In this Chapter, we so through the basic definitions and results are required in studying the defined the

concepts. We use notations G, (X; ) and (X; ; I) for generalized topology, topological space and ideal space

respectively on the set X throughout this paper.

Definition 1.2.1 [53] Let G be a generalized topology on topological space X and B be a subset of X. Then

G interior [5]of B is denoted by iG (B) and is defined to be the union of all G -open sets contained in B.

The

G-closure of B is denoted by cG (B) and is defined to be the intersection of all G -closed sets containing B.

Remark[53] Since arbitrary union of G-open sets is a G-open set and arbitrary intersection of G-closed sets

is a G-closed set, it follows that iG (B)is a G-open set and cG (B) is a G-closed set. Thus iG (B)is the largest

G-open set contained in B and cG(B) is the smallest G-closed set containing B.

Definition 1.2.2 [127] Let (X; ; I) be an ideal space and B X, then the set B?=fx2 X : BU< I for each

neighborhood U of xg is called the local function of B with respect to ideal I and topology :

Theorem 1.2.1 [127] Let (X; ; I) be an Ideal topological space and B X, then the map cl? : }(X) !

}(X), defined by cl?(B) = B [ B?, B? denotes local function of B. Then cl is called kuratowski closure

## operator.

Definition 1.2.3 [146] Let X be a non-void set. Then a Kuratowski closure operator cl : }(X) ! }(X)

### satisfies following conditions:

1. cl?(;) = ;

2. B cl?(B); B 2 }(X)

3. cl?(B [ A) = cl?(B) [ cl?(A); for B; A 2 }(X)

4. cl?(cl?(B)) = cl?(B).

Theorem 1.2.2 [127] Let (X; ; I) be ideal space and fB g 2 be a family of subsets of X, then

1.

S

2 cl?(B ) =cl? (

S

2 B )

2. cl?(

T

2 B )

T

2 cl?(B ).

Definition 1.2.4 [228] let G be a generalized topology on a topological space (X; ), then a subset B of X is

called G -open set [11] if B cl(iG(B)).

Definition 1.2.5 [228] Let G be a generalized topology on an ideal topologicalspace(X, ; I), then a subset

B of X is called IG-open set if there exists a G-open set U such that UnB2I and Bncl(U)2I.

Definition 1.2.6 [228] Let G be a generalized topology on an ideal space (X; ; I), thena subset B of X is

called weakly IG-open set , if B = or if B , , then there is a non-empty G-open set U such that UnB2I.

The complement of a weakly IG-open set is called weakly IG-closed set.

2

1.3 p -Open Sets and their properties

Definition 1.3.1 Let (X; ) be a topological space with generalized topology G on X and any subset B of set

X is a p -open set, if B iG(cl(B)). The set XnB is p -closed set.

Theorem 1.3.1 Let (X; ) be a topological space with generalized topology G on X and let B be any G-open

set on set X then subset B is a p -open set on set X.

Proof Let (X; ) be a topological space with generalized topology G on X and let B be any G-open set on set

X. Then B = iG(B). we known that B cl(B), then we have B =iG(B) iG(cl(B)). It implies that subset B

### is a p -open set in X.

Theorem 1.3.2 Let (X; ) be a topological space with generalized topology G on X and Then B be any open

set on set X is p -open set if G.

Proof Let (X; ) be a topological space with generalized topology G on X and let B be any open set on set X.

Given that G , which implies that subset B is G-open. By Theorem 1.3.1, subset B is a p -open on set X.

In following example we observe that p -open set is neitherG-open nor open set on (X; ) with generalized

## topology G.

Example 1.3.1 Let us consider X = f d1; d2; d3; d4 g and topology =f ; fd1g; fd1; d2g; fd1; d3g; fd1; d2; d3g; Xg

with generalized topology G = f ; fd1; d2g; fd2; d3g; fd1; d2; d3g; Xg on X. If we consider B = fd1; d4g, then

clearly set B is a p -open set on (X; ) with generalized topology G but it is neither G-open nor open set.

Proposition 1.3.1 Let (X; ) be a topological space with generalized topology G on X and let subset B (,

be any p -open set on set X, then iG (cl (B) ) , ;.

## Proof Obviously.

Theorem 1.3.3 Let (X; ) be a topological space with generalized topology G on X and suppose B (, is

a p -open set on set X then 9 a G-open set V in X such that cl(B) V B.

Proof Let (X; ) be a topological space with generalized topology G on X and suppose B (, is a p –

open set on set X, then B iG(cl(B)) = V (say). It means that B V, V is G-open set in X. Since

V = iG(cl(B)) cl(B); then we obtain V cl(B). Finally, we find that there exist a G-open set V with

### property cl(B) V B:

Conversely, suppose B X and V is a G-open set in X with property cl(B) V B: Given that cl(B) V,

this means that iG(cl(B)) iG(V) = V, hence iG(cl(B)) V: Also B V, then we find B iG(cl(B)): Thus,

### B is a p -open set in G on (X; ).

Theorem 1.3.4 Let (X; ) be a topological space with generalized topology G on X. Then any arbitrary

union of p -open sets is a p -open set.