Exploring Types of Open Sets in Neutrosophic Over Soft Topological Spaces with an Application to Organic Catalyst Selection

Introduction

The introduction of fuzzy sets ²⁸ by Zadeh in 1965 created the foundation for fuzzy set theory, permitting partial membership of elements and thereby inducing a revolution in decision-making and control systems. Zadeh also extended this idea in 1978 by establishing the frame of reference of possibility theory²⁹ as a more intuitive way to deal with uncertainty than probability theory. Then, in 1970, Bellman and Zadeh ³ laid the first applications of fuzzy logic in decision-making environments on these theoretical foundations, giving birth to a host of real-world applications in fields such as economics, engineering, and operations research. In 1995 Bustince and Burillo ⁵ contributed to uncertainty modeling through the study of interval-valued intuitionistic fuzzy sets, extending the expressive power of fuzzy sets through the inclusion of both membership and non-membership functions with interval-valued arguments. Subsequently, in 1998, Atanassov and Shannon¹  expanded the concept of intuitionistic fuzzy logic by introducing logical operations and properties, thereby enlarging its theoretical horizon.

Molodtsov¹⁸ introduced in 1999 the soft set theory notion as an innovative mathematical tool capable of fighting against vagueness in situations parameterized with applications where normal models could not be utilized. In parallel, Smarandache²³ introduced neutrosophic logic in the same year as a powerful generalization of fuzzy and intuitionistic fuzzy logic, distinguishing between degrees of truth, indeterminacy, and falsity, and thus allowing a more sophisticated approach to modeling uncertainty. Maji et al. in¹⁵ in 2003 formalized a soft-set framework into a structured decision-making tool providing the groundwork for more integration with fuzzy and neutrosophic theories. Wang et al.²⁶ extended the entire neutrosophic theory through the introduction of single-valued neutrosophic sets in 2010, where the application of this theory could be better addressed in real-life decision support systems.

Ye²⁷ introduced some innovative single-valued neutrosophic correlation coefficients in 2013. Moreover, he proposed improved decision-making processes that would augment computational tools in dealing with uncertainty. In 2016, Smarandache²⁵ came up with oversets, undersets, and offsets to further develop neutrosophic theory. This expansion gave the theory more ways of structuring itself. Dhavaseelan et al.¹⁰, in 2019, introduced the idea of neutrosophic -continuity, adding to the study of generalized continuity in neutrosophic topologies. During the same year, RN Majeed and SA El-Sheikh²¹ dealt with fuzzy orbit topological spaces offering some new applications in the field of materials science and engineering systems. Correlation measures associated with pythagorean neutrosophic sets were actually put forward by Jansi et al.¹¹ in 2019 giving better decision-making tools for complex uncertainty. Mehmood et al. developed the concept of neutrosophic soft -open sets in 2020,¹⁷ marking another important step in soft set and neutrosophic topological theories.

In the same year, 2020, Saeed et al.²² took on medical diagnosis through applications of multi-polar neutrosophic soft sets which reveal the capability of neutrosopholic models in healthcare settings. The same year, Christianto et al.⁶ investigated some philosophical and scientific implications of neutrosophic logic to encourage its adoption in mainstream physical sciences. Smarandache and Pramanik²⁴ compiled a hefty edited book in 2020, which captures the most recent developments and future directions of neutrosophistic theory. Mallick and Pramanik [16] are also in line, proposing pentapartitioned neutrosophic sets which will accommodate much more assorted forms of uncertainty without extending them from classical partition.

Radha et al.²⁰ in 2021 presented neutrosophic Pythagorean sets with dependent components which improved correlation coefficients and hence considerably boosted the evaluation of such-related similarity in uncertain datasets as well as strengthened decision-making systems. In 2021, Madhumathi and Nirmala Irudayam^13,14 introduced neutrosophic orbit topological spaces, and in 2022, orbit continuous mappings, these are new instruments that studies dynamical and parametric uncertainty via topological structures. In 2022, Atanassov established the intuitionistic fuzzy modal topological structure.², which incorporated modal operators into fuzzy logic and paved the way for new possibilities in reasoning under graded necessity and possibility. Broumi⁴ has enriched the conceptualization of generalized neutrosophic soft sets in terms of the better analytical formulation.

Kumaravel et al.¹² applied fuzzy and neutrosophic cognitive maps for disease diagnosis and COVID variant modeling by Murugesan et al.¹⁹ This relevance is highlighted with respect to the real-world issues today. From 2024 to 2025, Devi and Parthiban^7-9 have incorporated neutrosophic approach over soft topological constructs in decision making problem.

This study focuses on the extension of Neutrosophic Over Soft Topological Spaces (N_s^o-spaces) by introducing new generalized open sets, namely N_s^o α-open, N_s^o β-open, N_s^o-semi open, and N_s^o-pre open sets. Various types of continuity corresponding to these open sets are investigated, including N_s^o-continuous, N_s^o α-continuous, N_s^o β-continuous, semi-continuous, and pre-continuous functions. These enhancements provide a deeper understanding of the structural behavior of N_s^o-spaces. Additionally, a numerical application related to optimal catalyst selection is presented to demonstrate the effectiveness of the proposed framework. By systematically exploring the interrelationships among the newly defined open sets and their associated continuity types , this study establishes a comprehensive view of their mutual dependencies . Such an investigation reveals how each class of set and function interacts with the others , leading to a clearer understanding of the underlying topological framework . Nevertheless, This integrated perspective strengthens the theoretical foundations of neutrosophic over soft topological spaces and highlights previously unexplored structural properties . As a result , the study contributes to building a more robust and versatile mathematical structure for -spaces.

Preliminaries

This section provides the fundamental concepts and definitions necessary for understanding Neutrosophic Over Soft Set[9] and Neutrosophic Over Soft Topological Spaces[9].

Definition 2.1 [9] Let H be an non-empty set and ∈ be a set of parameter on H. Then N_s^o-set is defined by a set valued function

where ρ(H) is an set of all N_s^o-set on H. N_s^o-set is an valued function from the set of parameter E on H is defined as

Definition 2.2 [9] A N_s^o-set ⊳ ={ e,{〈h,0,0,Ω〉:h ∈ H}: e ∈ E} is said to be a Null N_s^o-set and ▶ = {e, {〈h,Ω,Ω,0〉: h ∈ H}:e ∈ E} is said to be an universal N_s^o-set.

Definition 2.3 [9] A neutrosophic over soft topology(N_s^o-topology) τ_Ns^o on non-empty set H such that

⊳,▶∈τ_Ns^o.

The union of an arbitrary collection τ_Ns^o is in τ_Ns^o.

The finite intersection of subsets τ_Ns^o is in τ_Ns^o.

Then H, τ_Ns^o is called neutrosophic over soft topological space(N_s^o-topological space). An element of τ_Ns^o is called an neutrosophic over soft open set(N_s^o-openset) and complement of τ_Ns^o neutrosophic over soft closed set(N_s^o-closedset).

Definition 2.4 [9] An operators of N_s^o-set J ∈ τ_Ns^o, then neutrosophic over soft topological closure and interior are cl_Ns^o (J) and int_Ns^o(J) is defined as:

Note: In this paper, the Neutrosophic Over Soft Set  is defined initially as

in terms of elements e ∈ E and corresponding maps involving h ∈ H. Yet, in the main work throughout, J is always represented as

emphasizing the basic relations within the framework.

Characterization of Open Set Types in Neutrosophic Over Soft Sets

Definition 3.1 Let  be a -topological space and let J be a N_s^o-set of H then J is said to be

N_s^oa-open if J ⊆ int_Ns^o(cl_Ns^o(int_Ns^o(J)))

N_s^o β-open if J ⊆ cl_Ns^o(int_Ns^o(cl_Ns^o(J)))

N_s^o-semi open if J ⊆ cl_Ns^o(intNso(J))

N_s^o-pre open if J ⊆ int_Ns^o(cl_Ns^o(J))

Proposition 3.2  

Every N_s^o-open set is N_s^o-open.

Every N_s^o-open set is N_s^o-open.

Every N_s^o-open set is N_s^o-semi open.

Every N_s^o-open set is N_s^o-pre open.

Proof. Let  be a N_s^o-open set in a N_s^o-topological space.

Since is N_s^o-open, we have J = int_Ns^o  (J). Now consider

Hence, J ⊆ int_Ns^o(cl_Ns^o(J)), which implies that J is N_s^oa- open.

Since J is N_s^o-open, as above, we have J = int_Ns^o (J). Then

since cl_Ns^o (J) ⊇ J and int_Ns^o (cl_Ns^o (J)) ⊇ J because J is open. Hence J is N_s^o β-open.

Since J is N_s^o-open, then int_Ns^o(J) = J and so cl_Ns^o(int_Ns^o(J)) = cl_Ns^o (J) ⊇ J hence J ⊆ cl_Ns^o (int_Ns^o(J)) which proves J is N_s^o-semi open.

Similarly, since J = int_Ns^o(J), and cl_Ns^o (J) ⊇ J, we have

Hence J ⊆ int_Ns^o(cl_Ns^o (J)),  so J is N_s^o-pre open.

Therefore, every N_s^o-open set is also N_s^oa-open, N_s^o β-open, N_s^o-semi open and N_s^o-pre open.

Proposition 3.3 Let H,T_Ns^o be a N_s^o-topological space. Then

The union of any family of N_s^oa-open sets is N_s^oa-open.

The union of any family of N_s^o β-open sets is N_s^o β-open.

The union of any family of N_s^o-semi open sets isN_s^o-semi open.

The union of any family of N_s^o-pre open sets is N_s^o-pre open.

Proof. Let {J_λ}_λ∈Λ be a family of N_s^oa-open sets in a N_s^o-topological space H,T_Ns^o.

By definition of N_s^oa-open set, for each λ ∈ Λ, we have:

Now consider the union J =_λ∈Λ J_λ.

To prove that J ⊆ int_Ns^o(cl_Ns^o(int_Ns^o(J))).

Since the interior and closure operators are monotone and preserve unions over open sets,

we get:

Applying the interior operator again:

Now, since J_λ ⊆ int_Ns^o(cl_Ns^o(int_Ns^o(J))). for each λ, we have:

Hence,

Therefore, J is N_s^oα-open.

Let {J_λ}_λ∈Λ be a family of N_s^o β-open sets in a N_s^o-topological space (H, τ _Ns^o).

By definition of N_s^o β -open sets, for each λ ∈ Λ, we have:

Let J =_λ∈Λ  J_λ. We want to show:

First, since J =_λ∈Λ  J_λ, and the closure operator is monotonic and preserves unions:

Applying the interior operator (which is also monotonic):

Now apply closure again:

Therefore:

Hence,

Therefore, J is N_s^o β -open. For iii and iv It is obviously true.

Proposition 3.4 Let  be -topological space then

⊳ and ▶ are N_s^oα-open sets.

⊳ and ▶ are N_s^o β -open sets.

⊳ and ▶ are N_s^o-semi open sets.

⊳ and ▶ are N_s^o-pre open sets.

Proof. i. By definition of a N_s^o-topological space, the N_s^o null set ⊳  and the N_s^o universal set ▶ are elements of τ _Ns^o . Hence, they are N_s^o -open sets.

We now verify whether these sets satisfy the condition for being  N_s^oα-open. That is, we need to check whether:

for J = ⊳ and J = ▶.

Case 1: Let J = ⊳.

Since ⊳ is open, we have:

(because closure of the empty set is empty in any topological space)

Hence,

Therefore, ⊳ is N_s^oα-open.

Case 2: Let J = ▶.

Since ▶ is open, we have:

Hence,

Therefore, ▶ is N_s^oα-open.

ii, iii and iv are obviously true.

Proposition 3.5 Every N_s^oα-open set is N_s^o-pre open.

Proof. Let ((H, τ_Ns^o) be a N_s^o-topological space and let J be a N_s^oα-open set.

By definition of N_s^oα-open set, we have:

Let us denote A = int_Ns^o(J) .

Since A ⊆ J, it follows that:

Now apply the interior operator on both sides:

From 3.1 and 3.2

This is precisely the definition of a N_s^o-pre open set. Hence, every N_s^oα-open set is N_s^o-pre open.

Remark 3.6 Converse of proposition3.5 is need not to be true which is proven by the example 3.7

Example 3.7 Example:  Let H = {x₁, x₂, x₃} and define a N_s^o-topology on H by:

where

Define the neutrosophic over soft set:

where

From 3.3 and 3.4

From (3.5) and (3.6)

∴ K is not N_s^oα-open

Proposition 3.8 Every N_s^oα-open set is N_s^o-semi open.

Proof. Let (H, τ_Ns^o ) be a neutrosophic over soft topological space.

Let J be a N_s^oα-open set. By definition of N_s^oα-open set, we have:

Now let us denote K = int_Ns^o (J), which is a N_s^o-open set since the interior of any set in a topology is open by definition.

So we have:

Since K is open, its closure cl_Ns^o (K) is a superset of K and thus contains all points that are limit points or in K.

The interior of the closure, int_Ns^o (cl_Ns^o (K)), is therefore an open set in τ_Ns^o that contains J.

Now, recall the definition of a N_s^o-semi open set: A set J is said to be semi open if

But from the assumption that J is N_s^oα-open, we already have:

(because any set is contained in its interior of the closure implies it’s also in the closure itself).

Since K = int_Ns^o (J), it follows that:

which is the definition of a N_s^o-semi open set.

Hence, every N_s^oα-open set is indeed N_s^o-semi open.

Example 3.9 Let H = {x₁, x₂, x₃} and define a N_s^o-topology on H by:

where

Let the neutrosophic over soft set be:

Clearly,

So,

However,

∴ M is N_s^o-semi open but not N_s^oα-open

Proposition 3.10 Every N_s^oβ-open set is N_s^o-semi open.

Proof. Let (H, τ_Ns^o ) be a N_s^o-topological space, and let J be a N_s^oβ-open set.

By definition of N_s^oβ-openness, we have:

Now observe that:

Then,

Since

and from the above inclusion, we conclude that:

This is precisely the condition for J to be N_s^o-semi open.

Hence, every N_s^oβ-open set is N_s^o-semi open.

Example 3.11 Let {H, τ_Ns^o} and define a N_s^o-topology on  as:

where

Define the neutrosophic over soft set:

We observe:

Thus,

But now consider:

Since

we conclude that

Proposition 3.12 Every -N_s^oβ-open set is N_s^o-pre open.

Proof. Let {H, τ_Ns^o} be a neutrosophic over soft topological space. Let J be a N_s^oβ-open set in this space. By definition of a N_s^oβ-open set, we have:

On the other hand, a N_s^o-pre open set J is defined as:

Since the interior operator is monotonic, meaning that for any sets A ⊆ B, we have

and since closure is extensive, i.e., J ⊆ cl_Ns^o (J), applying closure and then interior again will only shrink or retain the set:

So if J ⊆ cl_Ns^o (int_Ns^o (cl_Ns^o (J))), then it follows that

which means that J is also N_s^o-pre open.

Example 3.13 Let  be a universe, and define a -topology by:

where

Now define the neutrosophic over soft set:

We observe:

So,

Now check:

Hence,

Definition 3.14 Let (H₁,τ ¹_Ns^o)  and (H₂,τ ²_Ns^o)  be two -topological spaces. A mapping f: H₁ → H₂  is said to be: 

N_s^o-continuous Function if the pre-image of every N_s^o-open set in (H₂,τ ²_Ns^o) is a

N_s^oα-continuous if the pre-image of every N_s^oα-open set in (H₂,τ ²_Ns^o) is a

N_s^o β-continuous if the pre-image of every N_s^o β-open set in (H₂,τ ²_Ns^o) is a

N_s^o-semi continuous if the pre-image of every N_s^o-semi open set in (H₂,τ ²_Ns^o) is a

N_s^o-pre continuous if the pre-image of every N_s^o-pre open set in (H₂,τ ²_Ns^o) is a

Proposition 3.15  

Every N_s^o-continuous function is N_s^oα-continuous.

Every N_s^o-continuous function is N_s^o β-continuous.

Every N_s^o-continuous function is N_s^o-pre continuous.

Every N_s^o-continuous function is N_s^o-semi continuous.

Proof. Let f: (H₁, τ_Ns^o) → (H₂, τ_Ns^o)  be a N_s^o-continuous function.

This means for every N_s^o-open set O in H₂, the preimage f^-1 (O) is a N_s^o-open set in H₁.

N_s^oα-continuity:

Let A be a N_s^oα-open set in H₂, i.e.,

Then

Using continuity of f, inverse image distributes over interior and closure:

which shows that f^-1 (A)  is N_s^oα-open. Hence, f is N_s^oα-continuous.

N_s^o β-continuity:

Let B be a N_s^o β-open set in H₂, i.e.,

Then

which implies f is N_s^o β-continuous.

N_s^o-pre continuity:

Let P be N_s^o-pre open in H₂,, i.e.,

Then

showing that f^-1 (P) is N_s^o-pre open and hence f is N_s^o-pre continuous.

N_s^o-semi continuity:

Let S be a N_s^o-semi open set in H₂, i.e.,

Then

so f is N_s^o-semi continuous.

Thus, f is N_s^oα-, β-, pre-, and semi-continuous.

Proposition 3.16 Every N_s^oα-continuous function is N_s^oβ-continuous.

Proof. Let f: (H₁, τ_Ns^o) → (H₂, τ_Ns^o) be a N_s^oα-continuous function. Let A be a N_s^oβ-open set in H₂.

Then by definition of N_s^oβ-open set, we have:

i.e.,

Take the preimage of both sides:

Using the assumption that f is N_s^oα-continuous, and the fact that preimage commutes with interior and closure, we get:

that is,

which means f^-1 (A) is a N_s^oβ-open set in H₁.

Hence, f is N_s^oβ-continuous.

Proposition 3.17 Every N_s^oβ-continuous function is N_s^o-pre continuous.

Proof. Let f: (H₁, τ_Ns^o) → (H₂, τ_Ns^o) be a N_s^oβ-continuous function.

Let P be a N_s^o-pre open set in H₂.

Then by the definition of N_s^o-pre open set, we have:

Now consider the preimage of both sides:

Since f is N_s^oβ-continuous, and since p is a subset of a composition involving a N_s^oβ-open set, we can apply:

Hence, f^-1 (P) is N_s^o-pre open in H₁, which shows that f is N_s^o-pre continuous.

Proposition 3.18 The composition of two N_s^o-continuous functions is N_s^o-continuous.

Proof. Let (H₁, τ ¹_Ns^o), (H₂, τ ²_Ns^o) and (H₃, τ ³_s^o) be N_s^o–-topological spaces.

Let f:H₁ → H₂ and f:H₂ → H₃ be two N_s^o-continuous functions.

We must show that the composition g ∘ f: H₁ → H₃ is also N_s^o-continuous.

Let  be any N_s^o-open set in H₃, i.e., O ∈ τ³_Ns^o.

Since g is N_s^o-continuous, we have

Since f is N_s^o-continuous, the preimage under f of this set is also N_s^o-open in H₁, i.e.,

Therefore, the preimage of every N_s^o-open set in H₃ under g ∘ f is N_s^o-open in H₁.

Hence, g ∘ f is N_s^o-continuous.

Proposition 3.19 The composition of two N_s^oα-continuous (respectively, β-, semi-, or pre-continuous) functions is again N_s^oα-continuous (respectively, β-, semi-, or pre-continuous).

Proof. Let (H₁, τ ¹_Ns^o), (H₂, τ ²_Ns^o) and (H₃, τ ³_s^o)  be N_s^o-topological spaces.

Let f: H₁ → H₂ and g: H₂ → H₃ be both N_s^oα-continuous (respectively, β-, semi-, or pre-continuous).

We aim to show that the composition g ∘ f: H₁ → H₃ is also N_s^oα-continuous (respectively, β-, semi-, or pre-continuous).

Let U be a N_s^oα-open (respectively, β-, semi-, or pre-open) set in H₃.

Since g is N_s^oα-continuous (respectively, β-, semi-, or pre-continuous), we have

Since f is N_s^oα-continuous (respectively, β-, semi-, or pre-continuous), we have

Hence, the composition g ∘ f is N_s^oα-continuous (respectively, β-, semi-, or pre-continuous).

Proposition 3.20 If  f is a constant function between any two N_s^o-topological spaces, then f is N_s^o-continuous.

Proof. Let (H₁, τ ¹_Ns^o) and (H₂, τ ²_Ns^o)  be two N_s^o-topological spaces, and let f: H₁ → H₂ be a constant function.

Then, there exists an element a ∈ H₂ such that f(h) = a for all H₁.

Now, let O ∈ τ ²_Ns^o be any N_s^o-open set in H₂.

We consider two cases:

If a ∉ O, then f ^-1 (O) = ⊳, which is N_s^o-open in H₁.

If a ∈ O, then f ^-1 (O) = H₁, which is also N_s^o-open in H₁.

In both cases, f ^-1 (O) ∈ τ ²_Ns^o, so f is N_s^o-continuous.

Proposition 3.21 A bijective mapping f: (H₁, τ ¹_Ns^o) → (H₂, τ ²_Ns^o)  is a N_s^o-homeomorphism if both f  and f ^-1 are N_s^o-continuous.

Proof. Let f: (H₁,τ ¹_Ns^o) → (H₂,τ ²_Ns^o)  be a bijective mapping.

Assume f is N_s^o-continuous. That is, for every O ∈ τ ²_Ns^o, we have:

Also, assume f ^-1: H₂ → H₁ is N_s^o-continuous. Then, for every U ∈  τ ¹_Ns^o, we have:

Since f is a bijection and both f and f ^-1 are N_s^o-continuous, the mapping f establishes a one-to-one, onto correspondence between the N_s^o-open sets of H₁ and H₂.

Hence, f is a N_s^o-homeomorphism.

Proposition 3.22  

A bijective mapping f: (H₁, τ ¹_Ns^o) → (H₂, τ ²_Ns^o) is said to be a N_s^oα-homeomorphism if both f and f ^-1 are N_s^oα-continuous,

A bijective mapping f: (H₁, τ ¹_Ns^o) → (H₂, τ ²_Ns^o) is said to be a N_s^oβ-homeomorphism if both f and f ^-1 are N_s^oβ-continuous,

A bijective mapping f: (H₁, τ ¹_Ns^o) → (H₂, τ ²_Ns^o) is said to be a N_s^o-semi homeomorphism if both f and f ^-1 are N_s^o-semi continuous,

A bijective mapping f: (H₁, τ ¹_Ns^o) → (H₂, τ ²_Ns^o) is said to be a N_s^o-pre homeomorphism if both f and f ^-1 are N_s^o-pre continuous.

Proof. i. Let f: (H₁, τ ¹_Ns^o) → (H₂, τ ²_Ns^o) be a bijective function.

We consider case (i) for N_s^oα-continuity (other cases follow similarly):

Suppose f and f ^-1 are both N_s^oα-continuous. Then for any O₂ ∈ τ ²_Ns^o, we have:

where τ ¹_Ns^o denotes the collection of all N_s^oα-open sets in H₁.

Similarly, for any O₁ ∈ τ ²_Ns^o,

Hence, f establishes a one-to-one correspondence between N_s^oα-open sets of H₁ and H₂. Therefore, f is a N_s^oα-homeomorphism.

The proofs for ii, iii, and iv follow similarly by replacing α-open sets with β-, semi-, and pre-open sets respectively.

Numerical Application: Selection of Optimal Catalyst for Organic Reaction Using Neutrosophic Over Soft Topological Spaces

Choosing the best catalyst is really important in organic synthesis of large-volume industrial reactions with many other factors in play, like reaction speed, yield, cost, recycling ability, and environmental consideration. Catalysts have an effect not just on the conversion efficiency of raw materials into the desired product, but also on the doability of the process in economic terms, and, of course, its sustainability. However, in real-world laboratory and industrial environments, data with concern to the performance of a catalyst mostly comes as imprecise and incomplete, and then subjected to human interpretation. Expert chemists may point to different directions of efficiency and long-term usability; there can be uncertainty with experimental conditions; and the possible behavior of a catalyst can differ even under slightly changed conditions.

It is known that conventional mathematical models and classical logic based frameworks fail to address this inherent uncertainty and vagueness. Hence, interfacing neutrosophic logic, which constitutes truthiness, falsity, and indeterminate information, with soft topological structures is a robust instrument for making decisions. Neutrosophic over soft topological spaces give a flexibility and intelligent way of evaluating alternatives under imprecise or uncertain conditions since such spaces allow not only binary membership but also partial and ambiguous truths associated with performance parameters. This methodology is particularly important in common applications in chemical and pharmaceutical industries, where a good catalyst might cause losses of material, energy, and time if it is not effective. Decision-makers can assess catalysts through this framework by multidimensional perspectives of degrees of truth (effectiveness), indeterminacy (experimental variability), and falsity (instability or toxicity), leading to more well-informed, comprehensive, and realistic decisions.

Perhaps, the methodology will facilitate the ranking of numerous candidates in terms of score functions which normalize and assign different weights to various aspects of uncertainty in situations where classical data analysis would fail. Consequently, the described neutrosophic over soft topology approach is mathematically sound and practically relevant, given the rise in interest of intelligent uncertain models in green chemistry and process optimization. By aiding researchers to select catalysts that are effective, safe, and sustainable, the proposed methodology will favour the advancement of cleaner technologies; knowledge-driven eco-conscious chemical manufacturing would further gain from this.

Consider the hydrogenation of alkenes, a fundamental organic reaction where alkenes are converted into alkanes using a metal catalyst and hydrogen gas. The general reaction is:

The choice of catalyst significantly influences the rate, yield, and sustainability of the reaction.

Let H = {h₁, h₂, h₃, h₄} be a set of catalysts:

h₁: Nickel (Ni)

h₂: Platinum (Pt)

h₃: Palladium (Pd)

h₄: Copper (Cu)

Define a neutrosophic over soft set J on H as:

Each triple {〈h, ℵ_J (h), ð_J (h), Υ_J (h)〉} denotes the degrees of:

Truth-membership (effectiveness/yield)

Indeterminacy-membership (experimental uncertainty)

Falsity-membership (side effects/cost/instability)

Now define a neutrosophic over soft topology on H:

Where:

Then  forms a neutrosophic over soft topological space.

Normalized Score Function for Catalyst Ranking

To prioritize the best catalyst, define a score function:

Raw Score:

Raw Scores:

Normalization:

Figure 1: Graphical Representation of Ranking of Elements. Click here to View Figure

 Ranking Based on Normalized Scores

This line graph0 illustrates the ranking of four elements based on their corresponding scores. Among them,  achieves the highest score of , followed by  with ,  with , and  with the lowest score of . The graph presents these values in descending order using distinct markers and labels for clarity. This visual representation helps in easily identifying the most optimal element among the given set.

Discussion

The results indicate that ​ is the most optimal element based on the normalized scores, followed by and​. This ranking aligns well with the expected performance hierarchy of the elements under the given criteria. The clear distinction in scores supports the effectiveness of the proposed neutrosophic over soft topological approach for reliable decision-making in uncertain environments.

Conclusion

An interesting advancement in the framework of Neutrosophic Over Soft Topological Spaces or -spaces delineated in the below work is the introduction and investigation of different generalized open sets such as -, -, semi-, and pre-open sets. Together with these, the corresponding types of continuity functions have also been defined, backed up by some propositions explaining their links. This theoretical insight into the new types of open sets enriches the already existing topological structure under  with potential development of finer classification and mapping schemes in uncertain situations. Moreover, various numerical instances like that of optimal catalyst selection for organic reactions are included in order to show practical usefulness of the newly proposed concepts, thus demonstrating relevance for application and real-life decision-making within the extended framework obtained.

Acknowledgement

This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.

Funding Sources

The author(s) received no financial support for the research, authorship, and/or publication of this article.

Conflict of Interest

The author(s) do not have any conflict of interest.

Data Availability Statement

This statement does not apply to this article.

Ethics Statement

This research did not involve human participants, animal subjects, or any material that requires ethical approval.

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