on Λ_μ-Generalized Closed Sets in Generalized Topological Spaces

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 µ- g-closed sets and µ- g-open sets in GTS. We also investigate several properties of such sets. It turns out that µ- g-closed sets and µ- g-open sets are weaker forms of closed sets and open sets, and stronger forms of g-µ-closed sets and g-µ-open sets, respectively.

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

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

1. INTRODUCTION

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

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

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

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

   Definition: 1.1[3]

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

   ( )

   ( )

   {: , }, , = { , .

   Definition: 1.2[3]

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

   Definition: 1.3[3]

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

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

   Lemma:1.1 [4]

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

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

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

2. µ-GENERALIZED CLOSED SETS Definition: 2.1

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

Definition: 2.2

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

Definition: 2.3

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

Remark: 2.4

From above definitions, we have the following

diagram

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

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

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

Proof.

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

µ-closed

-closed

a (, µ)-closed subset of cµ(A) \ A. Then A Sc. Since A

µ-g

µ-g-closed

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

(, µ)-closed Example: 2.5;

µ-g-closed

gµ-closed

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

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

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

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

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

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

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

Remark: 2.8

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

Example: 2.9

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

Theorem: 2.10

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

Let F be a µ-closed subset contained in cµ(A) \ A. Clearly A Fc where A is µ-g-closed and Fc is an µ-open set of X. Thus cµ(A) Fc or F (cµ(A))c. Then F (cµ(A))c (cµ(A) \ A) (cµ(A))c cµ(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 µ = {,{a},{a, b},{b, c},X}. If A = {a, c}, then cµ(A) \ A = {b} does not contain a non-empty µ-closed set. But A is not µ-g-closed in (X, µ). Corollary: 2.12

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

Proof.

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

Theorem: 2.13

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

Theorem: 2.14

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

Proof.

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

Theorem: 2.15

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

Proof.

Since A is (, µ)-open and µ-g-closed, cµ(A)

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

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

{x}c is µ-g-closed in (X, µ). Proof.

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

Definition: 2.17

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

Theorem: 2.18.

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

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

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

Definition: 2.19

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

Theorem: 2.20

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

Proof.

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

Proof.

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

Hence Gc A. By assumption Gc iµ(A) which implies that (iµ(A))c G, so cµ(Ac) G. Hence Ac is µ-g-closed. i.e., A is µ-g-open

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

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

   cµ(Ac) Fc. This implies that F (cµ(Ac))c = iµ(A). Thus F iµ(A)..

   Theorem: 2.21

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

   G.

   Proof.

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

   Theorem: 2.22

   If iµ(A) B A and A is µ-g-open in (X, µ), then B is µ-g-open in (X, µ).

   Proof.

   Suppose iµ(A) B A and A is µ-g-open in (X,

   µ). Then Ac Bc cµ(Ac) and Ac is µ-g- closed. Theorem: 2.23

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

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

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

   Theorem: 2.24

   For the subset A X the following properties are equivalent:

   1. A is µ-g-closed;

   2. cµ(A) \ A contains no nonempty is (, µ)-closed set;

(iii).cµ(A) \ A is µ-g-open Proof.

This follows from the theorem 2.10 and 2.23

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