New topologies derived from the old one via operators

Faical Yacine Issaka; Murad Özkoç

doi:10.1515/dema-2024-0094

Article Open Access

New topologies derived from the old one via operators

Faical Yacine Issaka and Murad Özkoç

Published/Copyright: April 9, 2025

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From the journal Demonstratio Mathematica Volume 58 Issue 1

Abstract

The main purpose of this work is to study the ideal topology defined by the minimal and maximal ideals on a topological space. Also, we define and investigate the concepts of ideal quotient and annihilator of any subfamily of 2 X , where 2 X is the power set of X . We obtain some of their fundamental properties. In addition, several relationships among the above notions have been discussed. Moreover, we define a new topology on an ideal topological space, called sharp topology, via the sharp operator defined in this study, which turns out to be finer than the original topology. Furthermore, a decomposition of open sets (in the original topology) has been obtained. Finally, we conclude our work with some interesting applications.

Keywords: maximal ideal; minimal ideal; ideal quotient; annihilator; sharp-operator; sharp-topology

MSC 2010: 54A10; 54C60; 47H04

1 Introduction

Structures such as filters [1], ideals [2], grills [3], and primals [4] are some of the topics studied extensively in general topology. An ideal ℐ on a topological space ( X , τ ) is a non-empty collection of subsets of X which satisfies (i) A ∈ ℐ and B ⊆ A imply B ∈ ℐ and (ii) A ∈ ℐ and B ∈ ℐ imply A ∪ B ∈ ℐ . A topological space with an ideal is called ideal topological space. The concept of the local function in general topology was introduced by Kuratowski [5] in 1933 and studied from very different aspects by many mathematicians. In an ideal topological space ( X , τ , ℐ ) , the local function ( ⋅ ) * [5] is defined as A * ( ℐ , τ ) = { x ∈ X ∣ ( ∀ U ∈ τ ( x ) ) ( U ∩ A ∉ ℐ ) } , where τ ( x ) is the collection of all open subsets containing x ∈ X . Vaidyanathaswamy [6] investigated properties of the local function in 1945 in detail. Thanks to the concept of the local function, the literature gained a new topology called ⁎-topology and was studied further by Hayashi [7] in 1964 and Njastad [8] in 1966, later by Samuels [9] in 1975 and many others. In 1986, Natkaniec [10] introduced the complement of the local function called Ψ -operator. In 1990, after a hiatus of about 4 years, this topic was revisited by Jankovic and Hamlett [11]. In that article, they have not only summarized all the known facts on this topic, but also presented some new results. In 2018, Islam and Modak [12] investigated the properties of ⁎ and Ψ operators.

Subsequently, many papers have been published on this topic. For instance, Arenas et al. [13] studied the idealization of some weak separation axioms, while Navaneethakrishnan and Joseph [14] in 2008 devoted their attention to investigate g -closed sets in ideal topological spaces. The others such as Hatır [15] and Ekici [16] have studied the decompositions of continuity in ideal topological spaces and I -Alexandroff topological spaces in ideal topological spaces, respectively.

Most of these approaches to the subject are similar. In one approach, a new topology with new properties is obtained by changing the definition of the local function, while in another approach, topologies arising from well known ideals are investigated.

On the other hand, since the main topic of this study is the use of ideals in topology, there are also practical applications of the use of ideals in topology in the processing of information systems. Al-shami et al. [17] introduced four new kinds of rough set models based on cardinality neighborhoods and two ideals. The significance of these methods lies in their reliance on ideals, which serve as topological tools. Hosny and Al-shami [18] have endeavored by rough set theory to enlarge the knowledge they obtain from the information systems for this reason. They have applied the abstract concept of ideal structures to build new generalized approximation spaces with less vagueness. Kaur et al. [19] introduced a novel frame of nano-topology using various covering-based neighborhoods via multiple ideals. They also discussed a medical application where multi-ideal nano-topology is used to find the key symptoms of Dengue disease.

In Section 3, we define the notions of the maximal and minimal ideals. Some characterizations of these concepts are obtained.

In Section 4, we introduce and study the notion of ideal quotient and investigate some of its fundamental properties. We also give a characterization for maximal ideals using the ideal quotients.

In Section 5, we define the concepts of the annihilator of a family of sets and faithful ideal. We obtain a characterization of the minimal ideal with the help of the annihilator. We also give a new characterization of density via annihilators.

In Section 6, we introduce a new operator called sharp operator and obtain some of its fundamental properties. Also, we define a new Kuratowski closure operator using the sharp operator. The topology obtained from this Kuratowski closure operator turns out to be finer than the original one. Moreover, we obtain a decomposition of open sets in the original topology.

In Section 7, we introduce the concepts of ⁎-continuity and ♯ -continuity. We give a relation between ⁎-continuity and ♯ -continuity. Furthermore, we obtain a new decomposition of continuity.

In the last section, we give some applications of sharp operator and prove the density of the set of all rational numbers using the notions of sharp topology and annihilator defined in the scope of this study. We are also looking for answers to the following questions:

Is there a Hausdorff topology on R such that the set of irrational numbers I is not dense, while the set of rational numbers Q is dense?
Is there a Hausdorff topology such that the set of rational numbers Q is clopen?
Is there a disconnected Hausdorff topology on R except discrete topology?

We prefer to use the universal symbolic language of mathematics in the proofs of the theorems given in the scope of this study throughout the article.

2 Preliminaries

Throughout this study, ( X , τ ) and ( Y , σ ) (or simply X and Y ) always mean topological spaces on which no separation axioms are assumed unless explicitly stated. For a subset A of a space X , the closure and the interior of A will be denoted by c l ( A ) and int ( A ) , respectively. For an ideal ℐ on a space X , the operator c l * : 2 X → 2 X defined by c l * ( A ) = A ∪ A * is a Kuratowski closure operator. The topology induced by the operator c l * is τ * ( ℐ , τ ) = { A ⊆ X ∣ c l * ( X \ A ) = X \ A } and called ⁎-topology which is finer than τ . Natkaniec introduced the complement of local function called Ψ -operator which is defined by Ψ ( A ) = X \ ( X \ A ) * for any subset A of X . An ideal topological space ( X , τ , ℐ ) is called a Hayashi-Samuel space [20] if τ ∩ ℐ = { ∅ } .

Lemma 2.1

Let ( X , τ , ℐ ) be an ideal topological space and A ⊆ X . If A ∈ ℐ , then A * = ∅ .

Lemma 2.2

[21] Let ( X , τ ) be a topological space and ℐ , J ⊆ 2 X be ideals on X. Then, the following hold for any subset A ⊆ X .

A * ( ℐ ∩ J , τ ) = A * ( ℐ , τ ) ∪ A * ( J , τ ) ,
τ * ( ℐ ∩ J , τ ) = τ * ( ℐ , τ ) ∩ τ * ( J , τ ) .

Definition 2.3

Let X be a non-empty set and A ⊆ X . Then, the ideal generated by A is defined as ℐ ( A ) ≔ { I ∣ I ⊆ A } .

Lemma 2.4

Let X be a non-empty set and A ⊆ X . Then, the family ℐ ε ( A ) = { I ⊆ X ∣ I ∩ A = ∅ } is an ideal on X.

Proof

This is obvious since ℐ ε ( A ) = ℐ ( X \ A ) .□

3 Maximal and minimal ideal on topological space

Definition 3.1

Let ℐ be a proper ideal on X i.e., ℐ ≠ 2 X . Then, ℐ is said to be a maximal ideal if for any ideal J with ℐ ⊆ J , ℐ = J or J = 2 X .

Proposition 3.2

Let ℐ be a proper ideal on X , i.e., ℐ ≠ 2 X . Then, ℐ is a maximal ideal if and only if ℐ = J for all ideal J ≠ 2 X with ℐ ⊆ J .

Proof

Let ℐ be a proper ideal on X .

□ ℐ is a maximal ideal on X ⇔ ( ∀ J ⊆ 2 X ) [ ℐ ⊆ J ⇒ ( ℐ = J ∨ J = 2 X ) ] ⇔ ( ∀ J ⊆ 2 X ) [ ℐ ⊈ J ∨ ( ℐ = J ∨ J = 2 X ) ] ⇔ ( ∀ J ⊆ 2 X ) [ ℐ ⊈ J ∨ ( J = 2 X ∨ ℐ = J ) ] ⇔ ( ∀ J ⊆ 2 X ) [ ( ℐ ⊈ J ∨ J = 2 X ) ∨ ℐ = J ] ⇔ ( ∀ J ⊆ 2 X ) [ ¬ ( ℐ ⊆ J ∧ J ≠ 2 X ) ∨ ℐ = J ] ⇔ ( ∀ J ⊆ 2 X ) [ ( ℐ ⊆ J ∧ J ≠ 2 X ) ⇒ ℐ = J ] .

Lemma 3.3

Let ℐ be an ideal on X. Then, the family J = { A ∩ B ∣ A ∈ ℐ ∨ B ∈ ℐ } is an ideal on X.

Theorem 3.4

Let ℐ be a proper ideal on X. Then, ℐ is a maximal ideal if and only if for all A , B ⊆ X , if A ∩ B ∈ ℐ , then A ∈ ℐ or B ∈ ℐ .

Proof

( ⇒ ) : Let ℐ be a maximal ideal and A ∩ B ∈ ℐ .

J ≔ { A ∩ B ∣ A ∈ ℐ ∨ B ∈ ℐ } ⇒ ( J is an ideal ) ( ℐ ⊆ J ) ℐ is maximal ideal ⇒ ℐ = J A ∩ B ∈ ℐ ⇒ A ∩ B ∈ J ⇒ A ∈ ℐ ∨ B ∈ ℐ

( ⇐ ): Let J be an ideal such that J ≠ 2 X and ℐ ⊆ J . We will prove that ℐ = J . Let A ∈ J .

ℐ is an ideal ⇒ ∅ ∈ ℐ ∅ = A ∩ ( X \ A ) ⇒ A ∩ ( X \ A ) ∈ ℐ ⇒ Hypothesis A ∈ ℐ ∨ ( X \ A ) ∈ ℐ ℐ ⊆ J ⇒ A ∈ ℐ ∨ ( X \ A ) ∈ J ⇒ A ∈ ℐ ∨ ( X \ A ) ∈ J A ∈ J ⇒ X = A ∪ ( X \ A ) ∈ J ∨ A ∈ ℐ J ≠ 2 X ⇒ A ∈ ℐ .

Thus, we have J ⊆ ℐ . Since ℐ ⊆ J , we obtain ℐ = J .□

Theorem 3.5

Let ℐ be a proper ideal on X. Then, ℐ is a maximal ideal if and only if A ∈ ℐ or X \ A ∈ ℐ for all A ⊆ X .

Proof

( ⇒ ) : Let ℐ be a maximal ideal and A ⊆ X .

ℐ is an ideal ⇒ ∅ ∈ ℐ A ⊆ X ⇒ ∅ = A ∩ ( X \ A ) ⇒ A ∩ ( X \ A ) ∈ ℐ ℐ is a maximal ideal ⇒ Theorem 3.3 A ∈ ℐ ∨ ( X \ A ) ∈ ℐ .

( ⇐ ): Let ℐ and J be two ideals such that J ≠ 2 X and ℐ ⊆ J . We will prove that ℐ = J . Now, let A ∈ J .

A ∈ J ⇒ A ⊆ X ⇒ Hypothesis A ∈ ℐ ∨ ( X \ A ) ∈ ℐ ℐ ⊆ J ⇒ A ∈ ℐ ∨ ( X \ A ) ∈ J A ∈ J ⇒ ⇒ X = A ∪ ( X \ A ) ∈ J ∨ A ∈ ℐ J ≠ 2 X ⇒ A ∈ ℐ .

Thus, we have J ⊆ ℐ . Since ℐ ⊆ J , we obtain ℐ = J .□

Corollary 3.6

Let ( X , τ , ℐ ) be an ideal topological space. If ℐ is a maximal ideal, then A * = ∅ or ( X \ A ) * = ∅ for all A ⊆ X .

Theorem 3.7

Let ( X , τ , ℐ ) be an ideal topological space and A ⊆ X . If ℐ is a maximal ideal, then A is τ * -closed or τ * -open.

Proof

Let ℐ be a maximal ideal on X and A ⊆ X .

□ ( ℐ is a maximal ideal ) ( A ⊆ X ) ⇒ A ∈ ℐ ∨ ( X \ A ) ∈ ℐ ⇒ A * = ∅ ∨ ( X \ A ) * = ∅ ⇒ A * ∪ A = A ∨ Ψ ( A ) = X \ ( X \ A ) * = X ⇒ c l * ( A ) = A ∨ A ⊆ Ψ ( A ) ⇒ A ∈ C ( X , τ * ) ∨ A ∈ τ * .

Corollary 3.8

Let ( X , τ , ℐ ) be an ideal topological space. If ℐ is a maximal ideal, then ( X , τ * ) is a T 0 space.

Proof

This follows from Theorem 3.7.□

Definition 3.9

Let ℐ be a proper ideal on X such that ℐ ≠ { ∅ } . Then, ℐ is said to be a minimal ideal if for any ideal J with J ⊆ ℐ , ℐ = J or J = { ∅ } .

Theorem 3.10

Let ℐ be a proper ideal on X . Then, ℐ is a minimal ideal if and only if A = B for all A , B ∈ ℐ \ { ∅ } .

Proof

( ⇒ ) : Let ℐ be a minimal ideal and A , B ∈ ℐ \ { ∅ } .

A , B ∈ ℐ \ { ∅ } ⇒ ( ℐ ( A ) ⊆ ℐ ) ( ℐ ( B ) ⊆ ℐ ) ℐ is a minimal ideal ⇒ ℐ ( A ) = ℐ ( B ) = ℐ ⇒ A = B .

( ⇐ ) : Let ℐ be an ideal and J ⊆ ℐ .

□ ℐ is an ideal ⇒ Hypothesis ∣ ℐ ∣ = 2 J ⊆ ℐ ⇒ ( ∣ J ∣ = 1 ∨ ∣ J ∣ = 2 ) ⇒ ( J = { ∅ } ∨ J = ℐ ) .

Corollary 3.11

Let ℐ be a proper ideal on X. Then, the following statements are equivalent:

ℐ is minimal ideal on X ;
∣ ℐ ∣ = 2 , where ∣ ℐ ∣ is the cardinality of ℐ ;
there exists a subset A of X such that ∣ A ∣ = 1 and ℐ = ℐ ( A ) .

Theorem 3.12

Let X be a non-empty set and A ⊆ X . Then, ℐ ( A ) is a minimal ideal on X if and only if ℐ ε ( A ) is a maximal ideal on X.

Proof

( ⇒ ) : Let ℐ ( A ) be a minimal ideal on X and B ⊆ X .

ℐ ( A ) is minimal ⇒ ( ∃ x ∈ X ) ( A = { x } ) ( ℐ = ℐ ( A ) ) B ⊆ X ⇒ A ∩ B = ∅ ∨ A ∩ ( X \ B ) = ∅ ⇒ B ∈ ℐ ε ( A ) ∨ X \ B ∈ ℐ ε ( A ) .

This means that ℐ ε ( A ) is maximal ideal by Theorem 3.5.

( ⇐ ) : Let ℐ ε ( A ) be a maximal ideal and x , y ∈ A .

□ x , y ∈ A ⇒ ( ℐ ε ( A ) ⊆ ℐ ε ( { x } ) ) ( ℐ ε ( A ) ⊆ ℐ ε ( { y } ) ) ℐ ε ( A ) is maximal ⇒ ℐ ε ( { x } ) = ℐ ε ( { y } ) = ℐ ε ( A ) ⇒ x = y x , y ∈ A ⇒ ⇒ ∣ A ∣ = 1 ⇒ ℐ ( A ) is minimal .

Corollary 3.13

Let ℐ be a proper ideal on X. If there exists a singleton subset A ⊆ X such that ℐ = ℐ ε ( A ) , then ℐ is a maximal ideal.

4 Ideal quotient

Lemma 4.1

Let ℐ be an ideal on X and J ⊆ 2 X . Then, the family ( ℐ : J ) = { A ⊆ X ∣ ( ∀ J ∈ J ) ( A ∩ J ∈ ℐ ) } is an ideal on X.

Proof

Let A ∈ ( ℐ : J ) and B ⊆ A . We will prove that ( ℐ : J ) is downward closed.

A ∈ ( ℐ : J ) ⇒ ( ∀ J ∈ J ) ( A ∩ J ∈ ℐ ) B ⊆ A ⇒ ( ∀ J ∈ J ) ( B ∩ J ⊆ A ∩ J ∈ ℐ ) ℐ is an ideal ⇒ ( ∀ J ∈ J ) ( B ∩ J ∈ ℐ ) ⇒ B ∈ ( ℐ : J ) .

Now, let A , B ∈ ( ℐ : J ) . We will prove that A ∪ B ∈ ( ℐ : J ) .

□ A , B ∈ ( ℐ : J ) ⇒ ( ∀ J ∈ J ) ( A ∩ J ∈ ℐ ) ( ∀ J ∈ J ) ( B ∩ J ∈ ℐ ) ⇒ ( ∀ J ∈ J ) ( A ∩ J ∈ ℐ ) ( B ∩ J ∈ ℐ ) ℐ is an ideal ⇒ ⇒ ( ∀ J ∈ J ) ( ( A ∪ B ) ∩ J = ( A ∩ J ) ∪ ( B ∩ J ) ∈ ℐ ) ⇒ A ∪ B ∈ ( ℐ : J ) .

Definition 4.2

Let ℐ be an ideal on X and J be any subfamily of 2 X . Then, the family ( ℐ : J ) is called ideal quotient.

Theorem 4.3

Let ℐ and ℐ ′ be two ideals on X and J , J ′ ⊆ 2 X . Then, the following properties hold:

ℐ ⊆ ( ℐ : J ) ;
J ⊆ ℐ if and only if ( ℐ : J ) = 2 X ;
if J ⊆ J ′ , then ( ℐ : J ′ ) ⊆ ( ℐ : J ) ;
if X ∈ J , then ( ℐ : J ) = ℐ ;
( ℐ ∩ ℐ ′ : J ) = ( ℐ : J ) ∩ ( ℐ ′ : J ) .

Proof

(a) Let A ∈ ℐ . We will show that A ∈ ( ℐ : J ) .

A ∈ ℐ ⇒ ( ∀ J ∈ J ) ( A ∩ J ⊆ A ∈ ℐ ) ℐ is an ideal ⇒ ( ∀ J ∈ J ) ( A ∩ J ∈ ℐ ) ⇒ A ∈ ( ℐ : J ) .

(b) ( ⇒ ) : We have always ( ℐ : J ) ⊆ 2 X . Now, we will prove that 2 X ⊆ ( ℐ : J ) . Let A ∈ 2 X .

A ∈ 2 X ⇒ Hypothesis ( ∀ J ∈ J ) ( J ∈ ℐ ) ( A ∩ J ⊆ J ) ⇒ ( ∀ J ∈ J ) ( A ∩ J ∈ ℐ ) ⇒ A ∈ ( ℐ : J ) .

Then, we have 2 X ⊆ ( ℐ : J ) . Thus, we obtain ( ℐ : J ) = 2 X .

( ⇐ ) : Let ( ℐ : J ) = 2 X . We will prove that J ⊆ ℐ . Now, let J ∈ J . If we show that J ∈ ℐ , then the proof is complete.

( ℐ : J ) = 2 X ⇒ X ∈ ( ℐ : J ) J ∈ J ⇒ J = X ∩ J ∈ ℐ .

(c) Let J ⊆ J ′ . We will prove that ( ℐ : J ′ ) ⊆ ( ℐ : J ) . Now, let A ∈ ( ℐ : J ′ ) . If we show that A ∈ ( ℐ : J ) , then the proof is complete.

A ∈ ( ℐ : J ′ ) ⇒ ( ∀ J ∈ J ′ ) ( A ∩ J ∈ ℐ ) J ⊆ J ′ ⇒ ( ∀ J ∈ J ) ( A ∩ J ∈ ℐ ) ⇒ A ∈ ( ℐ : J ) .

(d) From ( a ) , we have ℐ ⊆ ( ℐ : J ) . Now, we will prove that ( ℐ : J ) ⊆ ℐ . Let A ∈ ( ℐ : J ) . If we show that A ∈ ℐ , then the proof is complete.

A ∈ ( ℐ : J ) ⇒ ( ∀ J ∈ J ) ( A ∩ J ∈ ℐ ) Hypothesis ⇒ A = A ∩ X ∈ ℐ .

(e) Let A ⊆ X .

□ A ∈ ( ℐ ∩ ℐ ′ : J ) ⇔ ( ∀ J ∈ J ) ( A ∩ J ∈ ℐ ∩ ℐ ′ ) ⇔ ( ∀ J ∈ J ) ( A ∩ J ∈ ℐ ∧ A ∩ J ∈ ℐ ′ ) ⇔ ( ∀ J ∈ J ) ( A ∩ J ∈ ℐ ) ∧ ( ∀ J ∈ J ) ( A ∩ J ∈ ℐ ′ ) ⇔ A ∈ ( ℐ : J ) ∧ A ∈ ( ℐ ′ : J ) ⇔ A ∈ ( ℐ : J ) ∩ ( ℐ ′ : J ) .

Remark 4.4

The converse of the implications given in Theorem 4.3(c) and (d) need not always to be true as shown by the following examples.

Example 4.5

Let X = { a , b , c } and ℐ = { ∅ , { a } } . For the subfamilies J = { { a } , { a , c } } and J ′ = { { a , b } , { a , c } } of 2 X , we have ( ℐ : J ) = { ∅ , { a } , { b } , { a , b } } and ( ℐ : J ′ ) = { ∅ , { a } } .

( ℐ : J ′ ) ⊆ ( ℐ : J ) , but J ⊈ J ′ ;
( ℐ : J ′ ) = ℐ , but X ∉ J ′ .

Corollary 4.6

Let ℐ be an ideal on X and τ ⊆ 2 X . Then, the following statements hold.

( ℐ : ℐ ) = ( 2 X : ℐ ) = ( ℐ : { ∅ } ) = 2 X ;
if τ is a topology on X, then ( ℐ : τ ) = ℐ .

Remark 4.7

Let ℐ be an ideal on X and J ⊆ 2 X . Then, ℐ and ( ℐ : J ) need not be equal as shown by the following example.

Example 4.8

Let X = { a , b , c } with the ideal ℐ = { ∅ , { a } } on X . For the subfamily J = { { a } , { a , c } } of 2 X , we have ( ℐ : J ) = { ∅ , { a } , { b } , { a , b } } , which is not equal to ℐ .

Theorem 4.9

Let ℐ be a proper ideal on X. Then, ℐ is a maximal ideal if and only if ( ℐ : J ) = ℐ for each J ⊆ 2 X with the property J ⊈ ℐ .

Proof

( ⇒ ) : Let J ⊆ 2 X with the property J ⊈ ℐ . We will show that ( ℐ : J ) = ℐ . From Theorem 4.3(a), we have ℐ ⊆ ( ℐ : J ) . Now, we will prove that ( ℐ : J ) ⊆ ℐ . Let A ∈ ( ℐ : J ) . If we show that A ∈ ℐ , then the proof is complete.

A ∈ ( ℐ : J ) ⇒ ( ∀ J ∈ J ) ( A ∩ J ∈ ℐ ) ⇒ Hypothesis ( ∀ J ∈ J ) ( A ∈ ℐ ∨ J ∈ ℐ ) ⇒ A ∈ ℐ ∨ ( ∀ J ∈ J ) ( J ∈ ℐ ) ⇒ A ∈ ℐ ∨ J ⊆ ℐ J ⊈ ℐ ⇒ A ∈ ℐ .

( ⇐ ): Suppose that ℐ is not a maximal ideal on X .

ℐ is not a maximal ideal on X ⇒ ( ∃ A ⊆ X ) ( A ∉ ℐ ∧ X \ A ∉ ℐ ) J ≔ { X \ A } ⇒ ( A ∉ ℐ ) ( J ⊈ ℐ ) ( A ∈ ( ℐ : J ) ) ⇒ ( J ⊈ ℐ ) ( ( ℐ : J ) ≠ ℐ )

This contradicts our hypothesis.□

Corollary 4.10

Let ℐ be a proper ideal on X. Then, ℐ is a maximal ideal if and only if ( ℐ : J ) = ℐ or ( ℐ : J ) = 2 X for all J ⊆ 2 X .

Proof

This follows from Theorems 4.3 and 4.9.□

5 Annihilator of a family of sets

Definition 5.1

Let X be a non-empty set and J ⊆ 2 X . If ℐ = { ∅ } , then the ideal quotient ( { ∅ } : J ) is called the annihilator of J and denoted by Ann ( J ) . The notation Ann A will be used to denote Ann ( { A } ) , where A ⊆ X .

Corollary 5.2

Let X be a non-empty set and A ⊆ X . It is not difficult to see that Ann A = Ann ( ℐ ( A ) ) .

Definition 5.3

Let X be a non-empty set and J ⊆ 2 X . Then, J is said to be faithful if Ann ( J ) = { ∅ } .

Corollary 5.4

Every topology on X is faithful.

Lemma 5.5

Let ℐ be an ideal on X. Then, ℐ ∩ Ann ( ℐ ) = { ∅ } .

Proof

Let A ∈ ℐ ∩ Ann ( ℐ ) . We will show that A = ∅ .

□ A ∈ ℐ ∩ Ann ( ℐ ) ⇒ ( A ∈ ℐ ) ( A ∈ Ann ( ℐ ) ) ⇒ A ∩ A = ∅ ⇒ A = ∅ .

Theorem 5.6

Let X be an infinite set. Then, the family of all finite subsets of X, denoted by ℐ f , is a faithful ideal on X.

Proof

Suppose that Ann ( ℐ f ) ≠ { ∅ } .

Ann ( ℐ f ) ≠ { ∅ } ⇒ ( ∃ A ∈ Ann ( ℐ f ) ) ( A ≠ ∅ ) ⇒ ( ∃ x ∈ X ) ( x ∈ A ) ⇒ ( { x } ⊆ A ) ( ∣ { x } ∣ = 1 < ℵ 0 ) ⇒ ( { x } ∈ Ann ( ℐ f ) ) ( { x } ∈ ℐ f ) ⇒ { x } ∈ ℐ f ∩ Ann ( ℐ f ) .

This contradicts with Lemma 5.5.□

Theorem 5.7

Let ℐ be a faithful ideal on X. Then, Ann ( Ann ( ℐ ) ) = 2 X .

Proof

Let ℐ be a faithful ideal.

□ ℐ is faithful ⇒ Ann ( ℐ ) = { ∅ } ⇒ Ann ( Ann ( ℐ ) ) = Ann ( { ∅ } ) = ( { ∅ } : { ∅ } ) ⇒ Theorem 4.3(b) Ann ( Ann ( ℐ ) ) = 2 X .

Theorem 5.8

If ℐ is not a faithful ideal on X, then Ann ( Ann ( ℐ ) ) = ℐ .

Proof

Let A ∉ ℐ . Then, there are two cases. Now, we will consider them.

First case: Let A ∩ I = ∅ for all I ∈ ℐ .

( ∀ I ∈ ℐ ) ( A ∩ I = ∅ ) ⇒ A ∈ Ann ( ℐ ) A ∉ ℐ ⇒ A ≠ ∅ ⇒ Lemma 5.5 A ∉ Ann ( Ann ( ℐ ) ) … ( 1 )

Second case: Suppose that there exists I ∈ ℐ such that A ∩ I ≠ ∅ . Now, let T ≔ ⋂ { A \ I ∣ I ∩ A ≠ ∅ } .

T ≔ ⋂ { A \ I ∣ I ∩ A ≠ ∅ } ⇒ ( ∅ ≠ T ⊆ A ) ( ∀ I ∈ ℐ ) ( I ∩ T = ∅ ) ⇒ ( ∅ ≠ T ⊆ A ) ( T ∈ Ann ( ℐ ) ) ⇒ Lemma 5.5 ( ∅ ≠ T ⊆ A ) ( T ∉ Ann ( Ann ( ℐ ) ) ) ⇒ A ∉ Ann ( Ann ( ℐ ) ) … ( 2 )

Then, from (1) and (2), we have Ann ( Ann ( ℐ ) ) ⊆ ℐ … ( 3 ) .

Now, let A ∈ ℐ and J ∈ Ann ( ℐ ) . We will show that A ∈ Ann ( Ann ( ℐ ) ) .

J ∈ Ann ( ℐ ) ⇒ ( ∀ I ∈ ℐ ) ( I ∩ J = ∅ ) A ∈ ℐ ⇒ A ∩ J = ∅ .

Then, we have A ∈ Ann ( Ann ( ℐ ) ) and so ℐ ⊆ Ann ( Ann ( ℐ ) ) … ( 4 )

( 3 ) , ( 4 ) ⇒ Ann ( Ann ( ℐ ) ) = ℐ .□

Theorem 5.9

Let X be a set and A ⊆ X . Then, Ann ( ℐ ( A ) ) = Ann A = ℐ ε ( A ) .

Proof

It is obvious that Ann A = ℐ ε ( A ) . Also, we have

□ Ann ( ℐ ( A ) ) = { J ∣ ( ∀ I ∈ ℐ ( A ) ) ( J ∩ I = ∅ ) } = { J ∣ ( ∀ I ⊆ A ) ( J ∩ I = ∅ ) } = { J ∣ J ∩ A = ∅ } = ℐ ε ( A ) .

Corollary 5.10

Let ℐ be an ideal on X. Then, if ℐ is minimal ideal, then Ann ( ℐ ) is maximal ideal.

Proof

This follows from Theorems 3.12 and 5.9.□

Theorem 5.11

Let ( X , τ ) be a topological space and A ⊆ X . Then, A is a dense set in X if and only if ( X , τ , Ann A ) is a Hayashi-Samuel space.

Proof

( ⇒ ) : Let A be a dense set in X and J ∈ τ ∩ Ann A . We will show that J = ∅ .

J ∈ τ ∩ Ann A ⇒ ( J ∈ τ ) ( J ∈ Ann A ) ⇒ ( J ∈ τ ) ( A ∩ J = ∅ ) A is dense in X ⇒ c l ( A ) = X ⇒ ( ∀ x ∈ X ) ( ∀ U ∈ τ ( x ) ) ( U ∩ A ≠ ∅ ) ⇒ J = ∅ .

( ⇐ ): Let ( X , τ , Ann A ) be a Hayashi-Samuel space. We will show that c l ( A ) = X . Let x ∈ X and U ∈ τ ( x ) . We will show that U ∩ A ≠ ∅ .

( x ∈ X ) ( U ∈ τ ( x ) ) ⇒ U ∈ τ \ { ∅ } ( X , τ , Ann A ) is a Hayashi-Samuel space ⇒ τ ∩ Ann A = { ∅ } ⇒ U ∉ Ann A ⇒ U ∩ A ≠ ∅ .

Then, we have x ∈ c l ( A ) and so X ⊆ c l ( A ) . On the other hand, we have always c l ( A ) ⊆ X . Thus, c l ( A ) = X , i.e., A is a dense set in X .□

Example 5.12

Let U be the usual (metric) topology on R . For the set of rational numbers Q , we have Ann Q = 2 I , where I is the set of all irrational numbers. Since U ∩ Ann Q = ∅ , the ideal topological space ( R , U , Ann Q ) is a Hayashi-Samuel space. Q is also dense in R .

6 Sharp operator and sharp topology

Definition 6.1

Let ( X , τ , ℐ ) be an ideal topological space. Then, any subset A of X , A ♯ ( ℐ , τ ) ≔ { x ∈ X ∣ ( ∀ U ∈ τ ( x ) ) ( ∃ I ∈ ℐ \ { ∅ } ) ( I ⊆ U ∩ A ) } is called the sharp operator of A with respect to ℐ and τ . If there is no ambiguity, we will write A ♯ ( ℐ ) or simply A ♯ for A ♯ ( ℐ , τ ) .

Theorem 6.2

Let ( X , τ , ℐ ) be an ideal topological space and A ⊆ X . Then, A ♯ ( ℐ , τ ) = A * ( Ann ( ℐ ) , τ ) .

Proof

Let A ⊆ X .

□ x ∈ A ♯ ( ℐ , τ ) ⇔ ( ∀ U ∈ τ ( x ) ) ( ∃ I ∈ ℐ \ { ∅ } ) ( I ⊆ U ∩ A ) ⇔ ( ∀ U ∈ τ ( x ) ) ( I ∩ U ∩ A ≠ ∅ ) ⇔ ( ∀ U ∈ τ ( x ) ) ( U ∩ A ∉ Ann ( ℐ ) ) ⇔ x ∈ A * ( Ann ( ℐ ) , τ ) .

Theorem 6.3

Let ( X , τ , ℐ ) be an ideal topological space. Then, the following statements hold:

A ⊆ B ⇒ A ♯ ⊆ B ♯ ,
A ♯ = c l ( A ♯ ) ⊆ c l ( A ) ,
( A ∩ B ) ♯ ⊆ A ♯ ∩ B ♯ ,
( A ∪ B ) ♯ = A ♯ ∪ B ♯ ,
A ♯ \ B ♯ ⊆ ( A \ B ) ♯ ,
A ∈ Ann ( ℐ ) ⇒ A ♯ = ∅ ,
A ∈ Ann ( ℐ ) ⇒ ( A ∪ B ) ♯ = B ♯ = ( A \ B ) ♯ ,
if ℐ is f a i t h f u l , t h e n A ♯ = c l ( A ) .

Proof

This follows from the properties of local function and Theorem 6.2.□

Theorem 6.4

Let ( X , τ , ℐ ) be an ideal topological space and A ⊆ X . Then, A ♯ ( ℐ , τ ) ∪ A * ( ℐ , τ ) = c l ( A ) .

Proof

Let A ⊆ X .

A ⊆ X ⇒ A ♯ ( ℐ , τ ) ⊆ c l ( A ) A ⊆ X ⇒ A * ( ℐ , τ ) ⊆ c l ( A ) ⇒ A ♯ ( ℐ , τ ) ∪ A * ( ℐ , τ ) ⊆ c l ( A ) … ( 1 ) .

Now, let x ∉ A ♯ ( ℐ , τ ) ∪ A * ( ℐ , τ ) .

x ∉ A ♯ ( ℐ , τ ) ∪ A * ( ℐ , τ ) ⇒ ( x ∉ A ♯ ( ℐ , τ ) ) ( x ∉ A * ( ℐ , τ ) ) ⇒ ( ∃ U ∈ τ ( x ) ) ( U ∩ A ∈ ℐ ) ( ∃ V ∈ τ ( x ) ) ( V ∩ A ∈ Ann ( ℐ ) ) W ≔ U ∩ V ⇒

⇒ ( W ∈ τ ( x ) ) ( W ∩ A ∈ ℐ ) ( W ∩ A ∈ Ann ( ℐ ) ) ⇒ ( W ∈ τ ( x ) ) ( W ∩ A ∈ ℐ ∩ Ann ( ℐ ) ) ⇒ Lemma 5.5 ( W ∈ τ ( x ) ) ( W ∩ A ∈ { ∅ } ) ⇒ ( W ∈ τ ( x ) ) ( W ∩ A = ∅ ) ⇒ x ∉ c l ( A ) .

Then, we have c l ( A ) ⊆ A ♯ ( ℐ , τ ) ∪ A * ( ℐ , τ ) … ( 2 )

( 1 ) , ( 2 ) ⇒ A ♯ ( ℐ , τ ) ∪ A * ( ℐ , τ ) = c l ( A ) .□

Corollary 6.5

Let ( X , τ , ℐ ) be an ideal topological space and A ⊆ X . Then, the following properties hold:

If A ∈ ℐ , then A ♯ = c l ( A ) .
If A ∈ Ann ( ℐ ) , then A * = c l ( A ) .
If A ♯ = ∅ , then A * = c l ( A ) .
If A * = ∅ , then A ♯ = c l ( A ) .

Definition 6.6

Let ( X , τ , ℐ ) be an ideal topological space. We consider a map c l ♯ : 2 X → 2 X as c l ♯ ( A ) = A ∪ A ♯ , where A is any subset of X .

Theorem 6.7

Let ( X , τ , ℐ ) be an ideal topological space and A , B ⊆ X . Then, the following statements hold:

c l ♯ ( ∅ ) = ∅ ,
c l ♯ ( X ) = X ,
A ⊆ c l ♯ ( A ) ,
if A ⊆ B , then c l ♯ ( A ) ⊆ c l ♯ ( B ) ,
c l ♯ ( A ) ∪ c l ♯ ( B ) = c l ♯ ( A ∪ B ) ,
c l ♯ ( c l ♯ ( A ) ) = c l ♯ ( A ) .

Proof

Let A , B ⊆ X .

(a) Since ∅ ♯ = ∅ , we have c l ♯ ( ∅ ) = ∅ ∪ ∅ ♯ = ∅ .

(b) Since X ∪ X ♯ = X , we have c l ♯ ( X ) = X .

(d) Let A ⊆ B . We obtain from Theorem 4.3 ( e ) that A ♯ ⊆ B ♯ . Therefore, we have A ∪ A ♯ ⊆ B ∪ B ♯ , which means that c l ♯ ( A ) ⊆ c l ♯ ( B ) .

(e) This follows from the definition of operator c l ♯ and Theorem 6.3(d).

(f) It follows from ( c ) and ( d ) that c l ♯ ( A ) ⊆ c l ♯ ( c l ♯ ( A ) ) . On the other hand, since A ♯ is closed in X , we have ( A ♯ ) ♯ ⊆ A ♯ . Therefore,

c l ♯ ( c l ♯ ( A ) ) = c l ♯ ( A ) ∪ ( c l ♯ ( A ) ) ♯ = c l ♯ ( A ) ∪ ( A ∪ A ♯ ) ♯ = c l ♯ ( A ) ∪ A ♯ ∪ ( A ♯ ) ♯ ⊆ c l ♯ ( A ) ∪ A ♯ ∪ A ♯ = c l ♯ ( A ) .

Thus, we have c l ♯ ( c l ♯ ( A ) ) = c l ♯ ( A ) .□

Corollary 6.8

Let ( X , τ , ℐ ) be an ideal topological space. Then, the operator c l ♯ : 2 X → 2 X defined by c l ♯ ( A ) = A ∪ A ♯ is a Kuratowski closure operator.

Definition 6.9

Let ( X , τ , ℐ ) be an ideal topological space. Then, the family τ ♯ = { A ⊆ X ∣ c l ♯ ( X \ A ) = X \ A } is a topology called ♯ -topology on X induced by topology τ and ideal ℐ . We can also write τ ℐ ♯ instead of τ ♯ to specify the ideal if necessary.

Remark 6.10

We have the following diagram from the definitions of ⁎-topology and ♯ -topology. The following example shows that these implications are not reversible. Also, the notions of τ * -open set and τ ♯ -open set are independent.

τ * -open τ ♯ -open ↖ ↗ τ -open

Example 6.11

Let X = { a , b , c } and τ = { ∅ , X , { a , c } } with the ideal ℐ = { ∅ , { a } , { b } , { a , b } } . Simple calculations show that Ann ( ℐ ) = { ∅ , { c } } , τ * = { ∅ , X , { a , c } , { b , c } , { c } } and τ ♯ = { ∅ , X , { a , b } , { a , c } , { a } } .

The set { c } ∈ τ * , but { c } ∉ τ ;
The set { a } ∈ τ ♯ , but { a } ∉ τ ;
The set { c } ∈ τ * , but { c } ∉ τ ♯ ;
The set { a } ∈ τ ♯ , but { a } ∉ τ * .

Definition 6.12

Let ( X , τ , ℐ ) be an ideal topological space. We define the operator Ψ ♯ : 2 X → 2 X by Ψ ♯ ( A ) = X \ ( X \ A ) ♯ for any subset A of X . We can also write Ψ ♯ ( A ( ℐ , τ ) ) instead of Ψ ♯ ( A ) to specify the ideal and the topology if necessary.

Corollary 6.13

Let ( X , τ , ℐ ) be an ideal topological space and A ⊆ X . Then, Ψ ( A ( Ann ( ℐ ) , τ ) ) = Ψ ♯ ( A ( ℐ , τ ) ) .

Proof

This follows from the definition of Ψ -operator and Theorem 6.2.□

Theorem 6.14

Let ( X , τ , ℐ ) be an ideal topological space and A ⊆ X . Then, A is τ ♯ -open if and only if A ⊆ Ψ ♯ ( A ) .

Proof

Let A ⊆ X .

□ A ∈ τ ♯ ⇔ c l ♯ ( X \ A ) = ( X \ A ) ⇔ ( X \ A ) ∪ ( X \ A ) ♯ = X \ A ⇔ ( X \ A ) ♯ ⊆ X \ A ⇔ A ⊆ X \ ( X \ A ) ♯ = Ψ ♯ ( A ) ⇔ A ⊆ Ψ ♯ ( A ) .

Theorem 6.15

Let ( X , τ , ℐ ) be an ideal topological space and A ⊆ X . Then, Ψ ♯ ( A ) ∩ Ψ ( A ) = int ( A ) .

Proof

Let A ⊆ X .

□ Ψ ♯ ( A ) ∩ Ψ ( A ) = [ X \ ( X \ A ) ♯ ] ∩ [ X \ ( X \ A ) * ] = X \ [ ( X \ A ) ♯ ∪ ( X \ A ) * ] = Theorem 6.4 X \ c l ( X \ A ) = X \ ( X \ int ( A ) ) = int ( A ) .

Corollary 6.16

Let ( X , τ , ℐ ) be an ideal topological space and A ⊆ X .

If X \ A ∈ ℐ , then Ψ ♯ ( A ) = int ( A ) ,
If A ∈ Ann ( ℐ ) , then Ψ ( A ) = int ( A ) .

Theorem 6.17

Let ( X , τ , ℐ ) be an ideal topological space and A ⊆ X . Then, A is τ -open if and only if A is both τ * -open and τ ♯ -open.

Proof

It is clear from the equality below.

□ τ = τ * ( { ∅ } , τ ) = τ * ( ℐ ∩ Ann ( ℐ ) , τ ) = τ * ( ℐ , τ ) ∩ τ * ( Ann ( ℐ ) , τ ) = τ * ( ℐ , τ ) ∩ τ ♯ ( ℐ , τ ) = τ * ∩ τ ♯ .

Theorem 6.18

Let ℐ be a proper ideal on X. If ℐ is a minimal ideal, then A ♯ = ∅ or ( X \ A ) ♯ = ∅ for all A ⊆ X .

Proof

Let A ⊆ X .

□ ℐ is a minimal ideal Corollary (5.10) ⇒ Ann ( ℐ ) is a maximal ideal A ⊆ X ⇒ A ∈ Ann ( ℐ ) ∨ ( X \ A ) ∈ Ann ( ℐ ) ⇒ A ♯ = ∅ ∨ ( X \ A ) ♯ = ∅ .

Corollary 6.19

Let ℐ be a minimal ideal on X and A ⊆ X . Then, A is τ ♯ -closed or τ ♯ -open.

Theorem 6.20

Let ℐ be a proper ideal on X. If ℐ is a maximal ideal, then A ♯ = c l ( A ) or Ψ ♯ ( A ) = int ( A ) for all A ⊆ X .

Proof

Let A ⊆ X .

□ ℐ is a maximal ideal A ⊆ X ⇒ A ∈ ℐ ∨ ( X \ A ) ∈ ℐ ⇒ A * = ∅ ∨ ( X \ A ) * = ∅ A ⊆ X ⇒ Theorem 6.4 c l ( A ) = A * ∪ A ♯ ⇒ ⇒ c l ( A ) = A ♯ ∨ X \ int ( A ) = c l ( X \ A ) = ( X \ A ) ♯ ⇒ c l ( A ) = A ♯ ∨ int ( A ) = X \ ( X \ A ) ♯ = Ψ ♯ ( A ) .

Corollary 6.21

Let ℐ be a proper ideal on X. If ℐ is a minimal ideal, then A * = c l ( A ) or Ψ ( A ) = int ( A ) for all A ⊆ X .

Proof

This follows from Theorem 6.20.□

7 Decomposition of continuity

Definition 7.1

A function f : ( X , τ , ℐ ) → ( Y , σ ) is called ⁎-continuous ( ♯ -continuous) if f − 1 [ V ] ∈ τ * ( f − 1 [ V ] ∈ τ ♯ ) for each open set V of Y .

Corollary 7.2

A function f : ( X , τ , ℐ ) → ( Y , σ ) is ⁎-continuous if and only if f : ( X , τ * ( ℐ ) ) → ( Y , σ ) is continuous.

Corollary 7.3

A function f : ( X , τ , ℐ ) → ( Y , σ ) is ♯ -continuous if and only if f : ( X , τ ♯ ( ℐ ) ) → ( Y , σ ) is continuous.

Remark 7.4

Jankovic and Hamlett [11] showed that if f : ( X , τ ) → ( Y , σ ) is a continuous function and ℐ is an ideal on X , then f : ( X , τ * ) → ( Y , σ ) is also continuous. However, the converse need not always to be true as shown in [11].

Corollary 7.5

Let f : ( X , τ , ℐ ) → ( Y , σ ) be a function. If f is continuous, then it is also ♯ -continuous.

Remark 7.6

The converse of Corollary 7.5 need not to be true as shown by the following example.

Example 7.7

Let U be the usual (standard) topology on R with the ideal ℐ = { ∅ , { 0 } } . Let τ = { A ⊆ R ∣ 0 ∈ A } ∪ { ∅ } . Consider the identity function i : R → R . Now, let A ∈ τ \ { ∅ } .

A ∈ τ \ { ∅ } ⇒ 0 ∈ A ⇒ 0 ∉ X \ A ⇒ X \ A ∈ Ann ( ℐ ) ⇒ ( X \ A ) ♯ = ∅ ⇒ A ⊆ X = X \ ( X \ A ) ♯ = Ψ ♯ ( A ) ⇒ A ∈ U ♯ .

Hence, i : ( R , U ♯ ) → ( R , τ ) is continuous. However, i : ( R , U ) → ( R , τ ) is not continuous since A = { 0 } ∈ τ but { 0 } ∉ U .

Theorem 7.8

Let f : ( X , τ , ℐ ) → ( Y , σ ) be a function. Then, f is continuous if and only if f is ⁎-continuous and ♯ -continuous.

Proof

This follows from Theorem 6.17.□

Corollary 7.9

It is clear that the notions of ⁎-continuity and ♯ -continuity are independent from Remark 7.4, Corollary 7.5 and Theorem 7.8.

8 Some applications of the ♯ operator

Example 8.1

By using the sharp topology, we prove that the set of rational numbers Q is dense in ( R , U ) , where U is the usual topology on the set of real numbers R . For this, first we will prove that ( R , U , Ann ( Q ) ) is a Hayashi-Samuel space. This is obvious from the fact that ( a , b ) ∩ Q ≠ ∅ for all a , b ∈ R . Hence, by Theorem 5.11, Q is dense in R .

Question 8.2

Let R be the set of all real numbers. Is there a Hausdorff topology on R such that the set of irrational numbers I is not dense, while the set of rational numbers Q is dense?

Example 8.3

Let R be the set of all real numbers with the usual topology U and let ℐ = ℐ ( Q ) , where Q is the set of all rational numbers. Now, let I be the set of all irrational numbers. Since Q ∈ ℐ , by Lemma 2.1 and Corollary 6.5, we obtain Q * = ∅ and Q ♯ = c l ( Q ) = R . Thus, c l ♯ ( Q ) = Q ∪ Q ♯ = Q ∪ c l ( Q ) = R . Hence, Q is a dense set in ( R , U ♯ ) . On the other hand, we have I ♯ = ∅ since I ∈ Ann ( ℐ ) . Therefore, c l ♯ ( I ) = I ∪ I ♯ = I . In other word, I is not a dense set in ( R , U ♯ ) . Finally, it is obvious that ( R , U ♯ ) is Hausdorff since U ⊆ U ♯ and ( R , U ) is Hausdorff.

Question 8.4

Let R be the set of all real numbers. Is there a Hausdorff topology such that the set of rational numbers Q is clopen?

Example 8.5

Let R be the real line with the usual topology U and let ℐ = ℐ ( { 0 } ) = { ∅ , { 0 } } . Let Q be the set of all rational numbers and I be the set of all irrational numbers. By simple calculations, it is not difficult to see that if 0 ∈ A , then A ♯ = { 0 } . Thus, we have c l ♯ ( Q ) = Q ∪ Q ♯ = Q ∪ { 0 } = Q , i.e., Q is closed in ( R , U ♯ ) . On the other hand, it is easy to see that Ann ( ℐ ) = 2 R \ { 0 } , where 2 R \ { 0 } is the power set of R \ { 0 } . Now, let A ⊆ R .

First case: Let 0 ∈ A .

0 ∈ A ⇒ ( A ♯ = { 0 } ) ( A * = c l ( A ) ∨ A * = c l ( A ) \ { 0 } ) .

Second case: Let 0 ∉ A .

0 ∉ A ⇒ ( A ♯ = ∅ ) ( A * = c l ( A ) ) .

Then, c l ♯ ( Q ) = Q and c l ♯ ( I ) = I . Thus, Q is clopen in ( R , U ♯ ) .

Lemma 8.6

Let ( X , τ ) be a topological space. If F ∈ C ( X , τ ) \ { ∅ , X } , then the space ( X , τ ♯ , ℐ ( F ) ) is disconnected.

Proof

Let F ∈ C ( X , τ ) \ { ∅ , X } . We will prove that ( X , τ ♯ , ℐ ( F ) ) is disconnected. It is sufficient to show that F is clopen in ( X , τ ♯ , ℐ ( F ) ) .

□ F ∈ C ( X , τ ) \ { ∅ , X } ⇒ F c ∈ τ \ { ∅ , X } τ ⊆ τ ♯ ⇒ F c ∈ τ ♯ \ { ∅ , X } … ( 1 ) F c ∈ Ann F ⇒ ( F c ) ♯ = ∅ c l ♯ ( F c ) = ( F c ) ♯ ∪ F c ⇒ c l ♯ ( F c ) = F c ⇒ F c ∈ C ( X , τ ♯ , ℐ ( F ) ) \ { ∅ , X } … ( 2 ) ( 1 ) , ( 2 ) ⇒ F c ∈ ( τ ♯ \ { ∅ , X } ) ∩ ( C ( X , τ ♯ , ℐ ( F ) ) \ { ∅ , X } ) .

Example 8.7

In this example, we will build a disconnected Hausdorff space. Let R be the real line with the usual topology U . Let ℐ = ℐ ( N ) , where N is the set of all natural numbers. The set N is closed in ( R , U ) . By Lemma 8.6, ( R , U ♯ , ℐ ( N ) ) is disconnected.

9 Conclusion

In this study, we have defined and thoroughly examined the concepts of minimal and maximal ideals, as well as ideal quotients and annihilators, within the framework of the ideal topological spaces. Furthermore, a new topology, called sharp topology, which is finer than the original topology, has been defined, accompanied by a decomposition of open sets. These findings, supported by fundamental properties and interrelationships, make significant contributions to the theory of ideal topological spaces and provide a robust foundation for advanced research and applications in this field.

Some new topologies can be generated through primals and rough sets in the future. Furthermore, the properties of topologies generated by maximal and minimal ideals are other topics open for further research. We believe that this study will stimulate further research on primals as ideals as well.

murad.ozkoc@gmail.com

Acknowledgements

The authors would like to thank the anonymous referees and the editor for their helpful suggestions which improved the study. We would especially like to thank Prof. Dr. Doğan DÖNMEZ for his valuable linguistic and scientific suggestions. Also, this study has been supported by the Scientific Research Project Fund of Muğla Sıtkı Koçman University under the project number 24/173/04/1.

Funding information: Scholarship support for the PhD student and office supplies has been provided by the Scientific Research Project Fund of Muğla Sıtkı Koçman University under the project number 24/173/04/1.
Author contributions: All authors have accepted responsibility for the entire content of this manuscript and approved its submission.
Conflict of interest: The authors state no conflict of interest.
Ethical approval: The conducted research is not related to either human or animal use.
Data availability statement: No data were used to support the study.

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Received: 2024-09-09

Revised: 2024-12-04

Accepted: 2024-12-22

Published Online: 2025-04-09

This work is licensed under the Creative Commons Attribution 4.0 International License.

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https://doi.org/10.1515/dema-2024-0094

Keywords for this article

maximal ideal; minimal ideal; ideal quotient; annihilator; sharp-operator; sharp-topology

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