# on Î›Î¼-Generalized Closed Sets in Generalized Topological Spaces DOI : 10.17577/IJERTCONV3IS33012

Text Only Version

#### on -Generalized Closed Sets in Generalized Topological Spaces

1M. Jeyaraman,

1 Assistant Professor, Department of Mathematics, RDM Government Arts College,

Sivagangai.

2S. Chinthamani Vasthirani,

2Assosiate Professor, Department of Mathematics, Sree Sevugan Annamalai College,

Devakottai.

Abstract We introduce new classes of sets called Âµ- g-closed sets and Âµ- g-open sets in GTS. We also investigate several properties of such sets. It turns out that Âµ- g-closed sets and Âµ- g-open sets are weaker forms of closed sets and open sets, and stronger forms of g-Âµ-closed sets and g-Âµ-open sets, respectively.

2010 Mathematics Subject Classification: 57C10, 57C08, 57C05

Keywords and phrases: Âµ-g-closed set, locally closed sets, g-Âµ-closed sets.

1. INTRODUCTION

In 2010 E. Ekici and B. Roy introduced the notion of Âµ-sets in GTS. A Âµ-set is a set A which is equal to its kernel. In 2011 Bishwabhar and Ekici introduced and investigate (, Âµ)-closed sets by involving Âµ-sets and

Âµ-closed sets. The objective of this paper is to introduce new classes of sets called Âµ- g-closed sets and Âµ- g-open sets in GTS. It turns out that Âµ- g-closed sets and Âµ- g-open sets are weaker forms of closed sets and open sets, and stronger forms of g-Âµ-closed sets and g-Âµ-open sets, respectively.

We recall some notion defined in [ 1, 2]. Let X be a non-empty set and Âµ be a collection of subsets of

X. Then Âµ is called a generalized topology(briefly GT) on X if Âµ and Gi Âµ for i I implies G = iI Gi Âµ. We say Âµ is strong if X Âµ, and we call the pair (X, Âµ) a generalized topological space (briefly GTS) on X. The elements of Âµ are called Âµ-open sets and their complements are called Âµ-closed sets. For A X, we denote by cÂµ(A) the intersection of all Âµ-closed sets containing A, i.e., the smallest Âµ-closed set containing A; and by iÂµ(A) the union of all Âµ-open sets contained in A, i.e., the largest Âµ-open set contained in A.

Definition: 1.1

Let (X, Âµ) be a GTS and A X. Then the subset Âµ(A) is defined by

( )

( )

{: , }, , = { , .

Definition: 1.2

In a GTS (X, Âµ) a subset A is called a Âµ-set if A = Âµ(A).

Definition: 1.3

A subset A of a GTS (X, Âµ) is called (, Âµ)-closed set if A = L D, where L is a Âµ-sets and D is a Âµ-closed set.

The intersection of all (, Âµ)-closed sets containing a subset A of X is called the (, Âµ)-closure of A and is denoted by c(, Âµ)(A). The complement of (, Âµ)- closed set is called (, Âµ)-open set. We denote the collection of (, Âµ)-open sets (resp.(, Âµ)-closed sets) by Âµ-O(X, Âµ) (resp. Âµ-C(X, Âµ)).

Lemma:1.1 

For subsets Ai (i I) of a GTS (X, Âµ), the following properties hold:

1. If Ai is (, Âµ)-closed for each i I, then iIAi is (, Âµ)-closed.

2. If Ai is (, Âµ)-open for each i I, then iIAi is (, Âµ)-open.

2. Âµ-GENERALIZED CLOSED SETS Definition: 2.1

A subset A of a generalized topological space (X, Âµ) is called Âµ-generalized closed, briefly Âµ-g-closed, if cÂµ(A) U whenever A U and U is (, Âµ)-open.

Definition: 2.2

A subset A of a generalized topological space (X, Âµ) is called Âµ-g-closed, if c(, Âµ)(A) U whenever A U and U is (, Âµ)-open.

Definition: 2.3

A subset A of a generalized topological space (X, Âµ) is called gÂµ-closed, if c(, Âµ)(A) U whenever A U and U is Âµ-open.

Remark: 2.4

From above definitions, we have the following

diagram

1. Âµ-g-closed sets and (, Âµ)-closed sets are independent concepts.

2. Âµ-g-closed sets and Âµ-g-closed sets are independent concepts.

3. (, Âµ)-closed sets and Âµ-g-closed sets are also independent concepts

Proof.

Necessity: Suppose that A is Âµ-g-closed. Let S be

Âµ-closed

-closed

a (, Âµ)-closed subset of cÂµ(A) \ A. Then A Sc. Since A

Âµ-g

Âµ-g-closed

is Âµ-g-closed, we have cÂµ(A) Sc consequently S (cÂµ(A))c.. Hence S cÂµ(A) (cÂµ(A))c = . Therefore S is empty.

(, Âµ)-closed Example: 2.5;

Âµ-g-closed

gÂµ-closed

Sufficiency: Suppose that cÂµ(A) \ A contains no nonempty (, Âµ)-closed sets. Let A G and G be (, Âµ)-open. If cÂµ(A) G, then cÂµ(A) GC is a nonempty (, Âµ)-closed subset of cÂµ(A) \ A. Therefore, A is Âµ-g-closed.

Let X = {a, b, c,} and Âµ = {,{a},{a, b},{b, c},X}.

Thus Âµ-O(X, Âµ) = {,{a},{c},{a, b},{b, c},{a, c},X}. Take A = {a, c}. Then we obtained that A is a Âµ-g-closed set but it is not Âµ-g-closed.

Let X = {a, b, c,} and Âµ = {,{a},{a, b},{b, c},X}. Then A = {b} is a (, Âµ)-closed set but it is not Âµ-g-closed. Let X = {a, b, c,} and Âµ = {, {a}, X}. Then

A = {a, b} is Âµ-g-closed set but it is not (, Âµ)-closed Remark 2.6

The union of two Âµ-g-closed sets need not be Âµ-g-closed as can be verified from the following example Example: 2.7

Let X = {a, b, c} and Âµ = {,{a},{a, b},{b, c},X}. Now put A = {a} and B = {c} are two Âµ-g-closed sets. But A B = {a, c} is not a Âµ-g-closed set.

Remark: 2.8

The intersection of two Âµ-g-closed sets need not be Âµ-g-closed as can be verified from the following lemma

Example: 2.9

Let X = {a, b, c} and Âµ = {, {a}}. Now put A = {a} and B = {b} are two Âµ-g-closed. But A = is not a Âµ-g-closed set.

Theorem: 2.10

A subset A of GTS (X, Âµ) is Âµ-g-closed, then cÂµ(A) \ A contains no non empty Âµ-closed subset of (X, Âµ). Proof.

Let F be a Âµ-closed subset contained in cÂµ(A) \ A. Clearly A Fc where A is Âµ-g-closed and Fc is an Âµ-open set of X. Thus cÂµ(A) Fc or F (cÂµ(A))c. Then F (cÂµ(A))c (cÂµ(A) \ A) (cÂµ(A))c cÂµ(A) = . This shows that F = .

The converse of the above theorem is not true in general as it is shown in the following example.

Example: 2.11

Let X = {a, b, c,} and Âµ = {,{a},{a, b},{b, c},X}. If A = {a, c}, then cÂµ(A) \ A = {b} does not contain a non-empty Âµ-closed set. But A is not Âµ-g-closed in (X, Âµ). Corollary: 2.12

In a T1 space, every Âµ-g-closed set is Âµ-closed.

Proof.

Let A be a Âµ-g-closed set in a T1 space (X, Âµ). Let x cÂµ(A) \ A. Since (X, Âµ) is T1, {x} is a Âµ- closed set in (X, Âµ). By theorem 2.10, there exists no non empty Âµ- closed set in cÂµ(A) \ A = . Therefore cÂµ(A) = A. Hence A is Âµ-closed.

Theorem: 2.13

A set A is Âµ-g-closed if and only if cÂµ(A) \ A contains no non empty (, Âµ)-closed sets.

Theorem: 2.14

If A is a Âµ-g-closed set of (X, Âµ) and A B cÂµ(A), then B is a Âµ-g-closed set in (X, Âµ).

Proof.

Let A B and cÂµ(A) cÂµ(B). Hence cÂµ(B) \ B cÂµ(A) \ A. But by the theorem 2.13, cÂµ(A) \ A contains no non empty (, Âµ)-closed subsets of X and hence neither cÂµ(B) \ B. Again by theorem 2.13, B is a Âµ-g-closed set.

Theorem: 2.15

If A is (, Âµ)-open and Âµ-g-closed set in (X, Âµ), then A is Âµ-closed in (X, Âµ).

Proof.

Since A is (, Âµ)-open and Âµ-g-closed, cÂµ(A)

A and hence A is Âµ-closed. Theorem: 2.16

For each {x} X, either {x} is (, Âµ)-closed or

{x}c is Âµ-g-closed in (X, Âµ). Proof.

Suppose {x} is not (, Âµ)-closed in(X, Âµ) then{x}c is not (, Âµ)-open and the only (, Âµ)-open set containing{x}c is the space X itself. Therefore cÂµ({x}c) ) X and so {x}c is Âµ-g-closed in (X, Âµ).

Definition: 2.17

A subset S of X is said to be locally-Âµ- closed if S = U F, where U is Âµ-open F is Âµ-closed in (X, Âµ).

Theorem: 2.18.

Let A be locally-Âµ- closed subset of(X, Âµ). For the set A the following properties are equivalent:

(i). A is Âµ-closed; (ii). A is Âµ-g-closed; (iii). A is Âµ-g-closed. Proof.

By Remark 2.4, it suffices to prove that (iii) implies (i). By A (cÂµ(A)) c is Âµ-open in (X, Âµ) since A is locally-Âµ-closed. Now A (cÂµ(A)) c is an Âµ-open set of (X, Âµ) such that A A (cÂµ(A)) c. Since A is Âµ-g-closed, then cÂµ(A) A (cÂµ(A)) c . But cÂµ(A) (cÂµ(A))c = . Thus we have cÂµ(A) A and hence A is Âµ-closed.

Definition: 2.19

A subset A in (X, Âµ) is said to be Âµ-g-open in (X, Âµ) if and only if Ac is Âµ-g-closed.

Theorem: 2.20

A set A is said to be Âµ-g-open in (X, Âµ) if and only if F iÂµ(A) whenever F is (, Âµ)-closed in (X, Âµ) and F A.

Proof.

Suppose that F iÂµ(A) whenever F is , Âµ)- closed and F A. Let Ac G, where G is(, Âµ)-open.

Proof.

Necessity: Suppose that A is Âµ-g-closed in (X,

Hence Gc A. By assumption Gc iÂµ(A) which implies that (iÂµ(A))c G, so cÂµ(Ac) G. Hence Ac is Âµ-g-closed. i.e., A is Âµ-g-open

Conversely, let A be Âµ-g-open. Then Ac is Âµ-g-closed. Also let F be a (, Âµ)-closed set contained in

1. Then Fc is (, Âµ)-open. Therefore whenever Ac Fc,

cÂµ(Ac) Fc. This implies that F (cÂµ(Ac))c = iÂµ(A). Thus F iÂµ(A)..

Theorem: 2.21

A set A is said to be Âµ-g-open in (X, Âµ) if and only if G = X whenever G is (, Âµ)-open and iÂµ(A) Ac

G.

Proof.

Let A be Âµ-g-open, G be (, Âµ)-open and iÂµ(A) Ac G. This gives Gc (iÂµ(A))c (Ac)c = (iÂµ(A))c \ Ac = cÂµ(Ac) \ Ac. Since Ac is Âµ-g-closed and Gc is (, Âµ)-closed, by theorem 2.13, it follows that Gc = . Therefore X = G. Conversely, suppose that F is (, Âµ)-closed and F A. Then iÂµ(A) Ac iÂµ(A) Fc. It follows that iÂµ(A) Fc = X and hence F iÂµ(A). Therefore A is Âµ-g-open.

Theorem: 2.22

If iÂµ(A) B A and A is Âµ-g-open in (X, Âµ), then B is Âµ-g-open in (X, Âµ).

Proof.

Suppose iÂµ(A) B A and A is Âµ-g-open in (X,

Âµ). Then Ac Bc cÂµ(Ac) and Ac is Âµ-g- closed. Theorem: 2.23

A set A is Âµ-g-closed in (X, Âµ) if and only if cÂµ(A) \ A is Âµ-g-open in (X, Âµ).

Âµ). Let F cÂµ(A) \ A, where F is (, Âµ)-closed. By theorem 2.10, F = . Therefore F iÂµ(cÂµ(A) \ A) and by theorem 2.20, cÂµ(A) \ A is Âµ-g-open in (X, Âµ).

Sufficiency: Let A G, where G is (, Âµ)-open set. Then cÂµ(A) Gc cÂµ(A) Ac = cÂµ(A) \ A. Since cÂµ(A) Gc is (, Âµ)-closed and cÂµ(A) \ A is Âµ-g-open in by theorem 2.20, we have cÂµ(A) Gc iÂµ(cÂµ(A) \ A) = . Hence A is Âµ-g-closed in (X, Âµ).

Theorem: 2.24

For the subset A X the following properties are equivalent:

1. A is Âµ-g-closed;

2. cÂµ(A) \ A contains no nonempty is (, Âµ)-closed set;

(iii).cÂµ(A) \ A is Âµ-g-open Proof.

This follows from the theorem 2.10 and 2.23

REFERENES

1. A.CsÃ¡szÃ¡r, Generalized open sets, Acta Math. Hungar., 75(1- 2)(1997), 65-87.

2. A.CsÃ¡szÃ¡r, Generalized open sets in generalized topologies, Acta Math. Hungar., 106 (2005), 53-66.

3. B. Roy and Erdal Ekici, On (, )-closed sets in generalized topological spaces, Method of Functional Analysis and Topology, 17(2) (2011), 174-179.

4. E. Ekici and B. Roy, New generalized topologies on generalized topological spaces due to CsÃ¡szÃ¡r, Acta Math. Hungar., 132(1-2) (2011), 117-124.

5. B. Roy, On a type of generalized open sets, Applied General topology,12 (2) (2011), 163-173.

6. B. Roy, On generalized of R0 and R1 spaces, Acta Math. Hungar., 127(2010), 291-300.

7. M. Caldas, S. Jafari and T. Noiri, On – generalized closed sets in topological spaces, Acta. Math. Hungar., 118(4) (2008), 337 343.