International Journal of Analysis and Applications

Volume 18, Number 2 (2020), 149-160

URL: https://doi.org/10.28924/2291-8639

DOI: 10.28924/2291-8639-18-2020-149

IDEALS ON GENERALIZED TOPOLOGICAL SPACES

FAHAD ALSHARARI∗

Department of Mathematics, College of Science and Human Studies, Hotat Sudair, Majmaah University,

Majmaah 11952, Saudi Arabia

∗Corresponding author: f.alsharari@mu.edu.sa

Abstract. In this paper, we define the g-closure operator and investigate some of its crucial properties.

We also introduce and study the concept of ψg-classes and generalized compatibly of generalized topology

with ideal. This work is generalization of [4].

1. Introduction

The idea of ”idealizing” of a topological space can be found in some classical texts of Kuratowski ( [14], [15])

and Vaidyanathaswamy [18]. Some early applications of ideal topological spaces can be found in various

branches of mathematics, like a generalization of Cantor-Bendixson theorem by Freud [10], or in measure

theory by Scheinberg [17]. In 1990 Jankovic̀ and Hamlett [13] wrote a paper in which they, among their

results, included many other results in this area using modern notation, and logically and systematically

arranging them. This paper rekindled the interest in this topic, resulting in many generalizations of the ideal

topological space and many generalizations of the notion of open sets, like in papers of Jafari and Rajesh [11],

and Manoharan and Thangavelu [16].

In 1966 Velicko [19] introduced the notions of θ-open and θ-closed sets, and also a θ-closure, examining

H-closed spaces in terms of an arbitrary filterbase. A space X is called H-closed if every open cover of X

has a finite subfamily whose closures cover X. It turned out that θ-open sets completely correspond to the

Received 2019-11-22; accepted 2020-01-13; published 2020-03-02.

2010 Mathematics Subject Classification. 54A05.

Key words and phrases. generalized local function; g-closure operator; ψg-classes; generalized compatibly of generalized

topology with ideal.

c©2020 Authors retain the copyrights

of their papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License.

149

https://doi.org/10.28924/2291-8639
https://doi.org/10.28924/2291-8639-18-2020-149


Int. J. Anal. Appl. 18 (2) (2020) 150

already known notion of θ-continuity, introduced in 1943 by Fomin [9]. In 1975 Dickman and Porter [8]

continued the study of H-closed spaces using θ-closed sets proving that an H-closed space is not a countable

union of nowhere dense θ-closed sets. In 1980, Jankovic̀ [12] proved that a space is Hausdorff if and only if

every compact set is θ-closed. Recent applications of θ-open sets can be found in the paper of Caldas, Jafari

and Latif [6], and in the paper of Cammaroto, Catalioto, Pansera and Tsaban [7].

In [5], Al-Omari and Noiri introduced the local closure function as a generalization of the θ-closure and

the local function in an ideal topological space. They proved some basic properties for the local closure

function, and also introduced two new topologies obtained from the original one using the local closure

function. Afterwards, many properties in topological spaces have been explored by various researchers

( [20], [21], [22], [23], [24], [25])

Ideal topological space is a beautiful mixture of topology and geometry. A generalized topology (briefly,

GT) [1], µ on a nonempty set X is a collection of subsets of X such that φ ∈ µ and µ is closed under arbitrary

unions. Elements of µ will be called µ-open sets, and a subset A of (X,µ) will be called µ-closed if (X \A)

is µ-open. The pair (X,µ) will be called a generalized topological space (briefly, GTS) ( [2], [3]). By a space

X or (X,µ), we will always mean a GTS. Clearly, every topological space is a GTS.

If U is a subset of a space (X,µ), then the µ-closure Clµ(U) of µ is the intersection of all µ-closed sets

containing U and the µ-interior intµ(U) of U is the union of all µ-open sets contained in µ [2].

Let (X,τ) be a topological space, for each U ∈ P(X) and x ∈ X. Then, U is called generalized open

neighborhood of x if x ∈ U with U is µ-open set. That is Ng(x) = {U is µ-open: x ∈ U}.

2. Ideal Generalized Local Function

In this section we introduce and study the concept of ideal generalized topological spaces. We also

investigate some of its properties.

Definition 2.1. An ideal generalized topological space is a generalized topological space (X,µ) with an ideal I

on X and is denoted by (X,µ,I). For a subset A⊂ X, Ag(I,µ) = {x ∈ X : U ∩A /∈ I for each U ∈ Ng(x)}.

Clearly, every ideal topological space is ideal generalized topological space.

Lemma 2.1. For A⊆ (X,µ,I) we have, Clµ(Ag) = Ag.

Proof. One implication is immediate. Conversely, we show that Clµ(Ag) ⊆Ag.

Let x ∈ Clµ(Ag). Then for each µ-open set G containing x, G ∩Ag 6= φ. Thus, x ∈ Ag(I,µ) which

implies, for each generalized open neighborhood U of x, U ∩A /∈ I. Therefore, (G ∪ U) ∩A /∈ I, implies

x ∈Ag(I,µ). Hence, Clµ(Ag) ⊆Ag. which completes the proof �



Int. J. Anal. Appl. 18 (2) (2020) 151

Corollary 2.1. Let (X,µ,I) be an ideal generalized topological space. Then, Ag(φ,µ) = Clµ(A).

Proof. Follows from Lemma 2.1 �

Theorem 2.1. Let (X,µ,I) be an ideal generalized topological space and A,B ⊆ X. Then:

(1) Ag is µ-closed set,

(2) If I1 ⊆ I2, then Ag(I1) ⊇Ag(I2),

(3) If A⊆B, then Ag ⊆Bg,

(4) Ag ⊆ Clµ(A),

(5) (Ag)g ⊆Ag,

(6) Ag ∪Bg = (A∪B)g,

(7) Ag −Bg = (A−B)g,

(8) If B ∈ I, then (A∪B)g = (A−B)g = Ag,

(9) If B ∈ I, then (X −B)g = Xg,

(10) If U ∈ µ, then (U ∩Ag) ⊆ (U ∩A)g .

Proof. (1) If x /∈Ag, then for some µ-open set U, we have x ∈ U and U ∩A∈ I. This implies U ⊆ X −Ag,

which means that X −Ag is µ-open set. Thus, Ag is µ-closed set.

(2) It is clear.

(3) Let x /∈ Bg. Then, there exists U ∈ Ng(x) containing x such that U ∩ B ∈ I. This implies

U ∩A⊆ U ∩B ∈ I. Hence, x /∈Ag . Thus, Ag ⊆Bg.

(4) follows directly from (1).



Int. J. Anal. Appl. 18 (2) (2020) 152

(5) Since Ag ⊆ Clµ(A), (Ag)g ⊆ Clµ(Ag). By (1), (Ag)g ⊆ Clµ(Ag) = Ag.

(6) One implication is immediate from (2), that is (Ag ∪Bg) ⊂ (A∪B)g. To prove the reveres inclusion.

Let x /∈ (Ag ∪Bg), then x /∈ Ag or x /∈ Bg . Then there exists U1 ∈ Ng(x) and U2 ∈ Ng(x) such that

U1 ∩A ∈ I and U2 ∩B ∈ I. Since I is hereditary and additive, then (U1 ∩ U2) ∩ (A∪B) ∈ I. Thus,

x /∈ (A∪B)g . Hence, (Ag ∪Bg) ⊃ (A∪B)g.

(7) Since A = (A∩B) ∪ (A−B) for any A,B ⊆ X. Then by (2), we have

Ag −Bg = [(A∩B) ∪ (A−B)]g −Bg = (A−B)g.

(8) By using (6) and the fact that, if B ∈ I, then Bg = φ.

(9) Obvious.

(10) Let x ∈ U ∩Ag. Then x ∈ U and x ∈ Ag. This implies that there exists G ∈ Ng(x) such that

G∩A∈ I. Since, x ∈ U ∩G, U ∩ (G∩A) ∈ I. Hence, x ∈ (U ∩A)g . �

Theorem 2.2. Let (X,µ,I) be an ideal generalized topological space and {Ai}i∈J be a family of subsets of

X. Then:

(1) (
⋃

(Ai)g : i ∈ J) ⊆ (
⋃
Ai : i ∈ J)g,

(2) (
⋂
Ai : i ∈ J)g ⊆ (

⋂
(Ai)g : i ∈ J),

Proof. (1) Since Ai ⊆
⋃
Ai, for each i ∈ J, by Theorem 2.1(2), we have (Ai)g ⊆ (

⋃
Ai)g, for each i ∈ J.

This implies

(
⋃

(Ai)g : i ∈ J) ⊆ (
⋃
Ai : i ∈ J)g.

(2) Since
⋂
Ai ⊆Ai, (

⋂
Ai)g ⊆ (Ai)g , for each i ∈ J. Thus,

(
⋂
Ai : i ∈ J)g ⊆ (

⋂
(Ai)g : i ∈ J).

�

Now, we define the g-closure operator, denoted by Cl?µ for a generalized topology µ
?(I) finer than τ as

follows: Cl?µ(A) = A∪Ag for every A⊆ X. We will occasionally write Ag or Ag(I) for Ag(µ,I) and it will

cause no ambiguity.

We will denote by int?µ(A) and Cl?µ(A) the interior and closure of A⊆ (X,µ,I), respectively, with respect

to µ?.



Int. J. Anal. Appl. 18 (2) (2020) 153

Theorem 2.3. The class β(µ,I) = {U −E : U ∈ µ,E ∈ I} is a base for a generalized topology.

Proof. For every β1,β2 ∈ β, we have β1 = U1 −E1 and β2 = U2 −E2 where U1,U2 ∈ µ and E1,E2 ∈ I. Then

β1 ∩β2 = (U1 −E1) ∩ (U2 −E2) = (U1 ∩X −E1) ∩ (U2 ∩X −E2)

= (U1 ∩U2) ∩ (X −E1 ∩X −E2) = (U1 ∩U2) ∩X − (E1 ∪E2)

= (U1 ∩U2) − (E1 ∪E2) ∈ β

The generalized topology which have β(µ,I) as a base is called ?-generalized topology and is denoted by

µ?. �

Theorem 2.4. Let (X,µ,I) be an ideal generalized topological space. Then for each A,B ∈ (X,µ,I) we have:

(1) If A⊆B, then Cl?g(A) ⊆ Cl?g(B).

(2) Cl?µ(Cl
?
µ(A)) ⊆ Cl?µ(A).

(3) Cl?µ(A∪B) = Cl?µ(A) ∪Cl?µ(B).

(4) A⊆ Cl?µ(A) ⊆ Clµ(A).

(5) If I = φ, then, Ag = Clµ(A) ⊆ Cl(A).

(6) If I = PX, then, Ag = φ and A = Clµ(A).

Proof. We shall verify only the statements (2) and the remainder of this theorem can be proved similarly

(2) Form Theorem 2.1(5) we have

Cl?µ(Cl
?
µ(A)) = Cl

?
µ(A) ∪ (Cl

?
µ(A)µ = (A∪Aµ) ∪ (A∪Aµ)µ

= (A∪Aµ) = Cl?µ(A).

�

Theorem 2.5. If I1 and I2 are two ideals on (X,µ) such that I1 ⊆ I2, then:

(1) Ag(I1) ⊇Ag(I2).



Int. J. Anal. Appl. 18 (2) (2020) 154

(2) µ?(I1) ⊆ µ?(I2).

Proof. (1) Since A is a generalized open local function of I1 at x, it must also be a generalized open local

function of I2 at x (since every I1 is I2). Hence, Ag(I1) ⊇Ag(I2).

(2) Since I1 ⊆ I2 and by Theorem 2.1(2), Aµ(I1) ⊇ Ag(I2). This implies that Cl?µ(A)(I2,µ) ⊆

Cl?µ(A)(I1,µ)). Therefore, µ?(I1) ⊆ µ?(I2). �

Theorem 2.6. If (X,µ,I) is an ideal generalized topological space and A⊆ X. Then Ag−(Ag)g ⊆ (A−Ag)g.

Proof. Let x ∈ Ag − (Ag)g. Then x ∈ Ag ∩ (X − (Ag)g). Thus, x ∈ Ag that is, there exists U ∈ Ng(x)

such that U ∩A /∈ I. Hence, for each U ∈ Ng(x), U ∩ (A−Ag) /∈ I, which implies x ∈ (A−Ag) and this

completes the proof. �

Theorem 2.7. Let (X,µ) be a generalized topological space with I1 and I2 two ideals on X and A ⊆ X.

Then:

(1)Ag(I1 ∩ I2,µ) = Ag(I1,µ) ∪Ag(I2,µ).

(2) Ag(I1 ∪ I2,µ) = Ag(I1,µ?(I2)) ∩Ag(I2,µ?(I1)).

Proof. (1) Since I1 ∩ I2 ⊆ I1, then from Theorem 2.1 (2) we have Ag(I1,µ) ⊆ Ag(I1 ∩ I2,µ). Similarly,

Ag(I2,µ) ⊆Ag(I1 ∩ I2,µ). Hence,

Ag(I1,µ) ∪Ag(I2,µ) ⊆Ag(I1 ∩ I2,µ) (2.1)

To prove the reverse inclusion, let x /∈Ag(I1,µ)∪Ag(I2,µ), then x /∈Ag(I1,µ) and x /∈Ag(I2,µ), implies

that there exists U1 ∈ Ng(x) such that U1 ∩A∈ I1. Again, x /∈Ag(I2,µ), implies there exists U2 ∈ Ng(x)

such that U2 ∩A∈ I2.. Therefore, (U1 ∩U2) ∩A∈ (I1 ∩ I2). Hence, x /∈Ag(I1 ∩ I2,µ), implies that

Ag(I1 ∩ I2,µ) ⊆Ag(I1,µ) ∪Ag(I2,µ). (2.2)

Therefore, equations (2.1) and (2.2) establish the result.

(2) Assume that x /∈ Ag(I1 ∪ I2,µ), then there exists U ∈ Ng(x) such that U ∩A ∈ I1 ∪ I2. Let E ∈ I1

and H∈ I2 such that U ∩A = E∪H, because of the heredity of I, we may assume E∪H = φ. Thus we have

U ∩A−E = H and U ∩A−H = E. Thus, (U −E) ∩A = H ∈ I1 and (U −H) ∩A = E ∈ I2. Therefore,

x /∈Ag(I2,µ?(I1)) or x /∈Ag(I1,µ?(I2)) and hence,

Ag(I1,µ?(I2)) ∩Ag(I2,µ?(I1)) ⊆Ag(I1 ∪ I2,µ). (2.3)



Int. J. Anal. Appl. 18 (2) (2020) 155

Now assume that x /∈ Ag(I1,µ?(I2)). This implies that there exist U ∈ Ng(x) and H ∈ I2 such that

(U−H)∩A∈ I1. We may assume, because of the heredity of I2, that H⊆A. Put E = (U−H)∩A and we have

U ∩A = E∪H∈ I1 ∪I2. Thus, x /∈Ag(I1 ∪I2,µ). Thus we have shown that Ag(I1 ∪I2,µ) ⊆Ag(I1,µ?(I2)).

Similarly, we have that Ag(I1 ∪ I2,µ) ⊆Ag(I2,µ?(I1)). Therefore,

Ag(I1 ∪ I2,µ) ⊆Ag(I1,µ?(I2)) ∩Ag(I2,µ?(I1)). (2.4)

From (2.3) and (2.4) the result is establised. �

Remark 2.1. Put I1 = I2 in the above theorem, the following Corollary answers about the relationship

between µ? and [µ?]?.

Corollary 2.2. Let (X,µ) be a generalized topological space with I an ideal on X and A⊆ X. Then:

(1) Ag(I1,µ) = Ag(I1,µ?).

(2) µ? = [µ?]?

Definition 2.2. A subset A of the space (X,µ,I) is said to be µ-closed set iff Ag ⊆ A. Equivalently, A is

said to be µ?-closed iff Cl?µ(A) = A.

Theorem 2.8. The following statements are equivalent for a subset A of a space (X,µ,I).

(1) A∈ µ? .

(2) A is µ?-closed set.

(3) (X −A)g ⊆ (X −A).

(4) A⊆ (X −A)g .

Proof. It is clear. �

3. Ψg-Classes

Definition 3.1. If (X,µ,I) is ideal generalized topological spaces, we define an operator Ψg(µ,I) : PX → µ

as follows: for every A⊆ X, Ψg(µ,I)(A) = {x: there exists U ∈ Ng(x) such that U −A∈ I}. Equivalently,

Ψg(µ,I)(A) = X − (X −A)g for each A⊆ X. We denote Ψg(µ,I) by Ψg when no ambiguity.

Lemma 3.1. For A⊆ (X,µ,I) we have:

(1) If I = φ, then Ψg(A) = intµ(A).

(2) If I = PX, then Ψg(A) = X.



Int. J. Anal. Appl. 18 (2) (2020) 156

Proof. Obvious. �

Theorem 3.1. For A⊆ (X,µ,I) we have:

:

(1) Ψg(A) = ∪{U ∈ µ : U −A∈ I}.

(2) If U ∈ µ, then Ψg(U) = ∪{M∈ µ : (M−U) ∪ (U −M) ∈ I}.

Proof. (1) Follows immediately form Definition 3.1.

(2) ) If we denote ∪{M ∈ µ : (M − A) ∪ (U − M) ∈ I} by Ψg′ (A)1. By heredity of I we have

{M∈ µ : (M−A) ∪ (U −M) ∈ I}⊆{M∈ µ : (M−A) ∈ I} and hence by (1) we have for every A⊆ X

Ψg′ (A) ⊆ Ψg(A). (3.1)

Now assume that U ∈ µ and x ∈ Ψg(U). Then there exists M ∈ µ such that x ∈ (M− U) ∈ I. Let

N = M∪U . Then, N ∈ µ and x ∈ (N −U) ∪ (U −N) = (M −U) ∪φ = (M−U) ∈ I. Thus, x ∈ Ψg′ (U).

Hence

Ψg(U) ⊆ Ψg′ (U) (3.2)

From (3.1) and (3.2) we have Ψg(U) = Ψg′ (U), for every U ∈ µ. �

Theorem 3.2. Let A⊆ (X,µ,I) be a generalized topological space.

(1) If A⊆ X, then Ψg(A) is µ-open set.

(2) If A⊆B, then Ψg(A) ⊆ Ψg(B).

(3) If A,B ∈ PX, then Ψg(A∩B) = Ψg(A) ∩ Ψg(B).

(4) If U ∈ µ, then U ∈ Ψg(U).

(5) If A⊆ X, then Ψg(A) ⊆ Ψg(Ψg(A)).

(6) If A⊆ X, then Ψg(A) = Ψg(Ψg(A)) iff (X −A)g = ((X −A)g)g.

(7) If A⊆ I, then Ψg(A) = (X −Ag).

(8) If A⊆ X and E ⊆ I, then Ψg(A−E) = Ψg(A).

(9) If A⊆ X and E ⊆ I, then Ψg(A∪E) = Ψg(A).

(10) If (A−B) ∪ (B−A) ∈ I, then Ψg(A) = Ψg(B).

Proof. (1) Follows from Theorem 3.2(1).



Int. J. Anal. Appl. 18 (2) (2020) 157

(2) Since Ψg(µ,I)(A) = X − (X −A)g ⊆ X − (X −B)g = Ψg(µ,I)(B).

(3) One implication is immediate, i.e. Ψg(A∩B) ⊆ Ψg(A) ∩ Ψg(B). Conversely, let x ∈ Ψg(A) ∩ Ψg(B),

then x ∈ Ψg(A) and x ∈ Ψg(B) from Definition 3.1, there exists U1,U2 ∈ Ng(x) such that U1 −A ∈ I

and U2 −B ∈ I. Let U3 = U1 ∩ U2 and we have U3 −A ∈ I and U3 −B ∈ I (by heredity of I), Thus

U3 − (A∩B) = (U3 −A) ∪ (U3 −B) ∈ I (by additivity of I) and hence x ∈ Ψg(A∩B) which implies

Ψg(A) ∩ Ψg(B) ⊆ Ψg(A∩B) and this completes the proof.

(4) If U ∈ µ, then X −U is µ-cloed set which implies (X −U)g ⊆ (X −U) (Theorem 2.8) and hence by

Definition 3.1, we have U ⊆ X − (X −U)g = Ψg(U).

(5) Since Ψg(A) is µ-open set, then by (4) we have the result.

(6) Since Ψg(µ,I)(A) = X − (X −A)g and by hypothesis,

Ψg(Ψg(A)) = Ψg[X − (X −A)g] = X − [X − (X − (X −A)g)]g

= X − [(X −A)g]g = X − [(X −A)g]g

= X − (X −A)g = Ψg(A)

(7) Follows from Definition 3.1 and Theorem 2.1(9).

(8) From Theorem 2.1(9) and Definition 3.1, we have,

Ψg(A−E) = X − [X − (A−E)]g = X − [X − (A∩X −E)]g

= X − [(X −A) ∪E]g = X − [(X −A)]g = Ψg(A).

(9) From Theorem 2.1(9) and Definition 3.1, we have,

Ψg(A∪E) = X − [X − (A∪E)]g = X − [(X −A) ∪E]g

= X − [(X −A)]g = Ψg(A).

(10) Follows immediately from (8) and (9). �

Corollary 3.1. Let (X,µ,I) be an ideal generalized topological space for every A ⊆ X and E ∈ I, then,

Ψg(A−E) = Ψg(A) = Ψg(A∪E).

Proof. Follows immediately from Theorem 3.2 (8) and 3.2 (9). �

Theorem 3.3. Let (X,µ,I) be an ideal generalized topological space, then µ? = {A⊆ X : A⊆ Ψg(A)}.



Int. J. Anal. Appl. 18 (2) (2020) 158

Proof. Let δ = {A⊆ X : A⊆ Ψg(A)}.

Firstly, we show that δ is a generalized topology. Observe that φ ⊆ Ψg(φ) and ⊆ Ψg(X). Now, if

A,B ⊆ δ, then by Theorem 3.2(3) we have A∩B ⊆ Ψg(A∩B) = Ψg(A)∩Ψg(B), this implies that A∩B ∈ δ.

If {Ai}i∈Γ ⊆ δ, then Ai ⊆ Ψg(∪Ai) for every i ∈ Γ. This implies ∪Ai ⊆ Ψg(∪Ai) and we have shown that

δ is a topology.

Secondly, we show that µ? = δ. Now if U ∈ µ? and x ∈ U , there exists V ∈ µ? and E ∈ I such that

x ∈ V −E ⊆ U . Clearly, V − U ∈ E,. so that V − U ∈ I by heredity and hence x ∈ Ψg(U). Thus,

U ∈ Ψg(U) and we have shown µ? ⊆ δ. Now, let A ∈ δ we have by Definition 3.1 that A ∈ Ψg(A), implies

A ⊆ X − [(X −A)]g, which implies (X −A)g ⊆ (X −A), which also implies X −A is µ?-open and hence

A∈ µ?. Thus, µ? = δ. �

4. Some Forms of ?-Compatible

Definition 4.1. If (X,µ,I) is an ideal generalized topological space. Then I is said to be ?-compatible with

µ denoted by I ∼ µ, if for every A ⊆ X and for every x ∈ A, there exists U ⊆ Ag(x) such that U ∩A ∈ I,

then A∈ I.

Theorem 4.1. Let (X,µ,I) be an ideal generalized topological space. If I ∼ µ, then the base β(µ,I) for

µ?(I) is a generalized topology and hence β(µ,I) = µ?(I) and all µ-open in µ?(I) are of simple form, i.e.,

µ?(I) = {U −M : U ∈ µ}.

Proof. Follows immediately from the definition. �

Theorem 4.2. If I ∼ µ, then the following statements are equivalent for a subset A of a space (X,µ,I).

(1) A∈ I.

(2) Ag = φ.

(3) A∩Ag = φ.

Proof. (1) ⇒ (2): Let A ∈ I. Then from hypothesis for every x ∈ X and every U ∈ Ng(x), we have

U ∩A∈ I. Thus x /∈Ag which means Ag = φ.

(2) ⇒ (3): Obvious.

(3) ⇒ (1): A /∈ I. Then according to ?-compatible of µ with I, there is an x ∈ A such that for every

U ∈ Ng(x), U ∩A /∈ I, so x ∈Ag which means A∩Ag 6= φ, which is a contradiction. �



Int. J. Anal. Appl. 18 (2) (2020) 159

Theorem 4.3. Let (X,µ,I) be an ideal generalized topological space. Then I ∼ µ. iff Ψg(A) −A ∈ I, for

every A⊆ X.

Proof. Necessity Let I ∼ µ and A ⊆ X. We observe that x ∈ Ψg(A) −A iff x /∈ A and x /∈ (X −A)g

iff x /∈ A and there exists U ∈ Ng(x), such that U −A ∈ I, iff there exists x ∈ U ∈ Ng(x), such that

x ∈ (U −A) ∈ I. Now for each x ∈ Ψg(A)−A and U ∈ Ng(x) and U ∩(Ψg(A)−A) ∈ I, by heredity , since

I ∼ µ, then (Ψg(A) −A) ∈ I.

Sufficiency: Let A ⊆ X, and let for every x ∈ A there exists U ∈ Ng(x) such that (U ∩ A) ∈ I

and now we prove that A ∈ I. We observe that Ψg(X −A) − (X −A) = {x : ∃U ∈ Ng(x) such that

x ∈{[U−(X−A)] ∈ I}}. This implies Ψg(X−A)−(X−A) = {x : ∃U ∈ Ng(x) such that x ∈ (U∩A) ∈ I}.

Thus we have A⊆ Ψg(X −A) − (X −A) ∈ I and hence A∈ I by the heredity of I. �

Theorem 4.4. Let (X,µ,I) be an ideal generalized topological space with I ∼ µ. Then

(1)(A−Ag) ∈ I, for every A⊆ X.

(2) (Ag)g = Ag , for every A⊆ X.

Proof. Assume that x ∈ (A − A)g , then for every U ∈ Ng(x), we have (A − Ag) ∩ U /∈ I or

(U ∩A) −Ag /∈ I. This implies (A∩U) /∈ I and so x ∈ Ag . Thus, x ∈ A−Ag from which we conclude

that (A−Ag) ∩ (A−Ag)g = φ, from Theorem 4.2 we have A−Ag ∈ I.

(2) Let I ∼ µ. Then for every A⊆ X, A−Ag ∈ I. This implies A−Ag = φ, by using Theorem 2.1(7), we

have Ag − (Ag)g ⊆ (A−Ag)g = φ. Hence, Ag ⊆ (Ag)g. However, by Theorem 2.1(4) we have Ag ⊆ (Ag)g.

Thus, Ag = (Ag)g . �

Theorem 4.5. Let (X,µ,I) be an ideal generalized topological space with I ∼ µ. Then Ψg(Ψg(A)) = Ψg(A)

for every A⊆ X.

Proof. From Theorem 3.2(5) we have Ψg(A) ⊆ Ψg(Ψg(A)). Since I ∼ µ, then from Corollary 3.1, Ψg(A) =

A∪E for some E ∈ I and hence by Theorem 3.2(9) Ψg(Ψg(A)) = Ψg(A∪E) = Ψg(A). �

Acknowledgement: The author would like to express their sincere thanks to the referees for their useful

suggestions and comments.

Conflicts of Interest: The author declares that there are no conflicts of interest regarding the publication

of this paper.



Int. J. Anal. Appl. 18 (2) (2020) 160

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	1. Introduction
	2. Ideal Generalized Local Function
	3. g-Classes
	4. Some Forms of -Compatible
	References