In this Chapter we so through the basic definitions and results are Essay

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

2 cl?(B ) =cl? (

2 B )

2. cl?(

2 B )

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.

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.