Engineering Research, Volume 3, Issue 6, June-2012 4

ISSN 2229-5518

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) .

### 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.

### 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.

### 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

-1 -1

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

-1 -1

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.

### 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 = ,

-1 -1

# 3. PRESERVATION THEOREMS CONCERNING G-

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

-1 -1

# 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.

International Journal of Scientific & Engineering Research, Volume 3, Issue 6, June-2012 5

ISSN 2229-5518

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

## REFERENCES

1. I.Arockiarani, “Studies on Generalizations of Generalised Closed Sets and Maps in Topological Spaces”, Ph.D Thesis, Bharathiar University, Coimbatore, (1997).

2. S.P. Arya and R.Gupta,” On strongly Continous Mappings”, Kyungpook

Math., J.14 (1974), 131-143.

3. K.Balachandran, P. Sundaram and H.Maki, “On Generalized Contionous

Maps in Topological spaces”, Mem, I ac Sci. Kochi Uni.Math., 12 (1991),

5-13.

4. P.Bhattacharya and B.K.Lahiri, “Semi-genralized Closed sets in Topology”, Indian J.Math., 29(1987), 376 -382.

5. Y.Gnanambal, “On Generalized Pre-regular closed sets in Topological

Spaces”, Indian J. Pure Appl. Math., 28(1997), 351-360.

6. N.Levine, “Generalized Closed Sets in Topology”, Rend. Circ. Mat.

Palermo, 19 (1970), 89-96.

7. S.R.Malghan, “Generalized Closed Maps” J Karnatak Uni. Sci., 27(1982),

82-88.

8. N.Nagaveni,” Studies on Generalizations of Homeomorphisms in

Topological Spaces”, Ph.D., Thesis, Bharathiar University,(1999).

9. N.Palaniappan and K.C.Rao, “Regular Generalized Closed sets”,

Kyungpook, Math.J.,33(1993), 211-219.