On Some New Sets Via Local Closure Function in Ideal Topological Space

On Some New Sets Via Local Closure Function in Ideal Topological Space

Ali Bulama mamman, Haruna Usman Idriss, Bashir Mai Umar, Buba M.T Hambadga and Jamilu Adamu

Department of Mathematics, Federal University Gashua, Yobe State, Nigeria.

DOI: https://doi.org/10.51584/IJRIAS.2025.10040049

Received: 25 March 2025; Accepted: 29 March 2025; Published: 03 May 2025

ABSTRACT

In this paper, we introduce \( L \)-perfect set, \( R \)-perfect set and \( C \)-perfect set in ideal topological space and study their properties.

We investigate the relationship between the existing \( R^\ast \)-perfect sets and \( R \)-perfect set and also \( L^\ast \)-perfect sets and \( L \)-perfect set. We construct a topology \( \beta \) by using Kuratowski closure operator.

Keywords: \( L \)-perfect, \( R \)-perfect and \( C \)-perfect sets

INTRODUCTION

The concept of ideal in topological space was introduced by K. Kuratowski in (1930) as a nonempty collection \( I \) of subsets of a topological space \( (X, \beta) \) that satisfy the following conditions:

1. If \( A \in I \) and \( B \subseteq A \), then \( B \in I \) (heredity)
2. If \( A, B \in I \), then \( A \cup B \in I \) (finite additivity)

The space \( (X, \beta, I) \) is called an ideal topological space.

In 1933, Kuratowski introduced the notion of local function:

$$ A^{*}(I, \beta) := \{x \in X : U \cap A \notin I \text{ for every open set } U \ni x \} $$

In 2013, Ahmed Al-Omari and Takashi Noiri introduced the local closure function:

$$ \lambda(A)(I, \beta) := \{x \in X : A \cap \text{Cl}(U) \notin I \text{ for every } U \in \beta(x) \} $$

If there is no ambiguity, we write \( A^{*} \) and \( \lambda(A) \) instead of \( A^{*}(I, \beta) \) and \( \lambda(A)(I, \beta) \).

In 2013, R. Manoharan and P. Thangavelu introduced \( R^{*} \)-perfect, \( L^{*} \)-perfect and \( C^{*} \)-perfect sets.

In 2018, Lawrence et al. introduced \( R \)-perfect, \( L \)-perfect and \( C \)-perfect sets in ideal topological space.

In this paper, we introduce \( L \)-perfect, \( R \)-perfect and \( C \)-perfect sets as a generalisation of \( R^{*} \)-perfect, \( L^{*} \)-perfect and \( C^{*} \)-perfect sets respectively.

PRELIMINARIES

The following definitions, lemmas, and theorems are very important in this research.

Definition 2.1

If \( (X, \beta, I) \) is an ideal topological space and \( A \subseteq X \), then the following hold:

1. \( A \) is \( \beta^{*} \)-closed if \( A^{*} \subseteq A \)
2. \( A \) is \( * \)-dense-in-itself if \( A \subseteq A^{*} \)
3. \( A \) is \( I \)-dense if \( A = X \)
4. \( A \) is \( I \)-open if \( A \subseteq (\text{int}(A))^{*} \)
5. \( A \) is regular \( I \)-closed if \( A = (\text{int}(A))^{*} \)
6. \( A \) is almost \( I \)-open if \( A \subseteq \text{cl}((\text{int}(A))^{*}) \)

Definition 2.2

If \( (X, \beta, I) \) is an ideal topological space, then a topology \( \beta \) is compatible with ideal \( I \) if for every \( A \subseteq X \):

If for every \( x \in A \), there exists \( U \in \beta(x) \) such that \( U \cap A \in I \), then \( A \in I \).

This is denoted by \( \beta \Vdash I \).

Lemma 2.3

Let \( (X, \beta) \) be a topological space and \( I_1 \) and \( I_2 \) be two ideals on \( X \). If \( A, B \subseteq X \), then:

1. If \( A \subseteq B \), then \( A^{*} \subseteq B^{*} \)
2. If \( I_1 \subseteq I_2 \), then \( A^{*}_{I_2} \subseteq A^{*}_{I_1} \)
3. \( A^{*} = \text{cl}(A^{*}) \subseteq \text{cl}(A) \)
4. \( (A^{*})^{*} \subseteq A^{*} \)
5. \( (A \cup B)^{*} = A^{*} \cup B^{*} \)
6. \( A^{*} – B^{*} = (A \setminus B)^{*} – B^{*} \subseteq (A \setminus B)^{*} \)
7. For every \( I \in \mathcal{I} \), \( (A \cup I)^{*} = A^{*} = (A \cup I)^{*} \)

Theorem 2.4

Let \( (X, \beta, I) \) be an ideal topological space. Then the following are equivalent:

1. \( \beta \Vdash I \)
2. Every subset \( A \subseteq X \) having an open cover whose intersection with \( A \) belongs to \( I \)
3. If \( A \cap A^{*} = \emptyset \), then \( A \in I \)
4. If \( A – A^{*} \in I \)
5. If \( A \) contains no nonempty subset \( B \subseteq B^{*} \), then \( A \in I \)

Theorem 2.5

Let \( (X, \beta, I) \) be an ideal topological space. Then the following are equivalent:

1. \( \beta \Vdash \lambda I \)
2. If \( A \subseteq X \) has a cover of sg-open sets whose intersection with \( A \) belongs to \( I \)
3. If \( A \cap \lambda(A) = \emptyset \), then \( A \in I \)
4. If \( A – \lambda(A) \in I \)
5. If \( A \) contains no nonempty subset \( B \subseteq \lambda(B) \), then \( A \in I \)

Theorem 2.6

If \( (X, \beta, I) \) is an ideal topological space and \( A, B \subseteq X \), then the following hold:

1. \( \lambda(\emptyset) = \emptyset \)
2. \( \lambda(A) \cup \lambda(B) = \lambda(A \cup B) \)
3. If \( A \subseteq B \), then \( \lambda(A) \subseteq \lambda(B) \)

THE OPEN SETS OF \( \tau_{\mathbb{R}} \)

In this section, we investigate \( \beta^\lambda \), finer than \( \beta^* \), called the Kuratowski local closure operator. That is:

$$ \text{Cl}^\lambda(A) = A \cup \lambda(A) $$

A subset of an ideal topological space \( (X, \beta, I) \) is said to be \( \beta^\lambda \)-closed if \( \lambda(A) = A \). That is, if \( U \in \beta^\lambda \), then \( X \setminus U \) is \( \beta^\lambda \)-closed.

i.e.,

$$ \lambda(X \setminus U) = X \setminus U \iff U \subseteq X \setminus \lambda(X \setminus U) $$

Therefore, \( x \in U \Rightarrow x \notin \lambda(X \setminus U) \Rightarrow \) there exists \( V \in N(x) \) such that \( V \cap (X \setminus U) \in I \).

Let \( I = V \cap (X \setminus U) \) and we have \( x \in V \setminus I \subseteq U \), which is a basis for \( \beta^\lambda \), denoted by:

$$ \mathcal{B}(I; \beta) = \{ V \setminus I : V \in \beta, I \in \mathcal{I} \} $$

Theorem 3.1

Let \( (X, \beta, I) \) be an ideal topological space, and \( A, B \subseteq X \). Let \( \text{Cl}^\lambda(A) = \lambda(A) \cup A \), then the following hold:

1. \( \text{Cl}^\lambda(\emptyset) = \emptyset \)
2. \( A \subseteq \text{Cl}^\lambda(A) \)
3. \( \text{Cl}^\lambda(A \cup B) = \text{Cl}^\lambda(A) \cup \text{Cl}^\lambda(B) \)
4. \( \text{Cl}^\lambda(\text{Cl}^\lambda(A)) = \text{Cl}^\lambda(A) \)

Proof:

1. By Theorem 2.6(1), \( \lambda(\emptyset) = \emptyset \). Therefore, \( \text{Cl}^\lambda(\emptyset) = \lambda(\emptyset) \cup \emptyset = \emptyset \).
2. \( A \subseteq A \cup \lambda(A) = \text{Cl}^\lambda(A) \)
3. \[
   \begin{align*}
   \text{Cl}^\lambda(A \cup B) &= \lambda(A \cup B) \cup (A \cup B) \\
   &= \lambda(A) \cup \lambda(B) \cup A \cup B \\
   &= (\lambda(A) \cup A) \cup (\lambda(B) \cup B) \\
   &= \text{Cl}^\lambda(A) \cup \text{Cl}^\lambda(B)
   \end{align*}
   \]
4. \[
   \begin{align*}
   \text{Cl}^\lambda(\text{Cl}^\lambda(A)) &= \lambda(\lambda(A) \cup A) \cup (\lambda(A) \cup A) \\
   &= (\lambda(\lambda(A) \cup A) \cup \lambda(A)) \cup A \\
   &= \lambda(A) \cup A = \text{Cl}^\lambda(A)
   \end{align*}
   \]

\( L_{\Gamma}\text{-PERFECT},\ R_{\Gamma}\text{-PERFECT AND } C_{\Gamma}\text{-PERFECT SETS} \)

Definition 4.1
Let \( (X, \beta, I) \) be an ideal topological space. Then a subset \( A \subseteq X \) is said to be:

1. \( L^\lambda \)-perfect if \( A – \lambda(A) \in I \)
2. \( R^\lambda \)-perfect if \( \lambda(A) – A \in I \)
3. \( C^\lambda \)-perfect if it is both \( L^\lambda \)-perfect and \( R^\lambda \)-perfect

Lemma 4.2

If \( (X, \beta, I) \) is an ideal topological space, then \( A^* \subseteq \lambda(A) \).

Proof:

Let \( x \in A^* \). Then for every open set \( U \ni x \), we have \( A \cap U \notin I \).

Since \( A \cap U \subseteq A \cap \text{Cl}(U) \), clearly \( A \cap \text{Cl}(U) \notin I \). Thus, \( x \in \lambda(A) \), and so \( A^* \subseteq \lambda(A) \).

Example:

Let \( X = \{a, b, c\} \), \( \beta = \{\emptyset, X, \{a\}, \{a, b\}\} \), and \( I = \{\emptyset, \{a\}\} \).

If \( A = \{a, b\} \), then:

• \( A^* = \{b\} \)
• \( \lambda(A) = \{a, b\} \)

Proposition 4.3

If a subset \( A \) of an ideal topological space \( (X, \beta, I) \) is \( C^\lambda \)-perfect, then:

$$ A \, \Delta \, \lambda(A) \in I \quad \text{(where } \Delta \text{ denotes symmetric difference)} $$

Proof:

Since \( A \) is both \( L^\lambda \)-perfect and \( R^\lambda \)-perfect:

• \( A – \lambda(A) \in I \)
• \( \lambda(A) – A \in I \)

By finite additivity of \( I \), their union (i.e., symmetric difference) is also in \( I \):

$$ A \, \Delta \, \lambda(A) = (A – \lambda(A)) \cup (\lambda(A) – A) \in I $$

Example:

Let \( X = \{a, b, c\} \), \( \beta = \{\emptyset, X, \{b\}, \{a, b\}\} \), \( I = \{\emptyset, \{c\}\} \), \( A = \{a, b, c\} \), and \( \lambda(A) = \{a, b\} \).

Then:

• \( \lambda(A) \, \Delta \, A = \{c\} \in I \)

Proposition 4.4

Every \( \beta^\lambda \)-closed set is \( R^\lambda \)-perfect in \( (X, \beta, I) \).

Proof:

If \( A \) is \( \beta^\lambda \)-closed, then \( \lambda(A) \subseteq A \Rightarrow \lambda(A) – A = \emptyset \in I \). Hence \( A \) is \( R^\lambda \)-perfect.

Proposition 4.5

If \( A \in I \), then \( A \) is \( C^\lambda \)-perfect.

Proof:

Since \( A \in I \), \( \lambda(A) = \emptyset \). Thus:

• \( A – \lambda(A) = A \in I \)
• \( \lambda(A) – A = \emptyset \in I \)

So \( A \) is both \( L^\lambda \)-perfect and \( R^\lambda \)-perfect.

Example:

Let \( X = \{a, b, c\} \), \( \beta = \{\emptyset, X, \{a\}, \{a, b\}\} \), and \( I = \{\emptyset, \{a, b\}\} \).

If \( A = \{a, b\} \), then \( \lambda(A) = \emptyset \). Thus:

$$ A \, \Delta \, \lambda(A) = \{a, b\} \in I \quad \Rightarrow A \text{ is } C^\lambda \text{-perfect} $$

Corollary 4.6

If \( A \subseteq X \), then:

1. If \( A \in I \), then every subset of \( A \) is \( C^\lambda \)-perfect.
2. If \( A \) is \( R^\lambda \)-perfect, then \( \lambda(A) – A \) is \( C^\lambda \)-perfect.
3. If \( A \) is \( L^\lambda \)-perfect, then \( A – \lambda(A) \) is \( C^\lambda \)-perfect.
4. If \( A \) is \( C^\lambda \)-perfect, then \( A \, \Delta \, \lambda(A) \) is \( C^\lambda \)-perfect.

\( \text{RELATIONSHIP BETWEEN } L_{*}\text{-PERFECT},\ R_{*}\text{-PERFECT AND } L_{\Gamma}\text{-PERFECT},\ R_{\Gamma}\text{-PERFECT} \)

In this section, we investigate the relationship between the existing \( L^* \)-perfect, \( R^* \)-perfect sets and the newly introduced \( L^\lambda \)-perfect, \( R^\lambda \)-perfect sets.

Proposition 5.1

Every \( L^* \)-perfect set is \( L^\lambda \)-perfect, but the converse is not necessarily true.

Proof:

If \( A \) is \( L^* \)-perfect, then:

$$ A – A^* \in I $$

Since \( A^* \subseteq \lambda(A) \) (by Lemma 4.2), then:

$$ A – \lambda(A) \subseteq A – A^* \in I $$

By the hereditary property of ideals, \( A – \lambda(A) \in I \). Hence \( A \) is \( L^\lambda \)-perfect.

Proposition 5.2

Every \( R^\lambda \)-perfect set is \( R^* \)-perfect, but the converse is not necessarily true.

Proof:

If \( A \) is \( R^\lambda \)-perfect, then:

$$ \lambda(A) – A \in I $$

Since \( A^* \subseteq \lambda(A) \), then:

$$ A^* – A \subseteq \lambda(A) – A \in I $$

By heredity, \( A^* – A \in I \). So \( A \) is \( R^* \)-perfect.

Proposition 5.3

If \( A \) and \( B \) are \( R^\lambda \)-perfect, then \( A \cup B \) is also \( R^\lambda \)-perfect.

Proof:

Since \( \lambda(A) – A \in I \) and \( \lambda(B) – B \in I \), then:

$$ (\lambda(A) \cup \lambda(B)) – (A \cup B) \subseteq (\lambda(A) – A) \cup (\lambda(B) – B) \in I $$

But \( \lambda(A \cup B) \subseteq \lambda(A) \cup \lambda(B) \), so:

$$ \lambda(A \cup B) – (A \cup B) \subseteq I $$

Thus \( A \cup B \) is \( R^\lambda \)-perfect.

Proposition 5.4

If \( A \) and \( B \) are \( L^\lambda \)-perfect, then \( A \cup B \) is \( L^\lambda \)-perfect.

Proof:

Given \( A – \lambda(A) \in I \) and \( B – \lambda(B) \in I \), we know:

\( (A \cup B) – \lambda(A \cup B) \subseteq (A – \lambda(A)) \cup (B – \lambda(B)) \in I \)

Hence \( A \cup B \) is \( L^\lambda \)-perfect.

Corollary 5.5

In an ideal topological space:

1. Finite union of \( R^\lambda \)-perfect sets is \( R^\lambda \)-perfect.
2. Finite union of \( L^\lambda \)-perfect sets is \( L^\lambda \)-perfect.

Proposition 5.6

If \( A \) and \( B \) are \( R^\lambda \)-perfect, then \( A \cap B \) is also \( R^\lambda \)-perfect.

Proof:

Since \( \lambda(A) – A \in I \) and \( \lambda(B) – B \in I \), we have:

$$ \lambda(A \cap B) \subseteq \lambda(A) \cap \lambda(B) $$

So,
\[
\lambda(A \cap B) – (A \cap B) \subseteq (\lambda(A) – A) \cup (\lambda(B) – B) \in I
\]

Thus \( A \cap B \) is \( R^\lambda \)-perfect.

Proposition 5.7

If \( A \) and \( B \) are \( L^\lambda \)-perfect, then \( A \cap B \) is also \( L^\lambda \)-perfect.

Proof:

Given \( A – \lambda(A) \in I \) and \( B – \lambda(B) \in I \), we know:

$$ (A \cap B) – \lambda(A \cap B) \subseteq (A – \lambda(A)) \cup (B – \lambda(B)) \in I $$

Therefore, \( A \cap B \) is \( L^\lambda \)-perfect.

Corollary 5.8

In an ideal topological space \( (X, \beta, I) \):

1. Finite intersection of \( R^\lambda \)-perfect sets is \( R^\lambda \)-perfect.
2. Finite intersection of \( L^\lambda \)-perfect sets is \( L^\lambda \)-perfect.

Proof: The proof follows from propositions ?? and ?? above.

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