Homotopy Theory via Neutrosophic Multi-Valued Functions

1. Introduction

Classical homotopy theory studies continuous mappings

$$f:X⟶Y$$

and identifies two maps when one can be continuously deformed into the other. This fundamental idea leads to homotopy groups, fundamental groupoids, homology, cohomology, and the homotopy category of topological spaces.

In many applications, however, a point does not have a unique possible image. Instead, one may have a collection of possible images. This leads to a multi-valued function

$$F:X⊸Y,x⟼F\left(x\right)\subseteq Y.$$

Multi-valued mappings have been extensively studied in topology, selection theory, fixed point theory, and nonlinear analysis. Classical developments include the theories of Vietoris maps, admissible mappings, selectors, and multivalued homotopy [9,10].

A different extension of classical mathematics is provided by neutrosophic set theory. Neutrosophic sets associate with an object three components describing degrees of truth, indeterminacy, and falsity. The neutrosophic approach was introduced by Smarandache and subsequently developed in numerous directions, including topology [1,2,3].

Several authors have developed neutrosophic topological structures, including neutrosophic crisp topologies and neutrosophic continuous functions [4,5]. The relationship between neutrosophic sets and topology has also been investigated from several viewpoints [6].

More recently, Jafari, Rajesh, and Nordo introduced basic notions of neutrosophic homotopic functions and neutrosophic homotopy [8]. The present paper develops a related but different direction: instead of considering only single-valued neutrosophic mappings, we combine neutrosophic information with genuinely multi-valued mappings.

The principal object of the paper is therefore a neutrosophic multi-valued function. Informally, such a mapping assigns to each point $x\in X$ a nonempty family of possible points of Y, together with three numerical profiles indicating the truth, indeterminacy, and falsity associated with the possible images.

Our purpose is not to claim that this construction is the only possible definition. Rather, we introduce a mathematically explicit model and study its basic homotopy theory.

The main questions are:

1.

How should continuity be defined for neutrosophic multi-valued functions?

2.

What should a neutrosophic homotopy between two such functions be?

3.

Can admissible neutrosophic multi-valued functions induce homology morphisms?

4.

Can one construct a neutrosophic homotopy category?

5.

Under what conditions do classical and neutrosophic homotopy theories coincide?

2. Neutrosophic Sets

We first recall the basic neutrosophic framework.

Definition 2.1.

Let X be a nonempty set. A neutrosophic set A in X is described by three functions

$$T_A,I_A,F_A:X⟶[0,1],$$

where

$$T_A\left(x\right),I_A\left(x\right),F_A\left(x\right)$$

represent respectively the degrees of truth, indeterminacy, and falsity of x with respect to A.

In the original neutrosophic philosophy, these components need not satisfy the same restrictions imposed in intuitionistic fuzzy set theory. This independence of the three components is one of the characteristic features of the neutrosophic approach [1,2].

For the purposes of this paper, we work with the standard unit interval representation.

Definition 2.2.

A single-valued neutrosophic set on X is a triple

$$A=(T_A,I_A,F_A)$$

of functions

$$T_A,I_A,F_A:X⟶[0,1].$$

The value of A at x will be written

$$A\left(x\right)=\left(T_A\left(x\right),I_A\left(x\right),F_A\left(x\right)\right).$$

Neutrosophic topology extends classical topological ideas by allowing open sets and continuity to be expressed in terms of such neutrosophic structures. Various formulations of neutrosophic topology have been investigated in the literature [3,6].

3. Neutrosophic Multi-Valued Functions

We now combine set-valued mappings and neutrosophic information.

Definition 3.1.

Let X and Y be nonempty sets. A neutrosophic multi-valued function from X to Y is a mapping

$$F:X{⊸}_{N}Y$$

such that, for every $x\in X$,

$$F\left(x\right)=\left(M_F\left(x\right);T_F(x,·),I_F(x,·),F_F(x,·)\right),$$

where

$$⌀\ne M_F\left(x\right)\subseteq Y$$

and

$$T_F,I_F,F_F:\{(x,y):y\in M_F\left(x\right)\}⟶[0,1].$$

Thus $M_F\left(x\right)$ is the set of possible images of x, while

$$T_F(x,y),I_F(x,y),F_F(x,y)$$

describe the truth, indeterminacy, and falsity degrees of the statement

$$“y\mathrm{is}\mathrm{an}\mathrm{image}\mathrm{of}x\mathrm{under}F\text{”}.$$

Remark 3.2.

If

$$T_F(x,y)=1,I_F(x,y)=0,F_F(x,y)=0$$

for every $y\in M_F\left(x\right)$, then the neutrosophic information becomes classical set-valued information. If in addition every $M_F\left(x\right)$ is a singleton, then we recover an ordinary function.

Example 3.3.

Let

$$X=Y=[0,1]$$

and define

$$M_F\left(x\right)=\{x,1-x\}.$$

Suppose

$$T_F(x,x)=0.9,I_F(x,x)=0.1,F_F(x,x)=0.0,$$

while

$$T_F(x,1-x)=0.7,I_F(x,1-x)=0.2,F_F(x,1-x)=0.1.$$

Then F is a neutrosophic two-valued mapping.

4. Neutrosophic Graphs

The graph of a classical multi-valued map is

$$Gr\left(M_F\right)=\{(x,y)\in X\times Y:y\in M_F\left(x\right)\}.$$

For a neutrosophic multi-valued function we enrich the graph by the three neutrosophic components.

Definition 4.1.

The neutrosophic graph of

$$F:X{⊸}_{N}Y$$

is the quadruple

$${Gr}_N\left(F\right)=\left(Gr\left(M_F\right),T_F,I_F,F_F\right).$$

This gives a useful way of separating the set-valued part from the neutrosophic part.

Proposition 4.2.

Let

$$F:X{⊸}_{N}Y.$$

If

$$I_F(x,y)=0$$

and

$$F_F(x,y)=1-T_F(x,y)$$

for all $(x,y)\in Gr\left(M_F\right)$, then the neutrosophic information is determined by a single membership function.

Proof. 

The second and third components are completely determined by $T_F$. Hence only one independent numerical function remains. □

5. Neutrosophic Continuity

We now introduce a useful continuity concept.

Definition 5.1.

Let

$$F:X{⊸}_{N}Y$$

be a neutrosophic multi-valued function. We say that F is neutrosophically upper semicontinuous at $x_0\in X$ if:

1.

the underlying set-valued map $M_F$ is upper semicontinuous at $x_0$;

2.

whenever $y_0\in M_F\left(x_0\right)$ and $y_n\in M_F\left(x_n\right)$ with $x_n\to x_0$ and $y_n\to y_0$, then

$$\underset{n\to \infty }{\mathrm{lim\; inf}}{T}_F(x_n,y_n)\ge T_F(x_0,y_0),$$

$$\underset{n\to \infty }{\mathrm{lim\; sup}}{I}_F(x_n,y_n)\le I_F(x_0,y_0),$$

and

$$\underset{n\to \infty }{\mathrm{lim\; sup}}{F}_F(x_n,y_n)\le F_F(x_0,y_0).$$

The inequalities express the principle that truth should not disappear abruptly, while indeterminacy and falsity should not increase abruptly under a suitable upper-continuity convention.

Other conventions are possible. The choice above is designed to extend ordinary upper semicontinuity while respecting the three-component neutrosophic structure.

Definition 5.2.

A neutrosophic multi-valued function is called neutrosophically continuous if it is neutrosophically upper semicontinuous and its underlying set-valued map is lower semicontinuous.

Remark 5.3.

Neutrosophic continuity is not unique. Alternative definitions can be obtained from a chosen neutrosophic topology, for example by requiring the inverse image of every neutrosophically open set to be neutrosophically open. Such approaches are compatible with the general literature on neutrosophic topological spaces [3,4,6].

6. Neutrosophic Homotopy

We now reach the central concept.

Definition 6.1.

Let

$$F,G:X{⊸}_{N}Y$$

be neutrosophic multi-valued functions. A neutrosophic homotopy from F to G is a neutrosophic multi-valued function

$$H:X\times [0,1]{⊸}_{N}Y$$

such that

$$H(x,0)=F\left(x\right)$$

and

$$H(x,1)=G\left(x\right)$$

for every $x\in X$.

We write

$$F{\simeq }_{N}G.$$

The underlying set-valued part of H is a multi-valued homotopy, while its neutrosophic components vary continuously with the parameter t.

More explicitly, write

$$H(x,t)=\left(M_H(x,t);T_H(x,t,·),I_H(x,t,·),F_H(x,t,·)\right).$$

Then the endpoint conditions mean

$$M_H(x,0)=M_F\left(x\right),M_H(x,1)=M_G\left(x\right),$$

and

$$T_H(x,0,y)=T_F(x,y),$$

$$I_H(x,0,y)=I_F(x,y),$$

$$F_H(x,0,y)=F_F(x,y),$$

with analogous equalities at $t=1$.

7. Basic Properties of Neutrosophic Homotopy

Theorem 7.1.

The relation

$${\simeq }_N$$

is an equivalence relation on the class of neutrosophically continuous multi-valued functions.

Proof. 

We verify reflexivity, symmetry, and transitivity.

For reflexivity, define

$$H(x,t)=F\left(x\right)$$

for all $(x,t)\in X\times [0,1]$. Thus

$$F{\simeq }_{N}F.$$

For symmetry, suppose H is a homotopy from F to G. Define

$$\overline{H}(x,t)=H(x,1-t).$$

Then

$$\overline{H}(x,0)=G\left(x\right)$$

and

$$\overline{H}(x,1)=F\left(x\right).$$

Therefore

$$G{\simeq }_{N}F.$$

For transitivity, suppose $H_1$ is a homotopy from F to G and $H_2$ is a homotopy from G to K. Define

$$H(x,t)=\left\{\begin{array}{cc}{H}_1(x,2t),\hfill & 0\le t\le {\displaystyle \frac{1}{2}},\hfill \\ H_2(x,2t-1),\hfill & {\displaystyle \frac{1}{2}}\le t\le 1.\hfill \end{array}\right.$$

At $t=1/2$ the two definitions agree because

$$H_1(x,1)=G\left(x\right)=H_2(x,0).$$

Hence

$$F{\simeq }_{N}K.$$

□

8. Neutrosophic Selectors

Selectors are important in ordinary multi-valued topology. We now extend them to the neutrosophic setting.

Definition 8.1.

Let

$$F:X{⊸}_{N}Y.$$

A continuous function

$$f:X⟶Y$$

is called a neutrosophic selector of F if

$$f\left(x\right)\in M_F\left(x\right)$$

for every $x\in X$.

One may additionally associate the selector with the neutrosophic profile

$$x⟼\left(T_F(x,f\left(x\right)),I_F(x,f\left(x\right)),F_F(x,f\left(x\right))\right).$$

Definition 8.2.

A selector f is called maximally truthful if

$$T_F(x,f\left(x\right))=\underset{y\in M_F\left(x\right)}{\sup }{T}_F(x,y)$$

for every $x\in X$.

Similarly, one may define a minimally indeterminate selector by minimizing $I_F$, or a minimally false selector by minimizing $F_F$.

Remark 8.3.

These selector notions do not assert that such selectors always exist. Their existence is a selection problem and is related to the classical theory of continuous selections [11].

9. Neutrosophic Admissible Functions

The theory of admissible multi-valued mappings provides a useful framework for assigning algebraic invariants to set-valued maps [9,10]. We now add the neutrosophic information.

Definition 9.1.

A neutrosophic multi-valued function

$$F:X{⊸}_{N}Y$$

is called neutrosophically admissible if there exist a topological space Γ, a Vietoris map

$$p:\mathsf{\Gamma }⟶X$$

and a continuous map

$$q:\mathsf{\Gamma }⟶Y$$

such that

$$M_F\left(x\right)=q\left(p^{-1}\left(x\right)\right)$$

for every $x\in X$, together with neutrosophic functions

$$T_F,I_F,F_F$$

defined continuously on the corresponding representation.

The triple

$$(X\stackrel{p}{\leftarrow }\mathsf{\Gamma }\stackrel{q}{\to }Y;T_F,I_F,F_F)$$

is called a neutrosophic Vietoris representation of F.

The ordinary Vietoris representation is recovered when all neutrosophic information is trivial.

10. Neutrosophic Vietoris Maps

Recall that a Vietoris map is a continuous surjection

$$p:\mathsf{\Gamma }\to X$$

with suitable compactness and acyclicity conditions on its fibers [9,10].

Definition 10.1.

A neutrosophic Vietoris representation of

$$F:X{⊸}_{N}Y$$

is a diagram

$$X\stackrel{p}{\leftarrow }\mathsf{\Gamma }\stackrel{q}{\to }Y$$

where p is a Vietoris map, together with three continuous functions

$$T,I,F:\mathsf{\Gamma }⟶[0,1].$$

The three functions describe the neutrosophic status of a point of the representation.

Proposition 10.2.

If

$$X\stackrel{p}{\leftarrow }\mathsf{\Gamma }\stackrel{q}{\to }Y$$

is a Vietoris representation of the underlying multi-valued map, then the additional neutrosophic data do not alter the classical Vietoris property of p.

Proof. 

The Vietoris property depends on the topology of $\mathsf{\Gamma }$, the map p, and its fibers. The functions $T,I,F$ are additional structure and therefore do not change whether p is a Vietoris map. □

This separation is useful because the algebraic-topological part can be treated by ordinary homological methods, while the neutrosophic part records uncertainty and indeterminacy.

11. Induced Homology Maps

Suppose

$$F:X{⊸}_{N}Y$$

has a neutrosophic Vietoris representation

$$X\stackrel{p}{\leftarrow }\mathsf{\Gamma }\stackrel{q}{\to }Y.$$

By the Vietoris theorem, under the usual hypotheses,

$$p_{*}:H_{*}\left(\mathsf{\Gamma }\right)⟶H_{*}\left(X\right)$$

is an isomorphism. We may therefore define

$$F_{*}=q_{*}\circ p_{*}^{-1}.$$

Definition 11.1.

The underlying induced homology map of an admissible neutrosophic multi-valued function F is

$$F_{*}=q_{*}\circ p_{*}^{-1}.$$

The neutrosophic data may then be regarded as additional information attached to this homological morphism.

For example, define the three numerical summaries

$${\mathsf{\Theta }}_T\left(F_{*}\right),{\mathsf{\Theta }}_I\left(F_{*}\right),{\mathsf{\Theta }}_F\left(F_{*}\right)$$

by suitable integrals, suprema, or averages of the corresponding neutrosophic profiles. The precise choice depends on the intended application.

Remark 11.2.

The underlying homology map is classical, while the neutrosophic profile supplies additional information. This avoids changing the definition of singular homology itself.

12. Homotopy Invariance of Homology

Theorem 12.1.

Let

$$F,G:X{⊸}_{N}Y$$

be neutrosophically admissible maps. If

$$F{\simeq }_{N}G,$$

then

$$F_{*}=G_{*}$$

whenever the homological realization is independent of the chosen Vietoris representations.

Proof. 

A neutrosophic homotopy between F and G gives, after forgetting the numerical neutrosophic components, an ordinary multi-valued homotopy between the underlying admissible maps.

The standard homotopy invariance argument for admissible mappings then gives equality of the induced homology morphisms. The neutrosophic components vary continuously throughout the homotopy and therefore do not change the underlying homological morphism. □

This result is the fundamental bridge between neutrosophic homotopy and ordinary algebraic topology.

13. Composition

Let

$$F:X{⊸}_{N}Y$$

and

$$G:Y{⊸}_{N}Z.$$

The underlying multi-valued composition is

$$(G\circ F)\left(x\right)=\bigcup _{y\in M_F\left(x\right)}{M}_G\left(y\right).$$

For the neutrosophic components we define, for example,

$$T_{G\circ F}(x,z)=\underset{y\in M_F\left(x\right)}{\sup }\min \{T_F(x,y),T_G(y,z)\},$$

$$I_{G\circ F}(x,z)=\underset{y\in M_F\left(x\right)}{\inf }\max \{I_F(x,y),I_G(y,z)\},$$

and

$$F_{G\circ F}(x,z)=\underset{y\in M_F\left(x\right)}{\inf }\max \{F_F(x,y),F_G(y,z)\}.$$

These formulas are one possible neutrosophic composition rule; other N-norm/N-conorm structures can be considered. Smarandache has studied N-norms and N-conorms in connection with neutrosophic sets and neutrosophic topologies [2].

Proposition 13.1.

Under the above composition convention, the neutrosophic composition is compatible with the ordinary set-valued composition.

Proof. 

The underlying set of possible images is precisely

$$\bigcup _{y\in M_F\left(x\right)}{M}_G\left(y\right).$$

The three additional functions assign neutrosophic degrees to elements of this union according to the specified aggregation rules. □

14. Homotopy and Composition

Theorem 14.1.

Suppose

$$F_0{\simeq }_N{F}_1:X{⊸}_{N}Y$$

and

$$G_0{\simeq }_N{G}_1:Y{⊸}_{N}Z.$$

Assume that all four maps and their compositions are admissible. Then

$$G_0\circ F_0{\simeq }_N{G}_1\circ F_1.$$

Proof. 

Let

$$H_F:X\times [0,1]{⊸}_{N}Y$$

be a homotopy from $F_0$ to $F_1$ and let

$$H_G:Y\times [0,1]{⊸}_{N}Z$$

be a homotopy from $G_0$ to $G_1$.

Define a first homotopy by

$$K_1(x,t)=G_0\circ H_F(x,t)$$

and a second by

$$K_2(x,t)=H_G(·,t)\circ F_1\left(x\right).$$

Concatenating $K_1$ and $K_2$ gives a neutrosophic homotopy from

$$G_0\circ F_0$$

to

$$G_1\circ F_1.$$

□

15. Neutrosophic Homotopy Equivalence

Definition 15.1.

Topological spaces X and Y are called neutrosophically homotopy equivalent if there exist admissible neutrosophic multi-valued functions

$$F:X{⊸}_{N}Y,G:Y{⊸}_{N}X$$

such that

$$G\circ F{\simeq }_N{id}_X$$

and

$$F\circ G{\simeq }_N{id}_Y.$$

We write

$$X{\simeq }_{N}Y.$$

Theorem 15.2.

Neutrosophic homotopy equivalence is an equivalence relation on the class of spaces for which admissible neutrosophic compositions are defined.

Proof. 

Reflexivity follows from the ordinary identity map with trivial neutrosophic data.

Symmetry follows by reversing the two maps.

For transitivity, suppose

$$X{\simeq }_{N}Y$$

via $F,G$ and

$$Y{\simeq }_{N}Z$$

via $P,Q$. Then

$$P\circ F:X{⊸}_{N}Z$$

and

$$G\circ Q:Z{⊸}_{N}X$$

provide the required maps. Associativity and compatibility of neutrosophic homotopy with composition give the required homotopies to the identity. □

16. Neutrosophic Products

Let

$$F:X{⊸}_{N}Y$$

and

$$G:Z{⊸}_{N}W.$$

Define

$$F\times G:X\times Z{⊸}_{N}Y\times W$$

by

$$M_{F\times G}(x,z)=M_F\left(x\right)\times M_G\left(z\right).$$

For $(y,w)$ in this set define

$$T_{F\times G}((x,z),(y,w))=\min \{T_F(x,y),T_G(z,w)\},$$

$$I_{F\times G}((x,z),(y,w))=\max \{I_F(x,y),I_G(z,w)\},$$

and

$$F_{F\times G}((x,z),(y,w))=\max \{F_F(x,y),F_G(z,w)\}.$$

Proposition 16.1.

If F and G are neutrosophically continuous, then their product is neutrosophically continuous under the corresponding product topology and product aggregation rules.

Proof. 

The underlying set-valued map is the product of the underlying set-valued maps. The continuity of the three numerical components follows from continuity of the component functions and continuity of the minimum and maximum operations on ${[0,1]}^2$. □

17. The Neutrosophic Homotopy Category

The preceding constructions suggest a categorical formulation.

Let

$$\mathbf{NMultTop}$$

denote the category whose objects are suitable topological spaces and whose morphisms are admissible neutrosophic multi-valued functions.

Definition 17.1.

Theneutrosophic homotopy categoryis the category

$$\mathbf{HoNMultTop}$$

whose objects are those of $\mathbf{NMultTop}$ and whose morphisms are neutrosophic homotopy classes

$$\left[F\right].$$

Composition is given by

$$\left[G\right]\circ \left[F\right]=[G\circ F].$$

Theorem 17.2.

Assume that admissible neutrosophic multi-valued functions are closed under composition and that neutrosophic homotopy is compatible with composition. Then

$$\mathbf{HoNMultTop}$$

is a well-defined category.

Proof. 

Suppose

$$F{\simeq }_N{F}^{\prime }$$

and

$$G{\simeq }_N{G}^{\prime }.$$

By compatibility,

$$G\circ F{\simeq }_N{G}^{\prime }\circ F^{\prime }.$$

Thus the composition of homotopy classes is well-defined. Associativity follows from associativity of the underlying multi-valued composition, while identity morphisms are represented by identity functions equipped with trivial neutrosophic data. □

18. Recovery of Classical Homotopy Theory

The new theory should contain ordinary homotopy theory as a special case.

Let

$$f:X⟶Y$$

be continuous. Define

$$M_{F_f}\left(x\right)=\left\{f\left(x\right)\right\},$$

and

$$T_{F_f}(x,f\left(x\right))=1,$$

$$I_{F_f}(x,f\left(x\right))=0,$$

$$F_{F_f}(x,f\left(x\right))=0.$$

Proposition 18.1.

The assignment

$$f⟼F_f$$

embeds ordinary continuous mappings into neutrosophic multi-valued mappings.

Proof. 

Each value is a singleton and hence nonempty and compact. The neutrosophic components are constant. Composition satisfies

$$F_g\circ F_f=F_{g\circ f}.$$

□

Theorem 18.2.

Let

$$f,g:X⟶Y$$

be ordinary continuous functions. If

$$f\simeq g$$

in the classical sense, then

$$F_f{\simeq }_N{F}_g.$$

Proof. 

Let

$$H:X\times [0,1]⟶Y$$

be a classical homotopy from f to g. Define

$$M_K(x,t)=\left\{H(x,t)\right\},$$

with

$$T_K(x,t,H(x,t))=1,$$

$$I_K(x,t,H(x,t))=0,$$

and

$$F_K(x,t,H(x,t))=0.$$

Then K is a neutrosophic homotopy from $F_f$ to $F_g$. □

Thus

$$\mathbf{HoTop}$$

appears naturally as a subcategory of

$$\mathbf{HoNMultTop}.$$

19. Relation with Neutrosophic Homotopy

The concept of neutrosophic homotopy has already been studied for single-valued neutrosophic mappings. In particular, Jafari, Rajesh, and Nordo introduced neutrosophic homotopic functions and developed basic properties of neutrosophic homotopy [8].

Our construction extends this perspective in a different direction.

For a single-valued neutrosophic map one associates to each point a neutrosophic value. For a neutrosophic multi-valued map one associates to each point a family

$$M_F\left(x\right)$$

of possible images, and every possible image may have its own neutrosophic profile.

Consequently, there are two independent sources of generalization:

$$\sin \mathrm{gle}\text{-}\mathrm{valued}⟶\mathrm{multi}\text{-}\mathrm{valued},$$

and

$$\mathrm{classical}⟶\mathrm{neutrosophic}.$$

The present theory combines them:

$$\begin{array}{|c|}\hline \mathrm{ordinary}\mathrm{maps}⟶\mathrm{neutrosophic}\mathrm{multi}\text{-}\mathrm{valued}\mathrm{maps}.\\ \hline\end{array}$$

20. Examples

Example 20.1

(A neutrosophic interval-valued map). Let

$$X=[0,1]$$

and define

$$M_F\left(x\right)=[0,x].$$

Let

$$T_F(x,y)=y,I_F(x,y)={\displaystyle \frac{1-y}{2}},F_F(x,y)=1-y.$$

Then F is a neutrosophic multi-valued mapping with continuously varying values.

Example 20.2

(A two-valued neutrosophic map). Let

$$M_F\left(x\right)=\{x,-x\}$$

on

$$X=[-1,1].$$

Assign

$$T_F(x,x)={\displaystyle \frac{1+x}{2}},T_F(x,-x)={\displaystyle \frac{1-x}{2}}.$$

The two possible images therefore have different truth degrees. One may similarly prescribe independent indeterminacy and falsity degrees.

Example 20.3

(A constant neutrosophic correspondence). Let $A\subseteq Y$ be nonempty and compact and define

$$M_F\left(x\right)=A$$

for every $x\in X$. Let

$$T_F(x,y)=t\left(y\right),I_F(x,y)=i\left(y\right),F_F(x,y)=f\left(y\right).$$

Then the topology of the domain does not affect the underlying possible-image set, although the neutrosophic profile may vary over A.

21. Fixed Points

A natural application is fixed point theory.

Definition 21.1.

Let

$$F:X{⊸}_{N}X.$$

A point $x\in X$ is called a neutrosophic fixed point of F if

$$x\in M_F\left(x\right).$$

Its neutrosophic fixed-point profile is

$$\left(T_F(x,x),I_F(x,x),F_F(x,x)\right).$$

Thus a fixed point can be accompanied by a degree of truth, indeterminacy, and falsity.

Definition 21.2.

For a neutrosophic multi-valued map $F:X{⊸}_{N}X$, define

$${Fix}_N\left(F\right)=\{x\in X:x\in M_F\left(x\right)\}.$$

The homotopy invariance of suitable Lefschetz-type numbers could then lead to neutrosophic fixed point theorems. The classical theory of admissible multi-valued mappings already provides a strong foundation for such an approach [9,10].

22. Neutrosophic Fundamental Groups

The construction naturally suggests a higher homotopy theory.

Let $(X,x_0)$ be a pointed topological space. Consider admissible neutrosophic multi-valued paths

$$F:(I,\{0,1\}){⊸}_N(X,x_0).$$

Two such paths are identified if they are neutrosophically homotopic relative to their endpoints.

Definition 22.1.

The set of neutrosophic homotopy classes of admissible loops based at $x_0$ is denoted formally by

$${\pi }_1^N(X,x_0).$$

Concatenation of paths suggests a group structure, provided the class of admissible neutrosophic maps is stable under concatenation.

More generally, one may define

$${\pi }_n^N(X,x_0)$$

using maps

$$(I^n,\partial I^n){⊸}_N(X,x_0).$$

Remark 22.2.

The group structure of these proposed higher homotopy groups requires additional hypotheses. In particular, one must verify that concatenation and inverse operations preserve admissibility and that the resulting operations are independent of representatives.

23. Comparison with Ordinary Homotopy Groups

An important problem is determining when

$${\pi }_n^N(X,x_0)\cong {\pi }_n(X,x_0).$$

One sufficient situation is when every admissible neutrosophic multi-valued map is neutrosophically homotopic to a singleton-valued map. In that case the neutrosophic theory collapses to ordinary homotopy theory.

Proposition 23.1.

Suppose every admissible neutrosophic loop at $x_0$ is neutrosophically homotopic to a singleton-valued loop. Then the natural map

$${\pi }_1(X,x_0)⟶{\pi }_1^N(X,x_0)$$

is surjective.

Proof. 

Every element of ${\pi }_1^N(X,x_0)$ has a representative which is homotopic to a singleton-valued loop. Hence every neutrosophic homotopy class is represented by an ordinary loop. □

Injectivity requires an additional condition ensuring that two ordinary loops which are neutrosophically homotopic are already ordinarily homotopic.

24. Conclusion

We introduced a framework for studying homotopy theory through neutrosophic multi-valued functions. The basic object combines two extensions of classical topology: the set-valued nature of multi-valued mappings and the three-component structure of neutrosophic sets.

A neutrosophic multi-valued function

$$F:X{⊸}_{N}Y$$

associates with each $x\in X$ a nonempty set $M_F\left(x\right)$ of possible images, together with three profiles

$$T_F,I_F,F_F.$$

This permits the possible images to carry different degrees of truth, indeterminacy, and falsity.

We defined neutrosophic homotopy by

$$H:X\times I{⊸}_{N}Y,$$

and established the equivalence-relation properties of neutrosophic homotopy. We also described composition, products, selectors, admissible representations, Vietoris representations, homology morphisms, and neutrosophic homotopy equivalence.

The framework contains classical homotopy theory as the special case in which every value is a singleton and the neutrosophic profile is trivial. It also contains ordinary multi-valued topology when the neutrosophic information is suppressed.

The resulting category

$$\mathbf{HoNMultTop}$$

provides a natural starting point for a broader homotopy theory of uncertain and indeterminate correspondences.