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[[lambda].sub.j]-closed sets in generalized topological spaces.

[section]1. Introduction and preliminaries

A. Csaszar [1] has introduced and studied generalized open sets of a set X denned in terms of monotonic functions [gamma] : P(X) [right arrow] P(X). [mu] = {A [subset or equal to] X/A [subset] [gamma](A)} is called the family of [gamma]-open sets which is closed under arbitrary union and [phi] [member of] [mu], then [mu] is called a generalized topology. Let X be a non-empty set and [mu] be a collection of subsets of X. Then [mu] is called a generalized topology (briefly GT) on X iff [phi] [member of] [mu] and [G.sub.i] [member of] [mu] for i [member of] I [not equal to] [phi] implies G = [U.sub.i [member of] I][G.sub.i] [member of] [mu]. Generalized topological spaces are important generalizations of topological spaces and many results have been obtained by many topologist [3,4,5,11,12]. A generalized topology is said to be strong [2] if X [member of] [mu].

A space (X, [mu]) is said to be quasi-topological space [7], if [mu] is closed under finite intersection. The generalized closure of a subset S of X, denoted by [c.sub.[mu]](S), is the intersection of generalized closed sets including S and the interior of S, denoted by [i.sub.[mu]](S), is the union of generalized open sets contained in S.

If [mu] is a GT on X, A [subset or equal to] X, x [member of] X, then x [member of] [c.sub.[mu]](A) iff x in M [member of] [mu] [right arrow] M [intersection] A [not equal to] [phi] and [c.sub.[mu]](X/A) = X/[i.sub.[mu]](A).

In this paper, We define a new class of sets called [[lambda].sub.J]-closed sets in generalized topological spaces and some of their properties are established.

Definition 1.1. [6] Let (X, [mu]) be a GTS and A [subset or equal to] X, then A is said to be

(1) [mu]-semi open if A [subset or equal to] [c.sub.[mu]]([i.sub.[mu]](A)),

(2) [mu]-pre open if A [subset or equal to] [i.sub.[mu]]([c.sub.[mu]](A)),

(3) [mu]-a open if A [subset or equal to] [i.sub.[mu]]([c.sub.[mu]][i.sub.[mu]](A)),

(4) [mu]-[beta] open if A [subset or equal to] [c.sub.[mu]]([i.sub.[mu]][c.sub.[mu]](A)).

Let us denote [sigma]([[mu].sub.X]) (briefly [[sigma].sub.X] or [sigma]) the class of all [mu]-semi open sets on X, by [pi]([[mu].sub.X] or [pi]) the class of all [mu]-pre open sets, by a ([[mu].sub.X] or a) the class of all [mu]-a open sets, by [beta]([[mu].sub.X] or [beta]) the class of all [mu]-[beta] open sets.

Definition 1.2. [9] Let (X, [mu]) be a GTS and A [subset or equal to] X, then A is said to be [mu]-J-open if A [subset or equal to] [i.sub.[mu]]([c.sub.[pi]](A)).

The closure of a subset S of X, denoted by [c.sub.J](A) is the intersection of generalized J-closed sets including S and the interior of S, denoted by [i.sub.J](S), is the union of generalized J-open sets contained in S. The set of all J-open sets is denoted by JO(X). The set all J-closed sets is denoted by JC(X).

Theorem 1.1. Let (X, [mu]) be a generalized topological space. Then

(1) [c.sub.[mu]](A) = X-[i.sub.[mu]](X-A),

(2) [i.sub.[mu]](A) = X-[c.sub.[mu]](X-A).

Definition 1.3. [9] Let (X, [lambda]) be a GTS and A [subset or equal to] X, then A is said to be [[lambda].sub.[alpha]]-J-closed if [c.sub.[alpha]](A) [subset or equal to] U whenever A [subset or equal to] U and U [member of] JO(X).

Lemma 1.1. [7] Let (X, [mu]) be a quasi-topological space. Then [c.sub.[mu]](A [union] B) = [c.sub.[mu]](A) [union] [c.sub.[mu]](B) for every subsets A and B of X.

[section]2. J-Closed sets in generalized topological spaces

Definition 2.1. A subset A of a space (X, [lambda]) is said to be gJ[lambda]-closed if [c.sub.[lambda]](A) [subset or equal to] U whenever A [subset or equal to] U and U [member of] JO(X). The complement of gJ[lambda]-closed set is called an gJ[lambda]-open set.

Definition 2.2. A subset A of a space (X, [lambda]) is said to be J[lambda]g-closed if [c.sub.J](A) [subset or equal to] U whenever A [subset or equal to] U and U [member of] JO(X). The complement of J[lambda]g-closed set is called an J[lambda]g-open set.

Theorem 2.1. Let (X, [lambda]) be a GTS and A [subset or equal to] X, then

(i) Every closed set is gJ[lambda]-closed,

(ii) Every closed set is J[lambda]g-closed,

(iii) Every J-closed set is J[lambda]g-closed,

(iv) Every gJ[lambda]-closed set is a [[lambda].sub.[alpha]]-J-closed set,

(v) Every J[lambda]g-closed set is a [[lambda].sub.[alpha]]-J-closed set.

Theorem 2.2. Let (X, [lambda]) be a quasi topological space and A [subset or equal to] X. If A and B are gJ[lambda]-closed subsets of X, then A [union] B is also a gJ[lambda]-closed set.

Proof. Suppose A and B are gJ[lambda]-closed. Let U be a J-open set such that A [union] B [subset or equal to] U. Since A and B are gJ[lambda]-closed set, [c.sub.[lambda]](A) [subset or equal to] U and [c.sub.[lambda]](B) [subset or equal to] U and so [c.sub.[lambda]](A) [union] [c.sub.[lambda]](B) [subset or equal to] U. Therefore [c.sub.[lambda]](A [union] B) [subset or equal to] U.

Remark 2.1. The intersection of two gJ[lambda]-closed sets is not a gJ[lambda]-closed set.

Example 2.1. Let X = {a, b, c} and [lambda] = {[phi], {b}, X}. If A = {a, b} and B = {b, c} are gJ[lambda]-closed sets. but A [intersection] B = {b} is not a gJ[lambda]-closed set.

Theorem 2.3. Let (X, [lambda]) be a GTS and A [subset or equal to] X. Then A is J[lambda]g-closed if and only if F [subset or equal to] [c.sub.J](A)-A and F is J-closed implies that F = [phi].

Proof. Let F be a subset of J-closed subset of [c.sub.J](A)-A. Since A [subset or equal to] X - F and A is J[lambda]g-closed set, [c.sub.J](A)-A [subset or equal to] X-F and so F [subset or equal to] X - [c.sub.J](A) - A. Therefore F = [phi].

Conversely, U is a J-open set such that A [subset or equal to] U. If [c.sub.J](A) not a subset in U, then [c.sub.J](A) [intersection] (X - U) is a non-empty J-closed subset of [c.sub.J](A) - A, which is a contradiction. Therefore, [c.sub.J](A) [subset or equal to] U, which implies that A is J[lambda]g-closed set.

Theorem 2.4. Let (X, [lambda]) be a GTS. Let A and B be subsets of X. If A [subset] B [subset] [c.sub.J](A) and A is J[lambda]g-closed, then B is J[lambda]g-closed.

Proof. If F is J-closed such that F [subset or equal to] [c.sub.J](B) - B, therefore by hypothesis, F [subset or equal to] [c.sub.J](A) - A. Since A is J[lambda]g-closed, by theorem 2.3, F = [phi] and so B is J[lambda]g-closed.

Theorem 2.5. Let (X, [lambda]) be a GTS. Let A [subset or equal to] X be a gJ[lambda]-closed set of X, then [c.sub.J](A)/A does not contain any non-empty J-closed set.

Proof. Let F be a J-closed set of X such that F [subset or equal to] [c.sub.J](A)/A. Then F [subset or equal to] X/A and hence A [subset or equal to] X/F [member of] JO(X). Since A is gJ[lambda]-closed [c.sub.J](A) [subset or equal to] X/F and hence F [subset or equal to] X/[c.sub.J](A). So F [subset or equal to] X/[c.sub.J](A) [intersection] (X/[c.sub.J](A)) = [phi].

Theorem 2.6. Let (X, [lambda]) be a GTS. Let A [subset or equal to] X is gJ[lambda]-open iff F [subset or equal to] [i.sub.[lambda]](A) whenever F is a J-closed set such that F [subset or equal to] (A).

Proof. Let A be a gJ[lambda]-open set of X and F be a J-closed set such that F [subset or equal to] (A). Then X/A is a gJ[lambda]-closed set and X/A [subset or equal to] X/F [member of] JO(X). So [c.sub.[lambda]](X/A) = X/[i.sub.[lambda]](A) [subset or equal to] X/F, thus F [subset or equal to] [i.sub.[lambda]](A).

Conversely, let F [subset or equal to] [i.sub.[lambda]](A), whenever F is J-closed such that F [subset or equal to] A. Let X/A [subset or equal to] U where U [member of] JO(X). Then X/U [subset or equal to] A and X/U is J-closed. By the assumption, X/U [subset or equal to] [i.sub.[lambda]](A) and hence [c.sub.[lambda]](X/A) = X/[i.sub.[lambda]](A) [subset or equal to] U. Hence X/A is gJ[lambda]-closed and hence A is gJ[lambda]-open.

Theorem 2.7. Let (X, [lambda]) be a GTS. If A is a gJ[lambda]-closed subset of X, then [c.sub.[lambda]](A)/A is gJ[lambda]-closed.

Proof. Let A be a gJ[lambda]-closed subset of (X, [lambda]). Let F be a J-closed set such that F [subset or equal to] [c.sub.[lambda]](A)/A, so by theorem 2.5, F = [phi] and thus F [subset or equal to] [i.sub.[lambda]]([c.sub.[lambda]](A)/A). So by theorem 2.6, [c.sub.[lambda]](A)/A is gJ[lambda]-closed.

[section]3. [[lambda].sub.J]-Closed sets in generalized topological spaces

Definition 3.1. A subset A of [M.sub.J] = [union] {B/B [member of] J} of a space (X, [lambda]) is said to be [[lambda].sub.J]-closed if [c.sub.J](A) [intersection] [M.sub.J] [subset or equal to] U whenever A [subset or equal to] U and U [member of] JO(X). The complement of [[lambda].sub.J]-closed set is called an [[lambda].sub.J]-open set.

Theorem 3.1. Let X ba a non-empty set and [lambda] be the generalized topology on X and A [subset or equal to] X. Then the following properties hold:

(i)(X-[M.sub.J]) is a J-closed set contained in every J-closed set,

(ii) [c.sub.J](A [intersection] [M.sub.J]) [intersection] [M.sub.J] = [c.sub.J](A) [intersection] [M.sub.J],

(iii) If A is J-closed, then [c.sub.J](A [intersection] [M.sub.J]) [intersection] [M.sub.J] = A [intersection] [M.sub.J],

(iv) [c.sub.J](A) = ([c.sub.J](A) [intersection] [M.sub.J]) [union] (X - [M.sub.J]),

(v) If A is J-closed, then A = (A [intersection] [M.sub.J]) [union] (X - [M.sub.J]).

Proof.

(i) If G is J-open, then G [subset or equal to] [M.sub.J].

(ii) We know that [c.sub.J] (An [M.sub.J]) [intersection] [M.sub.J] [subset or equal to] [c.sub.J](A) [intersection] [M.sub.J]. Let x [member of] [c.sub.J](A) [intersection] [M.sub.J]. Then x [member of] [c.sub.J](A) and x [member of] [M.sub.J]. Now x [member of] [c.sub.J](A) implies that G [intersection] A [not member of] [phi] for every J-open set G containing x and so G [intersection] (A [intersection] [M.sub.J]) [not member of] [phi] for every J-open set G containing x. Therefore, x [member of] [c.sub.J](A) [intersection] [M.sub.J], and so x [member of] [c.sub.J](A) [intersection] [M.sub.J] [intersection] [M.sub.J]. Hence [c.sub.J](A) [intersection] [M.sub.J] [subset or equal to] [c.sub.J](A [intersection] [M.sub.J]) [intersection] [M.sub.J].

(iii) A is J-closed i.e. [c.sub.J](A) = A. From (ii), we have [c.sub.J](A [intersection] [M.sub.J]) [intersection] [M.sub.J] = [c.sub.J](A) [intersection] [M.sub.J] = A [intersection] [M.sub.J].

(iv) [c.sub.J](A) = [c.sub.J](A) [intersection] X = [c.sub.J](A) [intersection] ([M.sub.J] [union] (X - [M.sub.J])) = [c.sub.J](A) [intersection] ([M.sub.J]) [union] [c.sub.J](A) [intersection] (X - [M.sub.J]) = [c.sub.J](A) [intersection] ([M.sub.J]) [union] (X - [M.sub.J]).

(v) Since A is J-closed, from (ii) we have, [c.sub.J](A) = ([c.sub.J](A) [intersection] [M.sub.J]) [union] (X - [M.sub.J]) i.e. A = (A [intersection] [M.sub.J]) [union] (X - [M.sub.J]).

Theorem 3.2. Let (X, [lambda]) be a GTS and A [subset or equal to] X, then the following properties hold:

(i) If A is J-closed subset of X, then A [intersection] [M.sub.J] is [[lambda].sub.J]-closed set,

(ii) [c.sub.J](A) [intersection] [M.sub.J] is a [[lambda].sub.J]-closed set for every subset A of X.

Proof.

(i) Let A [intersection] [M.sub.J] [subset or equal to] U and U be J-open. Since [c.sub.J](A [intersection] [M.sub.J]) [intersection] [M.sub.J] = [c.sub.J](A) [intersection] [M.sub.J], we have [c.sub.J](A [intersection] [M.sub.J]) [intersection] [M.sub.J] = A [intersection] [M.sub.J] [subset or equal to] U and so A [intersection] [M.sub.J] is [[lambda].sub.J]-closed set.

(ii) it follows from (i).

Theorem 3.3. Let (X, [lambda]) be a GTS. Then a subset A of [M.sub.J] is [[lambda].sub.J]-closed if and only if F [subset] [c.sub.J](A) - A and F is [lambda]-J-closed implies that F = X - [M.sub.J].

Proof. Let F be a J-closed subset of [c.sub.J](A) - A. Since A [subset] X - F and A is [[lambda].sub.J]-closed set, [c.sub.J](A) [intersection] [M.sub.J] [subset or equal to] X-F and so F [subset] X - ([c.sub.J](A) [intersection] [M.sub.J]) = (X - [c.sub.J](A)) [union] (X - [M.sub.J]). Since F [subset] [c.sub.J](A), we have F [subset] (X-[M.sub.J]). Therefore F = X-[M.sub.J].

Conversely, Let A [subset] U and U [member of] J. Suppose ([c.sub.J](A) [intersection] [M.sub.J]) [intersection] (X - U) is a non empty subset. Then ([c.sub.J](A) [intersection] [M.sub.J]) [intersection] (X - U) [subset] [c.sub.J](A) [intersection] (X - U) [subset] [c.sub.J](A) [intersection] (X-A) [subset] [c.sub.J](A) [intersection] A. Thus ([c.sub.J](A) [intersection] [M.sub.J]) [intersection] (X-U) is a non empty J-closed set contained in [c.sub.J](A) [intersection] A. Therefore, ([c.sub.J](A) [intersection] [M.sub.J]) [intersection] (X - U) = [phi], which is a contradiction. Therefore, ([c.sub.J](A) [intersection] [M.sub.J]) [subset] U which implies A is a [[lambda].sub.J]-closed set.

Theorem 3.4. Let (X, [lambda]) be a GTS. Then a [[lambda].sub.J]-closed subset A of [M.sub.J] is a J-closed set, if [c.sub.J](A)-A is a J-closed set.

Proof. [c.sub.J](A)-A = X-M = [phi]. Then [c.sub.J](A) = A [union] (X-M). Therefore A is a J-closed set.

Theorem 3.5. Let (X, [lambda]) be a quasi topological space and A [subset or equal to] X. If A and B are [[lambda].sub.J]-closed subsets of X, then A [union] B is also a [[lambda].sub.J]-closed set.

Example 3.1. Let X = {a, b, c} and [lambda] = {[phi], {a}, {b}, {a, b}, {a, c}, {b, c}, X}. Then [lambda] is a generalized topology but not a quasi topology. A = {a} and B = {b} are [[lambda].sub.J]-closed sets. but A [union] B = {a, b} is not a [[lambda].sub.J]-closed set.

Example 3.2. Consider the topological space (X, [tau]) where X = {a, b, c} and [lambda] = {[phi], {b}, X}. If A = {a, b} and B = {b, c} are [[lambda].sub.J]-closed sets, but A [intersection] B = {b} is not a [[lambda].sub.J]-closed set.

Theorem 3.6. Let (X, [lambda]) be a GTS. If A is [[lambda].sub.J]-closed subset of [M.sub.J] and B is J-closed, then A [intersection] B is a [[lambda].sub.J]-closed set.

Proof. Suppose A [intersection] B [subset or equal to] U where U is J-open, then A [subset or equal to] (U [union] (X - B)). Since A is [[lambda].sub.J]-closed, [c.sub.J](A) [intersection] [M.sub.J] [subset or equal to] (U [union] (X - B)) and so [c.sub.J](A) [intersection] B [intersection] [M.sub.J]) = ([c.sub.J](A) [union] [c.sub.J](B)) [intersection] [M.sub.J] [subset or equal to] U, which implies that ([c.sub.J](A [union] B)) [intersection] M [subset or equal to] U and so A [intersection] B is a [[lambda].sub.J]-closed set.

Definition 3.2. A subset A of M in a space (X, [lambda]) is said to be [[lambda].sub.J]-open if [M.sub.J] - A is [[lambda].sub.J]-closed.

Theorem 3.7. Let (X, [lambda]) be a GTS. Let A [subset or equal to] [M.sub.J] is [[lambda].sub.J]-open if and only if F [intersection] [M.sub.J] [subset or equal to] [i.sub.J](A) whenever F is J-closed and F [intersection] [M.sub.J] [subset or equal to] A.

Proof. Let A be a [[lambda].sub.J]-open subset of [M.sub.J] and F be a J-closed subset of X such that F [intersection] [M.sub.J] [subset or equal to] A. Then [M.sub.J] - A [subset or equal to] [M.sub.J] - (F [intersection] [M.sub.J]) = [M.sub.J]-F. Since [M.sub.J]-F is J-open and [M.sub.J]-A is [[lambda].sub.J]-closed, [c.sub.J]([M.sub.J]-A) [intersection] [M.sub.J] [subset or equal to] [M.sub.J] - F and so F [subset or equal to] [M.sub.J] - ([c.sub.J]([M.sub.J] - A) [intersection] [M.sub.J]) = [M.sub.J] [intersection] ([M.sub.J] - [c.sub.J]([M.sub.J] - A)) = [i.sub.J](A) [intersection] [M.sub.J] = [i.sub.J](A), which implies F [intersection] [M.sub.J] [subset or equal to] A.

Conversely, Let A be a subset of [M.sub.J] and F be a J-closed set such that F [intersection] [M.sub.J] [subset or equal to] A. By hypothesis, F [intersection] [M.sub.J] [subset or equal to] [i.sub.J](A) which implies that [M.sub.J] - [i.sub.J](A) [subset or equal to] [M.sub.J]-(F [intersection] [M.sub.J]) and so [c.sub.J]([M.sub.J] - A) [subset or equal to] [M.sub.J] - F. Then [c.sub.J]([M.sub.J] - A) [intersection] [M.sub.J] [subset or equal to] ([M.sub.J] - F) [intersection] [M.sub.J] = [M.sub.J] - F which implies that [M.sub.J]-A is [[lambda].sub.J]-closed and so A is [[lambda].sub.J]-open.

Theorem 3.8. Let (X, [lambda]) be a GTS. Let A [subset or equal to] [M.sub.J] is [[lambda].sub.J]-open if and only if U = [M.sub.J] whenever U is J-open and [i.sub.J](A) [union] ([M.sub.J] - A) [subset or equal to] U.

Proof. Suppose A is [[lambda].sub.J]-open and M is J-open such that [i.sub.J](A) [union] ([M.sub.J] - 7A) [subset or equal to] U. Then [M.sub.J] - U [subset or equal to] ([M.sub.J] - [i.sub.J](A) [intersection] A) = [c.sub.J]([M.sub.J] - A) [intersection] A = [c.sub.J]([M.sub.J] - A) - ([M.sub.J] - A) and so ([M.sub.J] - U) [union] (X - [M.sub.J]) [subset or equal to] [c.sub.J]([M.sub.J] - A) [intersection] A, by theorem 3.3, ([M.sub.J] - U) [union] (X - [M.sub.J]) and so ([M.sub.J]-U) = [phi] which implies [M.sub.J] = U.

Conversely, Let F be a J-closed set such that F [intersection] [M.sub.J] [subset or equal to] A. Since [i.sub.J](A) [union] ([M.sub.J] - A) [subset or equal to] [i.sub.J](A) [union] ([M.sub.J]-F) [union] ([M.sub.J] - [M.sub.J]) = [i.sub.J](A) [union] ([M.sub.J] - F) and [i.sub.J](A) [union] ([M.sub.J]-F) is J-open, by hypothesis, [M.sub.J] = [i.sub.J](A) [union] ([M.sub.J] - F) and so F [intersection] [M.sub.J] [subset or equal to] ([i.sub.J](A) [union] ([M.sub.J] - F) [intersection] F) = ([i.sub.J](A) [intersection] F) [union] ([M.sub.J] - F [intersection] F) = [i.sub.J](A) [intersection] F [subset or equal to] [i.sub.J](A). By definition 3.2, A is [[lambda].sub.J]-open.

Theorem 3.9. Let (X, [lambda]) be a GTS. Let A and B be subsets of [M.sub.J]. If [i.sub.J](A) [subset or equal to] B [subset or equal to] A and A is [[lambda].sub.J]-open, then B is [[lambda].sub.J]-open.

Theorem 3.10. Let (X, [lambda]) be a GTS. Let A [subset or equal to] [M.sub.J] is [[lambda].sub.J]-closed if and only ([c.sub.J](A) - A) [intersection] [M.sub.J] is [[lambda].sub.J]-open.

Proof. Suppose ([c.sub.J](A) - A) [intersection] [M.sub.J] is [[lambda].sub.J]-open. Let A [subset or equal to] U and U is J-open, since ([c.sub.J](A) [intersection] ([M.sub.J] - U) [subset or equal to] ([c.sub.J](A) [intersection] ([M.sub.J] - A) = ([c.sub.J](A) - A) [intersection] [M.sub.J]. ([c.sub.J](A) - A) [intersection] [M.sub.J] is [[lambda].sub.J]-open and ([c.sub.J](A) [intersection] ([M.sub.J] - A) is J-closed, by theorem 3.7, ([c.sub.J](A) [intersection] ([M.sub.J] - U) [subset or equal to] [i.sub.J](([c.sub.J](A) - A) [intersection] [M.sub.J]) [subset or equal to] [i.sub.J]([c.sub.J](A) [intersection] [i.sub.J]([M.sub.J]-A)) [subset or equal to] [i.sub.J]([c.sub.J](A) [intersection] [i.sub.J](X - A)) = [i.sub.J]([c.sub.J](A)) [intersection] (X - [c.sub.J](A)) = [phi]. Therefore [c.sub.J](A) [intersection] [M.sub.J] [subset or equal to] U which implies that A is [[lambda].sub.J]-closed.

Conversely, suppose A is [[lambda].sub.J]-closed and F [intersection] [M.sub.J] [subset or equal to] ([c.sub.J](A) - A) [intersection] [M.sub.J] where F is J-closed. Then F [subset or equal to] ([c.sub.J](A) - A) and by theorem 3.3, F = X - [M.sub.J] and so [phi] = (X - [M.sub.J]) [intersection] [M.sub.J] = F [intersection] [M.sub.J] [subset or equal to] ([c.sub.J](A) - A) [intersection] [M.sub.J] which implies that F [intersection] [M.sub.J] [subset or equal to] [i.sub.J]([c.sub.J](A) - A) [intersection] [M.sub.J]). Therefore, [c.sub.J](A)-A is [[lambda].sub.J]-open.

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I. Arockiarani ([dagger]) and D. Sasikala ([double dagger])

Nirmala college for women, Coimbatore, Tamil Nadu, India

E-mail: d.sasikala@rocketmail.com
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Author:Arockiarani, I.; Sasikala, D.
Publication:Scientia Magna
Article Type:Technical report
Date:Sep 1, 2013
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