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

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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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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.

obtain some related results.

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.

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).

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)

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}.

(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).

= X all sg open sets

= X.

(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).

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.

(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.

={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.

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

This implies that

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).

**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).

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).

**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) .

closure of A to be the intersection of all sg-closed sets containing A.

In symbols, sg-cl(A) = {F : A F sgc(X)}.

(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).

**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.

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).

for every sg-closed sets V containing x.

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).

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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).

**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) .

{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).

(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

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.

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.

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.

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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.

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

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In this section we investigate preservation theorems concerning sg- Hausdorff spaces.

x f (U) and y f (V). Hence X is sg-Hausdorff space.

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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