on ψg* – Closed Sets in Topological Spaces

on g* – Closed Sets in Topological Spaces

1. 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.

   1. 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.

   2. 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

1. a semi-open set[9] if A Cl(Int(A)).

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

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

4. an semi pre-open set[1] if A Cl(Int(Cl(A)).

   The complement of a semiopen (resp.open, regular-open and semi pre-open) set is called semi-closed (resp.closed, regular-closed,semi pre-closed) set.

   The intersection of all semi-closed (resp.-closed, regular- closed and semi pre-closed) sets of X containing A is called the semi-closure (resp.-closure, regular closure and semi pre-closure) of A and is denoted by sCl(A) (resp.Cl(A), rCl(A) and spCl(A)). The family of all semi-open (resp. – open, regular-open and semi pre-open) 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

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

   2. a sg-closed set[3] if sCl(A) U whenever A U and U is semi-open in X.

   3. a gs-closed set[2] if sCl(A) U whenever A U and U is open in X.

   4. a g-closed set[10] if Cl(A) U whenever A

      U and U is open in X.

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

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

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

   8. a g*s-closed set[13] if sCl(A) U whenever A

      U and U is gs-open in X.

   9. a (gs)*-closed set[6] if Cl(A) U whenever A

      U and U is gs-open in X.

   10. a gsp-closed set[5] if spCl(A) U whenever A

       U and U is open in X.

   11. a -closed set[17] if sCl(A) U whenever A U and U is sg-open in X.

   12. a g-closed set [14]if Cl(A) U whenever A

U and U is open in X.

The complement of a g-closed (resp. sg-closed, gs-closed, g- closed, gr*closed, g*-closed, g**-closed, g*s-closed, (gs)*- closed, gsp-closed, -closed and g-closed) set is called g- open (resp. sg-open, gs-open, g-open, gr*open, g*-open, g**-open, g*s-open, (gs)*-open, gsp-open, -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

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

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

3. a Tb space[4] if every g-closed set in X is closed.

Remark 2.5: r-closed(r-open) closed(open) -closed(- open) semi-closed(semi-open) -closed(-open) semi pre-closed(semi pre-open)

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 semi-closed set is g*-closed set. Proof: Let A be any semi-closed set in X such that A U where U is g*-open. Since A is semi-closed, sCl(A) = A. By Remark 2.5, every semi-closed 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 semi-closed 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}}. r-C(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 g-closed set is g*-closed set.

Proof: Let A be any g-closed 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}}. g-C(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 g-closed sets.

Proposition 3.14: Every g-closed set is g*-closed set. Proof: Let A be any g-closed 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}}.

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.

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 g-open 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*s-closed set is g*-closed set. Proof: Let A be any g*s-closed set in X. Let U be any g*- open set in X such that A U. Since Every g*-open set is gs-open set and A is g*s-closed, 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*s-C(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*s-closed 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 gs-open 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 gsp-closed 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 gsp-closed 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}}. gsp-C(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 gsp-closed sets but not g*-closed sets.

Proposition 3.28: Every g*-closed set is g-closed 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 g-closed 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}}. g-C(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 g-closed 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. semi-closed

4. -closed 5. regular-closed 6. g-closed

1. g-closed 8. gr*-closed 9. g*-closed 10.g**-closed 11.g*s-closed 12.(gs)*-closed 13.gsp-closed 14.g-closed.

   1. 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 non-empty 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.

   2. 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 g-closed.

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 g-closed.

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 g-closed set. Therefore, A is g-closed. 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}}

g-C(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 T1/2-space.

Proof: Let (X,) be Tg*-space. Let A be g-closed set in (X,). By proposition 3.12, A is g*-closed set. Since (X,) is Tg*- space, A is closed. Hence, (X,) is T1/2-space.

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}}

g-C(X) = {X,,{c},{b,c},{a,c}}

C(X) = {X,,{c},{b,c},{,a,c}}

Here, (X,) is T1/2-space but not Tg*-space.

Proposition 5.9: Every Tg*-space is T*1/2-space.

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*1/2-space.

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*1/2-space but not Tg*-space.

Proposition 5.11: Every Tg*-space is Tb-space.

Proof: Let (X,) be Tg*-space. Let A be g-closed set in (X,). By proposition 3.14, A is g*-closed set. Since (X,) is Tg*-space, A is closed. Hence, (X,) is Tb-space.

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}}

g-C(X) = {X,,{c},{b,c},{a,c}}

C(X) = {X,,{c},{b,c},{,a,c}}

Here, (X,) is Tb-space 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

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1. Tg*-space 2. gTg*-space 3. T1/2-space

4. T*1/2-space 5. Tb-space.