ON p*gp-LOCALLY CLOSED SETS IN TOPOLOGICAL SPACES

In this article, we consider a new class of sets which are called p*gp-locally closed sets and obtain some of their properties and also their relationships with some other classes of topological spaces. In addition, we found P*GPLC continuous function and P*GPLC irresolute function. Moreover, several examples are providing to illustrate the behavior of these new classes of sets.


INTRODUCTION
Kuratowski and Sierpinski [7] have been studied the notion of a locally closed sets in a topological space. Bourbaki [1] defined by locally closed sets in topological spaces. Ganster and Reilly [4] used locally closed sets to define LC-continuity and LC-irresoluteness. The concept of generalized closed sets was considered by Levine [8] plays a significant role in general topology. 1168 ON p*gp-LOCALLY CLOSED SETS IN TOPOLOGICAL SPACES Noiri, Maki, and Umehara [10] provided the class of pre generalized closed sets and used them to obtain properties of pre-T1/2 spaces. Selvi [11] further investigated pre*closed sets using the g-closure operator due to Dunham [2,3]. The notion of pre open set was discovered by Mashhour [9]. This characterization paved a new direction.
The authors [5,6] brings out the p*gp-closed sets and p*gp-open sets in topological spaces and established their relationships with some generalized sets in topological spaces. The purpose of this paper is to discuss about the concept of p*gp-locally closed sets in topological spaces and study their basic properties. Also, we provide P*GPLC continuous function, P*GPLC* continuous function and P*GPLC** continuous function and discuss P*GPLC irresolute function. We obtain many interesting results, to substantiate these result, suitable examples are given at the respective places. This paper is organized as follows. In the second section, a brief survey of basic concepts and results in topological spaces which are essentially needed are given Section 3, we consider the properties of p*gp-locally closed sets and some basic results, while section 4, introduces the classes of P*GPLC continuous function, P*GPLC* continuous function, P*GPLC** continuous function and P*GPLC irresolute function and some of the properties of these functions. Last section, we provide a brief summary of work done in this paper.

PRELIMINARIES
Throughout this paper (X, ) represents a topological space on which no separation axiom is assumed unless otherwise mentioned. (X, ) will be replaced by X if there are no changes of confusion. For a subset A of a topological space X, cl(A), int(A) and X\A denote the closure of A, the interior of A and the complement of A respectively. Further, we denote the collection of all locally closed subsets of (X, ) by LC(X, ). We recall the following definitions and results which are prerequisites for our present work.
Definition 2.4. [9] Let (X, ) be a topological space and A  X. The pre closure of A denoted by pcl(A) and is defined by the intersection of all pre closed sets containing A.
Definition 2.5. [5] A subset A of a topological space (X, τ) is said to be pre*generalized pre closed set (briefly p*gp-closed) if pcl(A)  U whenever A  U and U is pre*open in (X, τ). The collection of all p*gp-closed sets of X is denoted by p*gp-C(X). Lemma 2.6. [5] Let (X, τ) be a topological space. Then (i). Every closed set is p*gp-closed.
(ii). Intersection of any two p*gp-closed sets is p*gp-closed. Definition 2.9. A subset A of a topological space (X, τ) is called a locally closed (briefly lc) set [4] if A = U∩V where U is open and V is closed in (X, τ).
closed set in (X, ) for locally closed set F of (Y, ).

PRE*GENERALIZED PRE LOCALLY CLOSED SETS
In this section, p*gp-locally closed sets are introduced to obtain some of their properties and their relationships with other existing sets. The class of all p*gp-locally closed sets in (X, τ) is denoted by P*GPLC(X, τ).

Definition 3.2.
A subset A of a topological space (X, τ) is said to be p*gplc* if there exist a p*gp-open set V and a closed set F of (X, τ) such that A = V∩F.

Definition 3.3.
A subset A of a topological space (X, τ) is said to be p*gplc** if there exist an open set V and a p*gp-closed set F of (X, τ) such that A=V∩F.
The class of all p*gplc** sets in (X, τ) is denoted by P*GPLC**(X, τ). Theorem 3.4. If a subset A of (X, τ) is locally closed then it is a p*gplc set, p*gplc* set and p*gplc** set.
Proof. Let A be a locally closed subset of X. Then A=V∩F, where V is open and F is closed in (X, τ). By Lemma 2.8 and Lemma 2.6, A is a p*gplc set, p*gplc* set and p*gplc** set.  Theorem 3.7. If a subset A of (X, τ) is p*gplc** then it is a p*gplc set.
Proof. Let A be a p*gplc** set. Then by Definition 3.3, A=V∩F, where V is an open set in (X, τ) and F is a p*gp-closed set in (X, τ). By Lemma 2.8, A is p*gplc set.   Theorem 3.11. If A ∈ P*GPLC*(X, ) and B is closed in (X, ), then A∩B ∈ P*GPLC*(X, ).
(iii) g∘f is P*GPLC* continuous if f is P*GPLC* irresolute and g is P*GPLC* continuous.