Singularities of Fuzzy Friedmann–Lemaître–Robertson–Walker Space

Abstract

We obtain different categories of singularities of fuzzy retracts, fuzzy deformation retracts, and fuzzy foldings on fuzzy fundamental groups of the fuzzy Friedmann–Lemaître–Robertson–Walker Space ${\tilde{W}}^4$. The fuzzy fundamental groups of fuzzy submanifolds in ${\tilde{W}}^4$ are characterized combinatorially using these fuzzy geometrical transformations. Also, the fuzzy fundamental groups of the fuzzy geodesics and the limit fuzzy foldings of ${\tilde{W}}^4$ are described. New types of fuzzy singularity of some fuzzy geometrical transformations of ${\tilde{W}}^4$ are obtained. Finally, the regularity of some other fuzzy retract singularities are discussed.

1. Introduction

The concept of fuzzy sets, initially introduced by Zadeh [1], has been extensively employed in various scientific fields [2,3,4,5,6]. Fuzzy folding, fuzzy retraction, and fuzzy deformation retracts are well known as being among the most fascinating issues in both Euclidean and non-Euclidean spaces. Several branches of differential geometry and topology have examined these topics.

In a series of papers (see, e.g., [7,8,9,10]), El-Ahmady, El-Ghoul, and others studied fuzzy folding and fuzzy deformation retracts of fuzzy horocycles and horospheres. In their studies, a new type of connection between fuzzy spheres in fuzzy Minskowski space and fuzzy open ball in fuzzy Euclidean space are obtained by using some fuzzy geometric transformations. In addition, Types of fundamental groups of the Klein bottle, elastic Klein bottle, Buchdahi space, Minkowski space, and some geometric transformations are achieved. After that, they defined the folding of chaotic fractal space-time. Then, the variation of the density functions on chaotic spheres in chaotic space-like Minkowski space time was introduced. In addition, folding–retraction of chaotic dynamical manifold and the VAK of vacuum fluctuation are also discussed. In later studies, some operations on the chaotic graphs, such as the union and the intersection—also both of the chaotic incidence matrices and the chaotic adjacency matrices, representing the chaotic graphs induced from these operations—were studied. Moreover, a calculation of geodesics in chaotic flat space, Ricci space, Schwarzchild space, and folding are obtained. The deformation shrinkage and topological folding of the Buchdahi space, and the deformation shrinkage and topological folding of the Schwarzschild space are also obtained. On parallel ruled surfaces in Minkowski 3-space and their retractions, the folding of fuzzy hypertori and their retractions, limits of fuzzy retractions of fuzzy hyperspheres and their foldings, and also fuzzy retractions of fuzzy open flat Robertson–Walker space are presented and obtained. The geodesic deformation retract of the Klein bottle and its folding are deduced. Finally, the topological folding of the hyperbola in Minkowski 3-space is presented.

The Friedmann–Lemaître–Robertson–Walker space, denoted by $\tilde{W}$, is yet a significant and iconic discovery in the annals of geometry (see, e.g., [11]) for recent studies). In spite of their initial appearance for geometric purposes, they have become significant in numerous branches of mathematics and physics. The convergence of geometry and applied science produced a beneficial impact as applied science contributed to the validation of the Friedmann–Lemaître–Robertson–Walker space, enabling its formalization [12]. The study of ${\tilde{W}}^4$ space is linked with fractal-fractional calculus. In particular, the end of the limits of fuzzy foldings of fuzzy ${\tilde{W}}^4$ space is equivalent to a type of fuzzy fractals of ${\tilde{W}}^4$, which reduces the dimension from integer to fractal. Moreover, the types of fuzzy deformation retracts of ${\tilde{W}}^4$ represent types of fuzzy fractals.

Singularity theory is a field of deep study in modern mathematics, with relations to many branches, including differential geometry, algebra, and topology. In fact, the essential way to understand an object is to classify its type, i.e., whether it is regular or singular. The exploration of the geometry of singular objects and its applications has been greatly stimulated since the beginning of the 19th century. Among many applications in the field of differential geometry and symplectic geometry, the geometric features of singular surfaces can be understood when classifying functions or mappings on such surfaces with respect to the $\mathcal{R}(V)$-equivalence relation (see, e.g., [13,14,15,16,17]). One of the goals of the current paper is to classify the fuzzy retract singularities.

The main goal of this work is to provide a comprehensive analysis of the fuzzy fundamental groups of the fuzzy Friedmann–Lemaître–Robertson–Walker space ${\tilde{W}}^4$. The manuscript has been divided into four major sections. In Section 2, we introduce basic and new definitions and recall some other ones, illustrating the main techniques and approaches that are used. In Section 3, the main results are stated with full details. In Section 4, the connections of the fuzzy retracts and singularity theory are discussed. Finally, in Section 5, the most important applications are investigated.

2. Preliminary

In the current section, we establish the basic definitions from differential geometry and recall some concepts from singularity theory, which will be used consistently throughout the paper.

We refer to a manifold $\tilde{M}$ with some physical features expressed through the density function $s\in [0,1]$ as a fuzzy manifold, which will be denoted by $(\tilde{M},s)$.

Definition 1.

A subset $(\tilde{A},s)$ of $(\tilde{M},s)$ is classified as a fuzzy retract (or retraction) singularity if a continuous mapping $\tilde{r}:(\tilde{M},s)\to (\tilde{A},s)$ exists such that $\tilde{r}$ serves as the identity on $(\tilde{A},s)$ and at least one component of $\tilde{r}(b,s(b))$ is identically zero. The function $\tilde{r}$ is then termed a fuzzy retraction singularity.

Definition 2.

A subset $(\tilde{A},s)$ is designated as b fuzzy deformation retract (or retraction) singularity if there exists a fuzzy retraction singularity $\tilde{r}:(\tilde{M},s)\to (\tilde{A},s)$ and a fuzzy homotopy $\tilde{\phi }:(\tilde{M},s)\times I\to (\tilde{M},s)$ that satisfies the following conditions:

(1) 

$\tilde{\phi }((u,s),0)=(u,s),$

(2) 

$\tilde{\phi }((u,s),1)=\tilde{r}(u,s),$

(3) 

$\tilde{\phi }((b,s),t)=(b,s)$

for all $u\in \tilde{M}$, $b\in \tilde{A}$, and $t\in I$.

Definition 3.

A map $\tilde{\tau }:\tilde{M}\to \tilde{M}$ is referred to as b fuzzy isometric folding singularity of $\tilde{M}$ into itself if the generated path $\tilde{\tau }\circ \gamma :L\to \tilde{M}$ is b piecewise fuzzy geodesic and has the same length as γ (where $L=[0,1]$). If $\tilde{\tau }$ fails to preserve lengths, it is b topological fuzzy folding of $\tilde{M}$, and at least one of its components is identically zero. In other words, the stratification is represented by the fuzzy folds or the fuzzy singularity.

Example 1

([7]). Consider the distinct points ${\overline{\tilde{P}}}_i,i=1,2,3,4$, and $\tilde{P},{\overline{\tilde{P}}}_i\ne \tilde{P}$ of ${\overline{\tilde{M}}}_1^n$, then there are points ${\underline{\tilde{P}}}_i,i=1,2,3,4,{\underline{\tilde{P}}}_i\ne \tilde{P}$ of ${\underline{\tilde{M}}}_1^n$, where ${\overline{\tilde{P}}}_i,{\underline{\tilde{P}}}_i$ have the same membership degrees for each i and $\tilde{P}$ has maximum membership degrees, as illustrated in the following figure:

Now, let ${\tilde{f}}_i:{\overline{\tilde{M}}}_1^n⟶{\overline{\tilde{M}}}_1^n,i=1,2,3$ be fuzzy foldings of ${\overline{\tilde{M}}}_1^n$ into itself, such that

$$\begin{array}{cc}{\tilde{f}}_1\left({\overline{\tilde{P}}}_1,{\mu }_1\right)=\left(\tilde{P},{\mu }_{max}\right),\hfill & {\tilde{f}}_1\left({\overline{\tilde{P}}}_2,{\mu }_2\right)=\left({\overline{\tilde{P}}}_2,{\mu }_2\right)and\hfill \\ {\tilde{f}}_2\left({\overline{\tilde{P}}}_4,{\mu }_4\right)=\left(\tilde{P},{\mu }_{max}\right),\hfill & {\tilde{f}}_2\left({\overline{\tilde{P}}}_3,{\mu }_3\right)=\left({\overline{\tilde{P}}}_3,{\mu }_3\right)and\hfill \\ {\tilde{f}}_3\left({\overline{\tilde{P}}}_2,{\mu }_2\right)=\left({\overline{\tilde{P}}}_3,{\mu }_3\right),\hfill & {\tilde{f}}_3\left(\tilde{P},{\mu }_{max}\right)=\left(\tilde{P},{\mu }_{max}\right).\hfill \end{array}$$

Then, the limit of the fuzzy folding is $P{\tilde{P}}_3$, which is a subset of ${\overline{\tilde{M}}}_i^n$, i.e., ${lim}_{j\to \infty }{\tilde{f}}_j\left({\overline{\tilde{M}}}_i^n\right)={\overline{\tilde{G}}}_i^n,{\overline{\tilde{G}}}_i^n$ has the same dimension as ${\overline{\tilde{M}}}_i^n$. Also, for any fuzzy folding up ${\tilde{M}}^n$ of ${\overline{\tilde{M}}}_i^n$ into itself, there are induced fuzzy foldings ${\tilde{M}}^n$ of ${\tilde{M}}_i^n$ into itself.

Example 2.

Let $\tilde{M}={\tilde{S}}^n-\left\{\tilde{p}\right\}$, where ${\tilde{S}}^n$ is a fuzzy n-sphere, $\tilde{p}$ is any point of ${\tilde{S}}^n$. Let ${\tilde{r}}_g:{\tilde{S}}^n-\left\{\tilde{p}\right\}⟶\tilde{N}$ be a geometric retraction that takes $\left\{{\tilde{S}}^n-\tilde{p}\right\}$ to the lower hemisphere, as illustrated in the following figures:

This retraction will condense the physical character in $\tilde{N}$. In this case, ${\tilde{r}}_g\left({\mu }_{{\tilde{S}}^n-p}\right)={\overline{\mu }}_{\tilde{N}}=m{\mu }_{{\tilde{S}}^n-\tilde{p}},m\ge 1$ which means that ${\tilde{r}}_g$ induces a fuzzy retraction of the physical character.

Now, if we start with a fuzzy retraction ${\tilde{r}}_f:\left\{{\tilde{S}}^n-\tilde{p}\right\}⟶\left\{{\tilde{S}}^n-\tilde{p}\right\}$, such that ${\tilde{r}}_f:\left\{{\tilde{S}}^n-p\right\}=\left\{{\tilde{S}}^n-\tilde{p}\right\},{\tilde{r}}_f\left({\mu }_{{\tilde{S}}^n-\tilde{p}}\right)={\overline{\mu }}_{\tilde{N}}$ where $\tilde{N}$ is the lower hemisphere. In this case, $\left\{{\tilde{S}}^n-\tilde{p}\right\}$ does not affect anything except the physical character, i.e., fuzzy retraction does not induce, in general, a geometric retraction.

Note that if we consider $\tilde{M}={\tilde{S}}^n-\left\{{\tilde{p}}_1,{\tilde{p}}_2\right\}$ and ${\tilde{r}}_g:{\tilde{S}}^n-\left\{{\tilde{p}}_1,{\tilde{p}}_1\right\}⟶\tilde{N}$ as a geometric retraction, which takes $\tilde{M}$ to circle $\tilde{N}$, then this retraction again induces a fuzzy retraction.

Example 3.

Let ${\tilde{S}}^1\times {\tilde{R}}^1$ be an infinite fuzzy cylinder, then a fuzzy folding of the fuzzy cylinder can be defined as follows:

This means that $\tilde{f}\left({\tilde{S}}^1\times {\tilde{R}}^1\right)=\tilde{f}\left({\tilde{S}}^1\right)\times {\tilde{R}}^1$.

This means that $\tilde{f}\left({\tilde{S}}^1\times {\tilde{R}}^1\right)={\tilde{S}}^1\times \tilde{f}\left({\tilde{R}}^1\right)$.

This means that $\tilde{f}\left({\tilde{S}}^1\times {\tilde{R}}^1\right)=\tilde{f}\left({\tilde{S}}^1\right)\times f\left({\tilde{R}}^1\right)$.

Let $\tilde{M}=\tilde{l},\tilde{N}=\tilde{l}$. Consider the following fuzzy folding of a fuzzy sheet of a fuzzy square paper, as shown in the next diagram.

Example 4.

Let ${\tilde{S}}^1\times {\tilde{R}}^1$ be an infinite fuzzy cylinder, then the limit of fuzzy folding can be introduced as follows:

Example 5.

Let $\tilde{C}={\tilde{S}}^1\times {\tilde{R}}^1$ be an infinite fuzzy cylinder. The fuzzy retraction and fuzzy folding of $\left({\tilde{S}}^1\times {\tilde{R}}^1\right)$ are defined as in the following diagram.

Example 6.

The fuzzy fundamental group of any fuzzy fobbing of fuzzy circle ${\tilde{S}}^1$ is either isomorphic to $\tilde{Z}$ or fuzzy identify group, i.e., if the fuzzy folding with singularity of ${\tilde{S}}^{\prime }$, then ${\pi }_1\left({\tilde{f}}_1\left({\tilde{S}}^1\right)\right)\cong$ to a fuzzy identity group. Also, if the fuzzy folding without singularity of ${\tilde{S}}^1$, then ${\tilde{f}}_2\left({\tilde{S}}^1\right)$ is a fuzzy manifold hemeomorphic to ${\tilde{S}}^1$, and ${\pi }_1\left({\tilde{f}}_2\left({\tilde{S}}^1\right)\cong \tilde{Z}\right)$. See the next diagram.

Let $F:N\to M$ be a differentiable map between two arbitrary manifolds M and N, with dimensions m and n, respectively. The derivative of a map at a point $u\in M$ is a linear map, denoted by $F_x^{*}$, of the tangent space $T_{x}N$ to the tangent space $T_{F(u)}M$:

$$F_x^{*}:T_{x}N\to T_{F(u)}M.$$

Definition 4 

([15]). A point $u\in M$ is known as a critical point or singularity point of F if the rank of the derivative meets the following condition:

$$rank(F_x^{*})<min(m,n),$$

and it is referred to as a “critical value.”

Let $(u_1,u_2,\dots ,x_n)$ be local coordinates in the neighborhood of u, and $(y_1,y_2,\dots ,y_m)$ be the local coordinates in the neighborhood of $F(u$). Let $y_i=f_i(u_1,\dots ,x_n)$, $i=1,\dots ,m$ be the smooth functions that define F locally at u.

Definition 5 

([18]). The Jacobian matrix of F, represented as $\mathbf{d}{F}_x$ or ${\mathbf{J}}_F$, is the $m\times n$ matrix with its $(i,j)$-th element defined as ${\mathbf{J}}_{ij}=\frac{\partial f_i}{\partial u_j},$ or explicitly

$$\mathbf{d}{F}_x=\left(\begin{array}{cccc}{\displaystyle \frac{\partial f_1}{\partial u_1}}& \dots & \dots & {\displaystyle \frac{\partial f_1}{\partial u_n}}\\ ⋮& ⋮& ⋮& ⋮\\ {\displaystyle \frac{\partial f_m}{\partial u_1}}& \dots & \dots & {\displaystyle \frac{\partial f_m}{\partial u_n}}\end{array}\right).$$

Remark 1

([18]).

1. 

Locally, point u is a critical point or singularity point of F if the rank of the Jacobian matrix of F at u is not maximum; that is, $rank(\mathbf{d}{f}_u)<min(n,m)$.

2. 

The critical set of F, denoted by ${\Sigma }^{*}F$, is the set of all critical points.

3. 

The image $F({\Sigma }^{*}F)$ is called the critical value of F.

4. 

Assume $n<m$. If F has critical points, then the image of F is singular or non-regular; otherwise, it is regular.

Our approach relies on the construction of a fuzzy deformation retract map on the fuzzy Friedmann–Lemaître–Robertson–Walker space ${\tilde{W}}^4$. This map is constructed by getting started with the fuzzy geodesic and then obtaining its spherical coordinates through the calculation of the Lagrangian equations. Also, the fuzzy retraction singularities of ${\tilde{W}}^4$ are obtained through the use of some fuzzy geometrical transformations and discussing their type via tools from singularity theory. These transformations are viewed from the perspective of the Remannian metric that pertains to ${\tilde{W}}^4$.

3. Main Results

Theorem 1.

The fuzzy geodesic retractions of ${\tilde{W}}^4$ from the viewpoint of Lagrangian equations are fuzzy spheres, fuzzy circles, and fuzzy subspaces.

Proof. 

Consider the fuzzy Friedmann–Lemaître–Robertson–Walker space ${\tilde{W}}^4$, in the fuzzy cylindrical coordinates $\varphi (s),\theta (s),\phi (s)$, and $t(s)$, with the following fuzzy metric:

$${\widetilde{ds}}^2=-d{t}^2(s)+b^2{cosh}^2{\displaystyle \frac{t(s)}{b}}\left[d{\phi }^2(s)+{sin}^2\phi (s)\left[d{\theta }^2(s)+{sin}^2\theta (s)d{\varphi }^2(s)\right]\right],$$

(1)

where $b,t(s)\in \mathbb{R}$, $\phi (s)\in [0,\pi ]$, $\theta (s)\in [0,\pi ]$, and $\varphi (s)\in [0,2\pi ].$

The fuzzy coordinates of ${\tilde{W}}^4$ are given by

$$\begin{array}{cc}\hfill {\tilde{u}}_1& =bcosh{\displaystyle \frac{t(s)}{b}}sin\phi (s)sin\theta (s)sin\varphi (s),\hfill \\ \hfill {\tilde{u}}_2& =bcosh{\displaystyle \frac{t(s)}{b}}sin\phi (s)sin\theta (s)cos\varphi (s),\hfill \\ \hfill {\tilde{u}}_3& =bcosh{\displaystyle \frac{t(s)}{b}}sin\phi (s)cos\theta (s),\hfill \\ \hfill {\tilde{u}}_4& =bcosh{\displaystyle \frac{t(s)}{b}}cos\phi (s),\hfill \\ \hfill {\tilde{u}}_5& =bsinh{\displaystyle \frac{t(s)}{b}}.\hfill \end{array}$$

(2)

Note that ${\tilde{u}}_1^2+{\tilde{u}}_2^2+{\tilde{u}}_3^2+{\tilde{u}}_4^2-{\tilde{u}}_5^2=b^2$.

Let $\tau ={\displaystyle \frac{1}{2}}\tilde{d}{s}^2$, $dt=t^{\prime }$, $d\phi ={\phi }^{\prime }$, $d\theta ={\theta }^{\prime }$ and $d\varphi ={\varphi }^{\prime }$. Then, (1) becomes

$$2\tau =t^{\prime 2}(s)+b^2{cosh}^2{\displaystyle \frac{t(s)}{b}}\left({\phi }^{\prime 2}(s)+{sin}^2\phi (s)\left({\theta }^{\prime 2}(s)+{sin}^2\theta (s){\varphi }^{\prime 2}(s)\right)\right).$$

(3)

We obtain the fuzzy geodesic singularities and fuzzy retract singularities in ${\tilde{W}}^4$ by solving the Lagrangian equations, as follows:

$${\displaystyle \frac{d}{ds}}({\displaystyle \frac{\partial \tau }{\partial {\psi }^{\prime }}})-{\displaystyle \frac{\partial \tau }{\partial \psi }}=0,$$

where $\psi \in \{t,\phi ,\theta ,\varphi \}.$ Thus, we obtain the following relations

$${\displaystyle \frac{d}{ds}}{t}^{\prime }(s)+bcosh{\displaystyle \frac{t(s)}{b}}sinh{\displaystyle \frac{t(s)}{b}}\left({\phi }^{\prime 2}(s)+{sin}^2\phi (s)({\theta }^{\prime 2}(s)+{sin}^2\theta (s){\varphi }^{\prime 2}(s))\right)=0,$$

(4)

$${\displaystyle \frac{d}{ds}}\left(b^2{cosh}^2{\displaystyle \frac{t(s)}{b}}{\phi }^{\prime }(s)\right)-b^2{cosh}^2{\displaystyle \frac{t(s)}{b}}sin\phi (s)cos\phi (s)\left({\theta }^{\prime 2}(s)+{sin}^2\theta (s){\varphi }^{\prime 2}(s)\right)=0,$$

(5)

$${\displaystyle \frac{d}{ds}}\left(b^2{cosh}^2{\displaystyle \frac{t(s)}{b}}{sin}^2\phi (s){\theta }^{\prime }(s)\right)-b^2{cosh}^2{\displaystyle \frac{t(s)}{b}}{sin}^2\phi (s)sin\theta (s)cos\theta (s){\varphi }^{\prime 2}(s)=0,$$

(6)

and

$${\displaystyle \frac{d}{ds}}\left(b^2{cosh}^2{\displaystyle \frac{t(s)}{b}}{sin}^2\phi (s){sin}^2\theta (s){\varphi }^{\prime }(s)\right)=0.$$

(7)

To be able to solve the Lagrangian equations and to find the fuzzy geodesics and the fuzzy retractions, we have to choose particular values. We will address the subsequent special cases:

(i)

From Equation (4), $t(s)=0$ is a particular solution, and therefore, (2) gives the fuzzy space

$${\tilde{S}}^3=\left\{bsin\phi (s)sin\theta (s)sin\varphi (s),bsin\phi (s)sin\theta (s)cos\varphi (s),bsin\phi (s)cos\theta (s),bcos\phi (s),0\right\},$$

(8)

which is a fuzzy 3-sphere; that is, the three-dimensional fuzzy closed space of constant positive curvature $\kappa =+1$ in ${\tilde{W}}^4$.

If $\phi (s)$ is considered to be a constant, say ${\displaystyle \frac{\pi }{2}}$, then the space (8) yields

$${\tilde{S}}_1^2=\left\{bsin\theta (s)sin\varphi (s),bsin\theta (s)cos\varphi (s),bcos\theta (s),0,0\right\},$$

(9)

which is a 2-fuzzy sphere of radius b, covered by the standard fuzzy spherical polar coordinates $\theta (s)$ and $\varphi (s)$.

On the other hand, if $\theta (s)$ is a constant equal ${\displaystyle \frac{\pi }{2}}$, then (8) gives

$${\tilde{S}}_2^2=\left\{bsin\phi (s)sin\varphi (s),bsin\phi (s)cos\varphi (s),0,bcos\phi (s),0\right\},$$

(10)

which is again a 2-fuzzy sphere of radius b covered by the standard fuzzy spherical polar coordinates $\phi (s)$ and $\varphi (s)$.

If $\phi (s)=\theta (s)={\displaystyle \frac{\pi }{2}}$, then (8) leads

$${\tilde{S}}_1^1=\left\{bsin\varphi (s),bcos\varphi (s),0,0,0\right\},$$

(11)

which is the fuzzy circle. Note that if $\theta (s)={\displaystyle \frac{\pi }{2}}$ and $\phi (s)$ is an arbitrary constant, then the fuzzy circles of constant latitude are transformed to fuzzy circles on a disc.

For if $\varphi (s)$ is constant and equal to ${\displaystyle \frac{\pi }{2}}$, (8) leads to

$${\tilde{S}}_3^2=\left\{bsin\phi (s)sin\theta (s),0,bsin\phi (s)cos\theta (s),bcos\phi (s),0\right\},$$

(12)

which is a 2-fuzzy sphere of radius b covered by the standard fuzzy spherical polar coordinates $\phi (s)$ and $\theta (s)$. Again, if $\theta$ is constant and equals 0, then (12) becomes

$${\tilde{S}}_2^1=\left\{0,0,bsin\phi (s),bcos\phi (s),0\right\},$$

(13)

which is a fuzzy circle in the ${\tilde{u}}_3{\tilde{u}}_4$-plane.

Also, if $\phi (s)={\displaystyle \frac{\pi }{2}}$, then (12) yields

$${\tilde{S}}_3^1=\left\{bsin\theta (s),0,bcos\theta (s),0,0\right\},$$

(14)

which is a fuzzy circle in the ${\tilde{u}}_1{\tilde{u}}_3$-plane.

Clearly, all of ${\tilde{S}}^3,{\tilde{S}}_1^2,{\tilde{S}}_2^2,{\tilde{S}}_1^1,{\tilde{S}}_2^1$, and ${\tilde{S}}_3^1$ are fuzzy geodesic and retraction singularities.

(ii)

From Equation (7), we have $b^2{cosh}^2{\displaystyle \frac{t(s)}{b}}{sin}^2\theta (s){\varphi }^{\prime }(s)=\xi$, where $\xi$ is a constant. When $\xi =0$, then $\varphi (s)=constant=\rho$. In particular, if $\rho =0$, then we obtain from (2)

$${\tilde{W}}^3=\left\{0,bcosh{\displaystyle \frac{t(s)}{b}}sin\phi (s)sin\theta (s),bcosh{\displaystyle \frac{t(s)}{b}}sin\phi (s)cos\theta (s),bcosh{\displaystyle \frac{t(s)}{b}}cos\phi (s),bsinh{\displaystyle \frac{t(s)}{b}}\right\},$$

(15)

which is a 3-dimensional fuzzy subspace. It is a fuzzy retraction singularity.

Also, from Equation (7), we have ${sin}^2\theta (s)=0$, where $\theta (s)\in [0,\pi ]$. Hence, $\theta (s)\in \{0,\pi \}$, and it follows from (2) that we obtain either

$${\tilde{W}}_1^2=\left\{0,0,bcosh{\displaystyle \frac{t(s)}{b}}sin\phi (s),bcosh{\displaystyle \frac{t(s)}{b}}cos\phi (s),bsinh{\displaystyle \frac{t(s)}{b}}\right\},$$

(16)

or

$${\tilde{W}}_2^2=\left\{0,0,-bcosh{\displaystyle \frac{t(s)}{b}}sin\phi (s),bcosh{\displaystyle \frac{t(s)}{b}}cos\phi (s),bsinh{\displaystyle \frac{t(s)}{b}}\right\}$$

(17)

which are both 2-dimensional fuzzy subspaces (see Figure 1), covered by the fuzzy coordinates $t(s)$ and $\phi (s)$. They are fuzzy retraction singularities.

Figure 1. The 2-dimensional cylinder.

(iii)

Finally, if we set $\phi (s)=0$ and $\phi (s)=\pi$ in (2), then we obtain

$${\tilde{W}}_1^1=\left\{0,0,0,bcosh{\displaystyle \frac{t(s)}{b}},bsinh{\displaystyle \frac{t(s)}{b}}\right\},$$

(18)

and

$${\tilde{W}}_2^1=\left\{0,0,0,-bcosh{\displaystyle \frac{t(s)}{b}},bsinh{\displaystyle \frac{t(s)}{b}}\right\}$$

(19)

which are both 1-dimensional fuzzy subspaces of ${\tilde{W}}^4$. They are fuzzy retraction singularities. Note that, ${\tilde{u}}_1={\tilde{u}}_2={\tilde{u}}_3=0$ are fuzzy coordinate singularities.

□

Theorem 2.

The fuzzy fundamental group of types of fuzzy deformation retracts of ${\tilde{W}}^4$ is either isomorphic to $\tilde{\mathbb{Z}}$ or it represents a fuzzy identity group.

Proof. 

We will show that all of ${\tilde{S}}^3,{\tilde{S}}_1^2,{\tilde{S}}_2^2,{\tilde{S}}_2^2,{\tilde{S}}_1^1,{\tilde{S}}_2^1,{\tilde{S}}_3^1,{\tilde{W}}^3,{\tilde{W}}_1^2,{\tilde{W}}_2^2,{\tilde{W}}_1^1$, and ${\tilde{W}}_2^1$ are the fuzzy deformation retract of open fuzzy Friedmann–Lemaître–Robertson–Walker ${\tilde{W}}^4$.

Let

$$\tilde{\psi }:({\tilde{W}}^4-\left\{{\tilde{\mu }}_i\right\})\times [0,1]\to ({\tilde{W}}^4-\left\{{\mu }_i\right\}),$$

being the fuzzy deformation retract of ${\tilde{W}}^4$, where $({\tilde{W}}^4-\left\{{\tilde{\mu }}_i\right\})$ is the open fuzzy Friedmann–Lemaître–Robertson–Walker space ${\tilde{W}}^4$.

Let

$$\tilde{\mathcal{R}}:({\tilde{W}}^4-\left\{{\tilde{\mu }}_i\right\})\to X,$$

being the fuzzy retractions of $({\tilde{W}}^4-\left\{{\tilde{\mu }}_i\right\})$, where $X\in \{{\tilde{S}}^3,{\tilde{S}}_1^2,{\tilde{S}}_2^2,{\tilde{S}}_1^1,{\tilde{S}}_2^1,{\tilde{S}}_3^1,{\tilde{W}}^3,{\tilde{W}}_1^2,{\tilde{W}}_2^2,{\tilde{W}}_1^1,{\tilde{W}}_2^1\}$.

Then, the fuzzy deformation retract of $({\tilde{W}}^4-\left\{{\tilde{\mu }}_i\right\})$ onto ${\tilde{S}}^3$ is defined as

$$\begin{gathered} \tilde{\psi }(\tilde{u},\lambda )=(1-\lambda )\{(bcosh{\displaystyle \frac{t(s)}{\alpha }}sin\phi (s)sin\theta (s)sin\varphi (s),bcosh{\displaystyle \frac{t(s)}{\alpha }}sin\phi (s)sin\theta (s)cos\varphi (s), \\ bcosh{\displaystyle \frac{t(s)}{\alpha }}sin\phi (s)cos\theta (s),bcosh{\displaystyle \frac{t(s)}{\alpha }}cos\phi (s),bsinh{\displaystyle \frac{t(s)}{\alpha }})-\left\{{\tilde{\mu }}_i\right\}\}+ \\ \lambda \left\{bsin\phi (s)sin\theta (s)sin\varphi (s),bsin\phi (s)sin\theta (s)cos\varphi (s),bsin\phi (s)cos\theta (s),bcos\phi (s),0\right\}. \end{gathered}$$

Hence, the ${\pi }_1({\tilde{W}}^4-\left\{{\tilde{\mu }}_i\right\})\cong {\pi }_1({\tilde{S}}^3)\cong \mathrm{fuzzy}\mathrm{identity}\mathrm{group}$.

The fuzzy deformation retract of $({\tilde{W}}^4-\left\{{\tilde{\mu }}_i\right\})$ onto ${\tilde{S}}_1^2$ is given by

$$\begin{array}{cc}\hfill \tilde{\psi }(\tilde{u},\lambda )=& cos{\displaystyle \frac{\pi \lambda }{2}}\{(bcosh{\displaystyle \frac{t(s)}{\alpha }}sin\phi (s)sin\theta (s)sin\varphi (s),bcosh{\displaystyle \frac{t(s)}{\alpha }}sin\phi (s)sin\theta (s)cos\varphi (s),\hfill \\ \hfill & bcosh{\displaystyle \frac{t(s)}{\alpha }}sin\phi (s)cos\theta (s),bcosh{\displaystyle \frac{t(s)}{\alpha }}cos\phi (s),bsinh{\displaystyle \frac{t(s)}{\alpha }})-\left\{{\tilde{\mu }}_i\right\}\}+\hfill \\ \hfill & sin{\displaystyle \frac{\pi \lambda }{2}}\left\{bsin\theta (s)sin\varphi (s),bsin\theta (s)cos\varphi (s),bcos\theta (s),0,0\right\}.\hfill \end{array}$$

So, ${\pi }_1({\tilde{W}}^4-\left\{{\tilde{\mu }}_i\right\})\cong {\tilde{S}}_1^2\cong \mathrm{fuzzy}\mathrm{identity}\mathrm{group}$.

For ${\tilde{S}}_2^2$, the fuzzy deformation retract of $({\tilde{W}}^4-\left\{{\tilde{\mu }}_i\right\})$ is given by

$$\begin{array}{cc}\hfill \tilde{\psi }(\tilde{u},\lambda )=& {\displaystyle \frac{1-\lambda }{1+\lambda }}\{(bcosh{\displaystyle \frac{t(s)}{\alpha }}sin\phi (s)sin\theta (s)sin\varphi (s),bcosh{\displaystyle \frac{t(s)}{\alpha }}sin\phi (s)sin\theta (s)cos\varphi (s),\hfill \\ \hfill & bcosh{\displaystyle \frac{t(s)}{\alpha }}sin\phi (s)cos\theta (s),bcosh{\displaystyle \frac{t(s)}{\alpha }}cos\phi (s),bsinh{\displaystyle \frac{t(s)}{\alpha }})-\left\{{\tilde{\mu }}_i\right\}\}+\hfill \\ \hfill & \lambda (2\lambda -1)\left\{bsin\phi (s)sin\varphi (s),bsin\phi (s)cos\varphi (s),0,bcos\phi (s),0\right\}.\hfill \end{array}$$

Therefore, ${\pi }_1({\tilde{W}}^4-\left\{{\tilde{\mu }}_i\right\})\cong {\tilde{S}}_2^2\cong \mathrm{is}\mathrm{a}\mathrm{fuzzy}\mathrm{identity}\mathrm{group}$.

Furthermore, the fuzzy deformation retract of $({\tilde{W}}^4-\left\{{\tilde{\mu }}_i\right\})$ onto ${\tilde{S}}_3^2$ is defined as

$$\begin{array}{cc}\hfill \tilde{\psi }(\tilde{u},\lambda )=& {\displaystyle \frac{1-\lambda }{1+\lambda }}\{(bcosh{\displaystyle \frac{t(s)}{\alpha }}sin\phi (s)sin\theta (s)sin\varphi (s),bcosh{\displaystyle \frac{t(s)}{\alpha }}sin\phi (s)sin\theta (s)cos\varphi (s),\hfill \\ \hfill & bcosh{\displaystyle \frac{t(s)}{\alpha }}sin\phi (s)cos\theta (s),bcosh{\displaystyle \frac{t(s)}{\alpha }}cos\phi (s),bsinh{\displaystyle \frac{t(s)}{\alpha }})-\left\{{\tilde{\mu }}_i\right\}\}+\hfill \\ \hfill & {\displaystyle \frac{2\lambda }{1+\lambda }}\left\{bsin\phi (s)sin\theta (s),0,bsin\phi (s)cos\theta (s),bcos\phi (s),0\right\}.\hfill \end{array}$$

Accordingly, ${\pi }_1({\tilde{W}}^4-\left\{{\tilde{\mu }}_i\right\})\cong {\tilde{S}}_3^2\cong \mathrm{is}\mathrm{a}\mathrm{fuzzy}\mathrm{identity}\mathrm{group}$.

The fuzzy deformation retract of $({\tilde{W}}^4-\left\{{\tilde{\mu }}_i\right\})$ onto ${\tilde{S}}_1^1$ is

$$\begin{array}{cc}\hfill \tilde{\psi }(\tilde{u},\lambda )=& (1-\lambda )\{(bcosh{\displaystyle \frac{t(s)}{\alpha }}sin\phi (s)sin\theta (s)sin\varphi (s),bcosh{\displaystyle \frac{t(s)}{\alpha }}sin\phi (s)sin\theta (s)cos\varphi (s),\hfill \\ \hfill & bcosh{\displaystyle \frac{t(s)}{\alpha }}sin\phi (s)cos\theta (s),bcosh{\displaystyle \frac{t(s)}{\alpha }}cos\phi (s),bsinh{\displaystyle \frac{t(s)}{\alpha }})-\left\{{\tilde{\mu }}_i\right\}\}+\hfill \\ \hfill & \lambda \left\{bsin\varphi (s),bcos\varphi (s),0,0,0\right\}.\hfill \end{array}$$

Thus, ${\pi }_1({\tilde{W}}^4-\left\{{\tilde{\mu }}_i\right\})\cong {\tilde{S}}_1^1\cong \tilde{\mathbb{Z}}.$

For ${\tilde{S}}_2^1$, the fuzzy deformation retract of $({\tilde{W}}^4-\left\{{\tilde{\mu }}_i\right\})$ is

$$\begin{array}{cc}\hfill \tilde{\psi }(\tilde{u},\lambda )=& {(1-\lambda )}^{{\displaystyle \frac{1}{2}}}\{(bcosh{\displaystyle \frac{t(s)}{\alpha }}sin\phi (s)sin\theta (s)sin\varphi (s),bcosh{\displaystyle \frac{t(s)}{\alpha }}sin\phi (s)sin\theta (s)cos\varphi (s),\hfill \\ \hfill & bcosh{\displaystyle \frac{t(s)}{\alpha }}sin\phi (s)cos\theta (s),bcosh{\displaystyle \frac{t(s)}{\alpha }}cos\phi (s),bsinh{\displaystyle \frac{t(s)}{\alpha }})-\left\{{\tilde{\mu }}_i\right\}\}+\hfill \\ \hfill & {\lambda }^{{\displaystyle \frac{1}{2}}}\left\{0,0,bsin\phi (s),bcos\phi (s),0\right\},\hfill \end{array}$$

where

$$\begin{array}{cc}\hfill \tilde{\psi }(\tilde{u},0)=& \{(bcosh{\displaystyle \frac{t(s)}{\alpha }}sin\phi (s)sin\theta (s)sin\varphi (s),bcosh{\displaystyle \frac{t(s)}{\alpha }}sin\phi (s)sin\theta (s)cos\varphi (s),\hfill \\ \hfill & bcosh{\displaystyle \frac{t(s)}{\alpha }}sin\phi (s)cos\theta (s),bcosh{\displaystyle \frac{t(s)}{\alpha }}cos\phi (s),bsinh{\displaystyle \frac{t(s)}{\alpha }})-\left\{{\tilde{\mu }}_i\right\}\},\hfill \end{array}$$

and

$$\begin{array}{c}\hfill \tilde{\psi }(\tilde{u},1)=\left\{0,0,bsin\phi (s),bcos\phi (s),0\right\}.\end{array}$$

Hence, ${\pi }_1({\tilde{W}}^4-\left\{{\tilde{\mu }}_i\right\})\cong {\pi }_1({\tilde{S}}_2^1)$ is isomorphic to $\tilde{\mathbb{Z}}.$

Finally, the fuzzy deformation retract of $({\tilde{W}}^4-\left\{{\tilde{\mu }}_i\right\})$ onto ${\tilde{S}}_3^1$ is

$$\begin{array}{cc}\hfill \tilde{\psi }(\tilde{u},\lambda )=& \{(bcosh{\displaystyle \frac{t(s)}{\alpha }}sin\phi (s)sin\theta (s)sin\varphi (s),bcosh{\displaystyle \frac{t(s)}{\alpha }}sin\phi (s)sin\theta (s)cos\varphi (s),\hfill \\ \hfill & bcosh{\displaystyle \frac{t(s)}{\alpha }}sin\phi (s)cos\theta (s),bcosh{\displaystyle \frac{t(s)}{\alpha }}cos\phi (s),bsinh{\displaystyle \frac{t(s)}{\alpha }})-\left\{{\mu }_i\right\}\}+\hfill \\ \hfill & \left\{0,bsin\theta (s),bcos\theta (s),0,0\right\},\hfill \end{array}$$

where

$$\begin{array}{cc}\hfill \tilde{\psi }(\tilde{u},0)=& \{(bcosh{\displaystyle \frac{t(s)}{\alpha }}sin\phi (s)sin\theta (s)sin\varphi (s),bcosh{\displaystyle \frac{t(s)}{\alpha }}sin\phi (s)sin\theta (s)cos\varphi (s),\hfill \\ \hfill & bcosh{\displaystyle \frac{t(s)}{\alpha }}sin\phi (s)cos\theta (s),bcosh{\displaystyle \frac{t(s)}{\alpha }}cos\phi (s),bsinh{\displaystyle \frac{t(s)}{\alpha }})-\left\{{\mu }_i\right\}\},\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill \tilde{\psi }(\tilde{u},1)=& \left\{0,bsin\theta (s),bcos\theta (s),0,0\right\}.\hfill \end{array}$$

Thus, ${\pi }_1({\tilde{W}}^4-\left\{{\tilde{\mu }}_i\right\})\cong {\pi }_1({\tilde{S}}_3^1)$ is isomorphic to $\tilde{\mathbb{Z}}.$

Similar arguments can be made for the remaining fuzzy retract singularities, and the proof is completed. □

Theorem 3.

The fuzzy fundamental group of the limit of the fuzzy foldings by cut of one of the fuzzy retract singularities ${\tilde{S}}_1^2,{\tilde{S}}_2^2$, and ${\tilde{S}}_1^2$ in ${\tilde{W}}^4$ is isomorphic to $\tilde{\mathbb{Z}}.$

Proof. 

Consider the fuzzy great sphere ${\tilde{S}}_3^2$, which is a 2-dimensional space, and is expressed explicitly as ${\tilde{u}}_1^2+{\tilde{u}}_2^2+{\tilde{u}}_4^2=b^2$. Clearly, it is fuzzy geodesic and a fuzzy retraction singularity in ${\tilde{W}}^4.$

Let ${\tilde{\gamma }}_{1_{cut}}:{\tilde{S}}_3^2\to {\tilde{S}}_3^2$ be b fuzzy folding by cut. Inductively, define the sequence of fuzzy folding by cut maps as follows:

$$\begin{array}{cc}\hfill {\tilde{\gamma }}_{1_{cut}}& :{\tilde{S}}_3^2\to {\tilde{S}}_3^2,\hfill \\ \hfill {\tilde{\gamma }}_{2_{cut}}& :{\tilde{\gamma }}_{1_{cut}}({\tilde{S}}_3^2)\to {\tilde{S}}_3^2,\hfill \\ \hfill {\tilde{\gamma }}_{3_{cut}}& :{\tilde{\gamma }}_{2_{cut}}({\tilde{\gamma }}_{1_{cut}}({\tilde{S}}_3^2))\to {\tilde{S}}_3^2,\hfill \\ \hfill ⋮& \\ \hfill {\tilde{\gamma }}_{n_{cut}}& :{\tilde{\gamma }}_{{(n-1)}_{cut}}({\tilde{\gamma }}_{{(n-2)}_{cut}}(\dots ({\tilde{\gamma }}_{2_{cut}}({\tilde{\gamma }}_{1_{cut}}({\tilde{S}}_3^2)))\dots ))\to {\tilde{S}}_3^2.\hfill \end{array}$$

Thus, ${\displaystyle \underset{b\to \infty }{lim}{\tilde{\gamma }}_{n_{cut}}({\tilde{\gamma }}_{{(n-1)}_{cut}}({\tilde{\gamma }}_{{(n-2)}_{cut}}(\dots ({\tilde{\gamma }}_{2_{cut}}({\tilde{\gamma }}_{1_{cut}}({\tilde{S}}_3^2)))\dots )))}$ is a fuzzy circle ${\tilde{S}}_4^1\subset {\tilde{W}}^4$ of dimension one. Therefore, ${\pi }_1({\tilde{S}}_4^1\subset {\tilde{W}}^4)\cong \tilde{\mathbb{Z}}.$ Sequentially, the fuzzy folding by cut of the fuzzy geodesic and retraction singularity ${\tilde{S}}_3^2$ is isomorphic to $\tilde{\mathbb{Z}}$ (see Figure 2, which represents a sequence of fuzzy foloding by cut such that the final result gives a fuzzy fundamental group that is isomorphic to $\tilde{\mathbb{Z}}$). Similar arguments hold for ${\tilde{S}}_1^2$ and ${\tilde{S}}_2^2$. □

Figure 2. Stages of fuzzy folding by cut maps ${\tilde{\gamma }}_{n_{cut}}$.

Remark 2.

In the proof of Theorem 3, it is observed in the folding by stages that the singularity points appear on the surface, while at the end of the fuzzy folding (by cuts), the surface turns into a smooth curve. Moreover, this type of fuzzy folding induces boundaries.

Theorem 4.

The fuzzy fundamental group of the fuzzy coordinates singularity is the fuzzy identity group.

Proof. 

The fuzzy poles of ${\tilde{S}}^3\subset {\tilde{W}}^4$ are the fuzzy single points $\phi (s)=0$ and $\phi (s)=\pi$. Since ${\tilde{u}}_1^2+{\tilde{u}}_2^2+{\tilde{u}}_3^2+{\tilde{u}}_4^2-{\tilde{u}}_5^2=b^2,$ the fuzzy points ${\tilde{u}}_1=\pm b,{\tilde{u}}_2={\tilde{u}}_3=0,{\tilde{u}}_4=bcosh{\displaystyle \frac{t(s)}{b}},$ and ${\tilde{u}}_5=bsinh{\displaystyle \frac{t(s)}{b}}.$ Hence, the fuzzy fundamental group of the fuzzy points $\phi (s)=0$ and $\phi (s)=\pi$ is the fuzzy identity group. □

Theorem 5.

The fuzzy ${\tilde{W}}^4$ induces the fuzzy $\overline{{\tilde{W}}^4}$ and ${\underline{\tilde{W}}}^4$, such that the fuzzy fundamental group of the fuzzy deformation retract of $\tilde{W}-{\tilde{\delta }}_i$ onto ${\tilde{S}}^3$ induces the fuzzy fundamental groups of the fuzzy deformation retracts of $\overline{{\tilde{w}}^4}-\overline{{\tilde{\delta }}_i}$ onto ${\overline{\tilde{S}}}^3$ and ${\underline{\tilde{W}}}^4-\underline{{\tilde{\delta }}_i}$ onto ${\underline{\tilde{S}}}^3$.

Proof. 

Consider a fuzzy ${\tilde{S}}_3^2$ with fuzzy retractions ${\tilde{S}}_2^1$ and ${\tilde{S}}_3^1$. There are nested fuzzy circles $\overline{{\tilde{S}}_i^1}$ and $\underline{{\tilde{S}}_i^1}$ inside and outside ${\tilde{S}}_2^1$ and ${\tilde{S}}_3^1$, respectively. Denote by $\tilde{D}·\tilde{R}$ the fuzzy deformation retract and consider the following diagram.

Note that from the above diagram that any fuzzy deformation retract of $\left({\tilde{S}}_2^1-\tilde{p}\right)$ onto ${\tilde{p}}^{\prime }$ induce fuzzy deformation retracts of ( ${\overline{\tilde{S}}}_i-{\overline{\tilde{p}}}_i$ ) onto ${\overline{{\tilde{p}}^{\prime }}}_i$ and $\left(\underline{{\tilde{\tilde{S}}}_i^{\prime }}-{\underline{\tilde{p}}}_i\right)$ onto $\underline{{\tilde{p}}_i^{\prime }}$. Thus, we have

$${\pi }_1\left(\tilde{D}·\tilde{R}\left({\tilde{S}}_3^2-\left({\tilde{\mu }}_i\right)\right)={\pi }_1\left(\tilde{D}·\tilde{R}\left({\overline{{\tilde{S}}^1}}_i-\left({\overline{\tilde{p}}}_1\right)\right)\simeq \right.\right.\mathrm{fuzzy} \mathrm{identity} \mathrm{group}.$$

Also, we have

$${\pi }_1\left(\tilde{D}·\tilde{R}\left({\tilde{S}}_3^2-\left({\tilde{\mu }}_i\right)\right)={\pi }_1\left(\tilde{D}·\tilde{R}\left(\underline{{\tilde{S}}_i^1}-\left(\underline{{\tilde{p}}_1}\right)\right)\simeq \right.\right.\mathrm{fuzzy} \mathrm{identity} \mathrm{group}.$$

This is a fuzzy deformation retract for ${\widetilde{S_3}}^2\subseteq {\tilde{W}}^4$. We conclude that the fuzzy deformation retract of ${\tilde{W}}^4$ is as follows:

□

Theorem 6.

The fuzzy fundamental group of ${\tilde{W}}^4$ generates the b couple of the fuzzy fundamental group of the two fuzzy systems of fuzzy $\bigcup \overline{{\tilde{W}}^4}$ and $\bigcup {\underline{\tilde{W}}}^4$.

Proof. 

Using Equation (8), ${\tilde{S}}^3$ is a fuzzy sphere and $s(b)=1$, for all points ${\tilde{\delta }}_i\in {\tilde{S}}^3$. Therefore, there are nested 2-chains of n-pure fuzzy spheres $\bigcup \overline{{\tilde{S}}_i^3}$ and $\bigcup {\underline{{\tilde{S}}_i}}^3$ induced. Thus, the fuzzy system consists of fuzzy spheres $\bigcup {\underline{{\tilde{S}}_i}}^3$ inside ${\tilde{S}}^3$ and $\bigcup \overline{{\tilde{S}}_i^3}$ outer ${\tilde{S}}^3$, with membership degree $\underline{s_i}=\underline{r_i}$, where $\underline{s_i}\to 0$ as $\underline{r_i}\to 0$ and ${\overline{s}}_i={\displaystyle \frac{1}{{\overline{r}}_i}}$, ${\overline{s}}_i\to 0$ as ${\overline{r}}_i\to \infty$ and ${\overline{s}}_i=$, $\forall i=1,2,\dots$.

Now, nested n-fuzzy spheres with common centers exist, as in Figure 3, so the nested fuzzy fundamental group of $\bigcup {\underline{{\tilde{S}}_i}}^3$ exists inside ${\tilde{S}}^3$, and nested fuzzy fundamental groups of $\bigcup \overline{{\tilde{S}}_i^3}$ outer ${\tilde{S}}^3$, apart from the fuzzy fundamental group of ${\tilde{S}}^3$. It follows that we have

$${\pi }_1({\tilde{W}}^4-{\delta }_i)\approx \pi (\bigcup \underline{\widetilde{S_i^3}}),$$

$${\pi }_1({\tilde{W}}^4-{\delta }_i)\approx \pi (\bigcup \overline{{\tilde{S}}_i^3},$$

and

$${\pi }_1({\tilde{W}}^4-{\delta }_i)\approx \pi ({\tilde{S}}^3).$$

Therefore, ${\pi }_1({\tilde{W}}^4-{\delta }_i)$ is isomorphic to fuzzy identity group. Moreover, the fuzzy identity group for $s_i$ is a chain of points up and down ${\tilde{S}}^3$ (see Figure 3). The proof is completed.

Figure 3. The n-fuzzy spheres with common center.

Figure 3. The n-fuzzy spheres with common center.

□

Theorem 7.

Let $\tilde{r}:\left({\tilde{S}}_3^2-({\tilde{\mu }}_i)\right)\to {\tilde{S}}_2^1,{\tilde{S}}_3^2\subset {\tilde{W}}^4,\tilde{f}:\left({\tilde{S}}_3^2-\left(\widetilde{{\mu }_i}\right)\right)\to \left({\tilde{S}}_3^2-\left(\widetilde{{\mu }_i}\right)\right),$

then there are induced 2-chains of fuzzy retractions and fuzzy foldings, such that

$${\pi }_1\left({\tilde{r}}_2\circ {\tilde{f}}_1\left({\tilde{S}}_3^2-\left({\tilde{\mu }}_r\right)\right)\right)={\pi }_1\left({\tilde{f}}_2\circ {\tilde{r}}_1\left({\tilde{S}}_3^2-\left({\tilde{\mu }}_i\right)\right)\right)\simeq \tilde{Z}.$$

Proof. 

Let the fuzzy retraction of ${\tilde{S}}_3^2$ be defined as

${\tilde{r}}_1:\left({\tilde{S}}_3^2-\left({\tilde{\mu }}_i\right)\right)\to {\tilde{S}}_2^1$, and the fuzzy foldings of ${\tilde{S}}_3^2,{\tilde{S}}_2^1$ are given by ${\tilde{f}}_1:\left({\tilde{S}}_3^2-\left({\tilde{\mu }}_i\right)\right)\to \left({\tilde{S}}_3^2-\left({\tilde{\mu }}_i\right)\right),{\tilde{f}}_2:{\tilde{S}}_2^1\to {\tilde{S}}_2^1$. Then, there are induced 2-chains of fuzzy retraction and fuzzy foldings given by

$$\begin{array}{cc}& {\overline{\tilde{r}}}_1:\left({\overline{\tilde{S}}}_3^2-\left({\overline{\tilde{\mu }}}_1\right)\right)\to {\overline{\tilde{S}}}_2^1,{\tilde{\underline{r}}}_1:\left({\underline{\tilde{S}}}_2^2-\left({\underline{\mu }}_i\right)\right)\to {\underline{\tilde{S}}}_2^1\hfill \\ & \overline{{\tilde{f}}_1}:\left({\overline{\tilde{S}}}_3^2-\left({\overline{\tilde{\mu }}}_i\right)\right)\to \left({\overline{\tilde{S}}}_3^2-\left({\overline{\tilde{\mu }}}_i\right)\right),\overline{{\tilde{f}}_2}:{\overline{\tilde{S}}}_2^1\to {\overline{\tilde{S}}}_2^1\hfill \\ & {\tilde{\underline{f}}}_1:\left({\underline{\tilde{S}}}_3^2-\left({\tilde{\mu }}_i\right)\right)\to \left({\underline{\tilde{S}}}_3^2-\left({\tilde{\mu }}_i\right)\right),{\tilde{\underline{f}}}_2:{\underline{\tilde{S}}}_2^1\to {\underline{\tilde{S}}}_2^1\hfill \end{array}$$

Hence, the following diagrams are commutative:

It follows that ${\pi }_1\left({\tilde{r}}_2\circ {\tilde{f}}_1\left({\tilde{S}}_3^2-({\tilde{\mu }}_i)\right)\right)={\pi }_1\left({\tilde{f}}_2\circ {\tilde{r}}_1\left({\tilde{S}}_3^2-\left({\tilde{\mu }}_i\right)\right)\right)\simeq \tilde{Z}$.

Also, we have

Therefore,

$${\pi }_1\left({\overline{\tilde{r}}}_2\circ {\overline{\tilde{f}}}_1\left({\overline{\tilde{S}}}_3^2-\left({\overline{\tilde{\mu }}}_i\right)\right)\right)={\pi }_1\left({\overline{\tilde{f}}}_2\circ {\overline{\tilde{r}}}_1\left({\overline{\tilde{S}}}_3^2-\left({\overline{\tilde{\mu }}}_i\right)\right)\right)\simeq \overline{\tilde{Z}}$$

and

$${\pi }_1\left({\underline{\tilde{r}}}_2\circ {\underline{\tilde{f}}}_1\left({\underline{\tilde{S}}}_3^2-\left({\underline{\tilde{\mu }}}_i\right)\right)\right)={\pi }_1\left({\underline{\tilde{f}}}_2\circ {\underline{\tilde{r}}}_1\left({\underline{\tilde{S}}}_3^2-\left({\underline{\tilde{\mu }}}_i\right)\right)\right)\simeq \underline{\tilde{Z}}.$$

This finishes the proof. □

Theorem 8.

If the fuzzy retraction of ${\tilde{S}}^3\subseteq {\tilde{W}}^4$ is ${\tilde{r}}_1:\left({\tilde{S}}^3-({\tilde{\mu }}_i)\right)\to \left({\tilde{S}}_1^2-\left({\tilde{\mu }}_{i-1}\right)\right)$ and the limit of the fuzzy foldings $\underset{n\to \infty }{lim}{\tilde{f}}_n$: $\left({\tilde{S}}_1^2-\left({\tilde{\mu }}_{i-1}\right)\right)\to {\tilde{S}}_1^1-\left({\tilde{\mu }}_{i-2}\right)$, then there are induced chains of fuzzy retractions and the limit of fuzzy folding, such that

$${\pi }_1\left({\tilde{r}}_2\circ \underset{n\to \infty }{lim}{\tilde{f}}_n\left({\tilde{S}}^3-\left({\tilde{\mu }}_i\right)\right)\right)={\pi }_1\left(\underset{n\to \infty }{lim}{\tilde{f}}_{n+1}\circ {\tilde{r}}_1\left({\tilde{S}}^3-\left({\tilde{\mu }}_i\right)\right)\right)\simeq \tilde{Z}$$

Proof. 

Let the fuzzy retractions of ${\tilde{S}}^3\subset {\tilde{W}}^4$ be defined by ${\tilde{r}}_1:\left({\tilde{S}}^3-\left({\tilde{\mu }}_i\right)\right)\to \left({\tilde{S}}_1^2-\left({\tilde{\mu }}_{i-1}\right)\right),{\tilde{r}}_2:\left({\tilde{S}}_1^2-\left({\tilde{\mu }}_{i-1}\right)\right)-\left({\tilde{S}}_1^1-\left({\tilde{\mu }}_{i-2}\right)\right)$ and the limit of fuzzy folding is given by

${\tilde{f}}_1:\left({\tilde{S}}^3-\left({\tilde{\mu }}_i\right)\to {\left({\tilde{S}}^3-\left({\tilde{\mu }}_i\right)\right)}^{\prime },{\left({\tilde{S}}^3-\left({\tilde{\mu }}_i\right)\right)}^{\prime }\subset \left({\tilde{S}}^3-\left({\tilde{\mu }}_i\right)\right),\right.$

${\tilde{f}}_2:{\tilde{f}}_1\left({\tilde{S}}^3-\left({\tilde{\mu }}_i\right)\right)\to {\tilde{f}}_1\left({\tilde{S}}^3-\left({\tilde{\mu }}_1\right)\right),{\tilde{f}}_2\left({\tilde{f}}_1\left({\tilde{S}}^3-\left({\tilde{\mu }}_i\right)\right)\subset {\tilde{f}}_1\left({\tilde{S}}^3-\left({\tilde{\mu }}_i\right)\right),\right.$

${\tilde{f}}_3:{\tilde{f}}_2\left({\tilde{f}}_1\left({\tilde{S}}^3-\left({\tilde{\mu }}_i,\right)\right)\to {\tilde{f}}_2\left({\tilde{f}}_1\left({\tilde{S}}^3-{\tilde{\mu }}_i,\right)\right)\right),{\tilde{f}}_3\left({\tilde{f}}_2\left({\tilde{f}}_1\left({\tilde{S}}^3-\left({\tilde{\mu }}_i\right)\right)\right)\subset f_2\left({\tilde{f}}_1\left({\tilde{S}}^3-{\tilde{\mu }}_i\right)\right)\right),$

⋮

${\tilde{f}}_n:{\tilde{f}}_{n-1}\left({\tilde{f}}_{n-2}\dots {\tilde{f}}_2\left({\tilde{f}}_1\left({\tilde{S}}^3-{\tilde{\mu }}_i\right)\right)\right)\to {\tilde{f}}_{n-1}\left({\tilde{f}}_{n-2}\dots {\tilde{f}}_2\left({\tilde{f}}_1\left({\tilde{S}}^3-\left({\tilde{\mu }}_i\right)\right)\right)\right..$

Note that ${\tilde{f}}_n\left({\tilde{f}}_{n-1}\left({\tilde{f}}_{n-2}\dots {\tilde{f}}_2\left({\tilde{f}}_1\left({\tilde{S}}^3-\left({\tilde{\mu }}_i\right)\right)\right)\right)\subset {\tilde{f}}_{n-1}\left({\tilde{f}}_{n-2}\dots {\tilde{f}}_2\left({\tilde{f}}_1\left({\tilde{S}}^3-\left({\tilde{\mu }}_1\right)\right)\right)\right)\right.$. Therefore, $\underset{n\to \infty }{lim}{\tilde{f}}_n\left({\tilde{f}}_{n-1}\left({\tilde{f}}_{n-2}\dots {\tilde{f}}_2\left({\tilde{f}}_1\left({\tilde{S}}^3-\left({\tilde{\mu }}_i\right)\right)\right)\right)=\left({\tilde{S}}^2-\left({\tilde{\mu }}_{i-1}\right)\right)\right.$.

Hence, the following diagrams are commutative:

Therfore,

$${\pi }_1\left({\overline{\tilde{r}}}_2\circ \underset{n\to \infty }{lim}{\overline{\tilde{f}}}_n\left({\tilde{S}}^3-\left({\overline{\tilde{\mu }}}_i\right)\right)={\pi }_1\left(\underset{n\to \infty }{lim}{\overline{\tilde{f}}}_{n+1}\circ {\underline{\tilde{r}}}_1\left({\overline{\tilde{S}}}^3-\left({\overline{\tilde{\mu }}}_i\right)\right)\right)\right.\simeq \overline{\tilde{Z}}$$

and

$${\pi }_1\left({\underline{\tilde{r}}}_2\circ \underset{n\to \infty }{lim}{\underline{\tilde{f}}}_n\left({\tilde{S}}^3-\left({\underline{\tilde{\mu }}}_i\right)\right)={\pi }_1\left(\underset{n\to \infty }{lim}{\underline{\tilde{f}}}_{n+1}\circ {\underline{\tilde{r}}}_1\left({\underline{\tilde{S}}}^3-\left({\underline{\tilde{\mu }}}_i\right)\right)\right)\right.\simeq \underline{\tilde{Z}}.$$

□

4. The Discriminants of the Fuzzy Retracts

In the current section, we discuss the discriminants of the fuzzy retract singularities, which were obtained in the previous section and are not well described. In particular, we shall consider the singularities ${\tilde{W}}^3,{\tilde{W}}_1^2$, and ${\tilde{W}}_2^2$.

Throughout this section and for simplicity, we shall set $t(s)=t,\phi (s)=\phi ,$ and $\theta (s)=\theta$.

Consider the fuzzy retract ${\tilde{W}}^3$, which is parametrized locally in ${\mathbb{R}}^4$ by

$$f:(t,\phi ,\theta )\mapsto \left(bcosh{\displaystyle \frac{t}{b}}sin\phi sin\theta ,bcosh{\displaystyle \frac{t}{b}}sin\phi cos\theta ,bcosh{\displaystyle \frac{t}{b}}cos\phi ,bsinh{\displaystyle \frac{t}{b}}\right).$$

Then, the Jacobian matrix of f is

$${\mathbf{J}}_f=\left(\begin{array}{ccc}sinh{\displaystyle \frac{t}{b}}sin\phi sin\theta & bcosh{\displaystyle \frac{t}{b}}cos\phi sin\theta & bcosh{\displaystyle \frac{t}{b}}sin\phi cos\theta \\ sinh{\displaystyle \frac{t}{b}}sin\phi cos\theta & bcosh{\displaystyle \frac{t}{b}}cos\phi cos\theta & -bcosh{\displaystyle \frac{t}{b}}sin\phi sin\theta \\ sinh{\displaystyle \frac{t}{b}}cos\phi & -bcosh{\displaystyle \frac{t}{b}}sin\phi & 0\\ cosh{\displaystyle \frac{t}{b}}& 0& 0\end{array}\right)$$

The critical points of f satisfy $rank({\mathbf{J}}_f)<min(4,3)=3$. So, we have to consider the $3\times 3$-matrix ${\mathbf{J}}_f(i)$ which is obtained from ${\mathbf{J}}_f$ by removing the ith row, and solve $|{\mathbf{J}}_f(i)|=0$, for all i. Thus, the critical points of f satisfy

$$|{\mathbf{J}}_f(1)|=-b^2{cosh}^3{\displaystyle \frac{t}{b}}sin\phi sin\theta =0,$$

(20)

$$|{\mathbf{J}}_f(2)|=b^{2}cosh{\displaystyle \frac{t}{b}}{sin}^2\phi cos\theta =0,$$

(21)

$$|{\mathbf{J}}_f(3)|=-b^2{cosh}^3{\displaystyle \frac{t}{b}}cos\phi sin\phi =0,$$

(22)

and

$$|{\mathbf{J}}_f(4)|=-b{cosh}^2{\displaystyle \frac{t}{b}}sinh{\displaystyle \frac{t}{b}}sin\phi =0.$$

(23)

Clearly, the common solutions of (20), (21), (22) and (23) is $\phi =0,\pi$. Therefore, the set of critical values is the union of

$${\Sigma }_1^{*}f=\left\{0,0,0,bcosh{\displaystyle \frac{t(s)}{b}},bsinh{\displaystyle \frac{t(s)}{b}}\right\},$$

(24)

and

$${\Sigma }_2^{*}f=\left\{0,bsin\phi sin\theta ,bsin\phi cos\theta ,cos\phi ,0\right\}.$$

(25)

Note that the set ${\Sigma }_1^{*}f$ is a parametrization of one of the two connected components of a hyperbola $u_4^2-u_5^2=b^2$ (see Figure 4). Moreover, it is a fuzzy retract singularity.

Figure 4. A connected component of a hyperbola $u_4^2-u_5^2=1$.

On the other hand, the set ${\Sigma }_2^{*}f$ represents a 2-fuzzy sphere, which is also a fuzzy retract singularity.

Next, consider the fuzzy retract ${\tilde{W}}_1^2$, which is parametrized locally in ${\mathbb{R}}^3$ by

$$h:(t,\phi )\mapsto \left(bcosh{\displaystyle \frac{t}{b}}sin\phi ,bcosh{\displaystyle \frac{t}{b}}cos\phi ,bsinh{\displaystyle \frac{t}{b}}\right).$$

The Jacobian matrix of h is

$${\mathbf{J}}_h=\left(\begin{array}{cc}sinh{\displaystyle \frac{t}{b}}sin\phi & bcosh{\displaystyle \frac{t}{b}}cos\phi \\ sinh{\displaystyle \frac{t}{b}}cos\phi & -bcosh{\displaystyle \frac{t}{b}}cos\phi \\ cosh{\displaystyle \frac{t}{b}}& 0\end{array}\right)$$

Note that the matrix ${\mathbf{J}}_f$ can have $rank({\mathbf{J}}_f)<min(3,2)=2$ if and only if the following are satisfied

$$\left|\begin{array}{cc}sinh{\displaystyle \frac{t}{b}}sin\phi & bcosh{\displaystyle \frac{t}{b}}cos\phi \\ sinh{\displaystyle \frac{t}{b}}cos\phi & -bcosh{\displaystyle \frac{t}{b}}cos\phi \end{array}\right|={\displaystyle \frac{11}{2}}bsinh{\displaystyle \frac{2t}{b}}=0,$$

(26)

$$\left|\begin{array}{cc}sinh{\displaystyle \frac{t}{b}}sin\phi & bcosh{\displaystyle \frac{t}{b}}cos\phi \\ cosh{\displaystyle \frac{t}{b}}& 0\end{array}\right|=b{cosh}^2{\displaystyle \frac{t}{b}}cos\phi =0,$$

(27)

and

$$\left|\begin{array}{cc}sinh{\displaystyle \frac{t}{b}}cos\phi & -bcosh{\displaystyle \frac{t}{b}}cos\phi \\ cosh{\displaystyle \frac{t}{b}}& 0\end{array}\right|=b{cosh}^2{\displaystyle \frac{t}{b}}cos\phi =0.$$

(28)

However, relations (27) and (28) can not hold simultaneously for all values of t and $\phi$. Hence, ${\tilde{W}}_2^2$ is regular and there is no further fuzzy retract singularity in it. Indeed, ${\tilde{W}}_2^2$ can be written explicitly as ${\tilde{u}}_3^2+{\tilde{u}}_4^2-{\tilde{u}}_5^2=b^2$, which represents a regular cylinder (see Figure 5a), unless b is identically zero, in which case ${\tilde{W}}_2^2$ becomes singular at the origin (see Figure 5b).

Figure 5. The cylinder ${\tilde{u}}_3^2+{\tilde{u}}_4^2-{\tilde{u}}_5^2=b^2$.

The above discussion implies the following.

Proposition 1.

The non-regular fuzzy retract singularities induce new ones in it. In particular, the ${\tilde{W}}^3$ singularity induces the two fuzzy retract singularities ${\Sigma }_1^{*}f$ and ${\Sigma }_2^{*}f$.

5. Applications

(i)

There are many applications for these density functions. In medicine, for example, a patient with cancer develops numerous other sub-diseases in the heart, kidneys, blood, prostate, skin, and virtually every other organ. People who are infected with the hepatitis virus suffer from many symptoms in their bodies as a result of liver problems. Also, in the field of medicine, there is a dizziness of the eyes: the eye will see fish in the air, and some real figures in the air, if the person concerned is unwell. The phenomenon is represented by one of the matrices $g_1,g_0{g}_2$, where $g_1$ is inside the cone, $g_0$ is on the cone, and $g_2$ is outside the cone.

Mathematically, these factors and symptoms are expressed by the density functions where $C({\lambda }_1,{\lambda }_2,\dots ,{\lambda }_n),{\mu }_1,{\mu }_2,\dots ,{\mu }_n)$, ${\mu }_1\in [0,1]$, ${\mu }_2,{\mu }_3,\dots ,{\mu }_n\in [-1,1]$ are the effective functions.

(ii)

The geometric metric function represents real phenomena if the metric is a Riemannian metric ($d{\tilde{s}}^2$ in our setting). Still, in the case of a pseudo-Riemannian metric, it will be applied in phenomena that are not real, like the complex potential function and complex radiation.

(iii)

The Ritz variational method [19], which was used to calculate the ground-state energy, is in a fuzzy framework. Consider Hamilton H and an arbitrary square integrable function $\Psi$, so that $⟨\Psi /\Psi ⟩=1$. Considering $\Psi$ as a fuzzy function and the ranking system, as defined in [19], it can be shown that $⟨\Psi /H/\Psi ⟩$ is a fuzzy upper bound on $E_0$ (ground-state energy). Now, $⟨\Psi /H/\Psi ⟩$ should be minimizing the distance between $E_0$ and with respect to a number of parameters $a_1,a_2,\dots$. This can be performed by minimizing the distance between $E_0$ and $⟨\Psi /H/\Psi ⟩$.

6. Conclusions

In this article, we studied the fuzzy Friedmann–Lemaître–Robertson–Walker space. By solving the Lagrangian equations, new types of fuzzy retract singularities in ${\tilde{W}}^4$ are obtained. Then, we described the fundamental groups of fuzzy deformation retracts of ${\tilde{W}}^4$, which appeared to be isomorphic to either the fuzzy identity group or $\tilde{\mathbb{Z}}$. Then, we discussed the fuzzy fundamental group of fuzzy deformation retracts of $({\tilde{W}}^4-\left\{{\tilde{\mu }}_i\right\})$, and we found that it induces the chains of fuzzy fundamental groups up and down $({\tilde{W}}^4-\left\{{\tilde{\mu }}_i\right\}).$ The fuzzy fundamental group of fuzzy fold boundaries was also determined by cutting some fuzzy contraction singularities, and proved to be isomorphic to $\tilde{\mathbb{Z}}.$

Moreover, we showed that two chains of fuzzy fundamental groups of the two fuzzy systems of fuzzy $\bigcup \overline{{\tilde{W}}^4}$ and $\bigcup {\underline{\tilde{W}}}^4$ are induced by the fuzzy fundamental group of ${\tilde{W}}^4$. In conclusion, we discussed the regularity of the fuzzy retract singularities and found that the non-regular one induces new ones.