Special Properties of Operators in Neutrosophic Crisp Topological Spaces

Hossam AL. Salman; Reyadh D. Ali

doi:10.12688/f1000research.173335.1

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Research Article

Special Properties of Operators in Neutrosophic Crisp Topological Spaces

[version 1; peer review: awaiting peer review]

Hossam AL. Salman ¹, Reyadh D. Ali²

PUBLISHED 09 Feb 2026

Author details Author details

¹ University of Kerbala, Karbala, Karbala Governorate, Iraq
² University of Kerbala, Karbala, Karbala Governorate, Iraq

Hossam AL. Salman
Roles: Conceptualization, Methodology, Writing – Original Draft Preparation

Reyadh D. Ali
Roles: Supervision, Validation, Writing – Review & Editing

OPEN PEER REVIEW

REVIEWER STATUS AWAITING PEER REVIEW

This article is included in the Fallujah Multidisciplinary Science and Innovation gateway.

Abstract

Neutrosophic crisp sets are an important and recent topic that has entered both pure and applied mathematics, especially general topology. A neutrosophic crisp topological space is defined as a generalization of classical topology in a broad sense, without specifying the type of neutrosophic crisp family or the algebraic operations of union and intersection, in addition to the kind of complement and the kinds of inclusion.

In this research on the neutrosophic crisp topological space, we constructed such a space in which the intersection is fixed as Type I, the union as Type II, and the complement as Type II for all kinds of neutrosophic crisp families, and we considered the two kinds of inclusion. Within this framework, we defined two kinds of closure for a neutrosophic crisp set and two kinds of interior for a neutrosophic crisp set.

We studied all relations, results, and theorems from general topology on them, as well as the properties related to them and the relationship between the closure and the interior of a neutrosophic crisp set. We proved all relations, results, and theorems that hold and gave examples of those that do not; however, many properties failed to hold.

We also defined continuity in the constructed neutrosophic crisp topological space and proved all corresponding relations, results, and theorems in topology that hold, while providing examples of those that do not hold.

Keywords

Neutrosophic Crisp sets, Neutrosophic Crisp Topological Spaces, Neutrosophic Crisp Closure, Neutrosophic Crisp interior, Neutrosophic crisp continuous

Corresponding author: Hossam AL. Salman

Competing interests: No competing interests were disclosed.

Grant information: The author(s) declared that no grants were involved in supporting this work.

Copyright: © 2026 AL. Salman H and D. Ali R. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

How to cite: AL. Salman H and D. Ali R. Special Properties of Operators in Neutrosophic Crisp Topological Spaces [version 1; peer review: awaiting peer review]. F1000Research 2026, 15:216 (https://doi.org/10.12688/f1000research.173335.1) First published: 09 Feb 2026, 15:216 (https://doi.org/10.12688/f1000research.173335.1) Latest published: 09 Feb 2026, 15:216 (https://doi.org/10.12688/f1000research.173335.1)

1. Introduction

A. A. Salama, Florentin Smarandache, and Valeri Kroumov^1,2 introduced the neutrosophic crisp set ( $Ɲ_{Մ} Ҫ R_{ʂ}$ ) and also defined the neutrosophic crisp topological space (Ɲ_ՄҪ $T$ ), and they presented three types of these neutrosophic crisp sets as follows:

Let $S^{ȵ} = < S_{1}, S_{2}, S_{3} >$ be a neutrosophic crisp set

Type I: $S_{1} \cap S_{2} = \emptyset, S_{1} \cap S_{3} = \emptyset, S_{2} \cap S_{3} = \emptyset .$

Type II: $S_{1} \cap S_{2} = \emptyset, S_{1} \cap S_{3} = \emptyset, S_{2} \cap S_{3} = \emptyset, and S_{1} \cup S_{2} \cup S_{3} =$ Ҳ

Type III: $S_{1} \cap S_{2} \cap S_{3} = \emptyset, and S_{1} \cup S_{2} \cup S_{3} =$ Ҳ

Two kinds of the union operation and two kinds of the intersection operation were also defined

Let $S^{ȵ} = < S_{1}, S_{2}, S_{3} >$ and $M^{ȵ} = < M_{1}, M_{2}, M_{3} >$ be two neutrosophic crisp sets in $Ҳ$ .

Type I: $S^{ȵ} \cup_{1} M^{ȵ}$ = $< S_{1} \cup M_{1}, S_{2} \cup M_{2}, S_{3} \cap M_{3} >$

Type II: $S^{ȵ} \cup_{2} M^{ȵ}$ = $< S_{1} \cup M_{1}, S_{2} \cap M_{2}, S_{3} \cap M_{3} >$

Type I: $S^{ȵ} \cap_{1} M^{ȵ}$ = $< S_{1} \cap M_{1}, S_{2} \cap M_{2}, S_{3} \cup M_{3} >$

Type II: $S^{ȵ} \cap_{2} M^{ȵ}$ = $< S_{1} \cap M_{1}, S_{2} \cup M_{2}, S_{3} \cup M_{3}$ >

In addition, Four kinds of the empty set and the universal set were also defined, namely,

\emptyset_{1}^{ȵ} = < \emptyset, \emptyset, Ҳ >, \emptyset_{2}^{ȵ} = < \emptyset, Ҳ, \emptyset >, \emptyset_{3}^{ȵ} = < \emptyset, Ҳ, Ҳ >, \emptyset_{4}^{ȵ} = < \emptyset, Ҳ, \emptyset >

Ҳ_{1}^{ȵ} = < Ҳ, \emptyset, \emptyset, >, Ҳ_{2}^{ȵ} = < Ҳ, Ҳ, \emptyset >, Ҳ_{3}^{ȵ} = < Ҳ, \emptyset, Ҳ >, Ҳ_{4}^{ȵ} = < Ҳ, Ҳ, Ҳ >

Moreover, they defined three kinds of complements, two kinds of inclusion, and equality, as follows.

Let $M^{ȵ} = < M_{1}, M_{2}, M_{3} >$ be a neutrosophic crisp set in $Ҳ, then$ complement of $M^{ȵ}$ is divided into three:

Type I $: {(M^{ȵ})}^{c_{1}} = < {M_{1}}^{C}, {M_{2}}^{C}, {M_{3}}^{C} >$

Type II: ${(M^{ȵ})}^{c_{2}} = < M_{3}, M_{2}, M_{1} >$

Type III $: {(M^{ȵ})}^{c_{3}} = < M_{3}, {M_{2}}^{C}, M_{1} >$

For any two a neutrosophic crisp sets $S^{ȵ} = < S_{1}, S_{2}, S_{3} >$ and $M^{ȵ} = < M_{1}, M_{2}, M_{3} >$ in $Ҳ,$ two subset relations are defined as:

Type I: $S^{ȵ} \subseteq_{1} M^{ȵ}$ ⟺ $S_{1} \subseteq M_{1}, S_{2} \subseteq M_{2}, M_{3} \subseteq S_{3}$

$Type II : S^{ȵ} \subseteq_{1} M^{ȵ} ⟺ S_{1} \subseteq M_{1}, M_{2} \subseteq S_{2}, M_{3} \subseteq S_{3}$

From the definition of set inclusion, two distinct forms of equality are obtained:

$Type I : S^{ȵ}$ =₁ $M^{ȵ}$ ⟺ $S^{ȵ} \subseteq_{1} M^{ȵ} and M^{ȵ} \subseteq_{1} S^{ȵ}$

$Type II : S$ =₂ $M^{ȵ} ⟺ S^{ȵ} \subseteq_{2} M^{ȵ} and M^{ȵ} \subseteq_{2} S^{ȵ}$

Subsequently, many researchers contributed additional relations, results, and theorems for neutrosophic crisp sets and neutrosophic crisp topological spaces. Among them, Qahtan, G. A., Jabar, L. A. A., Rasheed, I. M., and Ali, R. D.³ presented a generalization of both the ideal function and the local function via neutrosophic crisp sets, offering results and properties that reinforce the concept of the generalized local function; through its properties, their work was used to deduce the properties of the ψNC-operator that they generalized within neutrosophic crisp set spaces.

Furthermore, Ali, R. D., Jabar, L. A. A., Qahtan, G. A., and Shakir, A. Y.⁴ focused on the concept of the kernel within neutrosophic crisp sets and its relationship with the separation axioms in neutrosophic crisp topological spaces, highlighting the concordance among them and shedding light on the properties that characterize their structure.

In this research, we constructed a neutrosophic crisp topological space in which we fixed the intersection as Type I, the union as Type II, and the complement as Type II, while using the two kinds of inclusion. In this setting, we presented two kinds of closure for a neutrosophic crisp set and two kinds of interior for a neutrosophic crisp set, and we proved all related relations, results, and theorems. We also studied continuity on the constructed neutrosophic crisp topological space and proved all related relations, results, and theorems in general topology that hold, providing examples of those that do not hold.

The collection of all such neutrosophic crisp sets over Ҳ is usually denoted by Ɲ_ՄҪℜ_ʂ.

2. Methods

2.1 Preliminaries

Theorem 2.1:

¹

Let ${B_{j}^{ȵ} : j \in J}$ be a family of neutrosophic crisp subsets of a universe in Ҳ. Then

1. The intersection $\cap_{j \in J} B_{j}^{ȵ}$ can be defined in two ways:

i. $\cap_{1} B_{j}^{ȵ} = < \cap B_{j 1}, \cap B_{j 2}, \cup B_{j 3} >$ ii. $\cap_{2} B_{j}^{ȵ} = < \cap B_{j 1}, \cup B_{j 2}, \cup B_{j 3} >$

2. The union $\cup_{j \in J} B_{j}^{ȵ}$ can be defined in two ways:

i. $\cup_{1} B_{j}^{ȵ} = < \cup B_{j 1}, \cup B_{j 2}, \cap B_{j 3} >$ ii. $\cup_{2} B_{j}^{ȵ} = < \cup B_{j 1}, \cap B_{j 2}, \cap B_{j 3} >$

Theorem 2.2:

⁵

Let $S^{ȵ}$ and $M^{ȵ}$ be two neutrosophic crisp sets of any type in $Ҳ$ . Then:

i. ${(S^{ȵ} \cap_{1} M^{ȵ})}^{c_{2}} = {(S^{ȵ})}^{c_{2}} \cup_{2} {(M^{ȵ})}^{c_{2}}$
ii. ${(S^{ȵ} \cup_{2} M^{ȵ})}^{c_{2}} = {(S^{ȵ})}^{c_{2}} \cap_{1} {(M^{ȵ})}^{c_{2}}$
iii. ${(S^{ȵ} \subseteq_{1} M^{ȵ})}^{c_{2}} = {(M^{ȵ})}^{c_{2}} \subseteq_{2} {(S^{ȵ})}^{c_{2}}$
iv. ${(S^{ȵ} \subseteq_{2} M^{ȵ})}^{c_{2}} = {(M^{ȵ})}^{c_{2}} \subseteq_{1} {(S^{ȵ})}^{c_{2}}$

Definition 2.3:

⁶

1. Let (Ҳ, $T_{Ҳ}$ ) and ( $Y$ , $T_{Y}$ ) be two Ɲ_ՄҪ $T$ -spaces. If $ꭆ^{ȵ} = < ꭆ_{1}, ꭆ_{2}, ꭆ_{3} >$ is Ɲ_ՄҪℜ_ʂ in $T_{Y}$ , then the preimage of $ꭆ^{ȵ}$ under $f,$ denoted $f^{- 1} (ꭆ^{ȵ}) = < f^{- 1} (ꭆ_{1}), f^{- 1} (ꭆ_{2}), f^{- 1} (ꭆ_{3}) >,$ is Ɲ_ՄҪℜ_ʂ in $T_{Ҳ}$ .
2. Let (Ҳ, $T_{Ҳ}$ ) and ( $Y$ , $T_{Y}$ ) be two Ɲ_ՄҪ $T$ -spaces. If $S^{ȵ} = < S_{1}, S_{2}, S_{3} >$ is Ɲ_ՄҪℜ_ʂ in $T_{Ҳ}$ , then the image of $S^{ȵ}$ under $f,$ denoted $f (S^{ȵ}) = < f (S_{1}), f (S_{2}), f (S_{3}) >,$ is Ɲ_ՄҪℜ_ʂ in $T_{Y}$ .

Corollary 2.4:

²

Let $: Ҳ \to Y be a function, and S^{ȵ} = < S_{1}, S_{2}, S_{3} >, M^{ȵ} = < M_{1}, M_{2}, M_{3} >$ be a Ɲ_ՄҪℜ_ʂs in Ҳ.

i. If $S^{ȵ}$ $\subseteq_{1} M^{ȵ}, then f (S^{ȵ}) \subseteq_{1} f (M^{ȵ})$
ii. If $S^{ȵ}$ $\subseteq_{2} M^{ȵ}, then f (S^{ȵ}) \subseteq_{2} f (M^{ȵ})$

Moreover, if $f$ is injective, the converse (i and ii) holds.

Corollary 2.5:

²

Let $: Ҳ \to Y be a function, and ꭆ^{ȵ} = < ꭆ_{1}, ꭆ_{2}, ꭆ_{3} >, Ɉ^{ȵ} = < Ɉ_{1}, Ɉ_{2}, Ɉ_{3} >$ be a Ɲ_ՄҪℜ_ʂs in $Y$ .

i. If $ꭆ^{ȵ}$ $\subseteq_{1} Ɉ^{ȵ}, then f^{- 1} (ꭆ^{ȵ}) \subseteq_{1} f^{- 1} (Ɉ^{ȵ})$
ii. If $ꭆ^{ȵ}$ $\subseteq_{2} Ɉ^{ȵ}, then f^{- 1} (ꭆ^{ȵ}) \subseteq_{2} f^{- 1} (Ɉ^{ȵ})$

Moreover, if $f$ is surjective, the converse (i and ii) holds.

Corollary 2.6:

²

Let $f : Ҳ \to Y be an injective function, and let M^{ȵ} = < M_{1}, M_{2}, M_{3} >$ be a Ɲ_ՄҪℜ_ʂs in Ҳ. Then

i. $M^{ȵ} =_{1} f^{- 1} (f (M^{ȵ})) and Ҡ^{ȵ} =_{2} f^{- 1} (f (M^{ȵ}))$
ii. $f (\cap_{1} {M^{ȵ}}_{i}) = \cap_{1} f ({M^{ȵ}}_{i}) and f (\cap_{2} {M^{ȵ}}_{i}) = \cap_{2} f ({M^{ȵ}}_{i})$

Corollary 2.7:

¹

Let $f : Ҳ \to Y be a surjective function, and let ꭆ^{ȵ} = < ꭆ_{1}, ꭆ_{2}, ꭆ_{3} >$ be a Ɲ_ՄҪℜ_sʂ in $Y$ . Then

i. $f (f^{- 1} (ꭆ^{ȵ})) =_{1} ꭆ^{ȵ}$
ii. $f (f^{- 1} (ꭆ^{ȵ})) =_{2} ꭆ^{ȵ}$
iii. $f^{- 1} (\cup_{2} {ꭆ^{ȵ}}_{i}) = \cup_{2} f^{- 1} ({ꭆ^{ȵ}}_{i})$
iv. $f^{- 1} (\cup_{1} {ꭆ^{ȵ}}_{i}) = \cup_{1} f^{- 1} ({ꭆ^{ȵ}}_{i})$
v. $f^{- 1} (\cap_{1} {ꭆ^{ȵ}}_{i}) = \cap_{1} f^{- 1} ({ꭆ^{ȵ}}_{i})$
vi. $f^{- 1} (\cap_{2} {ꭆ^{ȵ}}_{i}) = \cap_{2} f^{- 1} ({ꭆ^{ȵ}}_{i})$

Corollary 2.8:

¹

Let $f : Ҳ \to Y be an injective function, and let S^{ȵ} = < S_{1}, S_{2}, S_{3} >$ be a Ɲ_ՄҪℜ_ʂ in Ҳ. Then

f ((S^{ȵ})^{c_{2}}) = {(f (S^{ȵ}))}^{c_{2}}

Corollary 2.9:

¹

Let $f : Ҳ \to Y be an injective function, and let ꭆ^{ȵ} = < ꭆ_{1}, ꭆ_{2}, ꭆ_{3} >$ be a Ɲ_ՄҪℜ_ʂ in $Y$ . Then

f^{- 1} ((ꭆ^{ȵ})^{c_{2}}) = {(f^{- 1} (ꭆ^{ȵ}))}^{c_{2}}

3. Neutrosophic crisp topological space ( Ɲ_ՄҪ $T$ _(1,2) -space)

In this section, we construct a neutrosophic crisp topological space in which the intersection is defined as Type I, while both the union and the complement are defined as Type II, considering all possible neutrosophic crisp families.

Definition 3.1:

The pair (Ҳ, $T$ ) is said to form a Neutrosophic Crisp Topological (Ɲ_ՄҪ $T$ _(1,2)-space) when the following criteria are satisfied:

a. $\emptyset_{1}^{ȵ}$ = < $\emptyset, \emptyset, Ҳ$ > $\in T$ , and $Ҳ_{1}^{ȵ}$ = < $Ҳ, \emptyset, \emptyset$ > $\in T$ .
b. For any $Ҥ^{ȵ}$ , $Ҡ^{ȵ}$ ∈ $T$ , the set $Ҥ^{ȵ}$ ∩₁ $Ҡ^{ȵ}$ $\in T$ .
c. If $Ҥ^{ȵ}$ _𝑗 ∈ $T$ ∀𝑗 ∈ 𝐽, then $\cup_{j 2} Ҥ^{ȵ}$ _𝑗 ∈ $T$ .

For any $Ҡ^{ȵ}$ ∈ $T$ , it is a neutrosophic crisp open set (Ɲ_ՄҪO − set), and ${(Ҡ^{ȵ})}^{c_{2}}$ is a neutrosophic crisp closed set (Ɲ_ՄҪҪ − set).

Example 3.2:

Let Ҳ = {𝑎, 𝑏, 𝑐, 𝑑} and $T$ = { $\emptyset_{1}^{ȵ}$ , $Ҳ_{1}^{ȵ}$ , $Ҥ^{ȵ}$ , $Ҡ^{ȵ}$ , $M^{ȵ}$ } such that $Ҥ^{ȵ}$ = < {𝑎}, {𝑑}, {𝑐} >, $Ҡ^{ȵ}$ = < {𝑎}, {𝑏}, {𝑐} >, $M^{ȵ}$ = < {𝑎}, $\emptyset$ , {𝑐} >. Then (Ҳ, $T$ ) is 𝑎 Ɲ_ՄҪ $T$ _(1,2)–space.

Example 3.3:

Let Ҳ = {𝑎, 𝑏, 𝑐, 𝑑} and $T$ = { $\emptyset_{1}^{ȵ}$ , $Ҳ_{1}^{ȵ}$ , $Ҥ^{ȵ}$ , $Ҡ^{ȵ}$ }, such that $Ҥ^{ȵ}$ = < {𝑎}, {𝑑}, {𝑐}> $Ҡ^{ȵ}$ = < {𝑎}, {𝑏}, {𝑐} >. Then $T$ is not Ɲ_ՄҪ $T$ _(1,2) – space. Since $Ҥ^{ȵ}$ ∩₁ $Ҡ^{ȵ} =$ $< {a}, {d}, {c} > \cap_{1}$ < {𝑎}, {b}, {𝑐} > = < {𝑎}, $\emptyset$ , {𝑐} > $\notin T .$

Corollary 3.4:

Let (Ҳ, $T$ ) be 𝑎 (Ɲ_ՄҪ $T$ _(1,2)–space), and let $ψ$ = {Ƒ: Ƒ 𝑖𝑠 a Ɲ_ՄҪҪ−set} satisfy the following conditions:

i. If $Ҥ^{ȵ}$ _𝑗 ∈ $ψ$ ∀ 𝑗 ∈, then $\cap_{1 j} Ҥ^{ȵ}$ _𝑗 ∈ $ψ$ .
ii. If $Ҥ^{ȵ}$ , $Ҡ^{ȵ}$ ∈ $ψ$ , then $Ҥ^{ȵ}$ ∪₂ $Ҡ^{ȵ}$ ∈ $ψ$ .

Proof:

i. Suppose $Ҥ_{1}^{ȵ}, Ҥ_{2}^{ȵ}, Ҥ_{3}^{ȵ}, \dots are$ Ɲ_ՄҪҪ− sets in a Ɲ_ՄҪ $T$ _(1,2)–space (Ҳ, $T$ ).

Hence, $Ҥ_{1}^{ȵ^{c_{2}}}, Ҥ_{1}^{ȵ^{c_{2}}}, Ҥ_{1}^{ȵ^{c_{2}}}, \dots are$ Ɲ_ՄҪO − sets. Therefore, $\cup_{2_{j}} Ҥ_{j}^{ȵ^{c_{2}}} is$ Ɲ_ՄҪO – set.

But ${(Ҥ^{ȵ} \cap_{1} Ҡ^{ȵ})}^{c_{2}} = {(Ҥ^{ȵ})}^{c_{2}} \cup_{2} {(Ҡ^{ȵ})}^{c_{2}} .$ So, ${(⋂_{1 j} Ҥ_{j}^{ȵ})}^{c_{2}} = ⋃_{2 j} Ҥ_{j}^{ȵ^{c_{2}}}$ .

Thus, ${(⋂_{1 j} Ҥ_{j}^{ȵ})}^{c_{2}} is$ Ɲ_ՄҪO – set. Therefore, $⋂_{1 j} Ҥ_{j}^{ȵ}$ Ɲ_ՄҪҪ − set. Hence, $⋂_{1 j} Ҥ_{j}^{ȵ} \in ψ$ .

ii. Suppose $Ҥ^{ȵ}$ , $Ҡ^{ȵ} are two$ Ɲ_ՄҪҪ – sets in a Ɲ_ՄҪ $T$ _(1,2) –space (Ҳ, $T$ ).

So, $Ҥ^{ȵ^{c_{2}}}$ , $Ҡ^{ȵ^{c_{2}}}$ $are two$ Ɲ_ՄҪO − sets, thus ${(Ҥ^{ȵ})}^{c_{2}} \cap_{1} {(Ҡ^{ȵ})}^{c_{2}}$ is Ɲ_ՄҪO – 𝑠𝑒𝑡. But ${(Ҥ^{ȵ} \cup_{2} Ҡ^{ȵ})}^{c_{2}} = {(Ҥ^{ȵ})}^{c_{2}} \cap_{1} {(Ҡ^{ȵ})}^{c_{2}}$ . Therefore ${(Ҥ^{ȵ} \cup_{2} Ҡ^{ȵ})}^{c_{2}}$ is Ɲ_ՄҪO – 𝑠𝑒𝑡.

$Thus, Ҥ^{ȵ} \cup_{2} Ҡ^{ȵ}$ is Ɲ_ՄҪҪ – 𝑠𝑒𝑡. $Hence, Ҥ^{ȵ} \cup_{2} Ҡ^{ȵ} \in ψ$ .

4. Neutrosophic crisp closure sets $(Ɲ_{Մ} {ҪҪԼ}_{(1, 2)}^{1}, Ɲ_{Մ} {ҪҪԼ}_{(1, 2)}^{2})$

In this section, we introduced two types of closures for neutrosophic crisp sets and examined all the related properties, discovering that most of these properties do not hold.

Definition 4.1:

Let (Ҳ, $T$ ) be 𝑎 Ɲ_ՄҪ $T$ _(1,2) –space. The neutrosophic crisp closure of neutrosophic crisp set $Ҥ^{ȵ},$ denoted by $Ɲ_{Մ} {ҪҪԼ}_{(1, 2)}^{1}$ ( $Ҥ^{ȵ}$ ), and we define the neutrosophic crisp closure of $Ҥ^{ȵ}$ by:

Ɲ_{Մ} {ҪҪԼ}_{(1, 2)}^{1} (Ҥ^{ȵ}) = ⋂_{1 j} {Ӻ_{j}^{ȵ} : Ӻ_{j}^{ȵ} is Ɲ_{Մ} ҪҪ - sets and Ҥ^{ȵ} \subseteq_{1} Ӻ_{j}^{ȵ} \forall j}

• $Ɲ_{Մ} {ҪҪԼ}_{(1, 2)}^{1}$ ( $Ҥ^{ȵ}$ ) is Ɲ_ՄҪҪ – set by Corollary 3.4.

Example 4.2:

Let Ҳ = {𝑎, 𝑏, 𝑐, 𝑑} and $T$ = { $\emptyset_{1}^{ȵ}$ , $Ҳ_{1}^{ȵ}$ , $Ҥ^{ȵ}$ , $Ҡ^{ȵ}$ , $M^{ȵ}$ } ∋ $Ҥ^{ȵ}$ = < {𝑎}, {𝑑}, {𝑐} >, $Ҡ^{ȵ}$ = < {𝑎}, {𝑏}, {𝑐} >,

$M^{ȵ}$ = < {𝑎}, $\emptyset$ , {𝑐} >. So, (Ҳ, $T$ ) is 𝑎 Ɲ_ՄҪ $T$ _(1,2)–space. Therefore:

i. $Ɲ_{Մ} {ҪҪԼ}_{(1, 2)}^{1} (Ҥ^{ȵ}$ = < {𝑎}, {𝑑}, {𝑐} >) does not exist since $∄$ Neutrosophic crisp closed set $Ӻ^{ȵ} ∋$ $Ҥ^{ȵ} \subseteq_{1} Ӻ^{ȵ}$ .
ii. $Ɲ_{Մ} {ҪҪԼ}_{(1, 2)}^{1}$ ( $A^{ȵ} = < {b}, \emptyset, \emptyset >$ )= $Ҳ_{1}^{ȵ} = < Ҳ, \emptyset, \emptyset >$ .
iii. $Ɲ_{Մ} {ҪҪԼ}_{(1, 2)}^{1}$ ( $M^{ȵ^{C_{2}}} = < {c}, \emptyset, {a} >$ ) = $M^{ȵ^{C_{2}}} = < {c}, \emptyset, {a}$ >.
iv. $Ɲ_{Մ} {ҪҪԼ}_{(1, 2)}^{1}$ ( $M^{ȵ} = < {a}, \emptyset, {c} >$ ) = $Ҳ_{1}^{ȵ} = < Ҳ, \emptyset, \emptyset >$
v. $Ɲ_{Մ} {ҪҪԼ}_{(1, 2)}^{1}$ ( $Ҥ^{ȵ^{C_{2}}}$ = < {c}, {d}, {a} >) = $Ҥ^{ȵ^{C_{2}}}$ = < {c}, {d}, {a} >.
vi. $Ɲ_{Մ} {ҪҪԼ}_{(1, 2)}^{1}$ ( $Ҡ^{ȵ^{C_{2}}}$ = < {c}, {b}, {a} >) = $Ҡ^{ȵ^{C_{2}}}$ = < {c}, {b}, {a} >.
vii. $Ɲ_{Մ} {ҪҪԼ}_{(1, 2)}^{1}$ ( $U^{ȵ^{C_{2}}}$ = < ${a}, {b},$ $\emptyset$ >) does not exist since $∄$ Neutrosophic crisp closed set $Ӻ^{ȵ} ∋$ $U^{ȵ^{C_{2}}} \subseteq_{1} Ӻ^{ȵ}$ .
viii. $Ɲ_{Մ} {ҪҪԼ}_{(1, 2)}^{1}$ ( $\emptyset_{1}^{ȵ}$ = < $\emptyset, \emptyset, Ҳ$ >) = $\emptyset_{1}^{ȵ} =$ < $\emptyset, \emptyset, Ҳ$ >
ix. $Ɲ_{Մ} {ҪҪԼ}_{(1, 2)}^{1}$ ( $Ҳ_{1}^{ȵ}$ = < Ҳ $, \emptyset, \emptyset$ > = $Ҳ_{1}^{ȵ}$ = < Ҳ $, \emptyset, \emptyset$ >

Remark 4.3:

In neutrosophic crisp topological spaces (Ɲ_ՄҪ $T$ _(1,2)–space), it is not always the case that every subset admits a $Ɲ_{Մ} {ҪҪԼ}_{(1, 2)}^{1}$ . Specifically, certain subsets may exist for which no neutrosophic crisp closed superset can be identified within the given topology. Such subsets are thus said to possess no closure with respect to this structure. For instance, as demonstrated in Example 4.2, the set $Ɲ_{Մ} {ҪҪԼ}_{(1, 2)}^{1} (Ҥ^{ȵ}$ =< {𝑎}, {𝑑}, {𝑐} >) fails to exist in this context.

Definition 4.4:

Let (Ҳ, $T$ ) be 𝑎 Ɲ_ՄҪ $T$ _(1,2)–space. The neutrosophic crisp closure of neutrosophic crisp set $Ҥ^{ȵ},$ denoted by $Ɲ_{Մ} {ҪҪԼ}_{(1, 2)}^{2}$ ( $Ҥ^{ȵ}$ ) and we define the neutrosophic crisp closure of $Ҥ^{ȵ}$ by: $Ɲ_{Մ} {ҪҪԼ}_{(1, 2)}^{2}$ ( $Ҥ^{ȵ}$ ) = $⋂_{1 j} {Ӻ_{j}^{ȵ}$ : $Ӻ_{j}^{ȵ}$ is Ɲ_ՄҪҪ – sets and $Ҥ^{ȵ}$ ⊆₂ $Ӻ_{j}^{ȵ} \forall j}$

• $Ɲ_{Մ} {ҪҪԼ}_{(1, 2)}^{2}$ ( $Ҥ^{ȵ}$ ) 𝑖𝑠 Ɲ_ՄҪҪ – 𝑠𝑒𝑡 by Corollary 3.4.

Example 4.5:

Let Ҳ = {𝑎, 𝑏, 𝑐, 𝑑} and $T$ = { $\emptyset_{1}^{ȵ}$ , $Ҳ_{1}^{ȵ}$ , $Ҥ^{ȵ}$ , $Ҡ^{ȵ}$ , $M^{ȵ}$ } such that $Ҥ^{ȵ}$ = < {𝑎}, {𝑑}, {𝑐} >, $Ҡ^{ȵ}$ = < {𝑎}, {𝑏}, {𝑐}>, $M^{ȵ}$ = < {𝑎}, $\emptyset$ , {𝑐} >. So, (Ҳ, $T$ ) is 𝑎 Ɲ_ՄҪ $T$ _(1,2)–space. Therefore:

i. $Ɲ_{Մ} {ҪҪԼ}_{(1, 2)}^{2}$ ( $Ҥ^{ȵ}$ = < {𝑎}, {𝑑}, {𝑐} >) = $Ҳ_{1}^{ȵ}$ = < Ҳ $, \emptyset, \emptyset$ >
ii. $Ɲ_{Մ} {ҪҪԼ}_{(1, 2)}^{2}$ ( $A^{ȵ}$ = < {b}, $\emptyset, \emptyset$ >) = $Ҳ_{1}^{ȵ}$ = < Ҳ $, \emptyset, \emptyset$ >
iii. $Ɲ_{Մ} {ҪҪԼ}_{(1, 2)}^{2} (U^{ȵ}$ = < $\emptyset, {b}, {a}$ >) = $Ҳ_{1}^{ȵ}$ = < Ҳ $, \emptyset, \emptyset$ >
iv. $Ɲ_{Մ} {ҪҪԼ}_{(1, 2)}^{2}$ ( $M^{ȵ^{C_{2}}}$ = < {c}, $\emptyset$ , {a} >) = < {c}, $\emptyset$ , {a} >
v. $Ɲ_{Մ} {ҪҪԼ}_{(1, 2)}^{2}$ ( $M^{ȵ}$ = < {𝑎}, $\emptyset$ , {𝑐}>) = $Ҳ_{1}^{ȵ}$ = < Ҳ $, \emptyset, \emptyset$ >
vi. $Ɲ_{Մ} {ҪҪԼ}_{(1, 2)}^{2}$ ( $Ҡ^{ȵ^{C_{2}}}$ = < {c}, {b}, {a} >) = $M^{ȵ^{C_{2}}}$ = < {c}, $\emptyset$ , {a} >
vii. $Ɲ_{Մ} {ҪҪԼ}_{(1, 2)}^{2}$ ( $\emptyset_{1}^{ȵ}$ = < $\emptyset, \emptyset, Ҳ$ >) = $\emptyset_{1}^{ȵ}$ = < $\emptyset, \emptyset, Ҳ$ >
viii. $Ɲ_{Մ} {ҪҪԼ}_{(1, 2)}^{2}$ ( $Ҳ_{1}^{ȵ}$ = < Ҳ $, \emptyset, \emptyset$ >) = $Ҳ_{1}^{ȵ}$ = < Ҳ $, \emptyset, \emptyset$ >

Corollary 4.6:

i. Let (Ҳ, $T$ ) be Ɲ_ՄҪ $T$ _(1,2)–space and $M^{ȵ}$ ⊆₁ $Ҳ_{1}^{ȵ}$ . Then $M^{ȵ}$ ⊆₁ $Ɲ_{Մ} {ҪҪԼ}_{(1, 2)}^{1}$ ( $M^{ȵ}$ ).
ii. Let (Ҳ, $T$ ) be 𝑎 Ɲ_ՄҪ $T$ _(1,2)–space and $S^{ȵ} any neutrosophic crisp set .$ Then $S^{ȵ}$ ⊆₂ $Ɲ_{Մ} {ҪҪԼ}_{(1, 2)}^{1}$ ( $S^{ȵ}$ ).

Remark 4.7:

For instance, consider Example 4.2, where $Ҥ^{ȵ}$ = <{𝑎},{𝑑},{𝑐}> and $Ҥ^{ȵ}$ ⊈₁ $Ҳ_{1}^{ȵ}$ .

In this case, $Ɲ_{Մ} {ҪҪԼ}_{(1, 2)}^{1}$ ( $Ҥ^{ȵ}$ ) does not exist; therefore, $Ҥ^{ȵ}$ ⊈₁ $Ɲ_{Մ} {ҪҪԼ}_{(1, 2)}^{1}$ ( $Ҥ^{ȵ}$ ), which shows that the condition of Corollary 4.6 (i) fails.

Theorem 4.8:

Let (Ҳ, $T$ ) be Ɲ_ՄҪ $T$ _(1,2)–space. Then $Ҥ^{ȵ}$ is Ɲ_ՄҪҪ– 𝑠𝑒𝑡 if and only if $Ɲ_{Մ} {ҪҪԼ}_{(1, 2)}^{1}$ ( $Ҥ^{ȵ}$ ) = $Ҥ^{ȵ}$ .

Proof:

If $Ɲ_{Մ} {ҪҪԼ}_{(1, 2)}^{1}$ ( $Ҥ^{ȵ}$ ) = $Ҥ^{ȵ}$ , then $Ҥ^{ȵ}$ Ɲ_ՄҪҪ – 𝑠𝑒𝑡 by Definition 4.1.

Conversely, Let { $Ƒ^{ȵ}$ _𝑗 = < $Ƒ$ _𝑗1, $Ƒ$ _𝑗2, $Ƒ$ _𝑗3 >: 𝑗 ∈ 𝐽} be the family of Ɲ_ՄҪҪ – sets such that $Ҥ^{ȵ}$ = < Ҥ₁, Ҥ₂, Ҥ₃ > ⊆₁ $Ƒ^{ȵ}$ _𝑗 ∀𝑗 ∈ 𝐽. Since Ҥ₁ ⊆ $Ƒ$ _𝑗1 ∀𝑗 ∈ 𝐽, Ҥ₂ ⊆ $Ƒ$ _𝑗2 ∀𝑗 ∈ 𝐽, Ҥ₃ ⊇ $Ƒ$ _𝑗3 ∀𝑗 ∈ 𝐽.

$Hence, Ɲ_{Մ} {ҪҪԼ}_{(1, 2)}^{1}$ ( $Ҥ^{ȵ}$ ) = $\cap_{1 j} {Ƒ^{ȵ}}_{j}$ =< ∩_𝑗 $Ƒ$ _𝑗1, ∩_𝑗 $Ƒ$ _𝑗2, ∪_𝑗 $Ƒ$ _𝑗3 > = < Ҥ₁, Ҥ₂, Ҥ₃ > = $Ҥ^{ȵ} .$

Remark 4.9:

For instance, consider Example 4.5, where $Ҡ^{ȵ^{c_{2}}}$ is NCC−𝑠𝑒𝑡. In this case $, Ɲ_{Մ} {ҪҪԼ}_{(1, 2)}^{2}$ ( $Ҡ^{ȵ^{C_{2}}}$ ) = $M^{ȵ^{c_{2}}} \neq Ҡ^{ȵ^{C_{2}}};$ which shows that the condition of Theorem 4.8 fails.

Theorem 4.10:

Let (Ҳ, $T$ ) be 𝑎 Ɲ_ՄҪ $T$ _(1,2)–space, and $M^{ȵ}$ is any neutrosophic crisp set. Then $M^{ȵ}$ ⊆₂ $Ɲ_{Մ} {ҪҪԼ}_{(1, 2)}^{2}$ ( $M^{ȵ}$ ).

Proof:

Let { $Ƒ^{ȵ}$ _𝑗 = <Ƒ_𝑗1, Ƒ_𝑗2, Ƒ_𝑗3 >: 𝑗 ∈ 𝐽} be the family of Ɲ_ՄҪҪ – 𝑠𝑒𝑡s such that $M^{ȵ} =$ < $M$ ₁, $M$ ₂, $M$ ₃ > ⊆₂ $Ƒ^{ȵ}$ _𝑗 ∀𝑗 ∈ 𝐽. Since, $M$ ₁ ⊆ Ƒ_𝑗1 ∀𝑗 ∈ 𝐽, $M$ ₂ ⊇ Ƒ_𝑗2 ∀𝑗 ∈ 𝐽, $M$ ₃ ⊇ Ƒ_𝑗3 ∀ 𝑗 ∈ 𝐽. Therefore, $M$ ₁ ⊆ $\cap$ Ƒ_𝑗1, $M$ ₂ ⊇ $\cap$ Ƒ_𝑗2, $M$ ₃ ⊇ ∪Ƒ_𝑗3. Hence, $M^{ȵ}$ ⊆₂ $Ɲ_{Մ} {ҪҪԼ}_{(1, 2)}^{2}$ ( $M^{ȵ}$ ).

Remark 4.11:

For instance, consider Example 4.2, where $Ҥ^{ȵ}$ = <{𝑎},{𝑑},{𝑐}>. In this case, $Ɲ_{Մ} {ҪҪԼ}_{(1, 2)}^{1}$ ( $Ҥ^{ȵ}$ ) does not exist; therefore, $Ҥ^{ȵ}$ ⊈₁ $Ɲ_{Մ} {ҪҪԼ}_{(1, 2)}^{1}$ ( $Ҥ^{ȵ}$ ) $,$ which shows that the condition of Theorem 4.10 fails.

Theorem 4.12:

Let (Ҳ, $T$ ) be 𝑎 Ɲ_ՄҪ $T$ _(1,2)–space and $Ҥ^{ȵ} =$ <Ҥ₁, Ҥ₂, Ҥ₃> be a Ɲ_ՄҪҪ – 𝑠𝑒𝑡. If $U^{ȵ} = < U_{1}, U_{2}, U_{3} > is any neutrosophic crisp set ∋ U^{ȵ}$ $\subseteq_{1} Ҥ^{ȵ}, then Ɲ_{Մ} {ҪҪԼ}_{(1, 2)}^{1}$ ( $U^{ȵ}$ ) $\subseteq_{1} Ҥ^{ȵ} .$

Proof:

Since $, Ҫ_{Մ} {ҪҪԼ}_{(1, 2)}^{1}$ ( $U^{ȵ}$ ) = $\cap_{1_{j}} Ӻ^{ȵ}$ _𝑗 = <∩ $Ӻ$ _𝑗1, ∩ $Ӻ$ _𝑗2, $\cup Ӻ$ _𝑗3> ∋ $Ӻ_{j}^{ȵ}$ is Ɲ_ՄҪҪ – sets and $U^{ȵ}$ ⊆₁ $Ӻ_{j}^{ȵ} \forall j .$

Therefore, $U_{1} \subseteq \cap Ӻ_{j 1} \forall j, U_{2} \subseteq \cap Ӻ_{j 2} \forall j, U_{3} \supseteq \cup Ӻ_{j 3} \forall j$ . Since, $U$ ₁ ⊆ Ҥ₁, $U$ ₂ ⊆ Ҥ₂, U₃ ⊇ Ҥ₃ and $Ҥ^{ȵ}$ is Ɲ_ՄҪҪ – 𝑠𝑒𝑡. Thus, $\cap Ӻ_{j 1} \subseteq Ҥ_{1} \forall j, \cap Ӻ_{j 2} \subseteq Ҥ_{2} \forall j, \cup Ӻ_{j 3} \supseteq Ҥ_{3} \forall j . Hence, Ɲ_{Մ} {ҪҪԼ}_{(1, 2)}^{1}$ ( $U^{ȵ}$ ) $\subseteq_{1} Ҥ^{ȵ}$ .

Remark 4.13:

For instance, consider Example 4.5, where $Ҡ^{ȵ^{C_{2}}}$ = < {c}, {b}, {a} > is a Ɲ_ՄҪҪ –𝑠𝑒𝑡 and $L^{ȵ} = < \emptyset,$ {𝑏},{𝑎}> $\subseteq_{2} Ҡ^{ȵ^{C_{2}}}$ . In this case, $Ҫ_{Մ} {ҪҪԼ}_{(1, 2)}^{2} (L^{ȵ}) = < {c}, \emptyset, {a} > ⊈_{2} Ҡ^{ȵ^{C_{2}}}, which shows that the$ condition of Theorem 4.12 fails.

Corollary 4.14:

Let (Ҳ, $T$ ) be 𝑎 Ɲ_ՄҪ $T$ _(1,2)–space and $Ҥ^{ȵ} =$ < Ҥ₁, $\emptyset$ , Ҥ₃ > be a Ɲ_ՄҪҪ –𝑠𝑒𝑡. If $U^{ȵ} = < U_{1}, \emptyset, U_{3} > is any neutrosophic crisp set ∋ U^{ȵ}$ $\subseteq_{2} Ҥ^{ȵ}, then Ɲ_{Մ} {ҪҪԼ}_{(1, 2)}^{1}$ ( $U^{ȵ}$ ) $\subseteq_{2} Ҥ^{ȵ} .$

Proof:

Since $, Ҫ_{Մ} {ҪҪԼ}_{(1, 2)}^{1}$ ( $U^{ȵ}$ ) = $\cap_{1_{j}} Ӻ^{ȵ}$ _𝑗 = <∩ $Ӻ$ _𝑗1, ∩ $Ӻ$ _𝑗2, $\cup Ӻ$ _𝑗3> ∋ $Ӻ_{j}^{ȵ}$ is Ɲ_ՄҪҪ – sets and $U^{ȵ}$ ⊆₂ $Ӻ_{j}^{ȵ} \forall j .$

Therefore, $U_{1} \subseteq \cap Ӻ_{j 1} \forall j, \emptyset \supseteq \cap Ӻ_{j 2} \forall j, U_{3} \supseteq \cup Ӻ_{j 3} \forall j$ . Since, $U$ ₁ ⊆ Ҥ₁, $U$ ₂ = Ҥ₂ = $\emptyset$ , U₃ ⊇ Ҥ₃ and $Ҥ^{ȵ}$ is Ɲ_ՄҪҪ –𝑠𝑒𝑡. Thus, $\cap Ӻ_{j 1} \subseteq Ҥ_{1} \forall j, \cap Ӻ_{j 2} = \emptyset, \cup Ӻ_{j 3} \supseteq Ҥ_{3} \forall j . Hence, Ɲ_{Մ} {ҪҪԼ}_{(1, 2)}^{1}$ ( $U^{ȵ}$ ) $\subseteq_{1} Ҥ^{ȵ}$ .

5. Neutrosophic crisp interior sets $(Ɲ_{Մ} {ҪԼȵȶ}_{(1, 2)}^{1}, Ɲ_{Մ} {ҪԼȵȶ}_{(1, 2)}^{2})$

In this section, we introduced two types of interiors of neutrosophic crisp sets and studied all the related properties, where it was found that most of these properties do not hold.

Definition 5.1:

Let (Ҳ, $T$ ) be 𝑎 Ɲ_ՄҪ $T$ _(1,2)–space then the neutrosophic crisp interior of neutrosophic crisp set $M^{ȵ},$ denoted by $Ɲ_{Մ} {ҪԼȵȶ}_{(1, 2)}^{1}$ ( $M^{ȵ}$ ), and we define the neutrosophic crisp interior of $Ҥ^{ȵ}$ by:

Ɲ_{Մ} {ҪԼȵȶ}_{(1, 2)}^{1} (M^{ȵ}) = \cup_{2 𝑗} {G_{j}^{ȵ} : G_{j}^{ȵ} is Ɲ_{Մ} ҪO - set and G_{j}^{ȵ} \subseteq_{1} M^{ȵ} \forall j} .

• $Ɲ_{Մ} {ҪԼȵȶ}_{(1, 2)}^{1}$ ( $M^{ȵ}$ ) 𝑖𝑠 Ɲ_ՄҪՕ – 𝑠𝑒𝑡

Example 5.2:

Let Ҳ = {𝑎, 𝑏, 𝑐, 𝑑} and = { $\emptyset_{1}^{ȵ}$ , $Ҳ_{1}^{ȵ}$ , $Ҥ^{ȵ}$ , $Ҡ^{ȵ}$ , $M^{ȵ}$ }, where $Ҥ^{ȵ}$ = < {𝑎}, {𝑑}, {𝑐} >, $Ҡ^{ȵ}$ = < {𝑎}, {𝑏}, {𝑐} >,

$M^{ȵ}$ = < {𝑎}, $\emptyset$ , {𝑐} >. So, (Ҳ, $T$ ) is 𝑎 Ɲ_ՄҪ $T$ _(1,2)–space. Therefore:

i. $Ɲ_{Մ} {ҪԼȵȶ}_{(1, 2)}^{1}$ ( $Ҥ^{ȵ}$ $=$ < {𝑎}, {𝑑}, {𝑐} >) = $M^{ȵ}$ = < {𝑎}, $\emptyset$ , {𝑐} >.
ii. $Ɲ_{Մ} {ҪԼȵȶ}_{(1, 2)}^{1}$ ( $A^{ȵ}$ $=$ < {b}, $\emptyset, \emptyset$ >) = $\emptyset_{1}^{ȵ}$ = < $\emptyset, \emptyset, Ҳ$ >.
iii. $Ɲ_{Մ} {ҪԼȵȶ}_{(1, 2)}^{1}$ ( $M^{ȵ^{C_{2}}}$ $=$ < {c}, $\emptyset$ , {a} >) $=$ $\emptyset_{1}^{ȵ}$ $=$ < $\emptyset, \emptyset, Ҳ$ >.
iv. $Ɲ_{Մ} {ҪԼȵȶ}_{(1, 2)}^{1}$ ( $M^{ȵ}$ $=$ < {𝑎}, $\emptyset$ , {𝑐}>) $=$ $M^{ȵ}$ $=$ < {𝑎}, $\emptyset$ , {𝑐}>.
v. $Ɲ_{Մ} {ҪԼȵȶ}_{(1, 2)}^{1}$ ( $Ҡ^{ȵ^{C_{2}}}$ $=$ < {c}, {b}, {a} >) $=$ $\emptyset_{1}^{ȵ}$ = < $\emptyset, \emptyset, Ҳ$ >.
vi. $Ɲ_{Մ} {ҪԼȵȶ}_{(1, 2)}^{1}$ ( $U^{ȵ}$ $=$ < $\emptyset, {b}, {a}$ >) = $\emptyset_{1}^{ȵ}$ $=$ < $\emptyset, \emptyset, Ҳ$ >.
vii. $Ɲ_{Մ} {ҪԼȵȶ}_{(1, 2)}^{1}$ ( $L^{ȵ}$ $= <$ {a},{d} $, \emptyset$ >) = $M^{ȵ}$ $=$ < {𝑎}, $\emptyset$ , {𝑐}>.
viii. $Ɲ_{Մ} {ҪԼȵȶ}_{(1, 2)}^{1}$ ( $\emptyset_{1}^{ȵ}$ $=$ < $\emptyset, \emptyset, Ҳ$ >) = $\emptyset_{1}^{ȵ}$ $=$ < $\emptyset, \emptyset, Ҳ$ >.
ix. $Ɲ_{Մ} {ҪԼȵȶ}_{(1, 2)}^{1}$ ( $Ҳ_{1}^{ȵ}$ $=$ < Ҳ $, \emptyset, \emptyset$ >) = $Ҳ_{1}^{ȵ}$ $=$ < Ҳ $, \emptyset, \emptyset$ >.

Definition 5.3:

Let (Ҳ, $T$ ) be 𝑎 Ɲ_ՄҪ $T$ _(1,2)–space. Then the neutrosophic crisp interior of neutrosophic crisp set $M,$ denoted by $Ɲ_{Մ} {ҪԼȵȶ}_{(1, 2)}^{2}$ ( $M^{ȵ}$ ), and we define the neutrosophic crisp interior of $M^{ȵ}$ by $:$

Ɲ_{Մ} {ҪԼȵȶ}_{(1, 2)}^{2} (M^{ȵ}) = \cup_{2 𝑗} {G_{j}^{ȵ} ∋ G_{j}^{ȵ} 𝑖𝑠 Ɲ_{Մ} ҪO - sets and G_{j}^{ȵ} \subseteq_{2} M^{ȵ} \forall j} .

• $Ɲ_{Մ} {ҪԼȵȶ}_{(1, 2)}^{2}$ ( $M^{ȵ}$ ) 𝑖𝑠 Ɲ_ՄҪO – 𝑠𝑒𝑡

Example 5.4:

Let Ҳ = {𝑎, 𝑏, 𝑐, 𝑑} and $T$ = { $\emptyset_{1}^{ȵ}$ , $Ҳ_{1}^{ȵ}$ , $Ҥ^{ȵ}$ , $Ҡ^{ȵ}$ , $M^{ȵ}$ }, where $Ҥ^{ȵ}$ = < {𝑎}, {𝑑}, {𝑐} >, $Ҡ^{ȵ}$ = < {𝑎}, {𝑏}, {𝑐} >, $M^{ȵ}$ = < {𝑎}, $\emptyset$ , {𝑐} >. So, (Ҳ, $T$ ) is 𝑎 Ɲ_ՄҪ $T$ _(1,2)–space. Therefore:

i. $Ɲ_{Մ} {ҪԼȵȶ}_{(1, 2)}^{2}$ ( $Ҥ^{ȵ}$ = < {𝑎}, {𝑑}, {𝑐} >) = $Ҥ^{ȵ}$ = < {𝑎}, {𝑑}, {𝑐} >.
ii. $Ɲ_{Մ} {ҪԼȵȶ}_{(1, 2)}^{2}$ ( $A^{ȵ}$ = < {b}, $\emptyset, \emptyset$ >) = $\emptyset_{1}^{ȵ}$ = < $\emptyset, \emptyset, Ҳ$ >.
iii. $Ɲ_{Մ} {ҪԼȵȶ}_{(1, 2)}^{2}$ ( $M^{ȵ^{C_{2}}}$ = < {c}, $\emptyset$ , {a} >) = $\emptyset_{1}^{ȵ}$ = < $\emptyset, \emptyset, Ҳ$ >.
iv. $Ɲ_{Մ} {ҪԼȵȶ}_{(1, 2)}^{2}$ ( $M^{ȵ}$ = < {𝑎}, $\emptyset$ , {𝑐}>) = $M^{ȵ}$ = < {𝑎}, $\emptyset$ , {𝑐}>.
v. $Ɲ_{Մ} {ҪԼȵȶ}_{(1, 2)}^{2}$ ( $Ҡ^{ȵ^{C_{2}}}$ = < {c}, {b}, {a} >) = $\emptyset_{1}^{ȵ}$ =< $\emptyset, \emptyset, Ҳ$ >.
vi. $Ɲ_{Մ} {ҪԼȵȶ}_{(1, 2)}^{2}$ ( $U^{ȵ^{C_{2}}}$ = < ${a}, {b},$ $\emptyset$ >) = $Ҡ^{ȵ}$ = < {𝑎}, {𝑏}, {𝑐} >.
vii. $Ɲ_{Մ} {ҪԼȵȶ}_{(1, 2)}^{2}$ ( $U^{ȵ}$ = < $\emptyset, {b}, {a}$ >) does not exist since $∄$ Neutrosophic Ɲ_ՄҪO – set $G^{ȵ} ∋$ $G^{ȵ} \subseteq_{1} Ҥ^{ȵ}$ .
viii. $Ɲ_{Մ} {ҪԼȵȶ}_{(1, 2)}^{2}$ ( $\emptyset_{1}^{ȵ}$ = < $\emptyset, \emptyset, Ҳ$ >) = $\emptyset_{1}^{ȵ}$ = < $\emptyset, \emptyset, Ҳ$ >.
ix. $Ɲ_{Մ} {ҪԼȵȶ}_{(1, 2)}^{2} (Ҳ_{1}^{ȵ}$ = < Ҳ $, \emptyset, \emptyset$ >) = $Ҳ_{1}^{ȵ}$ = < Ҳ $, \emptyset, \emptyset$ >.

Remark 5.5:

In neutrosophic crisp topological spaces (Ɲ_ՄҪ $T$ _(1,2)–space), it is not always the case that every subset admits a $Ɲ_{Մ} {ҪԼȵȶ}_{(1, 2)}^{2}$ . Specifically, certain subsets may exist for which no neutrosophic crisp open superset can be identified within the given topology. Such subsets are thus said to possess no interior with respect to this structure. For instance, as demonstrated in Example 5.4, the set $Ɲ_{Մ} {ҪԼȵȶ}_{(1, 2)}^{2}$ ( $U^{ȵ}$ =< $\emptyset, {b}, {a}$ >) fails to exist in this context.

Theorem 5.6:

Let (Ҳ, $T$ ) be 𝑎 Ɲ_ՄҪ $T$ _(1,2)–space. Then $Ҡ^{ȵ}$ is Ɲ_ՄҪO – 𝑠𝑒𝑡 if and only if $Ɲ_{Մ} {ҪԼȵȶ}_{(1, 2)}^{2}$ ( $Ҡ^{ȵ}$ ) = $Ҡ^{ȵ}$

Proof:

If $Ɲ_{Մ} {ҪԼȵȶ}_{(1, 2)}^{2}$ ( $Ҡ^{ȵ}$ ) = $Ҡ^{ȵ}$ , then $Ҡ^{ȵ}$ is Ɲ_ՄҪO – 𝑠𝑒𝑡 by Definition.5.3.

Conversely Let { ${G^{ȵ}}_{j}$ = < $G_{j 1}$ , $G_{j 2}$ , $G_{j 3}$ >: 𝑗 ∈ 𝐽} be the family of Ɲ_ՄҪO –𝑠𝑒𝑡s ∋ $G^{ȵ}$ _𝑗 $\subseteq_{2} Ҡ^{ȵ} = < Ҡ_{1}, Ҡ_{2}, Ҡ_{3} > \forall j \in J$ . So, $Ɲ_{Մ} {ҪԼȵȶ}_{(1, 2)}^{2}$ ( $Ҡ^{ȵ}$ ) = $⋃_{2_{j}} {G^{ȵ}}_{j} = < ⋃_{j} {G^{ȵ}}_{j 1}, ⋂_{j} {G^{ȵ}}_{j 2}, ⋂_{j} {G^{ȵ}}_{j 3} >$ = < $Ҡ_{1}$ , $Ҡ_{2}$ , $Ҡ_{3}$ > = $Ҡ^{ȵ}$ since 𝘎_𝑗1 ⊆ $Ҡ$ ₁ ∀𝑗 ∈ 𝐽, 𝘎_𝑗2 ⊇ $Ҡ$ ₂ ∀𝑗 ∈ 𝐽, 𝘎_𝑗3 ⊇ $Ҡ$ ₃ ∀𝑗 ∈𝐽.

Remark 5.7:

For instance, consider Example 5.2, where $Ҥ^{ȵ}$ is Ɲ_ՄҪO–𝑠𝑒𝑡. In this case, $Ɲ_{Մ} {ҪԼȵȶ}_{(1, 2)}^{1}$ ( $Ҥ^{ȵ}$ )= $M^{ȵ} \neq Ҥ^{ȵ}$ , which shows that the condition of Theorem 5.6 fails.

Theorem 5.8:

Let (Ҳ, $T$ ) be 𝑎 Ɲ_ՄҪ $T$ _(1,2)–space and $M^{ȵ}$ be any neutrosophic crisp set. Then $Ɲ_{Մ} {ҪԼȵȶ}_{(1, 2)}^{1}$ ( $M^{ȵ}$ ) ⊆₁ $M^{ȵ}$ .

Proof:

Let { ${G^{ȵ}}_{j}$ = < $G_{j 1}$ , $G_{j 2}$ , $G_{j 3}$ >: 𝑗 ∈ 𝐽} be the family of Ɲ_ՄҪՕ – 𝑠𝑒𝑡s ∋ $G^{ȵ}$ _𝑗⊆₁ $M^{ȵ}$ =< $M_{1}$ , $M_{2}$ , $M_{3}$ > ∀ 𝑗 ∈ 𝐽 since 𝐺_𝑗1 ⊆ $M$ ₁ ∀ 𝑗 ∈ 𝐽, 𝐺_𝑗2 ⊆ $M_{2}$ ∀ 𝑗 ∈ 𝐽, 𝐺 _𝑗3 ⊇ $M$ ₃ ∀ 𝑗 ∈ 𝐽. Therefore, ∪_𝑗 𝐺_𝑗1 ⊆ $M$ ₁, ∩_𝑗 𝐺_𝑗2 ⊆ $M$ ₂, ∩_𝑗 𝐺_𝑗3 ⊇ $M$ ₃. Hence, $Ɲ_{Մ} {ҪԼȵȶ}_{(1, 2)}^{1}$ ( $M^{ȵ}$ ) ⊆₁ $M^{ȵ}$

Remark 5.9:

For instance, consider Example 5.4, where $Ҡ^{ȵ^{C_{2}}} = < {c},$ {b} $, {a} >$ is any neutrosophic crisp set. In this case, $Ɲ_{Մ} {ҪԼȵȶ}_{(1, 2)}^{2}$ ( $Ҡ^{ȵ^{C_{2}}}$ ) = $< \emptyset, \emptyset, Ҳ > = \emptyset_{1}^{ȵ} ⊈_{2} Ҡ^{ȵ^{C_{2}}}$ . This example demonstrates that the condition of Theorem 5.8 fails.

Theorem 5.10:

Let (Ҳ, $T$ ) be 𝑎 Ɲ_ՄҪ $T$ _(1,2)–space, $Ҡ^{ȵ} = < Ҡ_{1}, Ҡ_{2}, Ҡ_{3} >$ be a Ɲ_ՄҪO–𝑠𝑒𝑡, and $U^{ȵ} = < U_{1}, U_{2}, U_{3} >$ be any neutrosophic crisp set, $∋ Ҡ^{ȵ} \subseteq_{2} U^{ȵ}, then Ҡ^{ȵ}$ $\subseteq_{2} Ɲ_{Մ} {ҪԼȵȶ}_{(1, 2)}^{2}$ ( $U^{ȵ}$ ).

Proof:

$Ɲ_{Մ} {ҪԼȵȶ}_{(1, 2)}^{2}$ ( $U^{ȵ}$ ) = $\cup_{2_{j}} G^{ȵ}$ _𝑗 = < $\cup$ G_𝑗1, ∩G_𝑗2, $\cap$ G_𝑗3 > ∋ { ${G^{ȵ}}_{j}$ = < $G_{j 1}$ , $G_{j 2}$ , $G_{j 3}$ >: 𝑗 ∈ 𝐽} be the family of Ɲ_ՄҪO – 𝑠𝑒𝑡s ∋ $G^{ȵ}$ _𝑗 $\subseteq_{2} U^{ȵ}$ . Therefore, $\cup G_{j 1} \subseteq U_{1} \forall j, \cap G_{j 2} \supseteq U_{2} \forall j, \cap G_{j 3} \supseteq U_{3} \forall j . Since,$ Ҡ₁⊆ U₁, Ҡ₂⊇ $U$ ₂, Ҡ₃⊇ $U$ ₃ and $Ҡ^{ȵ}$ is Ɲ_ՄҪO – 𝑠𝑒𝑡. Thus, $Ҡ_{1} \subseteq \cup G_{j 1} \forall j, Ҡ_{2} \supseteq \cap G_{j 2} \forall j, Ҡ_{3} \supseteq \cap G_{j 3} \forall j .$

Hence, $Ҡ^{ȵ}$ $\subseteq_{2} Ɲ_{Մ} {ҪԼȵȶ}_{(1, 2)}^{2}$ ( $U^{ȵ}$ ).

Remark 5.11:

For instance, consider Example 5.2, $Ҥ^{ȵ} =$ < {a}, {d}, {c} >, which is Ɲ_ՄҪO – 𝑠𝑒𝑡 $and Ҥ^{ȵ} \subseteq_{1} L^{ȵ} = <$ {a},{d} $, \emptyset$ > $.$ In this case, $Ҥ^{ȵ} ⊈_{1} Ɲ_{Մ} {ҪԼȵȶ}_{(1, 2)}^{1}$ ( $L^{ȵ}$ ) $= <$ {a}, $\emptyset, {c}$ >, which shows that the condition of Theorem 5.10 fails.

Corollary 5.12:

Let (Ҳ, $T$ ) be 𝑎 Ɲ_ՄҪ $T$ _(1,2)–space and $Ҡ^{ȵ} =$ < $Ҡ$ ₁, $\emptyset$ , $Ҡ$ ₃ be a Ɲ_ՄҪO–𝑠𝑒𝑡, and $U^{ȵ} = < U_{1}, \emptyset, U_{3} >$ be any neutrosophic crisp set, $∋ Ҡ^{ȵ} \subseteq_{1} U^{ȵ}, then Ҡ^{ȵ} \subseteq_{1}$ $Ɲ_{Մ} {ҪԼȵȶ}_{(1, 2)}^{1}$ ( $U^{ȵ}$ ) $.$

Proof:

$Ɲ_{Մ} {ҪԼȵȶ}_{(1, 2)}^{2}$ ( $U^{ȵ}$ ) = $\cup_{2_{j}} G^{ȵ}$ _𝑗 = < $\cup$ G_𝑗1, ∩G_𝑗2, $\cap$ G_𝑗3 > ∋ { ${G^{ȵ}}_{j}$ = < $G_{j 1}$ , $G_{j 2}$ , $G_{j 3}$ >: 𝑗 ∈ 𝐽} be the family of Ɲ_ՄҪO – 𝑠𝑒𝑡s ∋ $G^{ȵ}$ _𝑗 $\subseteq_{1} U^{ȵ}$ . Therefore, $\cup G_{j 1} \subseteq U_{1} \forall j, \cap G_{j 2} \supseteq \emptyset, \cap G_{j 3} \supseteq U_{3} \forall j . Since,$ Ҡ₁ ⊆ U₁, Ҡ₂ = $U$ ₂ = $\emptyset$ , Ҡ₃ ⊇ $U$ ₃ and $Ҡ^{ȵ}$ is a Ɲ_ՄҪO–𝑠𝑒𝑡. Thus, $Ҡ_{1} \subseteq \cup G_{j 1} \forall j, \emptyset = \cap G_{j 2} \forall j, Ҡ_{3} \supseteq \cap G_{j 3} \forall j$ . Hence, $Ҡ^{ȵ}$ $\subseteq_{1} Ɲ_{Մ} {ҪԼȵȶ}_{(1, 2)}^{2}$ ( $U^{ȵ}$ ).

Theorem 5.13:

Let (Ҳ, $T$ ) be 𝑎 Ɲ_ՄҪ $T$ _(1,2)–space and $Ҥ^{ȵ}$ be any neutrosophic crisp set. Then

i. [ $Ɲ_{Մ} {ҪҪԼ}_{(1, 2)}^{2} (Ҥ^{ȵ})]^{c_{2}} =$ $Ɲ_{Մ} {ҪԼȵȶ}_{(1, 2)}^{1}$ ( $Ҥ^{ȵ^{c_{2}}}$ ).
ii. [ $Ɲ_{Մ} {ҪҪԼ}_{(1, 2)}^{1} (Ҥ^{ȵ})]^{c_{2}} =$ $Ɲ_{Մ} {ҪԼȵȶ}_{(1, 2)}^{2}$ ( $Ҥ^{ȵ^{c_{2}}}$ ).
iii. [ $Ɲ_{Մ} {ҪԼȵȶ}_{(1, 2)}^{2} (Ҥ^{ȵ})]^{c_{2}} =$ $Ɲ_{Մ} {ҪҪԼ}_{(1, 2)}^{1}$ ( $Ҥ^{ȵ^{c_{2}}}$ ).
iv. [ $Ɲ_{Մ} {ҪԼȵȶ}_{(1, 2)}^{1} (Ҥ^{ȵ})]^{c_{2}} =$ $Ɲ_{Մ} {ҪҪԼ}_{(1, 2)}^{2}$ ( $Ҥ^{ȵ^{c_{2}}}$ ).
v. $Ɲ_{Մ} {ҪԼȵȶ}_{(1, 2)}^{1}$ ( $Ҥ^{ȵ}$ ) $=$ [ $Ɲ_{Մ} {ҪҪԼ}_{(1, 2)}^{2} (Ҥ^{ȵ^{c_{2}}})]^{c_{2}}$
vi. $Ɲ_{Մ} {ҪԼȵȶ}_{(1, 2)}^{2}$ ( $Ҥ^{ȵ}$ ) $=$ [ $Ɲ_{Մ} {ҪҪԼ}_{(1, 2)}^{1} (Ҥ^{ȵ^{c_{2}}})]^{c_{2}}$
vii. $Ɲ_{Մ} {ҪҪԼ}_{(1, 2)}^{1}$ ( $Ҥ^{ȵ}$ ) $=$ [ $Ɲ_{Մ} {ҪԼȵȶ}_{(1, 2)}^{2} (Ҥ^{ȵ^{c_{2}}})]^{c_{2}}$
viii. $Ɲ_{Մ} {ҪҪԼ}_{(1, 2)}^{2}$ ( $Ҥ^{ȵ}$ ) $=$ [ $Ɲ_{Մ} {ҪԼȵȶ}_{(1, 2)}^{1} (Ҥ^{ȵ^{c_{2}}})]^{c_{2}}$

Proof:

i. Let $Ҥ^{ȵ}$ = < Ҥ₁, Ҥ₂, Ҥ₃ > and { $G^{ȵ}$ _𝑗 = <𝘎_𝑗1, 𝘎_𝑗2, 𝘎_𝑗3>: 𝑗 ∈ 𝐽} be family of Ɲ_ՄҪO – 𝑠𝑒𝑡s such that ${G^{ȵ}}_{j}$ ⊆₁ $Ҥ^{ȵ^{c_{2}}}$ = <Ҥ₃, Ҥ₂, Ҥ₂ >. So, $Ɲ_{Մ} {ҪԼȵȶ}_{(1, 2)}^{1}$ ( $Ҥ^{ȵ^{c_{2}}}$ ) = $⋃_{2_{j}} G^{ȵ}$ _𝑗 = < $\cup$ 𝘎_𝑗1, ∩𝘎_𝑗2, ∩𝘎_𝑗3>…….(1)

Thus, ${G^{ȵ}}_{j}^{c_{2}} = < G_{j 3}, G_{j 2}, G_{j 1} > : j \in J} be$ the family of Ɲ_ՄҪҪ – 𝑠𝑒𝑡s.

∋ Ҥ^{ȵ} \subseteq_{2} {G^{ȵ}}_{j}^{c_{2}} . So, Ҥ_{1} \subseteq G_{j 3} \forall j \in J, Ҥ_{2} \supseteq G_{j 2} \forall j \in J, Ҥ_{3} \supseteq G_{j 1} \forall j \in J .

Therefore, $Ɲ_{Մ} {ҪҪԼ}_{(1, 2)}^{2} (Ҥ^{ȵ}) = \cap_{1_{j}} {G^{ȵ}}_{j}^{c_{2}} = < \cap_{j} G_{j 3}, \cap_{j} G_{j 2}, \cup_{j} G_{j 1} > .$

$Hence,$ ( $Ɲ_{Մ} {ҪҪԼ}_{(1, 2)}^{2} (Ҥ^{ȵ}))^{c_{2}} = < \cup_{j} G_{j 1}, \cap_{j} G_{j 2}, \cap_{j} G_{j 3} >$ …….(2)

From (1) and (2) we get [ $Ɲ_{Մ} {ҪҪԼ}_{(1, 2)}^{2} (Ҥ^{ȵ})]^{c_{2}} =$ $Ɲ_{Մ} {ҪԼȵȶ}_{(1, 2)}^{1}$ ( $Ҥ^{ȵ^{c_{2}}}$ ).

ii. Let $Ҥ^{ȵ}$ = < Ҥ₁, Ҥ₂, Ҥ₃ > and { $G^{ȵ}$ _𝑗 = <𝘎_𝑗1, 𝘎_𝑗2, 𝘎_𝑗3>: 𝑗 ∈ 𝐽} be family of Ɲ_ՄҪO – 𝑠𝑒𝑡s such that ${G^{ȵ}}_{j}$ ⊆₂ $Ҥ^{ȵ^{c_{2}}}$ = <Ҥ₃, Ҥ₂, Ҥ₂ >. So, $Ɲ_{Մ} {ҪԼȵȶ}_{(1, 2)}^{2}$ ( $Ҥ^{ȵ^{c_{2}}}$ ) = $\cup_{2_{j}} G^{ȵ}$ _𝑗 = < $\cup$ 𝘎_𝑗1, ∩𝘎_𝑗2, ∩𝘎_𝑗3>…….(1)

Thus, ${G^{ȵ}}_{j}^{c_{2}} = < G_{j 3}, G_{j 2}, G_{j 1} > : j \in J} be$ the family of Ɲ_ՄҪҪ – 𝑠𝑒𝑡s.

∋ Ҥ^{ȵ} \subseteq_{1} {G^{ȵ}}_{j}^{c_{2}} . So, Ҥ_{1} \subseteq G_{j 3} \forall j \in J, Ҥ_{2} \subseteq G_{j 2} \forall j \in J, Ҥ_{3} \supseteq G_{j 1} \forall j \in J .

Therefore, $Ɲ_{Մ} {ҪҪԼ}_{(1, 2)}^{1} (Ҥ^{ȵ}) = \cap_{1_{j}} {G^{ȵ}}_{j}^{c_{2}} = < \cap_{j} G_{j 3}, \cap_{j} G_{j 2}, \cup_{j} G_{j 1} > .$

$Hence,$ ( $Ɲ_{Մ} {ҪҪԼ}_{(1, 2)}^{1} (Ҥ^{ȵ}))^{c_{2}} = \cup_{2_{j}} G^{ȵ} j = < \cup_{j} G_{j 1}, \cap_{j} G_{j 2}, \cap_{j} G_{j 3} >$ …….(2)

From (1) and (2) we get [ $Ɲ_{Մ} {ҪҪԼ}_{(1, 2)}^{1} (Ҥ^{ȵ})]^{c_{2}} =$ $Ɲ_{Մ} {ҪԼȵȶ}_{(1, 2)}^{2}$ ( $Ҥ^{ȵ^{c_{2}}}$ ).

iii. $Let Ҥ^{ȵ} = < Ҥ 1, Ҥ 2, Ҥ 3 >$ and { $Ӻ^{ȵ}$ _𝑗 = <Ӻ_𝑗1, Ӻ _𝑗2, Ӻ _𝑗3>: 𝑗 ∈ 𝐽} be the family of Ɲ_ՄҪҪ – 𝑠𝑒𝑡s such that $Ҥ^{ȵ^{c_{2}}} = < Ҥ_{3}, Ҥ_{2}, Ҥ_{1} >$ ⊆₁ ${Ӻ^{ȵ}}_{j}$ . So, $Ɲ_{Մ} {ҪҪԼ}_{(1, 2)}^{1}$ ( $Ҥ^{ȵ^{c_{2}}}$ ) = $\cap_{1_{j}} Ӻ^{ȵ}$ _𝑗 = <∩Ӻ_𝑗1, ∩Ӻ_𝑗2, $\cup$ Ӻ_𝑗3>…….(1)

Now ${{Ӻ^{ȵ}}_{j}^{c_{2}} = < Ӻ_{j 3}, Ӻ_{j 2}, Ӻ_{j 1} > : j \in J} be$ the family of Ɲ_ՄҪO – 𝑠𝑒𝑡s,

∋ ${Ӻ^{ȵ}}_{j}^{c_{2}} \subseteq_{2} Ҥ^{ȵ} . So, Ӻ_{j 3} \subseteq Ҥ_{1} \forall j \in J, Ӻ_{j 2} \supseteq Ҥ_{2} \forall j \in J$ , $Ӻ_{j 1} \supseteq Ҥ_{3} \forall j \in J$ .

Therefore $Ɲ_{Մ} {ҪԼȵȶ}_{(1, 2)}^{2} (Ҥ^{ȵ}) = \cup_{2_{j}} {Ӻ^{ȵ}}_{j}^{c_{2}} = < \cup_{j} Ӻ_{j 3}, \cap_{j} Ӻ_{j 2}, \cap_{j} Ӻ_{j 1} > .$

$Hence,$ ( $Ɲ_{Մ} {ҪԼȵȶ}_{(1, 2)}^{2} (Ҥ^{ȵ}))^{c_{2}} = < \cap_{j} Ӻ_{j 1}, \cap_{j} Ӻ_{j 2}, \cup_{j} Ӻ_{j 3} >$ …….(2)

From (1) and (2) we get [ $Ɲ_{Մ} {ҪԼȵȶ}_{(1, 2)}^{2} (Ҥ^{ȵ})]^{c_{2}} =$ $Ɲ_{Մ} {ҪҪԼ}_{(1, 2)}^{1}$ ( $Ҥ^{ȵ^{c_{2}}}$ ).

iv. Proof: $Ҥ^{ȵ}$ =<Ҥ₁,Ҥ₂,Ҥ₃> and { $Ӻ^{ȵ}$ _𝑗 = <Ӻ_𝑗1, Ӻ _𝑗2, Ӻ _𝑗3>: 𝑗 ∈ 𝐽} be the family of Ɲ_ՄҪҪ – sets such that $Ҥ^{ȵ^{c_{2}}} = < Ҥ_{3}, Ҥ_{2}, Ҥ_{1} >$ ⊆₂ ${Ӻ^{ȵ}}_{j}$ . So, $Ɲ_{Մ} {ҪҪԼ}_{(1, 2)}^{2}$ ( $Ҥ^{ȵ^{c_{2}}}$ ) = $\cap_{1_{j}} Ӻ^{ȵ}$ _𝑗 = <∩Ӻ_𝑗1, ∩Ӻ_𝑗2, $\cup$ Ӻ_𝑗3>…….(1)

Now ${{Ӻ^{ȵ}}_{j}^{c_{2}} = < Ӻ_{j 3}, Ӻ_{j 2}, Ӻ_{j 1} > : j \in J} be$ the family of Ɲ_ՄҪO – 𝑠𝑒𝑡s,

∋ {Ӻ^{ȵ}}_{j}^{c_{2}} \subseteq_{1} Ҥ^{ȵ} . So, Ӻ_{j 3} \subseteq Ҥ_{1} \forall j \in J, Ӻ_{j 2} \subseteq Ҥ_{2} \forall j \in J, Ӻ_{j 1} \supseteq Ҥ_{3} \forall j \in J .

Therefore $Ɲ_{Մ} {ҪԼȵȶ}_{(1, 2)}^{1} (Ҥ^{ȵ}) = \cup_{2_{j}} {Ӻ^{ȵ}}_{j}^{c_{2}} = < \cup_{j} Ӻ_{j 3}, \cap_{j} Ӻ_{j 2}, \cap_{j} Ӻ_{j 1} > .$

$Hence,$ ( $Ɲ_{Մ} {ҪԼȵȶ}_{(1, 2)}^{1} (Ҥ^{ȵ}))^{c_{2}} {= \cap}_{1_{j}} Ӻ^{ȵ} j = < \cap_{j} Ӻ_{j 1}, \cap_{j} Ӻ_{j 2}, \cup_{j} Ӻ_{j 3} >$ …….(2)

From (1) and (2) we get [ $Ɲ_{Մ} {ҪԼȵȶ}_{(1, 2)}^{1} (Ҥ^{ȵ})]^{c_{2}} =$ $Ɲ_{Մ} {ҪҪԼ}_{(1, 2)}^{2}$ ( $Ҥ^{ȵ^{c_{2}}}$ ).

v. Let $Ҥ^{ȵ}$ = <Ҥ₁,Ҥ₂,Ҥ₃> and { $G^{ȵ}$ _𝑗 = <𝘎_𝑗1,𝘎_𝑗2,𝘎_𝑗3>: 𝑗 ∈ 𝐽} be the family of Ɲ_ՄҪO – sets such that ${G^{ȵ}}_{j}$ ⊆₁ $Ҥ^{ȵ}$ = <Ҥ₁,Ҥ₂,Ҥ₃>. So, $Ɲ_{Մ} {ҪԼȵȶ}_{(1, 2)}^{1}$ ( $Ҥ^{ȵ}$ ) = $⋃_{2_{j}} G^{ȵ}$ _𝑗 = < $\cup$ 𝘎_𝑗1, ∩𝘎_𝑗2, ∩𝘎_𝑗3>…….(1)

Now ${G^{ȵ}}_{j}^{c_{2}} = < G_{j 3}, G_{j 2}, G_{j 1} > : j \in J}$ the family of Ɲ_ՄҪҪ – 𝑠𝑒𝑡s

∋ Ҥ^{ȵ^{c_{2}}} \subseteq_{2} {G^{ȵ}}_{j}^{c_{2}} . So, Ҥ_{3} \subseteq G_{j 3} \forall j \in J, Ҥ_{2} \supseteq G_{j 2} \forall j \in J, Ҥ_{1} \supseteq G_{j 1} \forall j \in J .

Therefore, $Ɲ_{Մ} {ҪҪԼ}_{(1, 2)}^{2} (Ҥ^{ȵ^{c_{2}}}) = \cap_{1_{j}} {G^{ȵ}}_{j}^{c_{2}} = < \cap_{j} G_{j 3}, \cap_{j} G_{j 2}, \cup_{j} G_{j 1} > .$

$Hence,$ ( $Ɲ_{Մ} {ҪҪԼ}_{(1, 2)}^{2} (Ҥ^{ȵ^{c_{2}}}))^{c_{2}} = < \cup_{j} G_{j 1}, \cap_{j} G_{j 2}, \cap_{j} G_{j 3} >$ …….(2)

From (1) and (2) we get $Ɲ_{Մ} {ҪԼȵȶ}_{(1, 2)}^{1}$ ( $Ҥ^{ȵ}$ ). $=$ [ $Ɲ_{Մ} {ҪҪԼ}_{(1, 2)}^{2} (Ҥ^{ȵ^{c_{2}}})]^{c_{2}}$ .

The proof of parts (vi (to) viii) is demonstrated in a similar manner.

Remarks 5.14:

i. For instance, consider Examples 5.4 and 4.5, where $U^{ȵ}$ =< $\emptyset, {b}, {a}$ > is a Ɲ_ՄҪℜ_s.
In this case, ${[Ɲ_{Մ} {ҪҪԼ}_{(1, 2)}^{2} (U^{ȵ})]}^{c_{2}} = \emptyset_{1}^{ȵ}$ $\neq$ $Ɲ_{Մ} {ҪԼȵȶ}_{(1, 2)}^{2}$ ( $U^{ȵ^{C_{2}}}$ ) $= Ҡ^{ȵ}$ = < {𝑎}, {𝑏}, {𝑐} >, which shows that the condition of [ $Ɲ_{Մ} {ҪҪԼ}_{(1, 2)}^{2} (M^{ȵ})]^{c_{2}} =$ $Ɲ_{Մ} {ҪԼȵȶ}_{(1, 2)}^{2}$ ( $M^{ȵ^{c_{2}}})$ fails.
ii. For instance, consider Examples 4.2 and 5.2, where $Ҥ^{ȵ}$ = <{𝑎},{𝑑},{𝑐}> is a Ɲ_ՄҪℜ_s.
In this case, [ $Ɲ_{Մ} {ҪҪԼ}_{(1, 2)}^{1} (Ҥ^{ȵ})]^{c_{2}}$ does not exist, $and$ $Ɲ_{Մ} {ҪԼȵȶ}_{(1, 2)}^{1}$ ( $Ҥ^{ȵ^{c_{2}}}$ ) = $\emptyset_{1}^{ȵ}$ , which shows that the condition of [ $Ɲ_{Մ} {ҪҪԼ}_{(1, 2)}^{1} (M^{ȵ})]^{c_{2}} =$ $Ɲ_{Մ} {ҪԼȵȶ}_{(1, 2)}^{1}$ ( $M^{ȵ^{c_{2}}})$ fails.
iii. For instance, consider Examples 5.4 and 4.5, where $Ҥ^{ȵ}$ = <{𝑎},{𝑑},{𝑐}> is a Ɲ_ՄҪℜ_s.
In this case, [( $Ɲ_{Մ} {ҪԼȵȶ}_{(1, 2)}^{2} (Ҥ^{ȵ})]^{c_{2}}$ = $Ҥ^{ȵ^{c_{2}}} = < {c}, {d}, {a} > \neq$ $Ɲ_{Մ} {ҪҪԼ}_{(1, 2)}^{2}$ ( $Ҥ^{ȵ^{c_{2}}}$ ) $= M^{ȵ^{c_{2}}}$ , which shows that the condition of [ $Ɲ_{Մ} {ҪԼȵȶ}_{(1, 2)}^{2} (M^{ȵ})]^{c_{2}} =$ $Ɲ_{Մ} {ҪҪԼ}_{(1, 2)}^{2}$ ( $M^{ȵ^{c_{2}}}$ ).
iv. For instance, consider Examples 5.2 and 4.2, where $U^{ȵ}$ = < $\emptyset, {b}, {a}$ >.
In this case, [ $Ɲ_{Մ} {ҪԼȵȶ}_{(1, 2)}^{1} (U^{ȵ})]^{c_{2}}$ = $Ҳ_{1}^{ȵ} \neq$ $Ɲ_{Մ} {ҪҪԼ}_{(1, 2)}^{1}$ ( $U^{ȵ^{c_{2}}}$ ). Since $Ɲ_{Մ} {ҪҪԼ}_{(1, 2)}^{1}$ ( $U^{ȵ^{c_{2}}}$ ) does not exist, which shows that the condition of [ $Ɲ_{Մ} {ҪԼȵȶ}_{(1, 2)}^{1} (M^{ȵ})]^{c_{2}} =$ $Ɲ_{Մ} {ҪҪԼ}_{(1, 2)}^{1}$ ( $M^{ȵ^{c_{2}}}$ ) fails.
v. For instance, consider Examples 4.2 and 5.2, where $Ҥ^{ȵ}$ = <{𝑎},{𝑑},{𝑐}> is a Ɲ_ՄҪℜ_s.
In this case, [ $Ɲ_{Մ} {ҪҪԼ}_{(1, 2)}^{1} (Ҥ^{ȵ^{c_{2}}})]^{c_{2}} = Ҥ^{ȵ} \neq$ $Ɲ_{Մ} {ҪԼȵȶ}_{(1, 2)}^{1}$ ( $Ҥ^{ȵ}$ ) = $M^{ȵ}$ = <{𝑎}, $\emptyset$ ,{𝑐}>, which shows that the condition of ( $Ɲ_{Մ} {ҪԼȵȶ}_{(1, 2)}^{1}$ ( $M^{ȵ}$ ) $=$ [ $Ɲ_{Մ} {ҪҪԼ}_{(1, 2)}^{1} (M^{ȵ^{c_{2}}})]^{c_{2}}$ fails.
vi. For instance, consider Examples 5.4 and 4.5, where $U^{ȵ}$ = < $\emptyset, {b}, {a}$ > is a Ɲ_ՄҪℜ_s.
In this case, $Ɲ_{Մ} {ҪԼȵȶ}_{(1, 2)}^{2}$ ( $U^{ȵ}) does not exist, and$ ( $Ɲ_{Մ} {ҪҪԼ}_{(1, 2)}^{2} (U^{ȵ^{c_{2}}}))^{c_{2}} = Ҳ_{1}^{ȵ}$ , which shows that the condition of $Ɲ_{Մ} {ҪԼȵȶ}_{(1, 2)}^{2}$ ( $M^{ȵ}$ ) $=$ [ $Ɲ_{Մ} {ҪҪԼ}_{(1, 2)}^{2} (M^{ȵ^{c_{2}}})]^{c_{2}}$ fails.
vii. For instance, consider Examples 4.2 and 5.2, where $Ҥ^{ȵ}$ = <{𝑎},{𝑑},{𝑐}> is a Ɲ_ՄҪℜ_s.
In this case, $Ɲ_{Մ} {ҪҪԼ}_{(1, 2)}^{1}$ ( $Ҥ^{ȵ}$ ) $does not exsist, and$ [ $Ɲ_{Մ} {ҪԼȵȶ}_{(1, 2)}^{1} (Ҥ^{ȵ^{c_{2}}})]^{c_{2}} = Ҳ_{1}^{ȵ}$ , which shows that the condition of $Ɲ_{Մ} {ҪҪԼ}_{(1, 2)}^{1}$ ( $Ҥ^{ȵ}$ ) $=$ [ $Ɲ_{Մ} {ҪԼȵȶ}_{(1, 2)}^{1} (Ҥ^{ȵ^{c_{2}}})]^{c_{2}}$ fails.
viii. For instance, consider Examples 5.4 and 4.5, where $U^{ȵ}$ = < $\emptyset, {b}, {a}$ > is a Ɲ_ՄҪℜ_s.
In this case, [ $Ɲ_{Մ} {ҪԼȵȶ}_{(1, 2)}^{2} (U^{ȵ})]^{c_{2}} does not exist, and$ $Ɲ_{Մ} {ҪҪԼ}_{(1, 2)}^{2}$ ( $U^{ȵ}$ ) $= Ҳ_{1}^{ȵ}$ , which shows that the condition of $Ɲ_{Մ} {ҪҪԼ}_{(1, 2)}^{2}$ ( $M^{ȵ}$ ) $=$ [ $Ɲ_{Մ} {ҪԼȵȶ}_{(1, 2)}^{2} (M^{ȵ^{c_{2}}})]^{c_{2}}$ fails.

Corollary 5.15:

Let(Ҳ, $T$ ) be 𝑎 Ɲ_ՄҪ $T$ _(1,2)–space. Then:

i. $Ɲ_{Մ} {ҪҪԼ}_{(1, 2)}^{1}$ ( $Ҳ_{1}^{ȵ}$ ) $= Ҳ_{1}^{ȵ}$ = $< Ҳ, \emptyset, \emptyset >$
ii. $Ɲ_{Մ} {ҪԼȵȶ}_{(1, 2)}^{1}$ ( $Ҳ_{1}^{ȵ}$ ) $= Ҳ_{1}^{ȵ}$ = $< Ҳ, \emptyset, \emptyset >$
iii. $Ɲ_{Մ} {ҪҪԼ}_{(1, 2)}^{2}$ ( $Ҳ_{1}^{ȵ}$ ) $= Ҳ_{1}^{ȵ}$ = $< Ҳ, \emptyset, \emptyset >$
iv. $Ɲ_{Մ} {ҪԼȵȶ}_{(1, 2)}^{2}$ ( $Ҳ_{1}^{ȵ}$ ) $= Ҳ_{1}^{ȵ}$ = $< Ҳ, \emptyset, \emptyset >$
v. $Ɲ_{Մ} {ҪҪԼ}_{(1, 2)}^{1}$ ( $\emptyset_{1}^{ȵ}$ ) $= \emptyset_{1}^{ȵ}$ = $< \emptyset, \emptyset, Ҳ >$
vi. $Ɲ_{Մ} {ҪԼȵȶ}_{(1, 2)}^{1}$ ( $\emptyset_{1}^{ȵ}$ ) $= \emptyset_{1}^{ȵ}$ = $< \emptyset, \emptyset, Ҳ >$
vii. $Ɲ_{Մ} {ҪҪԼ}_{(1, 2)}^{2}$ ( $\emptyset_{1}^{ȵ}$ ) $= \emptyset_{1}^{ȵ}$ = $< \emptyset, \emptyset, Ҳ >$
viii. $Ɲ_{Մ} {ҪԼȵȶ}_{(1, 2)}^{2}$ ( $\emptyset_{1}^{ȵ}$ ) $= \emptyset_{1}^{ȵ}$ = $< \emptyset, \emptyset, Ҳ >$

Proof:

i. The only Ɲ_ՄҪℜ_s that satisfy $Ҳ_{1}^{ȵ}$ ⊆₁ $Ӻ^{ȵ} = < Ӻ_{1}, Ӻ_{2}, Ӻ_{3} >,$ $∋ Ӻ^{ȵ}$ are Ɲ_ՄҪҪ –𝑠𝑒𝑡s is $Ҳ_{1}^{ȵ} .$ Since, $Ҳ_{1}^{ȵ}$ is Ɲ_ՄҪҪ – 𝑠𝑒𝑡.
Hence, $Ɲ_{Մ} {ҪҪԼ}_{(1, 2)}^{1}$ ( $Ҳ_{1}^{ȵ}$ ) $= Ҳ_{1}^{ȵ}$ = $< Ҳ, \emptyset, \emptyset >$ .
ii. Let { $G^{ȵ}$ _𝑗 = <G_𝑗1, G _𝑗2, G _𝑗3>: 𝑗 ∈ 𝐽} be the family of Ɲ_ՄҪO – 𝑠𝑒𝑡s such that ${G^{ȵ}}_{j}$ ⊆₁ $Ҳ_{1}^{ȵ} .$ Since, $Ҳ_{1}^{ȵ}$ is Ɲ_ՄҪO – 𝑠𝑒𝑡. $Hence, Ɲ_{Մ} {ҪԼȵȶ}_{(1, 2)}^{1}$ ( $Ҳ_{1}^{ȵ}$ ) = $\cup_{1_{j}} G^{ȵ}$ _𝑗 = < $\cup$ G_𝑗1, ∩G_𝑗2, $\cup$ G_𝑗3> $= Ҳ_{1}^{ȵ}$ = $< Ҳ, \emptyset, \emptyset >$ .
iii. The proof is similar to part i.
iv. The proof is similar to part ii.
v. Let { $Ӻ^{ȵ}$ _𝑗 = <Ӻ_𝑗1, Ӻ _𝑗2, Ӻ _𝑗3>: 𝑗 ∈ 𝐽} be the family of Ɲ_ՄҪҪ – 𝑠𝑒𝑡s such that $\emptyset_{1}^{ȵ}$ ⊆₁ ${Ӻ^{ȵ}}_{j}$ .
So, $Ɲ_{Մ} {ҪҪԼ}_{(1, 2)}^{1}$ ( $\emptyset_{1}^{ȵ}$ ) = $\cap_{1_{j}} Ӻ^{ȵ}$ _𝑗 = <∩Ӻ_𝑗1, ∩Ӻ_𝑗2, $\cup$ Ӻ_𝑗3>. Since, $\emptyset_{1}^{ȵ}$ is Ɲ_ՄҪҪ – 𝑠𝑒𝑡. $Hence,$ $Ɲ_{Մ} {ҪҪԼ}_{(1, 2)}^{1}$ ( $\emptyset_{1}^{ȵ}$ ) = $\cap_{1_{j}} Ӻ^{ȵ}$ _𝑗 = $\emptyset_{1}^{ȵ} .$
vi. The only Ɲ_ՄҪℜ_s that satisfy $G^{ȵ} = < G_{1}, G_{2}, G_{3} >$ $\subseteq_{1} \emptyset_{1}^{ȵ}, G^{ȵ}$ are Ɲ_ՄҪO –𝑠𝑒𝑡s is $\emptyset_{1}^{ȵ} .$ Since, $\emptyset_{1}^{ȵ}$ is Ɲ_ՄҪO – 𝑠𝑒𝑡.
Hence, $Ɲ_{Մ} {ҪԼȵȶ}_{(1, 2)}^{1}$ ( $\emptyset_{1}^{ȵ}$ ) $= \emptyset_{1}^{ȵ}$ = $< \emptyset, \emptyset, Ҳ > .$
vii. Let { $Ӻ^{ȵ}$ _𝑗 = <Ӻ_𝑗1, Ӻ _𝑗2, Ӻ _𝑗3>: 𝑗 ∈ 𝐽} be the family of Ɲ_ՄҪҪ – 𝑠𝑒𝑡s such that $\emptyset_{1}^{ȵ}$ ⊆₂ ${Ӻ^{ȵ}}_{j}$ .
So, $Ɲ_{Մ} {ҪҪԼ}_{(1, 2)}^{2}$ ( $\emptyset_{1}^{ȵ}$ ) = $\cap_{1_{j}} Ӻ^{ȵ}$ _𝑗 = <∩Ӻ_𝑗1, ∩Ӻ_𝑗2, $\cup$ Ӻ_𝑗3>. Since, $\emptyset_{1}^{ȵ}$ is Ɲ_ՄҪҪ – 𝑠𝑒𝑡.
$Hence,$ $Ɲ_{Մ} {ҪҪԼ}_{(1, 2)}^{2}$ ( $\emptyset_{1}^{ȵ}$ ) = $\cap_{1_{j}} Ӻ^{ȵ}$ _𝑗 = $\emptyset_{1}^{ȵ}$ .
viii. The proof is similar to part vi.

6. Continuity _(1,2) Functions in Crisp Neutrosophic Topology (Ɲ_ՄҪ $T$ _(1,2)–space)

In this section, we investigate the concept of continuity within the constructed space (ƝՄҪT(1,2)–space). Furthermore, we examine and prove all relations, results, and theorems established in classical topology within this framework. Illustrative examples are also provided to demonstrate the relations, results, and theorems that fail to hold under the new setting.

Definition 6.1:

A function $f$ from a neutrosophic crisp topology spaces Ɲ_ՄҪ $T$ _(1,2)–space $(Ҳ, T_{Ҳ})$ into a neutrosophic crisp topology space Ɲ_ՄҪ $T$ _(1,2)–space $(Y, T_{Y})$ will be said to be a neutrosophic crisp ${continuous}_{(1, 2)}$ function if The inverse of every Ɲ_ՄҪO–𝑠𝑒𝑡 in $Y$ is a Ɲ_ՄҪO–𝑠𝑒𝑡 in Ҳ.

Example 6.2:

Let $Ҳ = {a, b, c}, T_{Ҳ} = {\emptyset_{1}^{ȵ}, Ҳ_{1}^{ȵ}, A^{ȵ}, B^{ȵ}} ∋ A^{ȵ} = < {a}, \emptyset . \emptyset >, B^{ȵ} = < {a}, \emptyset, {c} >, and Y = {s, r, t},$ $T_{Y} = {Φ_{1}^{ȵ}, Y_{1}^{ȵ}, M^{ȵ}, R^{ȵ}} ∋ M^{ȵ} = < {s}, \emptyset . \emptyset >, R^{ȵ} = < {s}, \emptyset, {t} >$ . So, $(Ҳ, T_{Ҳ}) and (Y, T_{Y}) are$ Ɲ_ՄҪ $T$ _(1,2) –spaces. Let $f : Ҳ \to Y$ be a function defined by $f (a) = s, f (b) = r, f (c) = t$ . $Then f is a neutrosophic crisp {continuous}_{(1, 2)}$ function since $f^{- 1} (Φ_{1}^{ȵ}) = \emptyset_{1}^{ȵ}, f^{- 1} (Y_{1}^{ȵ}) = Ҳ_{1}^{ȵ}$ , $f^{- 1} (M^{ȵ}) = A^{ȵ} and f^{- 1} (R^{ȵ}) = B^{ȵ}, such that \emptyset_{1}^{ȵ}, Ҳ_{1}^{ȵ}, A^{ȵ} and B^{ȵ}$ are Ɲ_ՄҪO– 𝑠𝑒𝑡s in Ҳ.

Example 6.3:

Let $Ҳ = {a, b, c}, T_{Ҳ} = {\emptyset_{1}^{ȵ}, Ҳ_{1}^{ȵ}, A^{ȵ}, B^{ȵ}} ∋ A^{ȵ} = < {a}, \emptyset . \emptyset >, B^{ȵ} = < {a}, \emptyset, {c} >, and Y = {s, r, t},$ $T_{Y} = {Φ_{1}^{ȵ}, Y_{1}^{ȵ}, M^{ȵ}, R^{ȵ}} ∋ M^{ȵ} = < {s}, \emptyset . \emptyset >, R^{ȵ} = < {s}, \emptyset, {t} >$ . So, $(Ҳ, T_{Ҳ}) and (Y, T_{V}) are$ Ɲ_ՄҪ $T$ _(1,2)–spaces. Let $f : Ҳ \to Y$ be a function defined by $f (a) = s, f (b) = t and f (c) = r . Then f is not a neutrosophic crisp {continuous}_{(1, 2)} function$ . Since, $f^{- 1} (R^{ȵ}) = < {a}, \emptyset, {b} >$ is not a Ɲ_ՄҪO– 𝑠𝑒𝑡 in Ҳ.

Corollary 6.4:

If $f : (Ҳ, T_{Ҳ}) \to (Y, T_{Y})$ is a bijective neutrosophic crisp. Then the following conditions stand in equivalence:

i. The inverse of every Ɲ_ՄҪO–𝑠𝑒𝑡 in $Y$ is a Ɲ_ՄҪO–𝑠𝑒𝑡 in Ҳ.
ii. The inverse of every Ɲ_ՄҪҪ –𝑠𝑒𝑡 in $Y$ is a Ɲ_ՄҪҪ–𝑠𝑒𝑡 in Ҳ.
iii. $f$ ( $Ɲ_{Մ} {ҪҪԼ}_{(1, 2)}^{1} (M^{ȵ})) \subseteq_{1} [$ $Ɲ_{Մ} {ҪҪԼ}_{(1, 2)}^{1} f (M^{ȵ})], \forall$ $M^{ȵ}$ ⊆₁ $Ҳ_{1}^{ȵ} .$

Proof:

(i→ii) Suppose $Ɉ^{ȵ}$ is Ɲ_ՄҪҪ–𝑠𝑒𝑡 in 𝚈 $. So, {(Ɉ^{ȵ})}^{c_{2}}$ is a Ɲ_ՄҪO–𝑠𝑒𝑡 in 𝚈. So, $f^{- 1} ({(Ɉ^{ȵ})}^{c_{2}})$ is a Ɲ_ՄҪO–𝑠𝑒𝑡 in Ҳ

By Corollary 2.8, $f^{- 1} ((Ɉ^{ȵ})^{c_{2}}) = {(f^{- 1} (Ɉ^{ȵ}))}^{c_{2}}$ . Therefore, $(f^{- 1} {(Ɉ^{ȵ})}^{c_{2}}$ is a Ɲ_ՄҪO–𝑠𝑒𝑡 in Ҳ. $Hence, f^{- 1} (Ɉ^{ȵ})$ is a Ɲ_ՄҪҪ–𝑠𝑒𝑡 in Ҳ.

Conversely, Suppose $ꭆ^{ȵ}$ is a Ɲ_ՄҪO–𝑠𝑒𝑡 in 𝚈 $. So, {(ꭆ^{ȵ})}^{c_{2}}$ is a Ɲ_ՄҪO–𝑠𝑒𝑡 in $Y$ . So $, f^{- 1} ({(ꭆ^{ȵ})}^{c_{2}})$ is a Ɲ_ՄҪO–𝑠𝑒𝑡 in Ҳ

By Corollary 2.8 $, f^{- 1} ((ꭆ^{ȵ})^{c_{2}}) = {(f^{- 1} (ꭆ^{ȵ}))}^{c_{2}}$ . Therefore, ${(f^{- 1} (ꭆ^{ȵ}))}^{c_{2}}$ is a Ɲ_ՄҪҪ–𝑠𝑒𝑡 in Ҳ. $Hence, f^{- 1} (ꭆ^{ȵ})$ is a Ɲ_ՄҪO–𝑠𝑒𝑡 in Ҳ.

(ii $\to$ iii):

$Since, M^{ȵ} \subseteq_{1} [$ $Ɲ_{Մ} {ҪҪԼ}_{(1, 2)}^{1} (M^{ȵ})], \forall$ $M^{ȵ}$ ⊆₁ $Ҳ_{1}^{ȵ}$ . By Corollary 4.6.(i). Therefore, $M^{ȵ} \subseteq_{1} f$ ⁻¹( $Ɲ_{Մ} {ҪҪԼ}_{(1, 2)}^{1} f (M^{ȵ}))$ By Corollary 2.6 (i) and Corollary 4.6 (i). Since, $Ɲ_{Մ} {ҪҪԼ}_{(1, 2)}^{1} f (M^{ȵ})$ is a Ɲ_ՄҪҪ–𝑠𝑒𝑡 in 𝚈 $.$ $So, f$ ⁻¹( $Ɲ_{Մ} {ҪҪԼ}_{(1, 2)}^{1} f (M^{ȵ}))$ is a Ɲ_ՄҪҪ–𝑠𝑒𝑡 in Ҳ. Therefore, $Ɲ_{Մ} {ҪҪԼ}_{(1, 2)}^{1} M^{ȵ}) \subseteq_{1} f$ ¹( $Ɲ_{Մ} {ҪҪԼ}_{(1, 2)}^{1} f (M^{ȵ})) .$

$So, f$ ( $Ɲ_{Մ} {ҪҪԼ}_{(1, 2)}^{1} (M^{ȵ}) \subseteq_{1} f [f$ ⁻¹( $Ɲ_{Մ} {ҪҪԼ}_{(1, 2)}^{1} f (M^{ȵ}))]$ . $Hence, f$ ( $Ɲ_{Մ} {ҪҪԼ}_{(1, 2)}^{1} (M^{ȵ})) \subseteq_{1}$ $[$ $Ɲ_{Մ} {ҪҪԼ}_{(1, 2)}^{1} f (M^{ȵ})]$ .

Conversely, Suppose $Ɉ^{ȵ} is a ƝՄҪҪ - set$ in 𝚈.

So, $f [$ ( $Ɲ_{Մ} {ҪҪԼ}_{(1, 2)}^{1} f^{- 1} (Ɉ^{ȵ}))] \subseteq_{1}$ $Ɲ_{Մ} {ҪҪԼ}_{(1, 2)}^{1} f [f^{- 1} (Ɉ^{ȵ})] \subseteq_{1} [$ $Ɲ_{Մ} {ҪҪԼ}_{(1, 2)}^{1} Ɉ^{ȵ}$ $]$ = $Ɉ^{ȵ}$ . By Theorem 4.8. Therefore, $Ɲ_{Մ} {ҪҪԼ}_{(1, 2)}^{1} f^{- 1} (Ɉ^{ȵ}) \subseteq_{1} f^{- 1} (Ɉ^{ȵ}) . So, Ɲ_{Մ} {ҪҪԼ}_{(1, 2)}^{1} f^{- 1} (Ɉ^{ȵ}) = f^{- 1} (Ɉ^{ȵ})$ . $Hence, f^{- 1} (Ɉ^{ȵ}) is a ƝՄҪҪ - set$ in Ҳ.

Corollary 6.5:

Let $f : (Ҳ, T_{Ҳ}) \to (Y, T_{Y})$ be bijective neutrosophic crisp function if The inverse of every $ƝՄҪҪ - set$ in 𝚈 is $a ƝՄҪҪ - set$ in Ҳ, then $f$ ( $Ɲ_{Մ} {ҪҪԼ}_{(1, 2)}^{2} (Ҥ^{ȵ})) \subseteq_{2}$ $[$ $Ɲ_{Մ} {ҪҪԼ}_{(1, 2)}^{2} f (Ҥ^{ȵ})], \forall$ $Ҥ^{ȵ}$ is $a ƝՄҪҪ - set$ in Ҳ.

Proof:

$Since, Ҥ^{ȵ} \subseteq_{2}$ $Ɲ_{Մ} {ҪҪԼ}_{(1, 2)}^{2} (Ҥ^{ȵ})$ By Theorem 4.10. Thus, $Ҥ^{ȵ} \subseteq_{2} f$ ⁻¹( $Ɲ_{Մ} {ҪҪԼ}_{(1, 2)}^{2} f (Ҥ^{ȵ}))$ By Corollary 2.6 (i) and Corollary 4.6 (i). Since, $Ɲ_{Մ} {ҪҪԼ}_{(1, 2)}^{2} f (Ҥ^{ȵ})$ $is a ƝՄҪҪ - set$ in 𝚈 $. So, f$ ⁻¹( $Ɲ_{Մ} {ҪҪԼ}_{(1, 2)}^{2} f (Ҥ^{ȵ}))$ $is a ƝՄҪҪ - set$ in Ҳ. Therefore, $Ɲ_{Մ} {ҪҪԼ}_{(1, 2)}^{2} Ҥ^{ȵ}) \subseteq_{2} f$ ⁻¹( $Ɲ_{Մ} {ҪҪԼ}_{(1, 2)}^{2} f (Ҥ^{ȵ})) . So, f$ ( $Ɲ_{Մ} {ҪҪԼ}_{(1, 2)}^{2} (Ҥ^{ȵ}) \subseteq_{2} f [f$ ⁻¹( $Ɲ_{Մ} {ҪҪԼ}_{(1, 2)}^{2} f (Ҥ^{ȵ}))]$ .

$Hence, f$ ( $Ɲ_{Մ} {ҪҪԼ}_{(1, 2)}^{2} (Ҥ^{ȵ})) \subseteq_{2}$ $[$ $Ɲ_{Մ} {ҪҪԼ}_{(1, 2)}^{2} f (Ҥ^{ȵ})]$ .

Remark 6.6:

For instance, consider example Let $Ҳ = {a, b, c}, T_{Ҳ} = {\emptyset_{1}^{ȵ}, Ҳ_{1}^{ȵ}, A^{ȵ}, B^{ȵ}} ∋ A^{ȵ} = < {a}, \emptyset . \emptyset >, B^{ȵ} = < {a}, \emptyset, {c} >, and Y = {s, r, t}, T_{Y} = {Φ_{1}^{ȵ}, Y_{1}^{ȵ}, M^{ȵ}, R^{ȵ}} ∋ M^{ȵ} = < {s}, \emptyset . \emptyset >, R^{ȵ} = < {s}, \emptyset, {t} >$ . So $(Ҳ, T_{Ҳ}) and (Y, T_{Y}) are$ Ɲ_ՄҪ $T$ _(1,2)–spaces. Let $f : Ҳ \to Y be a function defined by$

f (a) = t, f (b) = r, f (c) = s .

In this case, $f$ ( $Ɲ_{Մ} {ҪҪԼ}_{(1, 2)}^{2} (Ҥ^{ȵ})) \subseteq_{2}$ $[$ $Ɲ_{Մ} {ҪҪԼ}_{(1, 2)}^{2} f (Ҥ^{ȵ})], \forall$ $Ҥ^{ȵ}$ is a $ƝՄҪҪ - set$ in Ҳ, but ${M^{ȵ}}^{c_{2}} = < \emptyset, \emptyset, {s} > is a ƝՄҪҪ - set$ in 𝚈, so $f^{- 1} ({M^{ȵ}}^{c_{2}}) = < \emptyset, \emptyset, {c} >$ is not $a ƝՄҪҪ - set$ in Ҳ. Hence, it becomes clear that the converse of Corollary 6.5 is not valid.

Theorem 6.7:

The following statements are mutually equivalent:

i. $f : Ҳ \to Y is a neutrosophic crisp {continuous}_{(1, 2)}$ function.
ii. $f^{- 1}$ ( $Ɲ_{Մ} {ҪԼȵȶ}_{(1, 2)}^{2}$ ( $ꭆ^{ȵ}$ ) $) \subseteq_{2}$ $Ɲ_{Մ} {ҪԼȵȶ}_{(1, 2)}^{2}$ ( $f^{- 1} (ꭆ^{ȵ}$ )), $\forall ꭆ^{ȵ} is$ a Ɲ_ՄҪO–𝑠𝑒𝑡 in 𝚈.
iii. $Ɲ_{Մ} {ҪҪԼ}_{(1, 2)}^{1} (f^{- 1} (Ɉ^{ȵ})) \subseteq_{1} f^{- 1}$ ( $Ɲ_{Մ} {ҪҪԼ}_{(1, 2)}^{1} (Ɉ^{ȵ})), \forall Ɉ^{ȵ} is$ a $ƝՄҪҪ - set$ in 𝚈.

Proof:

(i $\to$ ii)

Suppose $ꭆ^{ȵ} is a ƝՄҪO - set$ in 𝚈. Since $Ɲ_{Մ} {ҪԼȵȶ}_{(1, 2)}^{2}$ ( $ꭆ^{ȵ}$ ) is a Ɲ_ՄҪO–𝑠𝑒𝑡 in 𝚈 by Definition 5.3, because of $f is a neutrosophic crisp {continuous}_{(1, 2)} . So, f^{- 1} ($ $Ɲ_{Մ} {ҪԼȵȶ}_{(1, 2)}^{2}$ ( $ꭆ^{ȵ}$ ) $)$ is Ɲ_ՄҪO– 𝑠𝑒𝑡 in Ҳ.

Since, $f^{- 1} ($ $Ɲ_{Մ} {ҪԼȵȶ}_{(1, 2)}^{2}$ ( $ꭆ^{ȵ}$ ) $) \subseteq_{2} f^{- 1} (ꭆ^{ȵ})$ by Theorom 5.6 and Corollary 2.5(ii).

$Hence, f^{- 1} ($ $Ɲ_{Մ} {ҪԼȵȶ}_{(1, 2)}^{2}$ ( $ꭆ^{ȵ}$ ) $) \subseteq_{2}$ $Ɲ_{Մ} {ҪԼȵȶ}_{(1, 2)}^{2}$ ( $f^{- 1} (ꭆ^{ȵ}$ )) by Theorem 5.10.

Conversely, let $ꭆ^{ȵ}$ be $a ƝՄҪO - set$ in 𝚈. Then $ꭆ^{ȵ} =$ $Ɲ_{Մ} {ҪԼȵȶ}_{(1, 2)}^{2}$ ( $ꭆ^{ȵ}$ ) $by Theorom 5.6$ .

$So, f^{- 1} (ꭆ^{ȵ}$ ) = $f^{- 1}$ $(Ɲ_{Մ} {ҪԼȵȶ}_{(1, 2)}^{2}$ ( $ꭆ^{ȵ}$ ) $) \subseteq_{2}$ $Ɲ_{Մ} {ҪԼȵȶ}_{(1, 2)}^{2}$ ( $f^{- 1} (ꭆ^{ȵ}$ )) $\subseteq_{2} f^{- 1} (ꭆ^{ȵ})$

$Thus, f^{- 1} (ꭆ^{ȵ})$ = $Ɲ_{Մ} {ҪԼȵȶ}_{(1, 2)}^{2} f^{- 1}$ ( $ꭆ^{ȵ}$ ). $Therefore f^{- 1} (ꭆ^{ȵ})$ is a Ɲ_ՄҪO–𝑠𝑒𝑡 in Ҳ. Hence, $f$ is a neutrosophic crisp continuous_(1,2) function.

(ii $\to$ iii)

Let $Ɉ^{ȵ}$ be a $ƝՄҪҪ - set$ in 𝚈.

Since, $Ɲ_{Մ} {ҪҪԼ}_{(1, 2)}^{1}$ ( $Ɉ^{ȵ}$ ) $=$ [ $Ɲ_{Մ} {ҪԼȵȶ}_{(1, 2)}^{2} (Ɉ^{ȵ^{c_{2}}})]^{c_{2}}$ by Theorem 5.13vii.

$So,$ $Ɲ_{Մ} {ҪҪԼ}_{(1, 2)}^{1} (f^{- 1}$ ( $Ɉ^{ȵ}$ ) $) =$ [ $(Ɲ_{Մ} {ҪԼȵȶ}_{(1, 2)}^{2} ({(f^{- 1} (Ɉ^{ȵ}))}^{c_{2}}]^{c_{2}}$ = [( $Ɲ_{Մ} {ҪԼȵȶ}_{(1, 2)}^{2}$ $(f^{- 1} {(Ɉ^{ȵ})}^{c_{2}}))]^{c_{2}}$ by Corollary 2.8.

$Therefore, Ɲ_{Մ} {ҪҪԼ}_{(1, 2)}^{1} (f^{- 1}$ ( $Ɉ^{ȵ}$ ) $) = {[(Ɲ_{Մ} {ҪԼȵȶ}_{(1, 2)}^{2} (f^{- 1} {(Ɉ^{ȵ})}^{c_{2}}))]}^{c_{2}} \subseteq_{1} [f^{- 1}$ ( $Ɲ_{Մ} {ҪԼȵȶ}_{(1, 2)}^{2} ({(Ɉ^{ȵ})}^{c_{2}}))]^{c_{2}}$ by using part (ii) and Theorem 2.2.iv.

Hence, $Ɲ_{Մ} {ҪҪԼ}_{(1, 2)}^{1} (f^{- 1}$ ( $Ɉ^{ȵ})) \subseteq_{1} f^{- 1}$ [ $Ɲ_{Մ} {ҪԼȵȶ}_{(1, 2)}^{2} ({(Ɉ^{ȵ})}^{c_{2}})]^{c_{2}}$ = $f^{- 1}$ ( $Ɲ_{Մ} {ҪҪԼ}_{(1, 2)}^{1} (Ɉ^{ȵ}))$ by Theorem 5.13.vii.

Conversely, let $ꭆ^{ȵ}$ be a a Ɲ_ՄҪO–𝑠𝑒𝑡 in𝚈. Since, $Ɲ_{Մ} {ҪԼȵȶ}_{(1, 2)}^{2}$ ( $ꭆ^{ȵ}$ ). $=$ ( $Ɲ_{Մ} {ҪҪԼ}_{(1, 2)}^{1} (ꭆ^{ȵ^{c_{2}}}))^{c_{2}}$ by Theorem 5.13.

$So, f^{- 1}$ $(Ɲ_{Մ} {ҪԼȵȶ}_{(1, 2)}^{2}$ ( $ꭆ^{ȵ}))$ = $f^{- 1}$ $(Ɲ_{Մ} {ҪҪԼ}_{(1, 2)}^{1} (ꭆ^{ȵ^{c_{2}}}))^{c_{2}}$ = $[f^{- 1} (Ɲ_{Մ} {ҪҪԼ}_{(1, 2)}^{1} (ꭆ^{ȵ^{c_{2}}})]^{c_{2}}$

$\subseteq_{2}$ $[($ $Ɲ_{Մ} {ҪҪԼ}_{(1, 2)}^{1} {(f^{- 1} (Ҥ^{ȵ^{c_{2}}})]}^{c_{2}}$ = $Ɲ_{Մ} {ҪԼȵȶ}_{(1, 2)}^{2} (f^{- 1} (ꭆ^{ȵ})$ .

$Therfore, f^{- 1} ($ $Ɲ_{Մ} {ҪԼȵȶ}_{(1, 2)}^{2}$ ( $ꭆ^{ȵ}$ ) $) \subseteq_{2}$ $Ɲ_{Մ} {ҪԼȵȶ}_{(1, 2)}^{2}$ ( $f^{- 1} (Ҥ^{ȵ}$ ))

Corollary 6.8:

i. if $f : Ҳ \to Y is a neutrosophic crisp {continuous}_{(1, 2)}$ function. Then $f^{- 1} ($ $Ɲ_{Մ} {ҪԼȵȶ}_{(1, 2)}^{1}$ ( $ꭆ^{ȵ}$ ) $) \subseteq_{1}$ $Ɲ_{Մ} {ҪԼȵȶ}_{(1, 2)}^{1}$ ( $f^{- 1} (ꭆ^{ȵ}$ )), $\forall ꭆ^{ȵ} is$ a Ɲ_ՄҪO–𝑠𝑒𝑡 in 𝚈.
ii. if $f^{- 1} (Ɲ_{Մ} {ҪԼȵȶ}_{(1, 2)}^{1}$ ( $ꭆ^{ȵ}$ ) $) \subseteq_{1}$ $Ɲ_{Մ} {ҪԼȵȶ}_{(1, 2)}^{1}$ ( $f^{- 1} (ꭆ^{ȵ}$ )), $\forall ꭆ^{ȵ} is$ a Ɲ_ՄҪO–𝑠𝑒𝑡 in 𝚈. Then $Ɲ_{Մ} {ҪҪԼ}_{(1, 2)}^{2} (f^{- 1} (Ɉ^{ȵ})) \subseteq_{2} f^{- 1}$ ( $Ɲ_{Մ} {ҪҪԼ}_{(1, 2)}^{2} (Ɉ^{ȵ})), and Ɉ^{ȵ} is$ a $ƝՄҪҪ - set$ in 𝚈.

Proof:

i. Suppose $ꭆ^{ȵ} is a ƝՄҪO - set$ in 𝚈. Since, $Ɲ_{Մ} {ҪԼȵȶ}_{(1, 2)}^{1}$ ( $ꭆ^{ȵ}$ ) is a Ɲ_ՄҪO–𝑠𝑒𝑡 in 𝚈 by Definition 5.1, because of $f is a neutrosophic crisp {continuous}_{(1, 2)} function . So, f^{- 1} (Ɲ_{Մ} {ҪԼȵȶ}_{(1, 2)}^{1})$ $(ꭆ^{ȵ}$ ) $)$ is Ɲ_ՄҪO– 𝑠𝑒𝑡 in Ҳ.
Since, $f^{- 1} ($ $Ɲ_{Մ} {ҪԼȵȶ}_{(1, 2)}^{1}$ ( $ꭆ^{ȵ}$ ) $) \subseteq_{1} f^{- 1} (ꭆ^{ȵ})$ by Theorom 5.8 and Corollary 2.5.
$Hence, f^{- 1} ($ $Ɲ_{Մ} {ҪԼȵȶ}_{(1, 2)}^{1}$ ( $ꭆ^{ȵ}$ ) $) \subseteq_{1}$ $Ɲ_{Մ} {ҪԼȵȶ}_{(1, 2)}^{1}$ ( $f^{- 1} (ꭆ^{ȵ}$ )) by Theorem 5.10.
ii. Let $Ɉ^{ȵ}$ be a $ƝՄҪҪ - set$ in 𝚈.
Since, $Ɲ_{Մ} {ҪҪԼ}_{(1, 2)}^{2}$ ( $Ɉ^{ȵ}$ ) $=$ [ $Ɲ_{Մ} {ҪԼȵȶ}_{(1, 2)}^{1} (Ɉ^{ȵ^{c_{2}}})]^{c_{2}}$ by Theorem 5.13.viii.
$So,$ $Ɲ_{Մ} {ҪҪԼ}_{(1, 2)}^{2} (f^{- 1}$ ( $Ɉ^{ȵ}$ ) $) =$ [ $Ɲ_{Մ} {ҪԼȵȶ}_{(1, 2)}^{1} {({(f^{- 1} (Ɉ^{ȵ}))}^{c_{2}}]}^{c_{2}}$ = [( $Ɲ_{Մ} {ҪԼȵȶ}_{(1, 2)}^{1}$ $(f^{- 1} {(Ɉ^{ȵ})}^{c_{2}}))]^{c_{2}}$ by Corollary 2.8.
$Thus, Ɲ_{Մ} {ҪҪԼ}_{(1, 2)}^{2} (f^{- 1}$ ( $Ɉ^{ȵ}$ ) $) = {[(Ɲ_{Մ} {ҪԼȵȶ}_{(1, 2)}^{1} (f^{- 1} {(Ɉ^{ȵ})}^{c_{2}}))]}^{c_{2}} \subseteq_{2} [f^{- 1}$ ( $Ɲ_{Մ} {ҪԼȵȶ}_{(1, 2)}^{1} ({(Ɉ^{ȵ})}^{c_{2}}))]^{c_{2}}$ by using part (i) and Theorem 2.2.iii.
Hence, $Ɲ_{Մ} {ҪҪԼ}_{(1, 2)}^{2} (f^{- 1}$ ( $Ɉ^{ȵ})) \subseteq_{2} f^{- 1}$ [ $Ɲ_{Մ} {ҪԼȵȶ}_{(1, 2)}^{1} ({(Ɉ^{ȵ})}^{c_{2}})]^{c_{2}}$ = $f^{- 1}$ ( $Ɲ_{Մ} {ҪҪԼ}_{(1, 2)}^{2} (Ɉ^{ȵ}))$ by Theorem 5.13.viii.

7. Conclusions

We succeeded in constructing a neutrosophic crisp topological space in which the intersection is fixed as Type I, the union as Type II, and the complement as Type II for all kinds of neutrosophic crisp families while considering the two kinds of inclusion. Within this framework, we defined two kinds of closure and interior for a neutrosophic crisp set, as well as continuity. We also established the relations, results, and theorems that are valid in general topology, while providing examples of those that are invalid, many of which we confirmed to be invalid.

In light of this, future work will study other general topological concepts—such as connectedness, compactness, and the separation axioms—on the neutrosophic crisp topological space that we constructed.

Ethical considerations

This study is entirely theoretical and does not involve any experiments on human participants or animals. Consequently, no ethical approval was required. All research procedures adhere to academic integrity standards, and all sources, theories, and previous works used in this study are properly cited and referenced.

Data availability

All data and information are included within the manuscript.

References

1. Salama AA, Smarandache F, Kroumov V: Neutrosophic crisp sets & neutrosophic crisp topological spaces. Infinite Study. 2014. Reference Source
2. Salama AA, Smarandache F, Kroumov V: Neutrosophic closed set and neutrosophic continuous functions. Neutrosophic Sets and Systems. 2014; 4: 2–8. Reference Source
3. Qahtan GA, Jabar LAA, Rasheed IM, et al.: On ψ NC-Operator in Neutrosophic Crisp Topological Spaces. International Journal of Neutrosophic Science (IJNS). 2025; 25(3). Publisher Full Text
4. Ali RD, Jabar LAA, Qahtan GA, et al.: Kernel Neutrosophic Crisp Sets. International Journal of Neutrosophic Science (IJNS). 2025; 25(2). Publisher Full Text
5. Zhang X, Li M, Lei T: On neutrosophic crisp sets and neutrosophic crisp mathematical morphology. Neutrosophic Sets and Systems. 2021; 43(1): 1. Reference Source
6. Hur K, Lim PK, Lee JG, et al.: The category of neutrosophic crisp sets. Annals of Fuzzy Mathematics and Informatics. 2017; 14(1):43–54. Publisher Full Text

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Version 1

VERSION 1 PUBLISHED 09 Feb 2026

Author details Author details

¹ University of Kerbala, Karbala, Karbala Governorate, Iraq
² University of Kerbala, Karbala, Karbala Governorate, Iraq

Hossam AL. Salman
Roles: Conceptualization, Methodology, Writing – Original Draft Preparation

Reyadh D. Ali
Roles: Supervision, Validation, Writing – Review & Editing

Competing interests

No competing interests were disclosed.

Grant information

The author(s) declared that no grants were involved in supporting this work.

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https://doi.org/10.12688/f1000research.173335.1

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AL. Salman H and D. Ali R. Special Properties of Operators in Neutrosophic Crisp Topological Spaces [version 1; peer review: awaiting peer review]. F1000Research 2026, 15:216 (https://doi.org/10.12688/f1000research.173335.1)

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[1] 1. Salama AA, Smarandache F, Kroumov V: Neutrosophic crisp sets & neutrosophic crisp topological spaces. Infinite Study. 2014. Reference Source

[2] 2. Salama AA, Smarandache F, Kroumov V: Neutrosophic closed set and neutrosophic continuous functions. Neutrosophic Sets and Systems. 2014; 4: 2–8. Reference Source

[3] 3. Qahtan GA, Jabar LAA, Rasheed IM, et al.: On ψ NC-Operator in Neutrosophic Crisp Topological Spaces. International Journal of Neutrosophic Science (IJNS). 2025; 25(3). Publisher Full Text

[4] 4. Ali RD, Jabar LAA, Qahtan GA, et al.: Kernel Neutrosophic Crisp Sets. International Journal of Neutrosophic Science (IJNS). 2025; 25(2). Publisher Full Text

[5] 5. Zhang X, Li M, Lei T: On neutrosophic crisp sets and neutrosophic crisp mathematical morphology. Neutrosophic Sets and Systems. 2021; 43(1): 1. Reference Source

[6] 6. Hur K, Lim PK, Lee JG, et al.: The category of neutrosophic crisp sets. Annals of Fuzzy Mathematics and Informatics. 2017; 14(1):43–54. Publisher Full Text

Special Properties of Operators in Neutrosophic Crisp Topological Spaces

Abstract

Keywords

1. Introduction

2. Methods

2.1 Preliminaries

Theorem 2.1:

Theorem 2.2:

Definition 2.3:

Corollary 2.4:

Corollary 2.5:

Corollary 2.6:

Corollary 2.7:

Corollary 2.8:

Corollary 2.9:

3. Neutrosophic crisp topological space ( ƝՄҪ T (1,2) -space)

Definition 3.1:

Example 3.2:

Example 3.3:

Corollary 3.4:

Proof:

4. Neutrosophic crisp closure sets (ƝՄҪҪԼ(1,2)1,ƝՄҪҪԼ(1,2)2)

Definition 4.1:

Example 4.2:

Remark 4.3:

Definition 4.4:

Example 4.5:

Corollary 4.6:

Remark 4.7:

Theorem 4.8:

Proof:

Remark 4.9:

Theorem 4.10:

Proof:

Remark 4.11:

Theorem 4.12:

Proof:

Remark 4.13:

Corollary 4.14:

Proof:

5. Neutrosophic crisp interior sets (ƝՄҪԼȵȶ(1,2)1,ƝՄҪԼȵȶ(1,2)2)

Definition 5.1:

Example 5.2:

Definition 5.3:

Example 5.4:

Remark 5.5:

Theorem 5.6:

Proof:

Remark 5.7:

Theorem 5.8:

Proof:

Remark 5.9:

Theorem 5.10:

Proof:

Remark 5.11:

Corollary 5.12:

Proof:

Theorem 5.13:

Proof:

Remarks 5.14:

Corollary 5.15:

Proof:

6. Continuity (1,2) Functions in Crisp Neutrosophic Topology (ƝՄҪ T (1,2)–space)

Definition 6.1:

Example 6.2:

Example 6.3:

Corollary 6.4:

Proof:

Corollary 6.5:

Proof:

Remark 6.6:

Theorem 6.7:

Proof:

Corollary 6.8:

Proof:

7. Conclusions

Ethical considerations

Data availability

References

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3. Neutrosophic crisp topological space ( Ɲ_ՄҪ $T$ _(1,2) -space)

4. Neutrosophic crisp closure sets $(Ɲ_{Մ} {ҪҪԼ}_{(1, 2)}^{1}, Ɲ_{Մ} {ҪҪԼ}_{(1, 2)}^{2})$

5. Neutrosophic crisp interior sets $(Ɲ_{Մ} {ҪԼȵȶ}_{(1, 2)}^{1}, Ɲ_{Մ} {ҪԼȵȶ}_{(1, 2)}^{2})$

6. Continuity _(1,2) Functions in Crisp Neutrosophic Topology (Ɲ_ՄҪ $T$ _(1,2)–space)