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.