 Open Access
 Total Downloads : 690
 Authors : K. Bala Deepa Arasi, G. Suganya
 Paper ID : IJERTV4IS120416
 Volume & Issue : Volume 04, Issue 12 (December 2015)
 DOI : http://dx.doi.org/10.17577/IJERTV4IS120416
 Published (First Online): 23122015
 ISSN (Online) : 22780181
 Publisher Name : IJERT
 License: This work is licensed under a Creative Commons Attribution 4.0 International License
on g* – Closed Sets in Topological Spaces

Bala Deepa Arasi1
Assistant Professor of Mathematics, A.P.C.Mahalaxmi College for Women, Thoothukudi, TN
G. Suganya2
M.Phil Scholar, A.P.C.Mahalaxmi College for Women,
Thoothukudi, TN
Abstract – In this paper, we introduce a new class of sets called g*closed sets in topological spaces. A subset A of X is said to
be g*closed if Cl(A) U whenever A U and U is g*open
in X. Also we study some of its basic properties and investigate the relationship with other existing closed sets in topological space. As an application, we introduce four new spaces namely Tg*space and gTg*space
Keywords: g*open sets, closure, closed sets, g*closed sets.

INTRODUCTION
Levine [8] introduced generalized closed sets (briefly g closed sets) in topological spaces and studied their basic properties. Veerakumar [17] introduced and studied closed sets. Veerakumar [16] introduced g*closed sets in topological spaces and studied their properties. The aim of this paper is to introduce a new class of generalized closed sets called g*closed sets. Applying these sets, we obtain four new spaces namely Tg*space and gTg*space.

PRELIMINARIES

Throughout this paper (X, ) (or simply X) represents topological spaces on which no separation axioms are assumed unless otherwise mentioned. For a subset A of (X,), Cl(A), Int(A) and Ac denote the closure of A, interior of A and the complement of A respectively. We are giving some definitions.
Definition 2.1: A subset A of a topological space (X,) is called

a semiopen set[9] if A Cl(Int(A)).

an open set[11] if A Int(Cl(Int(A))).

a regular open set[15] if A = Int(Cl(A)).

an semi preopen set[1] if A Cl(Int(Cl(A)).
The complement of a semiopen (resp.open, regularopen and semi preopen) set is called semiclosed (resp.closed, regularclosed,semi preclosed) set.
The intersection of all semiclosed (resp.closed, regular closed and semi preclosed) sets of X containing A is called the semiclosure (resp.closure, regular closure and semi preclosure) of A and is denoted by sCl(A) (resp.Cl(A), rCl(A) and spCl(A)). The family of all semiopen (resp. – open, regularopen and semi preopen) subsets of a space X is denoted by SO(X) (resp. O(X), rO(X) and spO(X)).
Definition 2.2: A subset A of a topological space (X,) is called

a generalized closed set (briefly gclosed)[8] if Cl(A) U whenever A U and U is open in X.

a sgclosed set[3] if sCl(A) U whenever A U and U is semiopen in X.

a gsclosed set[2] if sCl(A) U whenever A U and U is open in X.

a gclosed set[10] if Cl(A) U whenever A
U and U is open in X.

a gr*closed set[7] if rCl(A) U whenever A U and U is gopen in X.

a g*closed set[16] if Cl(A) U whenever A U and U is gopen in X.

a g**closed set[12] if Cl(A) U whenever A U and U is g*open in X.

a g*sclosed set[13] if sCl(A) U whenever A
U and U is gsopen in X.

a (gs)*closed set[6] if Cl(A) U whenever A
U and U is gsopen in X.

a gspclosed set[5] if spCl(A) U whenever A
U and U is open in X.

a closed set[17] if sCl(A) U whenever A U and U is sgopen in X.

a gclosed set [14]if Cl(A) U whenever A

U and U is open in X.
The complement of a gclosed (resp. sgclosed, gsclosed, g closed, gr*closed, g*closed, g**closed, g*sclosed, (gs)* closed, gspclosed, closed and gclosed) set is called g open (resp. sgopen, gsopen, gopen, gr*open, g*open, g**open, g*sopen, (gs)*open, gspopen, open and g open) set.
Definition 2.3: Cl(A) is defined as the intersection of all – closed sets containing A.
Definition 2.4: A space (X,) is called a

a T1/2 space[8] if every gclosed set in X is closed.

a T*1/2 space[16] if every g*closed set in X is closed.

a Tb space[4] if every gclosed set in X is closed.
Remark 2.5: rclosed(ropen) closed(open) closed( open) semiclosed(semiopen) closed(open) semi preclosed(semi preopen)
3. g*CLOSED SETS We introduce the following definition.
Definition 3.1: A subset A of a topological space (X,) is called a g*closed set if Cl(A) U whenever A U and U is g*open in X. The family of all g*closed sets of X are denoted by g*C(X).
Definition 3.2: The complement of a g*closed set is called g*open set. The family of all g*open sets of X are denoted by g*O(X).
Example 3.3: Let X = {a,b,c} and = {X,,{a},{b},{a,b},
{a,c}} then {X,,{b},{c},{a,c},{b,c}} are g*closed sets
and {X,,{b},{a},{a,c},{a,b}} are g*open sets in X.
Proposition 3.4: Every closed set is g*closed set.
Proof: Let A be any closed set in X and U be any g*open set in X such that A U. Since A is closed, Cl(A) = A for every subset A of X. By Remark 2.5, every closed set is closed, Cl(A) Cl(A) = A U. Therefore, Cl(A) U where U is g* open. Hence, A is g*closed set.
The following example shows that the converse of the above proposition need not be true.
Example 3.5: Let X = {a,b,c} and = {X, ,{a}}. g*C(X)
= {X,,{b},{c},{a,b},{b,c},{a,c}}. Here, {b},{c},{a,b},{a,c} are g*closed sets but not closed sets in X.
Proposition 3.6: Every semiclosed set is g*closed set. Proof: Let A be any semiclosed set in X such that A U where U is g*open. Since A is semiclosed, sCl(A) = A. By Remark 2.5, every semiclosed set is closed, Cl(A)
sCl(A) = A U. Therefore, Cl(A) U where U is g* open. Hence, A is g*closed set.
The converse of the above proposition need not be true as shown in the following example.
1
Example 3.7: Let X = {a,b,c} and = {X, ,{a},{a,b}}. s
C(X) = {X,,{b},{c},{b,c}} and g*C(X) = {X, ,{b},
{c},{b,c},{a,c}}. Here, {a,c} is g*closed set but not semi closed set in X.
Proposition 3.8: Every closed set is g*closed set.
Proof: The proof follows from the result that every closed set is semiclosed and by proposition 3.6.
The converse of the above proposition need not be true as shown in the following example.
Example 3.9: Let X = {a,b,c} and = {X,,{b},{a,c}}. –
C(X) = {X,,{b},{a,c}} and g*C(X) = {X,,{a},{b},
{c},{a,b},{b,c},{a,c}}. Here, {a},{c},{a,b},{b,c} are g* closed sets but not closed sets in X.
Proposition 3.10: Every regular closed set is g*closed set. Proof: The proof follows from the result that every regular closed set is closed and by proposition 3.4.
The reverse implication does not hold as shown in the following example.
Example 3.11: Let X = {a,b,c} and = {X, ,{b}}. rC(X) =
{X,} and g*C(X) = {X,,{a},{c},{a,b},{b,c},{a,c}}. Here, {a},{c},{a,b},{b,c},{a,c} are g*closed sets but not regular closed sets in X.
Proposition 3.12: Every gclosed set is g*closed set.
Proof: Let A be any gclosed set in X. Let U be any open set in X such that A U. Since Every open set is g*open set, we have Cl(A) Cl(A) U. Therefore, Cl(A) U where U is g*open. Hence, A is g*closed set.
The reverse implication does not hold as shown in the following example.
Example 3.13: Let X = {a,b,c,d} and = {X,,{d},{a,b}
,{a,b,d}}. gC(X) = {X,{c},{a,c},{b,c},{c,d},{a,b,c},
{b,c,d},{a,c,d}} and g*C(X) = {X,,{c},{d},{a,b},{a,c},
{b,c},{c,d},{a,b,c},{b,c,d},{a,c,d}}. Here, {d},{a,b} are g*–closed sets but not gclosed sets.
Proposition 3.14: Every gclosed set is g*closed set. Proof: Let A be any gclosed set in X. Let U be any open set in X such that A U. Since Every open set is g*open set, we have Cl(A) Cl(A) U. Therefore, Cl(A) U where U is g*open. Hence, A is g*closed set.
The reverse implication does not hold as shown in the following example.
Example 3.15: Let X = {a,b,c} and = {X,,{a},{b},{a,b}}.
gC(X) = {X,,{c},{b,c},{a,c}} and g*C(X) =
{X,,{a},{b},{c},{b,c},{a,c}}. Here, {a},{b} are g*– closed sets but not gclosed sets.
Proposition 3.16: Every gr*closed set is g*closed set. Proof: Let A be any gr*closed set in X. Let U be any g* open set in X such that A U. Since Every g*open set is g open set and A is gr*closed, rCl(A) U. For every subset A of X, Cl(A) rCl(A) and so Cl(A) U where U is g* open. Hence, A is g*closed set.
The reverse implication does not hold as shown in the following example.
Example 3.17: Let X = {a,b,c} and =
{X,,{b},{a,b},{b,c}}. gr*C(X) = {X,,{a,c}} and g*
C(X) = {X,,{a}, {c},{a,c}}. Here, {a},{c} are g*closed sets but not gr*closed sets.
Proposition 3.18: Every g*closed set is g*closed set. Proof: Let A be any g*closed set in X. Let U be any g*open set in X such that A U. Since Every g*open set is gopen set and A is g*closed, we have Cl(A) U where U is g* open. Hence, A is g*closed set.
The following example shows that the converse of the above proposition need not be true.
Example 3.19: Let X = {a,b,c} and = {X, ,{a}}. g*C(X) =
{X,,{b,c}} and g*C(X) = {X,,{b},{c},{a,b},
{b,c},{a,c}}. Here, {b},{c}, {a,b},{a,c} are g*closed sets but not g*closed sets in X.
Proposition 3.20: Every g**closed set is g*closed set. Proof: Let A be any g**closed set in X. Let U be any g* open set in X such that A U. Since A is g**closed, Cl(A)
U. For every subset A of X, Cl(A) Cl(A) and so Cl(A) U where U is g*open. Hence, A is g*closed set.
The following example shows that the converse of the above proposition need not be true.
Example 3.21: Let X = {a,b,c} and = {X, ,{a},{b},{a,b}}.
g**C(X) = {X,,{c},{b,c},{a,c}} and g*C(X) =
{X,,{a},{b},{c},{b,c},{a,c}}. Here, {a},{b} are g*closed sets but not g*closed sets in X.
Proposition 3.22: Every g*sclosed set is g*closed set. Proof: Let A be any g*sclosed set in X. Let U be any g* open set in X such that A U. Since Every g*open set is gsopen set and A is g*sclosed, sCl(A) U. For every subset A of X, Cl(A) sCl(A) and so Cl(A) U where U is g*open. Hence, A is g*closed set.
The converse of the above proposition need not be true as shown in the following example.
Example 3.23: Let X = {a,b,c} and = {X, ,{a}}. g*sC(X)
= {X, ,{b},{c},{b,c}} and g*C(X) = {X, ,{b},
{c},{a,b},{b,c},{a,c}}. Here, {a,b},{a,c} are g*closed sets but not g*sclosed sets in X.
Proposition 3.24: Every (gs)*closed set is g*closed set. Proof: Let A be any (gs)*closed set in X. Let U be any g* open set in X such that A U. Since Every g*open set is gsopen set and A is (gs)*closed, Cl(A) U. For every subset A of X, Cl(A) Cl(A) and so Cl(A) U where U is g*open. Hence, A is g*closed set.
The converse of the above proposition need not be true as shown in the following example.
Example 3.25: Let X = {a,b,c} and = {X, ,{a}}. (gs)*
C(X) = {X, ,{b,c}} and g*C(X) = {X, ,{b},
{c},{a,b},{b,c},{a,c}}. Here, {b},{c},{a,b},{a,c} are g* closed sets but not (gs)*closed sets in X.
Proposition 3.26: Every g*closed set is gspclosed set. Proof: Let A be any g*closed set in X. Let U be any open set such that A U. Since, Every open set is g*open set, Cl(A) U. For every subset A of X, spCl(A) Cl(A) and so spCl(A) U where U is open. Hence, A is gspclosed set.
The reverse implication does not hold as shown in the following example.
Example 3.27: Let X = {a,b,c,d} and = {X,,{a,c},{a,b,c},
{a,c,d}}. gspC(X) = {X,,{a},{b},{c},{d},{a,b},{a,d},{b,c},
{b,d},{c,d},{b,c,d},{a,b,d}} and g*C(X) = {X,,{b},{d},
{b,d},{b,c,d},{a,b,d}}. Here, {a},{c},{a,b},{a,d},
{b,c},{c,d} are gspclosed sets but not g*closed sets.
Proposition 3.28: Every g*closed set is gclosed set. Proof: Let A be any g*closed set in X. Let U be any open set such that A U. Since, Every open set is g*open set, Cl(A) U where U is open. Hence, A is gclosed set.
The reverse implication does not hold as shown in the following example.
Example 3.29: Let X = {a,b,c,d} and = {X,,{d},{a,b},
{a,b,d}}. gC(X) = {X,,{a},{b},{c},{d},{a,b},{a,c},{b,c},
{c,d},{a,b,c},{b,c,d},{a,c,d}} and g*C(X) = {X,,{c},{d},
{a,b},{a,c},{b,c},{c,d},{a,b,c},{b,c,d},{a,c,d}}. Here,
{a},{b} are gclosed sets but not g*closed sets.
Remark 3.30: The following diagram shows the relationship of g*closed sets with other known existing sets. A B represents A implies B but not conversely.
2 3
14 4
13 5
12 1 6
11 7
10 8
9
1. g*closed 2. Closed 3. semiclosed
4. closed 5. regularclosed 6. gclosed

gclosed 8. gr*closed 9. g*closed 10.g**closed 11.g*sclosed 12.(gs)*closed 13.gspclosed 14.gclosed.

CHARACTERIZATION
Lemma 4.1: The finite union of g*closed sets need not be g*closed set.
Example 4.2: Let X = {a,b,c} and = {X,,{a},{b},{a,b}}. g*C(X) = {X,,{a},{b},{c},{b,c},{a,c}}. Here, {b} =
{a,b} is not g*closed set.
Lemma 4.3: The finite intersection of g*closed sets need not be g*closed set.
Example 4.4: Let X = {a,b,c} and = {X,,{a}}. g*C(X) =
{X,,{b},{c},{a,b},{b,c},{a,c}}. Here, {a,c} = {a} is not g*closed set.
Proposition 4.5: Let A be a g*closed set of X. Then Cl(A)A does not contain a nonempty g*closed set.
Proof: Suppose A is a g*closed set. Let F be a g*closed set contained in Cl(A)A. Now Fc is a g*open set of X such that A Fc . Since A is g*closed, we have Cl(A) Fc. Hence, F Cl(A))c . Also, F Cl(A)A. Therefore, F
Cl(A)A) (Cl(A))c Cl(A) (Cl(A))c = . Hence, F must be .
Proposition 4.6: If A is both g*open and g*closed set of X, then A is closed.
Proof: Since A is both g*open and g*closed, we have Cl(A) A. Therefore, A = Cl(A). Hence, A is closed.
Proposition 4.7: The intersection of a g*closed set and a – closed set of X is always g*closed set.
Proof: Let A be a g*closed set and B be a closed set. Since A is g*closed, Cl(A) U whenever A U and U is g*open. Let B be such that B U where U is g*open. Now, B) Cl(A) Cl(B) U B U. Hence, A B is g*closed set. Therefore, intersection of any g* closed set and a closed set of X is always g*closed set.
Proposition 4.8: For X, the set X{x} is g*closed or g* open.
Proof: Suppose X{x} is not g*open then X is the only g* open set containing X{x} and Cl{X{x}} X. Hence X
{x} is g*closed in X.
Proposition 4.9: If A is g*closed and A Cl(A), then B is g*closed.
Proof: Let U be a g*open set of X such that B U then A
U. Since A is g*closed, then Cl(A) U. Now Cl(B)
Cl(A) U. Therefore B is g*closed in X.
Proposition 4.10: Let A Y X and suppose that A is g* closed in X, then A is g*closed relative to Y.
Proof: Given that A Y X and A is g*closed in X. To show that A is g*closed relative to Y. Let A U, where U is g*open in X. Since A is g*closed, A U, implies Cl(A) U. Therefore, Cl(A) U. Thus A is g*closed relative to Y.
Proposition 4.11: Suppose that B X,B is g*closed relative to A and that A is both g*open and g*closed set of X, then B is g*closed relative to X.
Proof: Let B G and G be an g*open set in X. But given that B A X, therefore B G. since B is g*closed relative to A, Cl(B) AG. Hence Cl(B) G. Thus A (Cl(B)) (Cl(B))c G Cl(B)c. Since A is both g*open and g*closed set in X, by proposition 4.6, A is closed, we have Cl(A) = A G Cl(B)c. Also B A implies Cl(B) Cl(A). Thus Cl(B) Cl(A) G Cl(B)c. Therefore, Cl(B) G. Since Cl(A) is not contained in (Cl(B))c. Thus B is g*closed relative to X.

APPLICATIONS

As an applications of g*closed sets, we introduce four new spaces namely, Tg*space and gTg*space.
Definition 5.1: A Space (X,) is called a Tg*space if every g*closed set in it is closed.
Definition 5.2: A Space (X,) is called a gTg*space if every g*closed set in it is gclosed.
Definition 5.3: A Space (X,) is called a g*Tg*space if every g*closed set in it is g*closed.
Definition 5.4: A Space (X,) is called a gTg*space if every g*closed set in it is gclosed.
Proposition 5.5: Every Tg*space is gTg*space.
Proof: Let (X,) be Tg*space. Let A be g*closed set in (X,). Since (X, ) is Tg*space, A is closed. But every closed set is gclosed set. Therefore, A is gclosed. Hence, (X,) is gTg*space.
The converse of the above proposition need not be true as shown in the following example.
Example 5.6: Let X = {a,b,c} and = {X,,{a,c},{a,b,c},{a,c, d}}
g*C(X) = {X,,{b},{d},{b,d},{b,c,d},{a,b,d}}
gC(X) = {X,,{b},{d},{b,d},{b,c,d},{a,b,d}} C(X) = {X,,{b},{d},{b,d}}
Here, (X,) is gTg*space but not gTg*space.
Proposition 5.7: Every Tg*space is T1/2space.
Proof: Let (X,) be Tg*space. Let A be gclosed set in (X,). By proposition 3.12, A is g*closed set. Since (X,) is Tg* space, A is closed. Hence, (X,) is T1/2space.
The converse of the above proposition need not be true as shown in the following example.
Example 5.8: Let X = {a,b,c} and = {X,,{a},{b},{a,b}}
g*C(X) = {X,,{a},{b},{c},{b,c},{a,c}}
gC(X) = {X,,{c},{b,c},{a,c}}
C(X) = {X,,{c},{b,c},{,a,c}}
Here, (X,) is T1/2space but not Tg*space.
Proposition 5.9: Every Tg*space is T*1/2space.
Proof: Let (X,) be Tg*space. Let A be g*closed set in (X,). By proposition 3.18, A is g*closed set. Since (X,) is Tg*space, A is closed. Hence, (X,) is T*1/2space.
The converse of the above proposition need not be true as shown in the following example.
Example 5.10: Let X = {a,b,c} and = {X,,{a},{b},{a,b}}
g*C(X) = {X,,{a},{b},{c},{b,c},{a,c}}
g*C(X) = {X,,{c},{b,c},{a,c}}
C(X) = {X,,{c},{b,c},{,a,c}}
Here, (X,) is T*1/2space but not Tg*space.
Proposition 5.11: Every Tg*space is Tbspace.
Proof: Let (X,) be Tg*space. Let A be gclosed set in (X,). By proposition 3.14, A is g*closed set. Since (X,) is Tg*space, A is closed. Hence, (X,) is Tbspace.
The converse of the above proposition need not be true as shown in the following example.
Example 5.12: Let X = {a,b,c} and = {X,,{a},{b},{a,b}}
g*C(X) = {X,,{a},{b},{c},{b,c},{a,c}}
gC(X) = {X,,{c},{b,c},{a,c}}
C(X) = {X,,{c},{b,c},{,a,c}}
Here, (X,) is Tbspace but not Tg*space.
Remark 5.13: The following diagram shows the relationship about Tg*space, gTg*space, g*Tg* and gTg*space with other known existing spaces. A B represents A implies B but not conversely.
2
3
1
6 REFERENCES

D. Andrijevic, Semipreopen sets, Mat. Vesnik., 38(1) (1986), 2432.

S.P.Arya and T.M.Nour, Characterizations of SNormal spaces, Indian J.Pure Appl. Math., Vol 21(1990).

P.Bhattacharya and B.K.Lahiri, Semigeneralized closed sets in Topology, Indian J. Math., 29(1987), 375382.

Devi R., Maki H., and Balachandran R., generalized closed maps and generalized closed maps, Indian J. Pure. Appl. Math., 14(1993), 4154.

J.Dontchev, on generalizing semipreopen sets, Mem. Fac. Sci. Kochi. Ser.A, Math., 16(1995), 3548.

Elvina Mary.L(2014), (gs)*closed sets in topological spaces, International Journal of Mathematics Trends and Technology,(7) 8393.

K.Indirani, P.sathishmohan, and V.Rajendran, On gr*closed sets in a topological spaces, International Journal of Mathematics Trends and Technology – Vol – 6, Feb 2014,(142 148).

N. Levine, Generalized closed sets in topology, Rend.Circ.Mat.Palermo, 19(2)(1970) 8996.

N. Levine, Semiopen sets and semicontinuity in topological spacemer.Math. Monthly, 70(1963), 3641.

Maki H., Devi R., and Balachandran R., Associated topologies of generalized closed sets and generalized closed sets, Mem. Fac. Sci. Kochi. Univ. Ser.A, Math., Vol15,pp.51 63,1994.

ONjastad, On some classes of nearly open sets, Pacific J Math., 15(1965).

Pauline Mary Helen M, Ponnuthai selvarani, Veronica Vijayan, g**closed sets in topological spaces, International Journal of Mathematical Archives, 3(5), (2012),115.

P.Pushpalatha and K.Aniitha, g*sclosed sets in topological spaces, Int. J. Contemp. Math. Sciences, Vol 6., March 2011, no 19, 917929.

Ramya N., and Parvathi A., closed sets in topological spaces, IJMA, Vol.2(10), PP.19921996, 2011.

Stone.M,Application of the theory of Boolean rings to general topology, Trans. Amer. Maths. Soc., 41(1937) 374481.

M.K.R.S. Veerakumar, Between closed sets and gclosed sets, Mem. Fac. Sci. Kochi. Univ. (Math), 21(2000), (119).

M.K.R.S. Veerakumar, Between Semiclosed sets and Semi pre closed sets, Rend, Instint. Univ. Trieste(Italy) XXX11, 25 41(2000).
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5
1. Tg*space 2. gTg*space 3. T1/2space
4. T*1/2space 5. Tbspace.