On p- Open Sets with Respect to an Ideal OPEN ACCESS

In this, paper we introduce and investigate, the notion of sets via idealization by using local function and studied their some properties.


Introduction
Ideal in topological space have been considered since 1930 by Kuratowski [1] and Vaidyanathaswamy [2]. After that ideal topology generalized in general topology by Jankovi and Hamleet [3]. In 2005 Hatir and Noir iintroduced the , , [4]. Finally in 2014 , se , sets are introuced by R.Shanthi and M.Rameshkumar [5]. In this paper we introduced the notion of , , set and studied some properties of their.

Preliminaries
Let (X, τ) be topological space with no separation properties assumed. For a subset of topologicalspace (X, τ), Cl (A) and Int (A) denote the closure and interior of A in (X, τ) resp. An ideal I of topological space is collection of non-empty subset of X together with the following.
(i) (ii) . The triplet forms. (X, τ, I) is called the ideal topological space where is topological space of X with an ideal I. Given a topological space (X, τ) with an ideal Ion X If P(x) is the set of all subset of X, a set operator (.) * , called a local function [5] of A with respect to τ and I is defi ned as follows: for for every Additionally, cl*(A) = AUA * defi nes kuratowski closure operator for a topology τ * (I, τ), called the *-topology and fi ner than τ.

OPEN ACCESS
Volume: 8

De nition 2.1
Let (X, τ) be a topological space. A subset A of X is said be a -open set [6] if there exists anopen set U in X such that . The complement of -open set is -closed. The collection of all -open sets in X is denoted by O(X) is called the -local function. The semi closure of A in (X,τ) is denoted by the intersection of all -closed setcontaining A and is denoted by .

De nition 2.2
For ( * , for every we write * ( * . .The closure operator for a topology is defi ned as follows * for a topology * and denotes the interior of the set A in .

De nition 2.3
A Subset of topological space X is said to be,

De nition 2.4
A Subset of topological space X is said to be, * * S * Lemma: For a subset of topological space, the following properties hold.
Lemma: let A be a topological space and A,B be subsets of X. then following properties hold: * * * * * * * In this we defi ne the sets, and studied some properties of their.

De nition 3.1
A Subset of topological space X is said to be.

B is C is
Every open set of an ideal topological space is an

Proof:
Let A be a . Thus,we have * .Then A is an .

Remark 3.4
Converse of the above proposition 3.3 need not be true as seen from the following example.

Proof
The proof is obvious.

Remark 3.8
Converse of the above proposition 3.7 need not be true as seen from the following example.

Proof:
The proof is obvious.

Remark 3.11
Converse of the proposition 3.10 need not be true. DFFD