Special Properties of Operators in Neutrosophic Crisp Topological Spaces

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Salman" }, { "@type": "Person", "name": "Reyadh D. Ali" } ], "publisher": { "@type": "Organization", "name": "F1000Research", "logo": { "@type": "ImageObject", "url": "https://f1000research.com/img/AMP/F1000Research_image.png", "height": 480, "width": 60 } }, "image": { "@type": "ImageObject", "url": "https://f1000research.com/img/AMP/F1000Research_image.png", "height": 1200, "width": 150 }, "description": "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." } { "@context": "http://schema.org", "@type": "BreadcrumbList", "itemListElement": [ { "@type": "ListItem", "position": "1", "item": { "@id": "https://f1000research.com/", "name": "Home" } }, { "@type": "ListItem", "position": "2", "item": { "@id": "https://f1000research.com/browse/articles", "name": "Browse" } }, { "@type": "ListItem", "position": "3", "item": { "@id": "https://f1000research.com/articles/15-216/v1", "name": "Special Properties of Operators in Neutrosophic Crisp Topological..." } } ] } Home Browse Special Properties of Operators in Neutrosophic Crisp Topological... ALL Metrics - Views Downloads Get PDF Get XML Cite How to cite this article AL. Salman H and D. Ali R. Special Properties of Operators in Neutrosophic Crisp Topological Spaces [version 1; peer review: 2 approved] . F1000Research 2026, 15 :216 ( https://doi.org/10.12688/f1000research.173335.1 ) NOTE: If applicable, it is important to ensure the information in square brackets after the title is included in all citations of this article. Close Copy Citation Details Export Export Citation Sciwheel EndNote Ref. Manager Bibtex ProCite Sente EXPORT Select a format first Track Share ▬ ✚ Research Article Special Properties of Operators in Neutrosophic Crisp Topological Spaces [version 1; peer review: 2 approved] Hossam AL. Salman https://orcid.org/0009-0002-2611-8624 1 , Reyadh D. Ali https://orcid.org/0000-0001-8464-5911 1 Hossam AL. Salman https://orcid.org/0009-0002-2611-8624 1 , Reyadh D. Ali https://orcid.org/0000-0001-8464-5911 1 PUBLISHED 09 Feb 2026 Author details Author details 1 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 DETAILS REVIEWER STATUS 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. READ ALL READ LESS Keywords Neutrosophic Crisp sets, Neutrosophic Crisp Topological Spaces, Neutrosophic Crisp Closure, Neutrosophic Crisp interior, Neutrosophic crisp continuous Corresponding Author(s) Hossam AL. Salman ( [email protected] ) Close 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: 2 approved] . 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 Մ Ҫ T ), and they presented three types of these neutrosophic crisp sets as follows: Let S ȵ = be a neutrosophic crisp set Type I: S 1 ∩ S 2 = ∅ , S 1 ∩ S 3 = ∅ , S 2 ∩ S 3 = ∅ . Type II: S 1 ∩ S 2 = ∅ , S 1 ∩ S 3 = ∅ , S 2 ∩ S 3 = ∅ , and S 1 ∪ S 2 ∪ S 3 = Ҳ Type III: S 1 ∩ S 2 ∩ S 3 = ∅ , and S 1 ∪ S 2 ∪ S 3 = Ҳ Two kinds of the union operation and two kinds of the intersection operation were also defined Let S ȵ = and M ȵ = be two neutrosophic crisp sets in Ҳ . Type I: S ȵ ∪ 1 M ȵ = Type II: S ȵ ∪ 2 M ȵ = Type I: S ȵ ∩ 1 M ȵ = Type II: S ȵ ∩ 2 M ȵ = In addition, Four kinds of the empty set and the universal set were also defined, namely, ∅ 1 ȵ = , ∅ 2 ȵ = , ∅ 3 ȵ = , ∅ 4 ȵ = Ҳ 1 ȵ = , Ҳ 2 ȵ = , Ҳ 3 ȵ = , Ҳ 4 ȵ = Moreover, they defined three kinds of complements, two kinds of inclusion, and equality, as follows. Let M ȵ = be a neutrosophic crisp set in Ҳ , then complement of M ȵ is divided into three: Type I : ( M ȵ ) c 1 = Type II: ( M ȵ ) c 2 = Type III : ( M ȵ ) c 3 = For any two a neutrosophic crisp sets S ȵ = and M ȵ = in Ҳ , two subset relations are defined as: Type I: S ȵ ⊆ 1 M ȵ ⟺ S 1 ⊆ M 1 , S 2 ⊆ M 2 , M 3 ⊆ S 3 Type II : S ȵ ⊆ 1 M ȵ ⟺ S 1 ⊆ M 1 , M 2 ⊆ S 2 , M 3 ⊆ S 3 From the definition of set inclusion, two distinct forms of equality are obtained: Type I : S ȵ = 1 M ȵ ⟺ S ȵ ⊆ 1 M ȵ and M ȵ ⊆ 1 S ȵ Type II : S = 2 M ȵ ⟺ S ȵ ⊆ 2 M ȵ and M ȵ ⊆ 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 Մ Ҫℜ ʂ . 2. Methods 2.1 Preliminaries Theorem 2.1: 1 Let { B j ȵ : j ∈ J } be a family of neutrosophic crisp subsets of a universe in Ҳ. Then 1. The intersection ∩ j ∈ J B j ȵ can be defined in two ways: i. ∩ 1 B j ȵ = ii. ∩ 2 B j ȵ = 2. The union ∪ j ∈ J B j ȵ can be defined in two ways: i. ∪ 1 B j ȵ = ii. ∪ 2 B j ȵ = Theorem 2.2: 5 Let S ȵ and M ȵ be two neutrosophic crisp sets of any type in Ҳ . Then: i. ( S ȵ ∩ 1 M ȵ ) c 2 = ( S ȵ ) c 2 ∪ 2 ( M ȵ ) c 2 ii. ( S ȵ ∪ 2 M ȵ ) c 2 = ( S ȵ ) c 2 ∩ 1 ( M ȵ ) c 2 iii. ( S ȵ ⊆ 1 M ȵ ) c 2 = ( M ȵ ) c 2 ⊆ 2 ( S ȵ ) c 2 iv. ( S ȵ ⊆ 2 M ȵ ) c 2 = ( M ȵ ) c 2 ⊆ 1 ( S ȵ ) c 2 Definition 2.3: 6 1. Let (Ҳ, T Ҳ ) and ( Y , T Y ) be two Ɲ Մ Ҫ T -spaces. If ꭆ ȵ = is Ɲ Մ Ҫℜ ʂ in T Y , then the preimage of ꭆ ȵ under f , denoted f − 1 ( ꭆ ȵ ) = , is Ɲ Մ Ҫℜ ʂ in T Ҳ . 2. Let (Ҳ, T Ҳ ) and ( Y , T Y ) be two Ɲ Մ Ҫ T - spaces. If S ȵ = is Ɲ Մ Ҫℜ ʂ in T Ҳ , then the image of S ȵ under f , denoted f ( S ȵ ) = , is Ɲ Մ Ҫℜ ʂ in T Y . Corollary 2.4: 2 Let : Ҳ → Y be a function , and S ȵ = , M ȵ = be a Ɲ Մ Ҫℜ ʂs in Ҳ. i. If S ȵ ⊆ 1 M ȵ , then f ( S ȵ ) ⊆ 1 f ( M ȵ ) ii. If S ȵ ⊆ 2 M ȵ , then f ( S ȵ ) ⊆ 2 f ( M ȵ ) Moreover, if f is injective, the converse (i and ii) holds. Corollary 2.5: 2 Let : Ҳ → Y be a function , and ꭆ ȵ = , Ɉ ȵ = be a Ɲ Մ Ҫℜ ʂs in Y . i. If ꭆ ȵ ⊆ 1 Ɉ ȵ , then f − 1 ( ꭆ ȵ ) ⊆ 1 f − 1 ( Ɉ ȵ ) ii. If ꭆ ȵ ⊆ 2 Ɉ ȵ , then f − 1 ( ꭆ ȵ ) ⊆ 2 f − 1 ( Ɉ ȵ ) Moreover, if f is surjective, the converse (i and ii) holds. Corollary 2.6: 2 Let f : Ҳ → Y be an injective function , and let M ȵ = be a Ɲ Մ Ҫℜ ʂs in Ҳ. Then i. M ȵ = 1 f − 1 ( f ( M ȵ ) ) and Ҡ ȵ = 2 f − 1 ( f ( M ȵ ) ) ii. f ( ∩ 1 M ȵ i ) = ∩ 1 f ( M ȵ i ) and f ( ∩ 2 M ȵ i ) = ∩ 2 f ( M ȵ i ) Corollary 2.7: 1 Let f : Ҳ → Y be a surjective function , and let ꭆ ȵ = be a Ɲ Մ Ҫℜ 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 ( ∪ 2 ꭆ ȵ i ) = ∪ 2 f − 1 ( ꭆ ȵ i ) iv. f − 1 ( ∪ 1 ꭆ ȵ i ) = ∪ 1 f − 1 ( ꭆ ȵ i ) v. f − 1 ( ∩ 1 ꭆ ȵ i ) = ∩ 1 f − 1 ( ꭆ ȵ i ) vi. f − 1 ( ∩ 2 ꭆ ȵ i ) = ∩ 2 f − 1 ( ꭆ ȵ i ) Corollary 2.8: 1 Let f : Ҳ → Y be an injective function , and let S ȵ = be a Ɲ Մ Ҫℜ ʂ in Ҳ. Then f ( ( S ȵ ) c 2 ) = ( f ( S ȵ ) ) c 2 Corollary 2.9: 1 Let f : Ҳ → Y be an injective function , and let ꭆ ȵ = be a Ɲ Մ Ҫℜ ʂ in Y . Then f − 1 ( ( ꭆ ȵ ) c 2 ) = ( f − 1 ( ꭆ ȵ ) ) c 2 3. Neutrosophic crisp topological space ( Ɲ Մ Ҫ 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 Մ Ҫ T (1,2) -space) when the following criteria are satisfied: a. ∅ 1 ȵ = ∈ T , and Ҳ 1 ȵ = ∈ T . b. For any Ҥ ȵ , Ҡ ȵ ∈ T , the set Ҥ ȵ ∩ 1 Ҡ ȵ ∈ T . c. If Ҥ ȵ 𝑗 ∈ T ∀𝑗 ∈ 𝐽, then ∪ j 2 Ҥ ȵ 𝑗 ∈ T . For any Ҡ ȵ ∈ T , it is a neutrosophic crisp open set (Ɲ Մ ҪO − set), and ( Ҡ ȵ ) c 2 is a neutrosophic crisp closed set (Ɲ Մ ҪҪ − set). Example 3.2: Let Ҳ = {𝑎, 𝑏, 𝑐, 𝑑} and T = { ∅ 1 ȵ , Ҳ 1 ȵ , Ҥ ȵ , Ҡ ȵ , M ȵ } such that Ҥ ȵ = , Ҡ ȵ = , M ȵ = . Then (Ҳ, T ) is 𝑎 Ɲ Մ Ҫ T (1,2) –space. Example 3.3: Let Ҳ = {𝑎, 𝑏, 𝑐, 𝑑} and T = { ∅ 1 ȵ , Ҳ 1 ȵ , Ҥ ȵ , Ҡ ȵ }, such that Ҥ ȵ = Ҡ ȵ = . Then T is not Ɲ Մ Ҫ T (1,2) – space. Since Ҥ ȵ ∩ 1 Ҡ ȵ = ∩ 1 = ∉ T . Corollary 3.4: Let (Ҳ, T ) be 𝑎 (Ɲ Մ Ҫ T (1,2) –space), and let ψ = {Ƒ: Ƒ 𝑖𝑠 a Ɲ Մ ҪҪ− set } satisfy the following conditions: i. If Ҥ ȵ 𝑗 ∈ ψ ∀ 𝑗 ∈, then ∩ 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 , … are Ɲ Մ ҪҪ− sets in a Ɲ Մ Ҫ T (1,2) –space (Ҳ, T ). Hence, Ҥ 1 ȵ c 2 , Ҥ 1 ȵ c 2 , Ҥ 1 ȵ c 2 , … are Ɲ Մ ҪO − sets . Therefore, ∪ 2 j Ҥ j ȵ c 2 is Ɲ Մ ҪO – set . But ( Ҥ ȵ ∩ 1 Ҡ ȵ ) c 2 = ( Ҥ ȵ ) c 2 ∪ 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 Մ ҪO – set . Therefore, ⋂ 1 j Ҥ j ȵ Ɲ Մ ҪҪ − set . Hence, ⋂ 1 j Ҥ j ȵ ∈ ψ . ii. Suppose Ҥ ȵ , Ҡ ȵ are two Ɲ Մ ҪҪ – sets in a Ɲ Մ Ҫ T (1,2) –space (Ҳ, T ). So, Ҥ ȵ c 2 , Ҡ ȵ c 2 are two Ɲ Մ ҪO − sets , thus ( Ҥ ȵ ) c 2 ∩ 1 ( Ҡ ȵ ) c 2 is Ɲ Մ ҪO – 𝑠𝑒𝑡. But ( Ҥ ȵ ∪ 2 Ҡ ȵ ) c 2 = ( Ҥ ȵ ) c 2 ∩ 1 ( Ҡ ȵ ) c 2 . Therefore ( Ҥ ȵ ∪ 2 Ҡ ȵ ) c 2 is Ɲ Մ ҪO – 𝑠𝑒𝑡. Thus , Ҥ ȵ ∪ 2 Ҡ ȵ is Ɲ Մ ҪҪ – 𝑠𝑒𝑡. Hence , Ҥ ȵ ∪ 2 Ҡ ȵ ∈ 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 Մ Ҫ 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 ⊆ 1 Ӻ j ȵ ∀ j } • Ɲ Մ ҪҪԼ ( 1 , 2 ) 1 ( Ҥ ȵ ) is Ɲ Մ ҪҪ – set by Corollary 3.4 . Example 4.2: Let Ҳ = {𝑎, 𝑏, 𝑐, 𝑑} and T = { ∅ 1 ȵ , Ҳ 1 ȵ , Ҥ ȵ , Ҡ ȵ , M ȵ } ∋ Ҥ ȵ = , Ҡ ȵ = , M ȵ = . So, (Ҳ, T ) is 𝑎 Ɲ Մ Ҫ T (1,2) –space. Therefore: i. Ɲ Մ ҪҪԼ ( 1 , 2 ) 1 ( Ҥ ȵ = ) does not exist since ∄ Neutrosophic crisp closed set Ӻ ȵ ∋ Ҥ ȵ ⊆ 1 Ӻ ȵ . ii. Ɲ Մ ҪҪԼ ( 1 , 2 ) 1 ( A ȵ = )= Ҳ 1 ȵ = . iii. Ɲ Մ ҪҪԼ ( 1 , 2 ) 1 ( M ȵ C 2 = ) = M ȵ C 2 = . iv. Ɲ Մ ҪҪԼ ( 1 , 2 ) 1 ( M ȵ = ) = Ҳ 1 ȵ = v. Ɲ Մ ҪҪԼ ( 1 , 2 ) 1 ( Ҥ ȵ C 2 = ) = Ҥ ȵ C 2 = . vi. Ɲ Մ ҪҪԼ ( 1 , 2 ) 1 ( Ҡ ȵ C 2 = ) = Ҡ ȵ C 2 = . vii. Ɲ Մ ҪҪԼ ( 1 , 2 ) 1 ( U ȵ C 2 = ) does not exist since ∄ Neutrosophic crisp closed set Ӻ ȵ ∋ U ȵ C 2 ⊆ 1 Ӻ ȵ . viii. Ɲ Մ ҪҪԼ ( 1 , 2 ) 1 ( ∅ 1 ȵ = ) = ∅ 1 ȵ = ix. Ɲ Մ ҪҪԼ ( 1 , 2 ) 1 ( Ҳ 1 ȵ = = Ҳ 1 ȵ = Remark 4.3: In neutrosophic crisp topological spaces (Ɲ Մ Ҫ 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 =) fails to exist in this context. Definition 4.4: Let (Ҳ, T ) be 𝑎 Ɲ Մ Ҫ 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 Մ ҪҪ – sets and Ҥ ȵ ⊆ 2 Ӻ j ȵ ∀ j } • Ɲ Մ ҪҪԼ ( 1 , 2 ) 2 ( Ҥ ȵ ) 𝑖𝑠 Ɲ Մ ҪҪ – 𝑠𝑒𝑡 by Corollary 3.4 . Example 4.5: Let Ҳ = {𝑎, 𝑏, 𝑐, 𝑑} and T = { ∅ 1 ȵ , Ҳ 1 ȵ , Ҥ ȵ , Ҡ ȵ , M ȵ } such that Ҥ ȵ = , Ҡ ȵ = , M ȵ = . So, (Ҳ, T ) is 𝑎 Ɲ Մ Ҫ T (1,2) –space. Therefore: i. Ɲ Մ ҪҪԼ ( 1 , 2 ) 2 ( Ҥ ȵ = ) = Ҳ 1 ȵ = ii. Ɲ Մ ҪҪԼ ( 1 , 2 ) 2 ( A ȵ = ) = Ҳ 1 ȵ = iii. Ɲ Մ ҪҪԼ ( 1 , 2 ) 2 ( U ȵ = ) = Ҳ 1 ȵ = iv. Ɲ Մ ҪҪԼ ( 1 , 2 ) 2 ( M ȵ C 2 = ) = v. Ɲ Մ ҪҪԼ ( 1 , 2 ) 2 ( M ȵ = ) = Ҳ 1 ȵ = vi. Ɲ Մ ҪҪԼ ( 1 , 2 ) 2 ( Ҡ ȵ C 2 = ) = M ȵ C 2 = vii. Ɲ Մ ҪҪԼ ( 1 , 2 ) 2 ( ∅ 1 ȵ = ) = ∅ 1 ȵ = viii. Ɲ Մ ҪҪԼ ( 1 , 2 ) 2 ( Ҳ 1 ȵ = ) = Ҳ 1 ȵ = Corollary 4.6: i. Let (Ҳ, T ) be Ɲ Մ Ҫ 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 Մ Ҫ 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 = 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 Մ Ҫ T (1,2) –space. Then Ҥ ȵ is Ɲ Մ ҪҪ– 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 Մ ҪҪ – 𝑠𝑒𝑡 by Definition 4.1 . Conversely, Let { Ƒ ȵ 𝑗 = : 𝑗 ∈ 𝐽} be the family of Ɲ Մ ҪҪ – set s such that Ҥ ȵ = ⊆ 1 Ƒ ȵ 𝑗 ∀𝑗 ∈ 𝐽. Since Ҥ 1 ⊆ Ƒ 𝑗1 ∀𝑗 ∈ 𝐽, Ҥ 2 ⊆ Ƒ 𝑗2 ∀𝑗 ∈ 𝐽, Ҥ 3 ⊇ Ƒ 𝑗3 ∀𝑗 ∈ 𝐽. Hence , Ɲ Մ ҪҪԼ ( 1 , 2 ) 1 ( Ҥ ȵ ) = ∩ 1 j Ƒ ȵ j = = = 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 ≠ Ҡ ȵ C 2 ; which shows that the condition of Theorem 4.8 fails. Theorem 4.10: Let (Ҳ, T ) be 𝑎 Ɲ Մ Ҫ 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 = : 𝑗 ∈ 𝐽} be the family of Ɲ Մ ҪҪ – 𝑠𝑒𝑡s such that M ȵ = ⊆ 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 ⊆ ∩ Ƒ 𝑗1 , M 2 ⊇ ∩ 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 = . 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 Մ Ҫ T (1,2) –space and Ҥ ȵ = be a Ɲ Մ ҪҪ – 𝑠𝑒𝑡. If U ȵ = is any neutrosophic crisp set ∋ U ȵ ⊆ 1 Ҥ ȵ , then Ɲ Մ ҪҪԼ ( 1 , 2 ) 1 ( U ȵ ) ⊆ 1 Ҥ ȵ . Proof: Since , Ҫ Մ ҪҪԼ ( 1 , 2 ) 1 ( U ȵ ) = ∩ 1 j Ӻ ȵ 𝑗 = ∋ Ӻ j ȵ is Ɲ Մ ҪҪ – sets and U ȵ ⊆ 1 Ӻ j ȵ ∀ j . Therefore, U 1 ⊆ ∩ Ӻ j 1 ∀ j , U 2 ⊆ ∩ Ӻ j 2 ∀ j , U 3 ⊇ ∪ Ӻ j 3 ∀ j . Since, U 1 ⊆ Ҥ 1 , U 2 ⊆ Ҥ 2 , U 3 ⊇ Ҥ 3 and Ҥ ȵ is Ɲ Մ ҪҪ – 𝑠𝑒𝑡. Thus, ∩ Ӻ j 1 ⊆ Ҥ 1 ∀ j , ∩ Ӻ j 2 ⊆ Ҥ 2 ∀ j , ∪ Ӻ j 3 ⊇ Ҥ 3 ∀ j . Hence , Ɲ Մ ҪҪԼ ( 1 , 2 ) 1 ( U ȵ ) ⊆ 1 Ҥ ȵ . Remark 4.13: For instance, consider Example 4.5 , where Ҡ ȵ C 2 = is a Ɲ Մ ҪҪ –𝑠𝑒𝑡 and L ȵ = ⊆ 2 Ҡ ȵ C 2 . In this case, Ҫ Մ ҪҪԼ ( 1 , 2 ) 2 ( L ȵ ) = ⊈ 2 Ҡ ȵ C 2 , which shows that the condition of Theorem 4.12 fails. Corollary 4.14: Let (Ҳ, T ) be 𝑎 Ɲ Մ Ҫ T (1,2) –space and Ҥ ȵ = be a Ɲ Մ ҪҪ –𝑠𝑒𝑡. If U ȵ = is any neutrosophic crisp set ∋ U ȵ ⊆ 2 Ҥ ȵ , then Ɲ Մ ҪҪԼ ( 1 , 2 ) 1 ( U ȵ ) ⊆ 2 Ҥ ȵ . Proof: Since , Ҫ Մ ҪҪԼ ( 1 , 2 ) 1 ( U ȵ ) = ∩ 1 j Ӻ ȵ 𝑗 = ∋ Ӻ j ȵ is Ɲ Մ ҪҪ – sets and U ȵ ⊆ 2 Ӻ j ȵ ∀ j . Therefore, U 1 ⊆ ∩ Ӻ j 1 ∀ j , ∅ ⊇ ∩ Ӻ j 2 ∀ j , U 3 ⊇ ∪ Ӻ j 3 ∀ j . Since, U 1 ⊆ Ҥ 1 , U 2 = Ҥ 2 = ∅ , U 3 ⊇ Ҥ 3 and Ҥ ȵ is Ɲ Մ ҪҪ –𝑠𝑒𝑡. Thus, ∩ Ӻ j 1 ⊆ Ҥ 1 ∀ j , ∩ Ӻ j 2 = ∅ , ∪ Ӻ j 3 ⊇ Ҥ 3 ∀ j . Hence , Ɲ Մ ҪҪԼ ( 1 , 2 ) 1 ( U ȵ ) ⊆ 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 Մ Ҫ 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 ) = ∪ 2 𝑗 { G j ȵ : G j ȵ is Ɲ Մ ҪO − set and G j ȵ ⊆ 1 M ȵ ∀ j } . • Ɲ Մ ҪԼȵȶ ( 1 , 2 ) 1 ( M ȵ ) 𝑖𝑠 Ɲ Մ ҪՕ – 𝑠𝑒𝑡 Example 5.2: Let Ҳ = {𝑎, 𝑏, 𝑐, 𝑑} and = { ∅ 1 ȵ , Ҳ 1 ȵ , Ҥ ȵ , Ҡ ȵ , M ȵ }, where Ҥ ȵ = , Ҡ ȵ = , M ȵ = . So, (Ҳ, T ) is 𝑎 Ɲ Մ Ҫ T (1,2) –space. Therefore: i. Ɲ Մ ҪԼȵȶ ( 1 , 2 ) 1 ( Ҥ ȵ = ) = M ȵ = . ii. Ɲ Մ ҪԼȵȶ ( 1 , 2 ) 1 ( A ȵ = ) = ∅ 1 ȵ = . iii. Ɲ Մ ҪԼȵȶ ( 1 , 2 ) 1 ( M ȵ C 2 = ) = ∅ 1 ȵ = . iv. Ɲ Մ ҪԼȵȶ ( 1 , 2 ) 1 ( M ȵ = ) = M ȵ = . v. Ɲ Մ ҪԼȵȶ ( 1 , 2 ) 1 ( Ҡ ȵ C 2 = ) = ∅ 1 ȵ = . vi. Ɲ Մ ҪԼȵȶ ( 1 , 2 ) 1 ( U ȵ = ) = ∅ 1 ȵ = . vii. Ɲ Մ ҪԼȵȶ ( 1 , 2 ) 1 ( L ȵ = ) = M ȵ = . viii. Ɲ Մ ҪԼȵȶ ( 1 , 2 ) 1 ( ∅ 1 ȵ = ) = ∅ 1 ȵ = . ix. Ɲ Մ ҪԼȵȶ ( 1 , 2 ) 1 ( Ҳ 1 ȵ = ) = Ҳ 1 ȵ = . Definition 5.3: Let (Ҳ, T ) be 𝑎 Ɲ Մ Ҫ 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 ) = ∪ 2 𝑗 { G j ȵ ∋ G j ȵ 𝑖𝑠 Ɲ Մ ҪO – sets and G j ȵ ⊆ 2 M ȵ ∀ j } . • Ɲ Մ ҪԼȵȶ ( 1 , 2 ) 2 ( M ȵ ) 𝑖𝑠 Ɲ Մ ҪO – 𝑠𝑒𝑡 Example 5.4: Let Ҳ = {𝑎, 𝑏, 𝑐, 𝑑} and T = { ∅ 1 ȵ , Ҳ 1 ȵ , Ҥ ȵ , Ҡ ȵ , M ȵ }, where Ҥ ȵ = , Ҡ ȵ = , M ȵ = . So, (Ҳ, T ) is 𝑎 Ɲ Մ Ҫ T (1,2) –space. Therefore: i. Ɲ Մ ҪԼȵȶ ( 1 , 2 ) 2 ( Ҥ ȵ = ) = Ҥ ȵ = . ii. Ɲ Մ ҪԼȵȶ ( 1 , 2 ) 2 ( A ȵ = ) = ∅ 1 ȵ = . iii. Ɲ Մ ҪԼȵȶ ( 1 , 2 ) 2 ( M ȵ C 2 = ) = ∅ 1 ȵ = . iv. Ɲ Մ ҪԼȵȶ ( 1 , 2 ) 2 ( M ȵ = ) = M ȵ = . v. Ɲ Մ ҪԼȵȶ ( 1 , 2 ) 2 ( Ҡ ȵ C 2 = ) = ∅ 1 ȵ =. vi. Ɲ Մ ҪԼȵȶ ( 1 , 2 ) 2 ( U ȵ C 2 = ) = Ҡ ȵ = . vii. Ɲ Մ ҪԼȵȶ ( 1 , 2 ) 2 ( U ȵ = ) does not exist since ∄ Neutrosophic Ɲ Մ ҪO – set G ȵ ∋ G ȵ ⊆ 1 Ҥ ȵ . viii. Ɲ Մ ҪԼȵȶ ( 1 , 2 ) 2 ( ∅ 1 ȵ = ) = ∅ 1 ȵ = . ix. Ɲ Մ ҪԼȵȶ ( 1 , 2 ) 2 ( Ҳ 1 ȵ = ) = Ҳ 1 ȵ = . Remark 5.5: In neutrosophic crisp topological spaces (Ɲ Մ Ҫ 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 =) fails to exist in this context. Theorem 5.6: Let (Ҳ, T ) be 𝑎 Ɲ Մ Ҫ T (1,2) –space. Then Ҡ ȵ is Ɲ Մ ҪO – 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 Մ ҪO – 𝑠𝑒𝑡 by Definition.5.3 . Conversely Let { G ȵ j = : 𝑗 ∈ 𝐽} be the family of Ɲ Մ ҪO –𝑠𝑒𝑡s ∋ G ȵ 𝑗 ⊆ 2 Ҡ ȵ = ∀ j ∈ J . So, Ɲ Մ ҪԼȵȶ ( 1 , 2 ) 2 ( Ҡ ȵ ) = ⋃ 2 j G ȵ j = = = 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 Մ ҪO–𝑠𝑒𝑡. In this case, Ɲ Մ ҪԼȵȶ ( 1 , 2 ) 1 ( Ҥ ȵ )= M ȵ ≠ Ҥ ȵ , which shows that the condition of Theorem 5.6 fails. Theorem 5.8: Let (Ҳ, T ) be 𝑎 Ɲ Մ Ҫ 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 = : 𝑗 ∈ 𝐽} be the family of Ɲ Մ ҪՕ – 𝑠𝑒𝑡s ∋ G ȵ 𝑗 ⊆ 1 M ȵ = ∀ 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 = is any neutrosophic crisp set. In this case, Ɲ Մ ҪԼȵȶ ( 1 , 2 ) 2 ( Ҡ ȵ C 2 ) = = ∅ 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 Մ Ҫ T (1,2) –space, Ҡ ȵ = be a Ɲ Մ ҪO–𝑠𝑒𝑡, and U ȵ = be any neutrosophic crisp set, ∋ Ҡ ȵ ⊆ 2 U ȵ , then Ҡ ȵ ⊆ 2 Ɲ Մ ҪԼȵȶ ( 1 , 2 ) 2 ( U ȵ ). Proof: Ɲ Մ ҪԼȵȶ ( 1 , 2 ) 2 ( U ȵ ) = ∪ 2 j G ȵ 𝑗 = ∋ { G ȵ j = : 𝑗 ∈ 𝐽} be the family of Ɲ Մ ҪO – 𝑠𝑒𝑡s ∋ G ȵ 𝑗 ⊆ 2 U ȵ . Therefore, ∪ G j 1 ⊆ U 1 ∀ j , ∩ G j 2 ⊇ U 2 ∀ j , ∩ G j 3 ⊇ U 3 ∀ j . Since , Ҡ 1 ⊆ U 1 , Ҡ 2 ⊇ U 2 , Ҡ 3 ⊇ U 3 and Ҡ ȵ is Ɲ Մ ҪO – 𝑠𝑒𝑡. Thus, Ҡ 1 ⊆ ∪ G j 1 ∀ j , Ҡ 2 ⊇ ∩ G j 2 ∀ j , Ҡ 3 ⊇ ∩ G j 3 ∀ j . Hence, Ҡ ȵ ⊆ 2 Ɲ Մ ҪԼȵȶ ( 1 , 2 ) 2 ( U ȵ ). Remark 5.11: For instance, consider Example 5.2 , Ҥ ȵ = , which is Ɲ Մ ҪO – 𝑠𝑒𝑡 and Ҥ ȵ ⊆ 1 L ȵ = . In this case, Ҥ ȵ ⊈ 1 Ɲ Մ ҪԼȵȶ ( 1 , 2 ) 1 ( L ȵ ) = , which shows that the condition of Theorem 5.10 fails. Corollary 5.12: Let (Ҳ, T ) be 𝑎 Ɲ Մ Ҫ T (1,2) –space and Ҡ ȵ = < Ҡ 1 , ∅ , Ҡ 3 be a Ɲ Մ ҪO–𝑠𝑒𝑡, and U ȵ = be any neutrosophic crisp set, ∋ Ҡ ȵ ⊆ 1 U ȵ , then Ҡ ȵ ⊆ 1 Ɲ Մ ҪԼȵȶ ( 1 , 2 ) 1 ( U ȵ ) . Proof: Ɲ Մ ҪԼȵȶ ( 1 , 2 ) 2 ( U ȵ ) = ∪ 2 j G ȵ 𝑗 = ∋ { G ȵ j = : 𝑗 ∈ 𝐽} be the family of Ɲ Մ ҪO – 𝑠𝑒𝑡s ∋ G ȵ 𝑗 ⊆ 1 U ȵ . Therefore, ∪ G j 1 ⊆ U 1 ∀ j , ∩ G j 2 ⊇ ∅ , ∩ G j 3 ⊇ U 3 ∀ j . Since , Ҡ 1 ⊆ U 1 , Ҡ 2 = U 2 = ∅ , Ҡ 3 ⊇ U 3 and Ҡ ȵ is a Ɲ Մ ҪO–𝑠𝑒𝑡. Thus, Ҡ 1 ⊆ ∪ G j 1 ∀ j , ∅ = ∩ G j 2 ∀ j , Ҡ 3 ⊇ ∩ G j 3 ∀ j . Hence, Ҡ ȵ ⊆ 1 Ɲ Մ ҪԼȵȶ ( 1 , 2 ) 2 ( U ȵ ). Theorem 5.13: Let (Ҳ, T ) be 𝑎 Ɲ Մ Ҫ 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 = and { G ȵ 𝑗 = : 𝑗 ∈ 𝐽} be family of Ɲ Մ ҪO – 𝑠𝑒𝑡s such that G ȵ j ⊆ 1 Ҥ ȵ c 2 = . So, Ɲ Մ ҪԼȵȶ ( 1 , 2 ) 1 ( Ҥ ȵ c 2 ) = ⋃ 2 j G ȵ 𝑗 = …….(1) Thus, G ȵ j c 2 = : j ∈ J } be the family of Ɲ Մ ҪҪ – 𝑠𝑒𝑡s. ∋ Ҥ ȵ ⊆ 2 G ȵ j c 2 . So , Ҥ 1 ⊆ G j 3 ∀ j ∈ J , Ҥ 2 ⊇ G j 2 ∀ j ∈ J , Ҥ 3 ⊇ G j 1 ∀ j ∈ J . Therefore, Ɲ Մ ҪҪԼ ( 1 , 2 ) 2 ( Ҥ ȵ ) = ∩ 1 j G ȵ j c 2 = . Hence , ( Ɲ Մ ҪҪԼ ( 1 , 2 ) 2 ( Ҥ ȵ ) ) c 2 = …….(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 = and { G ȵ 𝑗 = : 𝑗 ∈ 𝐽} be family of Ɲ Մ ҪO – 𝑠𝑒𝑡s such that G ȵ j ⊆ 2 Ҥ ȵ c 2 = . So, Ɲ Մ ҪԼȵȶ ( 1 , 2 ) 2 ( Ҥ ȵ c 2 ) = ∪ 2 j G ȵ 𝑗 = …….(1) Thus, G ȵ j c 2 = : j ∈ J } be the family of Ɲ Մ ҪҪ – 𝑠𝑒𝑡s. ∋ Ҥ ȵ ⊆ 1 G ȵ j c 2 . So , Ҥ 1 ⊆ G j 3 ∀ j ∈ J , Ҥ 2 ⊆ G j 2 ∀ j ∈ J , Ҥ 3 ⊇ G j 1 ∀ j ∈ J . Therefore, Ɲ Մ ҪҪԼ ( 1 , 2 ) 1 ( Ҥ ȵ ) = ∩ 1 j G ȵ j c 2 = . Hence , ( Ɲ Մ ҪҪԼ ( 1 , 2 ) 1 ( Ҥ ȵ ) ) c 2 = ∪ 2 j G ȵ j = …….(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 = and { Ӻ ȵ 𝑗 = : 𝑗 ∈ 𝐽} be the family of Ɲ Մ ҪҪ – 𝑠𝑒𝑡s such that Ҥ ȵ c 2 = ⊆ 1 Ӻ ȵ j . So, Ɲ Մ ҪҪԼ ( 1 , 2 ) 1 ( Ҥ ȵ c 2 ) = ∩ 1 j Ӻ ȵ 𝑗 = …….(1) Now { Ӻ ȵ j c 2 = : j ∈ J } be the family of Ɲ Մ ҪO – 𝑠𝑒𝑡s, ∋ Ӻ ȵ j c 2 ⊆ 2 Ҥ ȵ . So , Ӻ j 3 ⊆ Ҥ 1 ∀ j ∈ J , Ӻ j 2 ⊇ Ҥ 2 ∀ j ∈ J , Ӻ j 1 ⊇ Ҥ 3 ∀ j ∈ J . Therefore Ɲ Մ ҪԼȵȶ ( 1 , 2 ) 2 ( Ҥ ȵ ) = ∪ 2 j Ӻ ȵ j c 2 = . Hence , ( Ɲ Մ ҪԼȵȶ ( 1 , 2 ) 2 ( Ҥ ȵ ) ) c 2 = …….(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 = and { Ӻ ȵ 𝑗 = : 𝑗 ∈ 𝐽} be the family of Ɲ Մ ҪҪ – sets such that Ҥ ȵ c 2 = ⊆ 2 Ӻ ȵ j . So, Ɲ Մ ҪҪԼ ( 1 , 2 ) 2 ( Ҥ ȵ c 2 ) = ∩ 1 j Ӻ ȵ 𝑗 = …….(1) Now { Ӻ ȵ j c 2 = : j ∈ J } be the family of Ɲ Մ ҪO – 𝑠𝑒𝑡s, ∋ Ӻ ȵ j c 2 ⊆ 1 Ҥ ȵ . So , Ӻ j 3 ⊆ Ҥ 1 ∀ j ∈ J , Ӻ j 2 ⊆ Ҥ 2 ∀ j ∈ J , Ӻ j 1 ⊇ Ҥ 3 ∀ j ∈ J . Therefore Ɲ Մ ҪԼȵȶ ( 1 , 2 ) 1 ( Ҥ ȵ ) = ∪ 2 j Ӻ ȵ j c 2 = . Hence , ( Ɲ Մ ҪԼȵȶ ( 1 , 2 ) 1 ( Ҥ ȵ ) ) c 2 = ∩ 1 j Ӻ ȵ j = …….(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 = and { G ȵ 𝑗 = : 𝑗 ∈ 𝐽} be the family of Ɲ Մ ҪO – sets such that G ȵ j ⊆ 1 Ҥ ȵ = . So, Ɲ Մ ҪԼȵȶ ( 1 , 2 ) 1 ( Ҥ ȵ ) = ⋃ 2 j G ȵ 𝑗 = …….(1) Now G ȵ j c 2 = : j ∈ J } the family of Ɲ Մ ҪҪ – 𝑠𝑒𝑡s ∋ Ҥ ȵ c 2 ⊆ 2 G ȵ j c 2 . So , Ҥ 3 ⊆ G j 3 ∀ j ∈ J , Ҥ 2 ⊇ G j 2 ∀ j ∈ J , Ҥ 1 ⊇ G j 1 ∀ j ∈ J . Therefore, Ɲ Մ ҪҪԼ ( 1 , 2 ) 2 ( Ҥ ȵ c 2 ) = ∩ 1 j G ȵ j c 2 = . Hence , ( Ɲ Մ ҪҪԼ ( 1 , 2 ) 2 ( Ҥ ȵ c 2 ) ) c 2 = …….(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 = is a Ɲ Մ Ҫℜ s . In this case, [ Ɲ Մ ҪҪԼ ( 1 , 2 ) 2 ( U ȵ ) ] c 2 = ∅ 1 ȵ ≠ Ɲ Մ ҪԼȵȶ ( 1 , 2 ) 2 ( U ȵ C 2 ) = Ҡ ȵ = , 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 = is a Ɲ Մ Ҫℜ 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 ) = ∅ 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 = is a Ɲ Մ Ҫℜ s . In this case, [( Ɲ Մ ҪԼȵȶ ( 1 , 2 ) 2 ( Ҥ ȵ ) ] c 2 = Ҥ ȵ c 2 = ≠ 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 = . In this case, [ Ɲ Մ ҪԼȵȶ ( 1 , 2 ) 1 ( U ȵ ) ] c 2 = Ҳ 1 ȵ ≠ 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 = is a Ɲ Մ Ҫℜ s . In this case, [ Ɲ Մ ҪҪԼ ( 1 , 2 ) 1 ( Ҥ ȵ c 2 ) ] c 2 = Ҥ ȵ ≠ Ɲ Մ ҪԼȵȶ ( 1 , 2 ) 1 ( Ҥ ȵ ) = M ȵ = , 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 = is a Ɲ Մ Ҫℜ 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 = is a Ɲ Մ Ҫℜ 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 = is a Ɲ Մ Ҫℜ 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 Մ Ҫ T (1,2) –space. Then: i. Ɲ Մ ҪҪԼ ( 1 , 2 ) 1 ( Ҳ 1 ȵ ) = Ҳ 1 ȵ = ii. Ɲ Մ ҪԼȵȶ ( 1 , 2 ) 1 ( Ҳ 1 ȵ ) = Ҳ 1 ȵ = iii. Ɲ Մ ҪҪԼ ( 1 , 2 ) 2 ( Ҳ 1 ȵ ) = Ҳ 1 ȵ = iv. Ɲ Մ ҪԼȵȶ ( 1 , 2 ) 2 ( Ҳ 1 ȵ ) = Ҳ 1 ȵ = v. Ɲ Մ ҪҪԼ ( 1 , 2 ) 1 ( ∅ 1 ȵ ) = ∅ 1 ȵ = vi. Ɲ Մ ҪԼȵȶ ( 1 , 2 ) 1 ( ∅ 1 ȵ ) = ∅ 1 ȵ = vii. Ɲ Մ ҪҪԼ ( 1 , 2 ) 2 ( ∅ 1 ȵ ) = ∅ 1 ȵ = viii. Ɲ Մ ҪԼȵȶ ( 1 , 2 ) 2 ( ∅ 1 ȵ ) = ∅ 1 ȵ = Proof: i. The only Ɲ Մ Ҫℜ s that satisfy Ҳ 1 ȵ ⊆ 1 Ӻ ȵ = , ∋ Ӻ ȵ are Ɲ Մ ҪҪ –𝑠𝑒𝑡s is Ҳ 1 ȵ . Since, Ҳ 1 ȵ is Ɲ Մ ҪҪ – 𝑠𝑒𝑡. Hence, Ɲ Մ ҪҪԼ ( 1 , 2 ) 1 ( Ҳ 1 ȵ ) = Ҳ 1 ȵ = . ii. Let { G ȵ 𝑗 = : 𝑗 ∈ 𝐽} be the family of Ɲ Մ ҪO – 𝑠𝑒𝑡s such that G ȵ j ⊆ 1 Ҳ 1 ȵ . Since, Ҳ 1 ȵ is Ɲ Մ ҪO – 𝑠𝑒𝑡. Hence , Ɲ Մ ҪԼȵȶ ( 1 , 2 ) 1 ( Ҳ 1 ȵ ) = ∪ 1 j G ȵ 𝑗 = = Ҳ 1 ȵ = . iii. The proof is similar to part i. iv. The proof is similar to part ii. v. Let { Ӻ ȵ 𝑗 = : 𝑗 ∈ 𝐽} be the family of Ɲ Մ ҪҪ – 𝑠𝑒𝑡s such that ∅ 1 ȵ ⊆ 1 Ӻ ȵ j . So, Ɲ Մ ҪҪԼ ( 1 , 2 ) 1 ( ∅ 1 ȵ ) = ∩ 1 j Ӻ ȵ 𝑗 = . Since, ∅ 1 ȵ is Ɲ Մ ҪҪ – 𝑠𝑒𝑡. Hence , Ɲ Մ ҪҪԼ ( 1 , 2 ) 1 ( ∅ 1 ȵ ) = ∩ 1 j Ӻ ȵ 𝑗 = ∅ 1 ȵ . vi. The only Ɲ Մ Ҫℜ s that satisfy G ȵ = ⊆ 1 ∅ 1 ȵ , G ȵ are Ɲ Մ ҪO –𝑠𝑒𝑡s is ∅ 1 ȵ . Since, ∅ 1 ȵ is Ɲ Մ ҪO – 𝑠𝑒𝑡. Hence, Ɲ Մ ҪԼȵȶ ( 1 , 2 ) 1 ( ∅ 1 ȵ ) = ∅ 1 ȵ = . vii. Let { Ӻ ȵ 𝑗 = : 𝑗 ∈ 𝐽} be the family of Ɲ Մ ҪҪ – 𝑠𝑒𝑡s such that ∅ 1 ȵ ⊆ 2 Ӻ ȵ j . So, Ɲ Մ ҪҪԼ ( 1 , 2 ) 2 ( ∅ 1 ȵ ) = ∩ 1 j Ӻ ȵ 𝑗 = . Since, ∅ 1 ȵ is Ɲ Մ ҪҪ – 𝑠𝑒𝑡. Hence , Ɲ Մ ҪҪԼ ( 1 , 2 ) 2 ( ∅ 1 ȵ ) = ∩ 1 j Ӻ ȵ 𝑗 = ∅ 1 ȵ . viii. The proof is similar to part vi. 6. Continuity (1,2) Functions in Crisp Neutrosophic Topology (Ɲ Մ Ҫ 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 Մ Ҫ T (1,2) –space ( Ҳ , T Ҳ ) into a neutrosophic crisp topology space Ɲ Մ Ҫ 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 Մ ҪO–𝑠𝑒𝑡 in Y is a Ɲ Մ ҪO–𝑠𝑒𝑡 in Ҳ. Example 6.2: Let Ҳ = { a , b , c } , T Ҳ = { ∅ 1 ȵ , Ҳ 1 ȵ , A ȵ , B ȵ } ∋ A ȵ = , B ȵ = , and Y = { s , r , t } , T Y = { Φ 1 ȵ , Y 1 ȵ , M ȵ , R ȵ } ∋ M ȵ = , R ȵ = . So, ( Ҳ , T Ҳ ) and ( Y , T Y ) are Ɲ Մ Ҫ T (1,2) –spaces. Let f : Ҳ → 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 ) = ∅ 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 ∅ 1 ȵ , Ҳ 1 ȵ , A ȵ and B ȵ are Ɲ Մ ҪO– 𝑠𝑒𝑡s in Ҳ. Example 6.3: Let Ҳ = { a , b , c } , T Ҳ = { ∅ 1 ȵ , Ҳ 1 ȵ , A ȵ , B ȵ } ∋ A ȵ = , B ȵ = , and Y = { s , r , t } , T Y = { Φ 1 ȵ , Y 1 ȵ , M ȵ , R ȵ } ∋ M ȵ = , R ȵ = . So, ( Ҳ , T Ҳ ) and ( Y , T V ) are Ɲ Մ Ҫ T (1,2) –spaces. Let f : Ҳ → 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 ) = is not a Ɲ Մ ҪO– 𝑠𝑒𝑡 in Ҳ. Corollary 6.4: If f : ( Ҳ , T Ҳ ) → ( Y , T Y ) is a bijective neutrosophic crisp. Then the following conditions stand in equivalence: i. The inverse of every Ɲ Մ ҪO–𝑠𝑒𝑡 in Y is a Ɲ Մ ҪO–𝑠𝑒𝑡 in Ҳ. ii. The inverse of every Ɲ Մ ҪҪ –𝑠𝑒𝑡 in Y is a Ɲ Մ ҪҪ–𝑠𝑒𝑡 in Ҳ. iii. f ( Ɲ Մ ҪҪԼ ( 1 , 2 ) 1 ( M ȵ ) ) ⊆ 1 [ Ɲ Մ ҪҪԼ ( 1 , 2 ) 1 f ( M ȵ ) ] , ∀ M ȵ ⊆ 1 Ҳ 1 ȵ . Proof: (i→ii) Suppose Ɉ ȵ is Ɲ Մ ҪҪ–𝑠𝑒𝑡 in 𝚈 . So , ( Ɉ ȵ ) c 2 is a Ɲ Մ ҪO–𝑠𝑒𝑡 in 𝚈. So, f − 1 ( ( Ɉ ȵ ) c 2 ) is a Ɲ Մ ҪO–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 Մ ҪO–𝑠𝑒𝑡 in Ҳ. Hence , f − 1 ( Ɉ ȵ ) is a Ɲ Մ ҪҪ–𝑠𝑒𝑡 in Ҳ. Conversely, Suppose ꭆ ȵ is a Ɲ Մ ҪO–𝑠𝑒𝑡 in 𝚈 . So , ( ꭆ ȵ ) c 2 is a Ɲ Մ ҪO–𝑠𝑒𝑡 in Y . So , f − 1 ( ( ꭆ ȵ ) c 2 ) is a Ɲ Մ ҪO–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 Մ ҪҪ–𝑠𝑒𝑡 in Ҳ. Hence , f − 1 ( ꭆ ȵ ) is a Ɲ Մ ҪO–𝑠𝑒𝑡 in Ҳ. (ii → iii): Since , M ȵ ⊆ 1 [ Ɲ Մ ҪҪԼ ( 1 , 2 ) 1 ( M ȵ ) ] , ∀ M ȵ ⊆ 1 Ҳ 1 ȵ . By Corollary 4.6 .(i). Therefore, M ȵ ⊆ 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 Մ ҪҪ–𝑠𝑒𝑡 in 𝚈 . So , f −1 ( Ɲ Մ ҪҪԼ ( 1 , 2 ) 1 f ( M ȵ ) ) is a Ɲ Մ ҪҪ–𝑠𝑒𝑡 in Ҳ. Therefore, Ɲ Մ ҪҪԼ ( 1 , 2 ) 1 M ȵ ) ⊆ 1 f 1 ( Ɲ Մ ҪҪԼ ( 1 , 2 ) 1 f ( M ȵ ) ) . So , f ( Ɲ Մ ҪҪԼ ( 1 , 2 ) 1 ( M ȵ ) ⊆ 1 f [ f −1 ( Ɲ Մ ҪҪԼ ( 1 , 2 ) 1 f ( M ȵ ) ) ] . Hence , f ( Ɲ Մ ҪҪԼ ( 1 , 2 ) 1 ( M ȵ ) ) ⊆ 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 ) ) ] ⊆ 1 Ɲ Մ ҪҪԼ ( 1 , 2 ) 1 f [ f − 1 ( Ɉ ȵ ) ] ⊆ 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 ) ⊆ 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 ) → ( 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 ) ) ⊆ 2 [ Ɲ Մ ҪҪԼ ( 1 , 2 ) 2 f ( Ҥ ȵ ) ] , ∀ Ҥ ȵ is a ƝՄҪҪ – set in Ҳ. Proof: Since , Ҥ ȵ ⊆ 2 Ɲ Մ ҪҪԼ ( 1 , 2 ) 2 ( Ҥ ȵ ) By Theorem 4.10 . Thus, Ҥ ȵ ⊆ 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 ) ⊆ 2 f −1 ( Ɲ Մ ҪҪԼ ( 1 , 2 ) 2 f ( Ҥ ȵ ) ) . So , f ( Ɲ Մ ҪҪԼ ( 1 , 2 ) 2 ( Ҥ ȵ ) ⊆ 2 f [ f −1 ( Ɲ Մ ҪҪԼ ( 1 , 2 ) 2 f ( Ҥ ȵ ) ) ] . Hence , f ( Ɲ Մ ҪҪԼ ( 1 , 2 ) 2 ( Ҥ ȵ ) ) ⊆ 2 [ Ɲ Մ ҪҪԼ ( 1 , 2 ) 2 f ( Ҥ ȵ ) ] . Remark 6.6: For instance, consider example Let Ҳ = { a , b , c } , T Ҳ = { ∅ 1 ȵ , Ҳ 1 ȵ , A ȵ , B ȵ } ∋ A ȵ = , B ȵ = , and Y = { s , r , t } , T Y = { Φ 1 ȵ , Y 1 ȵ , M ȵ , R ȵ } ∋ M ȵ = , R ȵ = . So ( Ҳ , T Ҳ ) and ( Y , T Y ) are Ɲ Մ Ҫ T (1,2) –spaces. Let f : Ҳ → 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 ) ) ⊆ 2 [ Ɲ Մ ҪҪԼ ( 1 , 2 ) 2 f ( Ҥ ȵ ) ] , ∀ Ҥ ȵ is a ƝՄҪҪ – set in Ҳ, but M ȵ c 2 = is a ƝՄҪҪ – set in 𝚈, so f − 1 ( M ȵ c 2 ) = 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 → Y is a neutrosophic crisp continuous ( 1 , 2 ) function. ii. f − 1 ( Ɲ Մ ҪԼȵȶ ( 1 , 2 ) 2 ( ꭆ ȵ ) ) ⊆ 2 Ɲ Մ ҪԼȵȶ ( 1 , 2 ) 2 ( f − 1 ( ꭆ ȵ )), ∀ ꭆ ȵ is a Ɲ Մ ҪO–𝑠𝑒𝑡 in 𝚈. iii. Ɲ Մ ҪҪԼ ( 1 , 2 ) 1 ( f − 1 ( Ɉ ȵ ) ) ⊆ 1 f − 1 ( Ɲ Մ ҪҪԼ ( 1 , 2 ) 1 ( Ɉ ȵ ) ) , ∀ Ɉ ȵ is a ƝՄҪҪ – set in 𝚈. Proof: (i → ii) Suppose ꭆ ȵ is a ƝՄҪO – set in 𝚈. Since Ɲ Մ ҪԼȵȶ ( 1 , 2 ) 2 ( ꭆ ȵ ) is a Ɲ Մ ҪO–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 Մ ҪO– 𝑠𝑒𝑡 in Ҳ. Since, f − 1 ( Ɲ Մ ҪԼȵȶ ( 1 , 2 ) 2 ( ꭆ ȵ ) ) ⊆ 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 ) ) ⊆ 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 ) ) ⊆ 2 Ɲ Մ ҪԼȵȶ ( 1 , 2 ) 2 ( f − 1 ( ꭆ ȵ )) ⊆ 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 Մ ҪO–𝑠𝑒𝑡 in Ҳ. Hence, f is a neutrosophic crisp continuous (1,2) function. (ii → 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 5.13 vii. 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 ⊆ 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 ) ) ⊆ 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 Մ ҪO–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 ⊆ 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 ) ) ⊆ 2 Ɲ Մ ҪԼȵȶ ( 1 , 2 ) 2 ( f − 1 ( Ҥ ȵ )) Corollary 6.8: i. if f : Ҳ → Y is a neutrosophic crisp continuous ( 1 , 2 ) function. Then f − 1 ( Ɲ Մ ҪԼȵȶ ( 1 , 2 ) 1 ( ꭆ ȵ ) ) ⊆ 1 Ɲ Մ ҪԼȵȶ ( 1 , 2 ) 1 ( f − 1 ( ꭆ ȵ )), ∀ ꭆ ȵ is a Ɲ Մ ҪO–𝑠𝑒𝑡 in 𝚈. ii. if f − 1 ( Ɲ Մ ҪԼȵȶ ( 1 , 2 ) 1 ( ꭆ ȵ ) ) ⊆ 1 Ɲ Մ ҪԼȵȶ ( 1 , 2 ) 1 ( f − 1 ( ꭆ ȵ )), ∀ ꭆ ȵ is a Ɲ Մ ҪO–𝑠𝑒𝑡 in 𝚈. Then Ɲ Մ ҪҪԼ ( 1 , 2 ) 2 ( f − 1 ( Ɉ ȵ ) ) ⊆ 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 Մ ҪO–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 Մ ҪO– 𝑠𝑒𝑡 in Ҳ. Since, f − 1 ( Ɲ Մ ҪԼȵȶ ( 1 , 2 ) 1 ( ꭆ ȵ ) ) ⊆ 1 f − 1 ( ꭆ ȵ ) by Theorom 5.8 and Corollary 2.5 . Hence , f − 1 ( Ɲ Մ ҪԼȵȶ ( 1 , 2 ) 1 ( ꭆ ȵ ) ) ⊆ 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 ⊆ 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 ) ) ⊆ 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 Comments on this article Comments (0) Version 1 VERSION 1 PUBLISHED 09 Feb 2026 ADD YOUR COMMENT Comment Author details Author details 1 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. Article Versions (1) version 1 Published: 09 Feb 2026, 15:216 https://doi.org/10.12688/f1000research.173335.1 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. Download Export To Sciwheel Bibtex EndNote ProCite Ref. Manager (RIS) Sente metrics Views Downloads F1000Research - - PubMed Central info_outline Data from PMC are received and updated monthly. - - Citations open_in_new 0 open_in_new 0 open_in_new SEE MORE DETAILS CITE how to cite this article AL. Salman H and D. Ali R. Special Properties of Operators in Neutrosophic Crisp Topological Spaces [version 1; peer review: 2 approved] . F1000Research 2026, 15 :216 ( https://doi.org/10.12688/f1000research.173335.1 ) NOTE: If applicable, it is important to ensure the information in square brackets after the title is included in all citations of this article. COPY CITATION DETAILS track receive updates on this article Track an article to receive email alerts on any updates to this article. TRACK THIS ARTICLE Share Open Peer Review Current Reviewer Status: ? Key to Reviewer Statuses VIEW HIDE Approved The paper is scientifically sound in its current form and only minor, if any, improvements are suggested Approved with reservations A number of small changes, sometimes more significant revisions are required to address specific details and improve the papers academic merit. Not approved Fundamental flaws in the paper seriously undermine the findings and conclusions Version 1 VERSION 1 PUBLISHED 09 Feb 2026 Views 0 Cite How to cite this report: Thangaraj GT. Reviewer Report For: Special Properties of Operators in Neutrosophic Crisp Topological Spaces [version 1; peer review: 2 approved] . F1000Research 2026, 15 :216 ( https://doi.org/10.5256/f1000research.191141.r458670 ) The direct URL for this report is: https://f1000research.com/articles/15-216/v1#referee-response-458670 NOTE: it is important to ensure the information in square brackets after the title is included in this citation. Close Copy Citation Details Reviewer Report 06 Mar 2026 G.Thangaraj Thangaraj , Thiruvalluvar University, Vellore, Tamil Nadu, India Approved VIEWS 0 https://doi.org/10.5256/f1000research.191141.r458670 Neutrosophy introduced by F. Smarandache has many applications in different fields of sciences such as topology and laid the foundation for a whole family of new mathematical theories, generalizing both their crisp and fuzzy counterparts. In the manuscript “ Special Properties of ... Continue reading READ ALL Neutrosophy introduced by F. Smarandache has many applications in different fields of sciences such as topology and laid the foundation for a whole family of new mathematical theories, generalizing both their crisp and fuzzy counterparts. In the manuscript “ Special Properties of Operators in Neutrosophic Crisp Topological Spaces ”, the has constructed 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. In this setting, the authors introduced two types of closures for neutrosophic crisp sets , two types of interiors of neutrosophic crisp sets and examined all the related properties, discovering that most of these properties do not hold and showed by means of examples that in neutrosophic crisp topological spaces (ƝՄҪ T (1,2)–space), certain subsets may exist for which no neutrosophic crisp closed superset can be identified within the given topology and certain subsets may exist for which no neutrosophic crisp open subset can be identified within the given topology. The authors 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. This research article addresses specific types of complementation, intersection, or inclusion to better handle the "indeterminacy" component in neutrosophic sets and generalize topological properties for more robust mathematical modeling. I herewith confirm that the article is scientifically valid in its current form. The experimental design, including controls and methods, is adequate; results are presented accurately and the conclusions are justified and supported by the data. Is the work clearly and accurately presented and does it cite the current literature? Yes Is the study design appropriate and is the work technically sound? Yes Are sufficient details of methods and analysis provided to allow replication by others? Yes If applicable, is the statistical analysis and its interpretation appropriate? Not applicable Are all the source data underlying the results available to ensure full reproducibility? Yes Are the conclusions drawn adequately supported by the results? Yes Competing Interests: No competing interests were disclosed. Reviewer Expertise: Mathemtics - General Topology, Fuzzy Topology I confirm that I have read this submission and believe that I have an appropriate level of expertise to confirm that it is of an acceptable scientific standard. Close READ LESS CITE CITE HOW TO CITE THIS REPORT Thangaraj GT. Reviewer Report For: Special Properties of Operators in Neutrosophic Crisp Topological Spaces [version 1; peer review: 2 approved] . F1000Research 2026, 15 :216 ( https://doi.org/10.5256/f1000research.191141.r458670 ) The direct URL for this report is: https://f1000research.com/articles/15-216/v1#referee-response-458670 NOTE: it is important to ensure the information in square brackets after the title is included in all citations of this article. COPY CITATION DETAILS Report a concern Respond or Comment COMMENT ON THIS REPORT Views 0 Cite How to cite this report: Hamzah S. Reviewer Report For: Special Properties of Operators in Neutrosophic Crisp Topological Spaces [version 1; peer review: 2 approved] . F1000Research 2026, 15 :216 ( https://doi.org/10.5256/f1000research.191141.r458665 ) The direct URL for this report is: https://f1000research.com/articles/15-216/v1#referee-response-458665 NOTE: it is important to ensure the information in square brackets after the title is included in this citation. Close Copy Citation Details Reviewer Report 23 Feb 2026 Sattar Hamzah , University of Al-Qadisiyah, Al-Qadisiyah, Iraq Approved VIEWS 0 https://doi.org/10.5256/f1000research.191141.r458665 The manuscript "Special Properties of Operators in Neutrosophic Crisp Topological Spaces" study Neutrosophic crisp sets and recent topic that has entered both pure and applied mathematics, especially general topology. A neutrosophic crisp topological space is defined as a generalization ... Continue reading READ ALL The manuscript "Special Properties of Operators in Neutrosophic Crisp Topological Spaces" study Neutrosophic crisp sets 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.In this work 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 are constructed. Is the work clearly and accurately presented and does it cite the current literature? Yes Is the study design appropriate and is the work technically sound? Yes Are sufficient details of methods and analysis provided to allow replication by others? Yes If applicable, is the statistical analysis and its interpretation appropriate? Not applicable Are all the source data underlying the results available to ensure full reproducibility? Yes Are the conclusions drawn adequately supported by the results? Yes Competing Interests: No competing interests were disclosed. Reviewer Expertise: Mathematics General Topology I confirm that I have read this submission and believe that I have an appropriate level of expertise to confirm that it is of an acceptable scientific standard. Close READ LESS CITE CITE HOW TO CITE THIS REPORT Hamzah S. Reviewer Report For: Special Properties of Operators in Neutrosophic Crisp Topological Spaces [version 1; peer review: 2 approved] . F1000Research 2026, 15 :216 ( https://doi.org/10.5256/f1000research.191141.r458665 ) The direct URL for this report is: https://f1000research.com/articles/15-216/v1#referee-response-458665 NOTE: it is important to ensure the information in square brackets after the title is included in all citations of this article. COPY CITATION DETAILS Report a concern Respond or Comment COMMENT ON THIS REPORT Comments on this article Comments (0) Version 1 VERSION 1 PUBLISHED 09 Feb 2026 ADD YOUR COMMENT Comment keyboard_arrow_left keyboard_arrow_right Open Peer Review Reviewer Status info_outline Alongside their report, reviewers assign a status to the article: Approved The paper is scientifically sound in its current form and only minor, if any, improvements are suggested Approved with reservations A number of small changes, sometimes more significant revisions are required to address specific details and improve the papers academic merit. Not approved Fundamental flaws in the paper seriously undermine the findings and conclusions Reviewer Reports Invited Reviewers 1 2 Version 1 09 Feb 26 read read Sattar Hamzah , University of Al-Qadisiyah, Al-Qadisiyah, Iraq G.Thangaraj Thangaraj , Thiruvalluvar University, Vellore, India Comments on this article All Comments (0) Add a comment Sign up for content alerts Sign Up You are now signed up to receive this alert Browse by related subjects keyboard_arrow_left Back to all reports Reviewer Report 0 Views copyright © 2026 Thangaraj G. This is an open access peer review report 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. 06 Mar 2026 | for Version 1 G.Thangaraj Thangaraj , Thiruvalluvar University, Vellore, Tamil Nadu, India 0 Views copyright © 2026 Thangaraj G. This is an open access peer review report 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. format_quote Cite this report speaker_notes Responses (0) Approved info_outline Alongside their report, reviewers assign a status to the article: Approved The paper is scientifically sound in its current form and only minor, if any, improvements are suggested Approved with reservations A number of small changes, sometimes more significant revisions are required to address specific details and improve the papers academic merit. Not approved Fundamental flaws in the paper seriously undermine the findings and conclusions Neutrosophy introduced by F. Smarandache has many applications in different fields of sciences such as topology and laid the foundation for a whole family of new mathematical theories, generalizing both their crisp and fuzzy counterparts. In the manuscript “ Special Properties of Operators in Neutrosophic Crisp Topological Spaces ”, the has constructed 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. In this setting, the authors introduced two types of closures for neutrosophic crisp sets , two types of interiors of neutrosophic crisp sets and examined all the related properties, discovering that most of these properties do not hold and showed by means of examples that in neutrosophic crisp topological spaces (ƝՄҪ T (1,2)–space), certain subsets may exist for which no neutrosophic crisp closed superset can be identified within the given topology and certain subsets may exist for which no neutrosophic crisp open subset can be identified within the given topology. The authors 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. This research article addresses specific types of complementation, intersection, or inclusion to better handle the "indeterminacy" component in neutrosophic sets and generalize topological properties for more robust mathematical modeling. I herewith confirm that the article is scientifically valid in its current form. The experimental design, including controls and methods, is adequate; results are presented accurately and the conclusions are justified and supported by the data. Is the work clearly and accurately presented and does it cite the current literature? Yes Is the study design appropriate and is the work technically sound? Yes Are sufficient details of methods and analysis provided to allow replication by others? Yes If applicable, is the statistical analysis and its interpretation appropriate? Not applicable Are all the source data underlying the results available to ensure full reproducibility? Yes Are the conclusions drawn adequately supported by the results? Yes Competing Interests No competing interests were disclosed. Reviewer Expertise Mathemtics - General Topology, Fuzzy Topology I confirm that I have read this submission and believe that I have an appropriate level of expertise to confirm that it is of an acceptable scientific standard. reply Respond to this report Responses (0) Thangaraj GT. Peer Review Report For: Special Properties of Operators in Neutrosophic Crisp Topological Spaces [version 1; peer review: 2 approved] . F1000Research 2026, 15 :216 ( https://doi.org/10.5256/f1000research.191141.r458670) NOTE: it is important to ensure the information in square brackets after the title is included in this citation. The direct URL for this report is: https://f1000research.com/articles/15-216/v1#referee-response-458670 keyboard_arrow_left Back to all reports Reviewer Report 0 Views copyright © 2026 Hamzah S. This is an open access peer review report 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. 23 Feb 2026 | for Version 1 Sattar Hamzah , University of Al-Qadisiyah, Al-Qadisiyah, Iraq 0 Views copyright © 2026 Hamzah S. This is an open access peer review report 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. format_quote Cite this report speaker_notes Responses (0) Approved info_outline Alongside their report, reviewers assign a status to the article: Approved The paper is scientifically sound in its current form and only minor, if any, improvements are suggested Approved with reservations A number of small changes, sometimes more significant revisions are required to address specific details and improve the papers academic merit. Not approved Fundamental flaws in the paper seriously undermine the findings and conclusions The manuscript "Special Properties of Operators in Neutrosophic Crisp Topological Spaces" study Neutrosophic crisp sets 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.In this work 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 are constructed. Is the work clearly and accurately presented and does it cite the current literature? Yes Is the study design appropriate and is the work technically sound? Yes Are sufficient details of methods and analysis provided to allow replication by others? Yes If applicable, is the statistical analysis and its interpretation appropriate? Not applicable Are all the source data underlying the results available to ensure full reproducibility? Yes Are the conclusions drawn adequately supported by the results? Yes Competing Interests No competing interests were disclosed. Reviewer Expertise Mathematics General Topology I confirm that I have read this submission and believe that I have an appropriate level of expertise to confirm that it is of an acceptable scientific standard. reply Respond to this report Responses (0) Hamzah S. Peer Review Report For: Special Properties of Operators in Neutrosophic Crisp Topological Spaces [version 1; peer review: 2 approved] . F1000Research 2026, 15 :216 ( https://doi.org/10.5256/f1000research.191141.r458665) NOTE: it is important to ensure the information in square brackets after the title is included in this citation. The direct URL for this report is: https://f1000research.com/articles/15-216/v1#referee-response-458665 Alongside their report, reviewers assign a status to the article: Approved - the paper is scientifically sound in its current form and only minor, if any, improvements are suggested Approved with reservations - A number of small changes, sometimes more significant revisions are required to address specific details and improve the papers academic merit. Not approved - fundamental flaws in the paper seriously undermine the findings and conclusions Adjust parameters to alter display View on desktop for interactive features Includes Interactive Elements View on desktop for interactive features Competing Interests Policy Provide sufficient details of any financial or non-financial competing interests to enable users to assess whether your comments might lead a reasonable person to question your impartiality. 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