On the algebraic structure of neutrosophic ideal 𝓋-open sets in topological spaces

Introduction

After the presentation of open sets in topological spaces (T.P), each studies were presented for new types of open sets, some of which are based on each basic topological spaces, to mention but not limited to: in 2018 and showed them usefully and their properties, in addition to open sets that were used to build graph models, and clarify the topological structure of the distribution of these graphs into subsets by [1] , in 2019 by [2] , and the upper semi-continuous open topologies presented by [3] , open sets (⋏, 𝑏) presented by [4] in 2022, and *open sets, and at the same time (δ-open) were presented by [5] . These open sets helped each researchers to understand and present new topological spaces and explain their properties. Sharma et al. first introduced the quadrupole topology and developed it as in [6] , where a new open set termed 𝑞open was constructed and relied on. Many studies on these topologies have been presented, including fuzzy 𝑞-open sets and fuzzy q-b-open sets, as well as 𝑞-cont and 𝑞-𝑏-cont functions. Beyond introducing a new function for the quadrupole topology space in [7] , they designed completely new and different kinds of sets [8] . Moreover, Pang et al. proposed a topological framework of a quadrupole transmitter constituted by a first topology and a second topology, the first topology comprising a first bridge, a second phase shift element, and a third phase shift element [9] . These studies collectively offer insights into various topologies and their properties. The interplay between algebraic structures and topological spaces has been a fertile ground for mathematical research, yielding profound insights and applications across various domains. Classical topology, initiated by the fundamental concept of open sets, has evolved through numerous generalizations including 𝑠𝑒𝑚𝑖-open sets, 𝑝𝑟𝑒-open sets, 𝛼-open sets, and 𝑏-open sets, among others. More recently, the notion of 𝓋-open sets has emerged as a significant generalization [10] , defined in the context of multiple topologies on a common universe. These sets characterized by the existence of a non-empty set common to all topologies provide a robust framework for studying spaces with multiple compatible topological structures . Concurrently, neutr. logic, pioneered by Smarandache, has revolutionized uncertainty modeling by extending fuzzy and intuitionistic fuzzy logics through three independent components: truth (T), indeterminacy (I), and falsity (F) [11] . This tripartite structure has found rich applications in algebraic contexts through neutr. groups, rings, semigroups, and ideals. Neutr. algebraic structures incorporate the indeterminate element I, allowing for the mathematical representation of ambiguous, incomplete, or contradictory information . Despite significant advancements in both domains, a crucial gap remains: the integration of neutr. algebraic ideals with generalized topological structures. While neutr. topological spaces have been explored, and various generalizations of open s ets have been studied independently, the specific combination of neutr. ideals with v-open set theory represents unexplored mathematical territory. This integration is particularly compelling because Algebraic-topological synthesis: Combining the algebraic precision of ideals with the spatial intuition of topology creates a powerful framework for structured uncertainty. with multi-topological uncertainty: Real-world systems often operate under multiple topological perspectives simultaneously, each with inherent indeterminacy and Ideal-based perturbations: The concept of ideals naturally models "small" or "negligible" elements, allowing for controlled modifications of topological structures.

Fundamental concepts

2.1 Definition: [10] Let {𝜏 𝑘 } 𝑘∈𝐽 , 𝑘 ≥ 2 be a collection of a topological space on 𝑋, and 𝜘 ⊆ 𝑋. Then 𝜘 is termed 𝓋-open set (𝓋 𝑂set) in 𝑋 if any 𝑇 ∈ ⋂ 𝜏 𝑘 𝑘∈𝐽 under which ∅ ≠ 𝑇 ⊆ 𝜘 where 𝜘 ≠ ∅ and ∅ = 𝑇 where 𝜘 = ∅. [10] The set of all 𝓋 𝑂 -set in 𝑋 (𝓋 𝑜𝑋 ) and define as [10] Let (𝑋, 𝓋 𝑜𝑋 ) be a 𝓋-space. A 𝓋-closure of 𝜘 ⊆ 𝑋 (𝑐𝑙 𝓋 (𝜘)) its intersection of all 𝓋 𝐶 -set that include 𝜘.

Definition:

Definition:

2.4

Theorem. [10] Let (𝑋, 𝓋 𝑜𝑋 ) be a 𝓋-space and 𝜘 ⊆ 𝑋, then 𝜘 is a 𝓋 𝐶 -set that include 𝜘. [10] Assume (𝑋, 𝓋 𝑜𝑋 ) is a 𝓋-space and 𝜘 ⊆ 𝑋, then 𝜘 is a 𝓋 𝐶 -set if and only if 𝜘 = 𝑐𝑙 𝓋 (𝜘).

Corollary:

2.6

Remark: [10] Let {𝜏 𝑘 } 2≤𝑘∈𝐽 be a family of topological space on 𝑋 and 𝜏 be a proper non-empty set of 𝑋 under which 𝜏 ∈ 𝜏 𝑘 , for some 𝑘 ∈ 𝐽. However, this is not required 𝜏 ∈ 𝓋 𝑜𝑋 .

2.7

Remark: [10] Let {𝜏 𝑘 } 𝑘∈𝐽 be a family of topological space on 𝑋 and 𝜘 ∈ 𝓋 𝑜𝑋 . However, this is not required 𝜘 ∈ 𝜏 𝑘 for all 𝑘 ∈ 𝐽.

2.8 Proposition: [10] Let {𝜏 𝑘 } 𝑘∈𝐽 be a family of topological space on 𝑋. If 𝜘 ∈ 𝜏 𝑘 for all 𝑘 ∈ 𝐽, then 𝜘 ∈ 𝓋 𝑜𝑋 .In general, the converse is not true.

2.9 Definition: Neutr. logic generalizes fuzzy and intuitionistic fuzzy logics by evaluating properties 𝑃 via three independent components: 𝑃 = 〈𝑇, 𝐼, 𝐹〉 where • T as the membership of truth (certainty),

• 𝐼 as the membership of indeterminacy (ambiguity),

• 𝐹 as the membership of falsity (negativity). [11] A neutr. number is an element in a single-valued neutr. set, represent as 〈𝑇, 𝐼, 𝐹〉.

Definition:

2.11

Example: 𝑥(𝐼) = 5 + √2𝐼 has a determinate part 5 and in determinate part √2𝐼.

Definition:

Let 𝑆 be a universal set. A neutr. set 𝐾 is defined us form: 𝐾 = {(𝜆, 𝑇 𝐾 (𝜆), 𝐼 𝐾 (𝜆), 𝐹 𝐾 (𝜆))|𝜆 ∈ 𝑆} where 𝑇 𝐾 , 𝐼 𝐾 , 𝐹 𝐾: 𝑆 → [0,1], represent truth, indeterminacy, and falsity membership functions, respectively. [12] Let 𝑇, 𝐼, 𝐹 be standard or non-standard of [0,1]: 𝑠𝑢𝑝 𝑇 = 𝑡 𝑠𝑢𝑝 , 𝑖𝑛𝑓 𝑇 = 𝑡 𝑖𝑛𝑓 𝑠𝑢𝑝 𝐼 = 𝑖 𝑠𝑢𝑝 , 𝑖𝑛𝑓 𝐼 = 𝑖 𝑖𝑛𝑓 𝑠𝑢𝑝 𝐹 = 𝑓 𝑠𝑢𝑝 , 𝑖𝑛𝑓 𝐹 = 𝑓 𝑖𝑛𝑓 𝑛 𝑠𝑢𝑝 = 𝑡 𝑠𝑢𝑝 + 𝑖 𝑠𝑢𝑝 + 𝑓 𝑠𝑢𝑝 𝑛 𝑖𝑛𝑓 = 𝑡 𝑖𝑛𝑓 + 𝑖 𝑖𝑛𝑓 + 𝑓 𝑖𝑛𝑓 where 𝑇, 𝐼, 𝐹 are termed neutr. components 2.14 Definition: [13] If (𝐻, * ) be a group and let (𝐻 ∪ 𝐼) = {𝑛 + 𝑚𝐼 ∶ 𝑛, 𝑚 ∈ 𝐻}. 𝑁(𝐻) = ((𝐺 ∪ 𝐼), * ) is neutr. group by 𝐻 and 𝐼 with the binary operation * . 𝐼 is termed the neutr. element, 𝐼 2 = 𝐼. For example: (𝑁(ℂ), +), (𝑁(ℚ), +), and (𝑁(ℤ), +) are neutr. groups. [14] Let (𝑆, . ) be a semigroup. The neutr. semigroup 𝑆(𝐼) = 〈𝑆 ∪ 𝐼〉 is an algebraic structure generated by 𝑆 and the indeterminate 𝐼. A neutr. subsemigroup 𝑇(𝐼) of 𝑆(𝐼) is a subset closed under the semigroup operation and containing 𝐼. [15] Let (𝐺, * )be a group, and let 𝐺(𝐼) = 〈𝐺 ∪ 𝐼〉 be a neutr. group generated by 𝐺 and an indeterminate element 𝐼. A subset 𝐻(𝐼) of 𝐺(𝐼) is termed neutr. subgroup if 𝐻(𝐼) is neutr. group under operation * , the restraction of 𝐻(𝐼) to 𝐺 is a classical subgroup of 𝐺.

Definition:

Definition:

Definition:

2.17 Definition: [16] If (𝑅, +, ) be a ring. Then the set (𝑅 ∪ 𝐼) = {𝑐 + 𝑑𝐼 ∶ 𝑐, 𝑑 ∈ 𝑅} is termed a neutr. ring generated by 𝑅 and 𝐼 under the operations of 𝑅. For example: ℤ(𝐼) = {𝑎 + 𝑏𝐼|𝑎, 𝑏 ∈ ℤ}. [17] Let 𝑅(𝐼) be a neutr. ring, where 𝑅 is a classical ring and 𝐼 is an indeterminate. A non-empty subset 𝑆(𝐼) of 𝑅(𝐼) is termed a neutr. subring if 𝑆(𝐼) is a neutr. ring under the same operations, and ∩ 𝑆 = 𝑆(𝐼) ∩ 𝑅 is a classical subring of 𝑅. [18] Let 𝑅(𝐼) be a neutr. ring, where 𝑅 is a classical ring and 𝐼 is an indeterminate. A non-empty subring 𝐽(𝐼) of 𝑅(𝐼) is termed a neutr. ideal if for every 𝑟(𝐼) ∈ 𝑅(𝐼) and 𝑗(𝐼) ∈ 𝐽(𝐼) both 𝑟(𝐼). 𝑗(𝐼) and 𝑗(𝐼). 𝑟(𝐼) belong to 𝐽(𝐼).

Definition:

Definition:

Neutr. 𝓿-Open Set and Neutr. 𝓿-Ideal

This section introduces and develops the theory of neutr. v-open sets and their enhancement through algebraic ideals to form neutr. ideal 𝓋 𝑂 -sets. We establish:

Definition:

Let {𝜏 𝑘 } 2≤𝑘∈𝐽 be a collection of two or more topological spaces on the same universe 𝑋. Let 𝑁𝑆(𝑋) be the family of all neutr. sets over 𝑋, where a neutr. set 𝐴 is of the form:

, 𝑈 ≠ ∅ under which: 𝑈 ⊆ 𝑆𝑢𝑝𝑝 𝑇 (𝜂), and we define the neutr. membership for 𝜂 based on 𝑈 in the following: way:

Alternatively, more generally, we can say: 𝜂 is neutr. 𝓋 𝑂 -set if any a crisp 𝓋 𝑂 -set 𝑈 under which 𝜂 coincides with the neutr. characteristic function of 𝑈, i.e.: 𝜂 = ⟨𝜒 𝑈 , 0,1 -𝜒 𝑈 ⟩, where 𝜒 𝑈 is the crisp characteristic function of 𝑈 (1 if 𝑥 ∈ 𝑈, 0 otherwise), so: 𝑇 𝜂 = 𝜒 𝑈 , 𝐼 𝜂 = 0, 𝐹 𝜂 = 1 -𝜒 𝑈 . So: 𝜂 = {(𝑎, 1,0,0), (𝑏, 0,1,1), (𝑐, 0,1,1)}. This η is a 𝑁𝓋 𝑂 -set.

Example

Definition:

Let 𝑅(𝐼) be a neutr. ring. A non-empty subset 𝐽(𝐼) ⊆ 𝑅(𝐼) is a neutr. ideal of 𝑅(𝐼) (𝑁𝓋 𝐼 ) if: 𝐽(𝐼) is a neutr. subring (closed under subtraction and contains 𝐼), and For every 𝑟(𝐼) ∈ 𝑅(𝐼) and 𝑗(𝐼) ∈ 𝐽(𝐼), both 𝑟(𝐼) ⋅ 𝑗(𝐼) ∈ 𝐽(𝐼) and 𝑗(𝐼) ⋅ 𝑟(𝐼) ∈ 𝐽(𝐼). = {∅, {𝑎}, 𝑋},, hence its 𝓋 𝑂 -sets. Take 𝑈 = {𝑎}. Now we know the neutr. group 𝜂 corresponding to 𝑈 as 𝜂 = {(𝑎, 1,0,0), (𝑏, 0,1,1), (𝑐, 0,1,1)}, where 𝜂(𝑎) = (𝑇 = 1, 𝐼 = 0, 𝐹 = 0), 𝜂(𝑏) = (𝑇 = 0, 𝐼 = 1, 𝐹 = 1), and 𝜂(𝑐) = (𝑇 = 0, 𝐼 = 1, 𝐹 = 1), take 𝐽(𝐼) = {0, 𝐼}, then we get: 𝐼 ∈ 𝐽(𝐼), 0 + 𝐼 = 𝐼 ∈ 𝐽(𝐼), 𝐼 + 𝐼 = 0 ∈ 𝐽(𝐼), 𝐼. 𝐼 = 𝐼 ∈ 𝐽(𝐼), 0. 𝐼 = 0 ∈ 𝐽(𝐼).

To define 𝑁𝓋 𝐼 take 𝜉(𝐼) is 𝑁𝓋 𝐼 on 𝐽(𝐼) define by form:

Now we know 𝜈 = 𝜂 + 𝜉 (where the sum is the plural of neutrosophy):

Then 𝑣 = {(𝑎, 1, 𝐼, 0), (𝑏, 𝐼, 1,1 + 𝐼), (𝑐, 0,1,1 + 𝐼)} is 𝑁𝓋 𝐼 since 𝜂 is a 𝑁𝓋 𝑂 -set, 𝜉 takes its values from the neutr. example 𝐽(𝐼), and 𝜈 = 𝜂 + 𝜉.

Definition:

Let 𝑋 be a set with two or more topologies 𝜏 𝑘 , giving 𝓋 𝑂 -set structure 𝓋 𝑜𝑋 . Let 𝑁𝓋 𝑂 -set be the collection of neutr. 𝓋 𝑂 -sets. Let 𝐽(𝐼) be a neutr. ideal in some neutr. ring 𝑅(𝐼). We say a neutr. set 𝜈 ∈ 𝑁𝑆(𝑋) is a neutr. ideal 𝓋-open set (𝑁 𝐼 𝓋 𝑂 -set) with respect to 𝐽(𝐼) if for any 𝜂 𝑁𝓋 𝑂 -set, and 𝜉(𝐼) is a neutr. set whose membership values 𝑇 𝜉 (𝑥), 𝐼 𝜉 (𝑥), 𝐹 𝜉 (𝑥) are elements of the neutr. ideal 𝐽(𝐼) for each 𝑥 ∈ 𝑋, i.e., 𝑇 𝜉 (𝑥) = 𝑡 1 + 𝑡 2 𝐼 with 𝑡 1 , 𝑡 2 ∈ 𝑅, etc., but in practice for single-valued neutr. sets we take 𝑇 𝜉 (𝑥) ∈ [0,1], so we interpret: the triple (𝑇 𝜉 (𝑥), 𝐼 𝜉 (𝑥), 𝐹 𝜉 (𝑥)) belongs to some subset of [0,1] 3 that is isomorphic to a neutr. ideal in a formal way, then 𝜈 = 𝜂 + 𝜉(𝐼).

More concretely, we can define: Let 𝐽 be an ordinary ideal in a ring 𝑅 ⊆ 𝑅. Define a neutr. ideal 𝐽(𝐼) = { 𝑎 + 𝑏𝐼 | | 𝑎, 𝑏 ∈ 𝐽 }. A neutr. set 𝜈 is termed 𝑁 𝐼 𝓋 𝑂 -set if for each 𝑥 ∈ 𝑋, 𝑇 𝜈 (𝑥) = 𝑇 𝜂 (𝑥) + 𝑗 1 (𝑥), 𝐼 𝜈 (𝑥) = 𝐼 𝜂 (𝑥) + 𝑗 2 (𝑥), and 𝐹 𝜈 (𝑥) = 𝐹 𝜂 (𝑥) + 𝑗 3 (𝑥) where 𝜂 is 𝑁𝓋 𝑂 -set, and 𝑗 1 (𝑥), 𝑗 2 (𝑥), 𝑗 3 (𝑥) ∈ 𝐽(𝐼) (neutr. numbers from the ideal). = {∅, {𝑎, 𝑏}, 𝑋},, hence its 𝓋 𝑂 -sets. Take 𝑈 = {𝑎, 𝑏}. Now we know the neutr. group 𝜂 corresponding to 𝑈 as 𝜂 = {(𝑎, 1,0,0), (𝑏, 1,0,0), (𝑐, 0,1,1)}, where 𝜂(𝑎) = (𝑇 = 1, 𝐼 = 0, 𝐹 = 0), 𝜂(𝑏) = (𝑇 = 1, 𝐼 = 0, 𝐹 = 0), and 𝜂(𝑐) = (𝑇 = 0, 𝐼 = 1, 𝐹 = 1). Let 𝑅 = 𝑍 2 = {0,1} (ring mod 2). The neutr. ring: 𝑅(𝐼) = {0,1, 𝐼, 1 + 𝐼}, 𝐼 2 = 𝐼, take 𝐽(𝐼) = {0, 𝐼}. To define 𝑁𝓋 𝐼 take 𝜉(𝐼) is 𝑁𝓋 𝐼 on 𝐽(𝐼) define by form: 𝜉(𝑎) = (0, 𝐼, 0) 𝜉(𝑏) = (𝐼, 0,0) 𝜉(𝑐) = (0,0, 𝐼) Now we know 𝜈 = 𝜂 + 𝜉 (where the sum is the plural of neutr.):

Then 𝑣 = {(𝑎, 1, 𝐼, 0), (𝑏, 1 + 𝐼, 0,0), (𝑐, 0,1,1 + 𝐼)} is 𝑁 𝐼 𝓋 𝑂 -set.

Remark:

Define the family of 𝑁𝓋 𝑂 -set, s.t. 𝑁𝓋 𝑂 (𝑋) ⊆ 𝑁𝑆(𝑋) as follows: for each crisp 𝓋 𝑂 -set ∈ 𝜏 𝓋 , its corresponding neutr. 𝓋 𝑂 -set 𝜂 𝑈 ∈ 𝑁𝓋 𝑂 (𝑋) is defined by:

Since 𝜂 1 ∩ 𝜂 2 ∈ 𝑁𝓋 𝑂 (𝑋) (as crisp 𝓋 𝑂 -sets are closed under finite intersections) and 𝜉 1 (𝐼) + 𝜉 2 (𝐼) ∈ 𝐽(𝐼) (as 𝐽(𝐼) is closed under addition), we have 𝜈 1 ∩ 𝜈 2 ∈ 𝜏 𝑁𝓋 𝐼 3) Let {𝜈 𝛼 } 𝛼∈𝛬 ⊆ 𝜏 𝑁𝓋 𝐼 with 𝜈 𝛼 = 𝜂 𝛼 ⊕ 𝜉 𝛼 (𝐼). Then: Take 𝜈 = {(𝑎, 1, 𝐼, 0), (𝑏, 0,1,1), (𝑐, 0,1,1)} (corresponds to crisp {𝑎}). Then 𝑁𝑐𝑙 𝑁 𝐼 𝓋 (𝜈) = 𝜈 ∪ {(𝑏, 𝐼, 0,0), (𝑐, 0,0,1)} = {(𝑎, 1, 𝐼, 0), (𝑏, 𝐼, 1,1), (𝑐, 0,1,1 + 𝐼)}, (smallest 𝑁 𝐼 𝓋 𝐶 -set containing 𝜈).

Definition:

The neutr. interior of a neutr. set 𝑣 in 𝑁 𝐼 𝓋 𝑂 -space is the union of 𝑁 𝐼 𝓋 𝐶 -set containing 𝜈. Formally: Take 𝜈 = {(𝑎, 1, 𝐼, 0), (𝑏, 1 + 𝐼, 1,1), (𝑐, 0,1,1 + 𝐼)}. Then 𝑁𝑖𝑛𝑡 𝑁 𝐼 𝓋 (𝜈) = {(𝑎, 1,0,0), (𝑏, 1,0,0)}, (corresponds to crisp {𝑎, 𝑏} which is 𝓋 𝑂 -set).

Definition:

The neutr. exterior of a neutr. set 𝑣 is the interior of its complement Formally:

where 𝜈 𝑐 (𝑥) = (𝐹 𝜈 (𝑥), 𝐼 𝜈 (𝑥), 𝑇 𝜈 (𝑥)) for each 𝑥 ∈ 𝑋.

4.6

Example: let 𝑋 = {𝑎, 𝑏, 𝑐}, with two topologies: 𝜏 1 = {∅, {𝑎}, 𝑋}, 𝜏 2 = {∅, {𝑎, 𝑏}, 𝑋}

Then 𝜏 𝑣 = {∅, {𝑎}, 𝑋}.Let 𝑅 = 𝑍 2 , 𝐽 = {0} ⊆ 𝑅, so 𝐽(𝐼) = {0, 𝐼}.

Take 𝜈 = {(𝑎, 1, 𝐼, 0), (𝑏, 0,1,1), (𝑐, 0,1,1)}.

Then 𝜈 𝑐 = {(𝑎, 0, 𝐼, 1), (𝑏, 1,1,0), (𝑐, 1,1,0)} 𝑁𝑒𝑥𝑡 𝑁 𝐼 𝓋 (𝜈) = {(𝑏, 1,0,0), (𝑐, 1,0,0)}, (corresponds to crisp {𝑏, 𝑐} which is 𝓋 𝑂 -set) Take 𝜈 = {(𝑎, 1, 𝐼, 0), (𝑏, 1 + 𝐼, 0,0), (𝑐, 0,1,1)}.

𝑁𝑐𝑙 𝑁 𝐼 𝓋 (𝜈) = 𝜈 ∪ {(𝑐, 𝐼, 0,0)} = {(𝑎, 1, 𝐼, 0), (𝑏, 1 + 𝐼, 0,0), (𝑐, 𝐼, 1,1)}.

𝑁𝑖𝑛𝑡 𝑁 𝐼 𝓋 (𝜈) = {(𝑎, 1,0,0), (𝑏, 1,0,0)}.

𝑁𝑏𝑑 𝑁 𝐼 𝓋 (𝜈) = {(𝑎, 0, 𝐼, 0), (𝑏, 𝐼, 0,0), (𝑐, 𝐼, 1,1)}.

Definition:

A neutr. set 𝜔 is termed a 𝑁 𝐼 𝓋-neighborhood (𝑁 𝐼 𝓋-neigh) of a point 𝑥 ∈ 𝑋 if there exists 𝜂 ∈ 𝜏 𝑁𝓋 𝐼 under which: 𝜂(𝑥) = (1,0,0) and 𝜂 ⊆ 𝜔.

Example:

For 𝑥 = 𝑎, the neutr. set 𝜂 = {(𝑎, 1,0,0), (𝑏, 0,1,1), (𝑐, 0,1,1)} is a 𝑁 𝐼 𝓋 𝑂 -set. Any ω with 𝜂 ⊆ 𝜔 is a 𝑁 𝐼 𝓋-neigh of 𝑎.

Definition:

A point 𝑥 ∈ 𝑋 is a 𝑁 𝐼𝑣 -limit point of neutr. set 𝜈 if for every 𝑁 𝐼 𝓋-neigh 𝜂 of 𝑥, there exists 𝑦 ≠ 𝑥 under which: 𝑇 𝜂 (𝑦) > 0 and 𝑇 𝜈 (𝑦) > 0. The set of all 𝑁 𝐼𝑣 -limit point s of 𝜈 is denoted 𝑁𝑑 𝑁𝐼𝑣 (𝜈).

Example:

Let 𝜈 = {(𝑎, 0.8,0.2,0.1), (𝑏, 0.7,0.3,0.2), (𝑐, 0,1,1)}.

For 𝑥 = 𝑎, consider 𝑁 𝐼 𝓋-neigh 𝜂 = {(𝑎, 1,0,0), (𝑏, 0.5,0.5,0), (𝑐, 0,1,1)}.

Since 𝑇 𝜂 (𝑏) = 0.5 > 0 and 𝑇 𝜈 (𝑏) = 0.7 > 0, 𝑎 ∈ 𝑁𝑑 𝑁𝐼𝑣 (𝜈).

Remark:

In 𝑁 𝐼𝑣 -spaces, limit points depend on both the 𝓋-open structure and the neutr. ideal perturbations. Unlike classical topology where only truth membership matters, here indeterminacy and falsity components from 𝐽(𝐼) affect neighborhood structures.

By Definition of 𝑁𝑖𝑛𝑡 𝑁 𝐼 𝓋 (∅ 𝑁 ), ∅ 𝑁 is trivially 𝑁 𝐼 𝓋 𝑂 -set. Thus, its 𝑁 𝐼 𝓋 𝑂 -set is itself. For 𝑋 𝑁 , 𝑁𝑖𝑛𝑡 𝑁 𝐼 𝓋 (𝑋 𝑁 ) = 𝑋 𝑁 The idempotence follows because 𝑁 𝐼 𝓋 𝑂 -set are preserved under interior operations.

Monotonicity

Equality doesn't generally hold because a 𝑁 𝐼 𝑣 𝑂 -set contained in 𝜈 1 ∪ 𝜈 2 might intersect both 𝜈 1 and 𝜈 2 but not be entirely contained in either one alone. ∎ The union of 𝑁 𝐼 𝑣 𝑂 -sets is a 𝑁 𝐼 𝑣 𝑂 -set (as 𝜏 𝑁𝓋 𝐼 is closed under arbitrary unions, being a topology). Therefore, 𝜈 is a 𝑁 𝐼 𝑣 𝑂 -set. ∎ 4.14. Theorem: Under the same hypotheses as [Theorem 4.13], then:

1. 𝑁𝑐𝑙 𝑁 𝐼 𝓋 (∅ 𝑁 ) = ∅ 𝑁 and 𝑁𝑐𝑙 𝑁 𝐼 𝓋 (𝑋 𝑁 ) = 𝑋 𝑁 , and 𝑁𝑐𝑙 𝑁 𝐼 𝓋 (𝑁𝑐𝑙 𝑁 𝐼 𝓋 (𝑣)) = 𝑁𝑐𝑙 𝑁 𝐼 𝓋 (𝑣) (𝜈), if for every 𝑁 𝐼 𝑣-neigh 𝜂 of 𝑥, there exists 𝑦 ≠ 𝑥 under which: 𝑇 𝜂 (𝑦) > 0 and 𝑇 𝜈 (𝑦) > 0 Now, let 𝑥 ∈ 𝑁𝑑 𝑁𝐼𝑣 (𝜈 1 ). Then for every 𝑁 𝐼 𝑣-neigh 𝜂 of 𝑥, there exists 𝑦 ≠ 𝑥 with: 𝑇 𝜂 (𝑦) > 0 and 𝑇 𝜈 1 (𝑦) > 0 Since 𝜈 1 ⊆ 𝜈 2 , we have 𝑇 𝜈 1 (𝑦) ≤ 𝑇 𝜈 2 (𝑦) for all 𝑦 ∈ 𝑋 (because in neutr. set inclusion, 𝑇 𝜈 1 (𝑥) ≤ 𝑇 𝜈 2 (𝑥), 𝐼 𝜈 1 (𝑥) ≤ 𝐼 𝜈 2 (𝑥), 𝐹 𝜈 1 (𝑥) ≤ 𝐹 𝜈 2 (𝑥). Thus, if 𝑇 𝜈 1 (𝑦) > 0, then 𝑇 𝜈 2 (𝑦) > 0. Therefore, for every 𝑁 𝐼 𝑣-neigh 𝜂 of 𝑥, there exists 𝑦 ≠ 𝑥 with: 𝑇 𝜂 (𝑦) > 0 and 𝑇 𝜈 2 (𝑦) > 0 This means 𝑥 ∈ 𝑁𝑑 𝑁𝐼𝑣 (𝜈 2 ). Hence: 𝑁𝑑 𝑁𝐼𝑣 (𝜈 1 ) ⊆ 𝑁𝑑 𝑁𝐼𝑣 (𝜈 2 ). ∎

We prove both inclusions (⊆)𝑁𝑐𝑙 𝑁𝐼𝑣 (𝑣 ) ⊆ 𝑣 ∪ 𝑁𝑑 𝑁𝐼𝑣 (𝑣 ):

Let 𝑥 ∈ 𝑋 under which 𝑥 ∈ 𝑁𝑑 𝑁𝐼𝑣 (𝜈). Case 1: If 𝑥 ∈ 𝜈, then clearly 𝑥 ∈ 𝜈 ∪ 𝑁𝑑 𝑁𝐼𝑣 (𝜈). Case 2: If 𝑥 ∉ 𝜈, we need to show that 𝑥 ∈ 𝑁𝑑 𝑁𝐼𝑣 (𝜈). Since 𝑥 ∈ 𝑁𝑑 𝑁𝐼𝑣 (𝜈) and 𝑥 ∉ 𝜈, by the definition of closure, for every 𝑁 𝐼 𝑣-neigh 𝑈 of 𝑥, we have: 𝑈 ∩ 𝜈 ≠ ∅ 𝑁 Since 𝑥 ∉ 𝜈, the intersection contains at least one point 𝑦 ≠ 𝑥 with 𝑇 𝜈 (𝑦) > 0 Thus, for every 𝑁 𝐼 𝑣-neigh 𝑈 of 𝑥, there exists 𝑦 ≠ 𝑥 under which: 𝑇 𝑈 (𝑦) > 0and 𝑇 𝜈 (𝑦) > 0 This is exactly the definition of 𝑥 ∈ 𝑁𝑑 𝑁𝐼𝑣 (𝜈). Therefore, 𝑥 ∈ 𝜈 ∪ 𝑁𝑑 𝑁𝐼𝑣 (𝜈). (⊇) Show 𝑁𝑐𝑙 𝑁𝐼𝑣 (𝑣 ) ⊇ 𝑣 ∪ 𝑁𝑑 𝑁𝐼𝑣 (𝑣 ): a) Clearly 𝜈 ⊆ 𝑁𝑐𝑙 𝑁𝐼𝑣 (𝜈) by definition of closure. b) Let 𝑥 ∈ 𝑁𝑐𝑙 𝑁𝐼𝑣 (𝜈). We show 𝑥 ∈ 𝑁𝑐𝑙 𝑁𝐼𝑣 (𝜈). Suppose, for contradiction, that 𝑥 ∉ 𝑁𝑐𝑙 𝑁𝐼𝑣 (𝜈).

Then there exists a 𝑁 𝐼 𝓋 𝑐 -set 𝐶 under which 𝜈 ⊆ 𝐶 and 𝑥 ∉ 𝐶. Since 𝐶 is 𝑁 𝐼 𝓋 𝑐 -set, 𝐶 𝑐 is 𝑁 𝐼 𝓋 𝑂 -set and contains 𝑥. Now, 𝐶 𝑐 is a 𝑁 𝐼 𝑣-neigh of 𝑥. But since 𝜈 ⊆ 𝐶, we have 𝜈 ∩ 𝐶 𝑐 = ∅ 𝑁 . This means that for this particular 𝑁 𝐼 𝑣-neigh 𝐶 𝑐 of 𝑥, there is no y (and certainly no 𝑥 ≠ 𝑦 with 𝑇 𝐶 𝑐 (𝑦) > 0 and 𝑇 𝜈 (𝑦) > 0. This contradicts 𝑥 ∈ 𝑁𝑐𝑙 𝑁𝐼𝑣 (𝜈), which requires that every 𝑁 𝐼 𝑣-neigh of 𝑥 contains some 𝑥 ≠ 𝑦 with 𝑇 𝜈 (𝑦) > 0. Therefore, 𝑥 ∈ 𝑁𝑐𝑙 𝑁𝐼𝑣 (𝜈). Thus: 𝑁𝑐𝑙 𝑁𝐼𝑣 (𝜈) ⊆ 𝑁𝑐𝑙 𝑁𝐼𝑣 (𝜈). Since both inclusions hold, we have: 𝑁𝑐𝑙 𝑁𝐼𝑣 (𝑣 ) ⊇ 𝑣 ∪ 𝑁𝑑 𝑁𝐼𝑣 (𝑣 ) 3. Assume 𝒙 ∉ 𝑵𝒅 𝑵𝑰𝒗 (𝝂 𝟏 ) and 𝒙 ∉ 𝑵𝒅 𝑵𝑰𝒗 (𝝂 𝟐 ). Then Since 𝑥 ∉ 𝑁𝑑 𝑁𝐼𝑣 (𝜈 1 ), there exists a 𝑁 𝐼 𝑣-neigh 𝑈 1 of 𝑥 under which for all 𝑥 ≠ 𝑦 with 𝑇 𝑈 1 (𝑦) > 0, we have 𝑇 𝜈 1 (𝑦) = 0. By same way for 𝑥 ∉ 𝑁𝑑 𝑁𝐼𝑣 (𝜈 2 ) Let 𝑈 = 𝑈 1 ∩ 𝑈 2 . Since 𝜏 𝑁𝓋 𝐼 is a topology, 𝑈 is 𝑁 𝐼 𝓋 𝑂 -set. Also, 𝑥 ∈ 𝑈 (since 𝑥 ∈ 𝑈 1 and 𝑥 ∈ 𝑈 2 , so 𝑈 is a 𝑁 𝐼 𝑣-neigh of 𝑥. Now, for this neighborhood 𝑈: For any 𝑥 ≠ 𝑦 with 𝑇 𝑈 (𝑦) > 0, we have 𝑇 𝑈 1 (𝑦) > 0 and 𝑇 𝑈 2 (𝑦) > 0 (since 𝑈 ⊆ 𝑈 1 and 𝑈 ⊆ 𝑈 2 ) Then by (1): 𝑇 𝜈 1 (𝑦) = 0. And by (2): 𝑇 𝜈 2 (𝑦) = 0 So 𝑇 𝜈 1 (𝑦) ∪ 𝜈 2 (𝑦) = max(0,0) = 0 Thus, for the 𝑁 𝐼 𝑣-neigh 𝑈 of 𝑥, there is 𝑥 ≠ 𝑦 with 𝑇 𝑈 (𝑦) > 0 and 𝑇 𝜈 1 ∪ 𝜈 2 (𝑦) > 0. This contradicts 𝑥 ∈ 𝑁𝑑 𝑁𝐼𝑣 (𝜈 1 ∪ 𝜈 2 ). Therefore, our assumption is false, and we must have 𝑥 ∈ 𝑁𝑑 𝑁𝐼𝑣 (𝜈 1 ) or 𝑥 ∈ 𝑁𝑑 𝑁𝐼𝑣 (𝜈 2 ). Thus: 𝑥 ∈ 𝑁𝑑 𝑁𝐼𝑣 (𝜈 1 ) ∪ 𝑁𝑑 𝑁𝐼𝑣 (𝜈 2 ). Hence: 𝑁𝑑 𝑁𝐼𝑣 (𝜈 1 ∪ 𝜈 2 ) ⊆ 𝑁𝑑 𝑁𝐼𝑣 (𝜈 1 ) ∪ 𝑁𝑑 𝑁𝐼𝑣 (𝜈 2 ). (⊇) Show 𝑁𝑑 𝑁𝐼𝑣 (𝜈 1 ∪ 𝜈 2 ) ⊇ 𝑁𝑑 𝑁𝐼𝑣 (𝜈 1 ) ∪ 𝑁𝑑 𝑁𝐼𝑣 (𝜈 2 ): This follows from the monotonicity property (which can be proved separately). Since 𝜈 1 ⊆ 𝜈 1 ∪ 𝜈 2 , any limit point of 𝜈 1 is also a limit point of 𝜈 1 ∪ 𝜈 2: If 𝑥 ∈ 𝑁𝑑 𝑁𝐼𝑣 (𝜈 1 ), then for every 𝑁 𝐼 𝑣-neigh 𝑈 of 𝑥, there exists 𝑥 ≠ 𝑦 with 𝑇 𝑈 (𝑦) > 0 and 𝑇 𝜈 1 (𝑦) > 0. Since 𝑇 𝜈 1 (𝑦) > 0 implies 𝑇 𝜈 1 ∪ 𝜈 2 (𝑦) > 0, we have 𝑥 ∈ 𝑁𝑑 𝑁𝐼𝑣 (𝜈 1 ∪ 𝜈 2 ) Similarly, if 𝑥 ∈ 𝑁𝑑 𝑁𝐼𝑣 (𝜈 2 ), then 𝑁𝑑 𝑁𝐼𝑣 (𝜈 1 ∪ 𝜈 2 ) ⊇ 𝑁𝑑 𝑁𝐼𝑣 (𝜈 1 ) ∪ 𝑁𝑑 𝑁𝐼𝑣 (𝜈 2 ). Since both inclusions hold, we have: 𝑁𝑑 𝑁𝐼𝑣 (𝑣 1 ∪ 𝑣 2 ) = 𝑁𝑑 𝑁𝐼𝑣 (𝑣 1 ) ∩ 𝑁𝑑 𝑁𝐼𝑣 (𝑣 2 )

Conclusion

The results reveal that introducing the neutr. ideal in the structure of open sets of type 𝓋is not merely an addition, but also has a profound effect on changing the nature of the topological space. It can be inferred that the ideal neutr. topological space preserves the basic topology conditions (such as union closure and the retention of neutral elements), but achieves a more flexible approach to describing indeterminate processes. We show that the workings of inset, closure, and limitation in the case of this scenario are affected by the neutr. ideal, and that in this context, the accumulator and adjacency points depend not only on the right degree of belonging, but also to varying degrees of indeterminacy and falsehood. Moreover, the work demonstrates that many relations known in classical topology are true in a modified form while others are not, indicating the richness of the new structure. As you have previously seen, this provides an appropriate mathematical framework for modelling multi-perspective systems in which algebraic and topological structures overlap under circumstances of intrinsic uncertainty. Therefore, the contributions of this work provide some theoretical grounds for future works in neutr. continuity, topological separation, and use cases where you can use data in an imperfect or conflicting manner, such as intelligent modeling or complex decision systems.

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