Topological Analysis of Fibrations in Multidimensional ( C , R ) Space

: A holomorphically fibred space generates locally trivial bundles with positive dimensional fibers. This paper proposes two varieties of fibrations (compact and non ‐ compact) in the non ‐ uniformly scalable quasinormed topological ( C , R ) space admitting cylindrically symmetric continuous functions. The projective base space is dense, containing a complex plane, and the corresponding surjective fiber projection on the base space can be fixed at any point on real subspace. The contact category fibers support multiple oriented singularities of piecewise continuous functions within the topological space. A composite algebraic operation comprised of continuous linear translation and arithmetic addition generates an associative magma in the non ‐ compact fiber space. The finite translation is continuous on complex planar subspace under non ‐ compact projection. Interestingly, the associative magma resists transforming into a monoid due to the non ‐ commutativity of composite algebraic operation. However, an additive group algebraic structure can be admitted in the fiber space if the fibration is a non ‐ compact variety. Moreover, the projection on base space supports additive group structure, if and only if the planar base space passes through the real origin of the topological ( C , R ) space. The topological analysis shows that outward deformation retraction is not admissible within the dense topological fiber space. The comparative analysis of the proposed fiber space with respect to Minkowski space and Seifert fiber space illustrates that the group algebraic structures in each fiber spaces are of different varieties. The proposed topological fiber bundles are rigid, preserving sigma ‐ sections as compared to the fiber bundles on manifolds. in the corresponding topological base space, then the cross ‐ sections of an automorphic bundle within the subspace form an algebraic group structure. This paper proposes the construction and analysis of fiber space in the non ‐ uniformly scalable multidimensional topological space [16]. One of the interesting aspects of multidimensional topological space is that the space is quasinormed, admitting cylindrically symmetric continuous functions, and does not always preserve compactness under topological projections. Hence, the interesting and motivating questions are: what are the possible varieties of fibrations in such space, and is it possible to establish any algebraic structures within the respective fiber space? Moreover, what are the topological properties of the resulting fiber space within the quasinormed multidimensional the holomorphic condition of complex subspace is relaxed? These questions are addressed in this paper. The presented analysis considers algebraic as well as topological standpoints as required. The elements of functional analysis are employed whenever necessary.


Introduction
The Minkowski space is a four-dimensional topological vector space over reals (i.e., four-manifold admitting Poincare symmetry group of isometries) with applications in physical and mathematical sciences [1,2]. In general, the Minkowski space is not well behaved if the corresponding Euclidean topological space is considered to be a locally homogeneous space [3]. The reason is that the Minkowski topological space gets decomposed into two locally homogeneous Euclidean subspaces, where the two topological subspaces are separated in nature. Note that the finest topology in  n dimensional Minkowski space is Zeeman topology, which is separable, Hausdorff, locally noncompact, and also non-Lindeloff in nature. Moreover, the Zeeman topology generates a first countable topological space. On the other hand, the 1  n dimensional Minkowski space equipped with t-topology ( t M ) is not completely Euclidean in nature [4]. The topological space on t M is first countable, where the compactification and continuity of a function can be maintained through the Zeno sequences. In view of general topology, the 4D Minkowski space equipped with s-topology is not a normal topological space, and it is a non-compact Hausdorff space. The axial rotations of a Minkowski space generate various geometric hypersurfaces in space. For example, the three types of helicoidal hypersurfaces are generated by axial rotation of 4-dimensional Minkowski space [5]. In this paper, the construction of topological fiber space in a non-uniformly scaled quasinormed ) , ( R C space and the corresponding topological, as well as algebraic analysis of fibration varieties are presented. First, the brief descriptions about the related concepts, such as topological fiber spaces and immersion of manifolds are presented in Sections 1.1 and 1.2, respectively. Next, the motivation for this work and contributions made in this paper are explained in Section 1.3. In this paper, the symbols R , C and Z represent sets of extended real numbers, complex numbers, and integers, respectively.

Topological Fiber Spaces
The topological fiber bundles over a sphere exhibit a set of interesting topological properties if the respective fiber space is Euclidean. It is shown that if M is a closed and compact manifold maintaining Hausdorff topological property, then the function space X containing functions without interchanging the ends in R is a contractible space [6]. In view of algebraic topology, the structure n G represents a monoid of homotopy self-equivalences of   ) 1 (n spheres denoted by 1  n S [6]. Note that in this case, the topological space is in the compactopen category.
However, from the viewpoint of differential geometry, the structure of a fiber space can be considered to be rigid under specific conditions, such as orientations [7,8]. Specifically, a Seifert fiber space is profinitely rigid if it is an oriented variety [8]. Otherwise, the Seifert fiber space is based on three-manifold, which is classified depending upon a set of invariants, and it incorporates rigidity of infinite 1  with a hyperbolic two-orbifold base of fibration [9]. Similarly, the Haken orientable hyperbolic three-manifold is an irreducible variety, and it supports the rigidity of fibration [10].

Manifolds and Immersions
In view of geometric topology, the immersion of surfaces into the null submanifolds in 3D offers various interesting observations. If the structure ) , ( g M is a Lorentzian manifold under tensor h with M N i  : immersion function, then ) , ( h N is a degenerative null submanifold [11]. Interestingly, the Lorentzian manifolds are not metrizable, preserving isometry in reference to the space of immersion. It is known that the immersion of a hypersurface S into a Euclidean space with normal vector field N V is self-adjoint in the presence of a suitable shape operator. In general, a Lorentz space can be finitely covered by a circle bundle if it is a compact space [12]. Note that topologically the spaces with Lorentzian geometry (such as a torus bundle) are locally Hausdorff, and the corresponding manifold is not a globally Hausdorff topological space. The immersion of a  k dimensional manifold in a   l k dimensional Euclidean space is given by induces the one-to-one map in tangent space on the manifold [13]. The immersion space is considered to be a regular topological space, and the manifold is a connected topological space with orientation.

Motivation and Contributions
The fiber bundles are geometric as well as topological objects, which can be simulated and visualized in computer models. The computer visualization of fiber bundles as geometric objects has opened up a wide array of applications in various domains of physical sciences as well as computational sciences [14]. In the topological spaces of fiber bundles, the determination of equivalence between fiber bundles is a challenging task. The fiber bundle space reduction theorem indicates that two topological fiber bundle spaces are equivalent if and only if the corresponding Ehresmann bundles have cross-sections over common base space [15]. Interestingly, if we consider subspace X X  0 in the corresponding topological base space, then the cross-sections of an automorphic bundle within the subspace form an algebraic group structure. This paper proposes the construction and analysis of fiber space in the non-uniformly scalable multidimensional topological ) , ( R C space [16]. One of the interesting aspects of multidimensional topological ) , ( R C space is that the space is quasinormed, admitting cylindrically symmetric continuous functions, and does not always preserve compactness under topological projections. Hence, the interesting and motivating questions are: what are the possible varieties of fibrations in such space, and is it possible to establish any algebraic structures within the respective fiber space? Moreover, what are the topological properties of the resulting fiber space within the quasinormed multidimensional the holomorphic condition of complex subspace is relaxed? These questions are addressed in this paper. The presented analysis considers algebraic as well as topological standpoints as required. The elements of functional analysis are employed whenever necessary.
The main contributions made in this paper can be summarized as follows. The topological ) , ( R C space is a non-uniformly scalable and quasinormed space, where the cylindrical open sets form the topological basis. The proposed fibrations within the space can be constructed in two varieties, such as compact fibration and non-compact fibration. The fiber space is considered to be dense, and it can admit the concept of a special category of fibers called contact fibers. The fiber space is equipped with finite linear translation operation. The resulting fiber space in the topological ) , ( R C space supports the expansion and orientations of multiple singularities of a piecewise continuous function on contact fibers. It is shown that a composite algebraic operation comprised of linear translation, and arithmetic addition prepares an associative magma in the non-compact fiber space. The associative magma space is commutative under linear translation within the magma space of fibers, and it resists the formation of a monoid under composite algebraic operation. However, an additive group algebraic structure can be admitted in the fiber space and in projective base space under specific conditions. Interestingly, the proposed fiber space does not support outward deformation retraction in a dense subspace.
The rest of the paper is organized as follows. Section 2 presents a set of preliminary concepts enhancing the completeness of the paper. The definitions related to topological fiber spaces and fibrations are presented in Section 3. The algebraic and topological properties are presented in Section 4. Section 5 illustrates the concepts of expansion and singularities in the proposed topological fiber space. A detailed comparative analysis of this work with respect to other contemporary works in the domain is presented in Section 6. Finally, Section 7 concludes the paper.   [3].

Preliminary Concepts
The Steenrod formulation of fiber bundles is represented as an algebraic structure is a projection function, Y is a topological space called fiber, and G is a bundle group [15]. In the formulation, the elements j j V  , are coordinate neighborhoods and the corresponding coordinate functions, respectively. The coordinate functions must satisfy a condition which is given by Note that a four-dimensional Minkowski space is essentially a real vector space preserving Hausdorff topological property [17].

Topological Fiber Space and Fibrations
In this section, a set of definitions related to fibrations and the resulting generation of fiber spaces in a quasinormed topological In this paper, o A and A represent the interior and closure of an arbitrary set A such that define the projective base of a fiber space in the topological maintaining the ordering relation given by . The projection of corresponding tangent subspace onto topological base space at R r  is defined as:

Topological Fiber Bundle in
The topological fiber bundle in the corresponding tangent space

Fiber Space
The fiber space in a topological , ( X X  is formulated based on the sets of fiber bundles in the dense subspaces. Let us consider In the above definition a F is a local fiber bundle of corresponding topological subspace , ( X X  . The entire fiber space of topological ) , ( R C space can be generated from the sets of local fiber bundles as given below: Earlier it is mentioned that there are two varieties of fibrations, which can be admitted into the topological ) , ( R C space. However, it is interesting to observe that in any case, the fiber space in ) , ( X X  is not compact, which is presented in the following proposition.

Remark 2: The above proposition further indicates that the fiber space
In general, the Seifert fiber space on an oriented manifold maintains profinite rigidity [8].
However, in the case of a holomorphic map between two complex manifolds M and N , the rigidity of fibration allows sending a single fiber to another fiber, as along as the complex manifolds, are individually connected spaces [18]. In this paper, the topological ) , ( R C space is a connected space, and the associated fiber space supports finite translation of fibers within the space.

Remark 3: Note that, the translation
 within the open subspaces of fibers. However, the translation can be transformed into a strictly closed and convergent variety if the following restrictions are imposed on it: The properties of finite translations within the topological fiber space depend on the Baire categorization of a subspace (i.e., dense or meager) and compactness of the corresponding projective subspace. The following proposition presents such observation.
The proposed fibration in a topological ) , ( R C space is equipped with a special category of fiber called contact category, as defined below. A contact category fiber admits multiple oriented singularities of a function within the fibered topological space under specific conditions.

Contact Category Fiber
be two fiber subspaces in the topological is defined to be in the contact fiber category if the following conditions are maintained by it: Note that the restriction for a contact fiber to be maintained is given as, . This restriction is required to retain the finiteness of projection of fiber space into the topological base space of the contact fiber category. Once the contact fiber is defined, we can formulate the oriented singularities of a function in the topological space.

Oriented Singularities of Function
It is important to note that a planar and symmetric function with multiple oriented singularities in the fiber space of topological ) , ( R C space is a discontinuous variety (i.e., the function is piecewise continuous). Furthermore, the fiber admitting multiple singularities of a planar as well as symmetric function is in the contact fiber category, and the corresponding fibration is non-compact type.

Algebraic and Topological Properties
In this section, the algebraic and topological properties of fiber space in the quasinormed topological ) , ( R C space are presented. First, we show that the fiber space CR  is an associative magma under the algebraic composite operation ) ( T  as given below: The above representation indicates that ) ( T  is a composite algebraic operation, which is comprised of functional translation and arithmetic addition within the topological space. Note that the linearity of translation operation is considered to be maintained for generality. It is interesting to observe that the algebraic structure    (   CR  I  Tc  b  T  a  I  c  I  b  I  a   CR  I  Tc  Tb  a  I  c  I This indicates that the fiber space is associative if T T  Thus, further translation in the associative magma of fiber space leads to the following conclusions considering However, the algebraic structure . This leads to the following algebraic identity:    (   I  c  I  b  I  a  I  c  I  b  I  a   b  a  I  b  a  I  b  I  a (14) Moreover, at the origin 0  is an identity fiber (i.e., original fiber) because, Proof: The proof is relatively straightforward. Suppose we select an arbitrary to prepare a topological base space for projection. In this case, the and in addition the following condition is maintained,

Expansion and Singularity
The fiber space CR  in a quasinormed topological

Comparative Analysis
There are varieties in the construction of fiber bundles and associated singularities based on various parameters, such as the connectedness of topological spaces, locality of homeomorphism, and different types of projection maps. The comparison of various properties associated with different varieties of fiber spaces is summarized in Table 1.
In the case of Seifert fiber space, the fiber bundles are The fibration and fiber bundles in Minkowski space consider four-manifold ( 4 M ) structure [20,21]. Interestingly the fibration in Minkowski space is possible if the space is Hausdorff and connected topological space, which is locally Euclidean on  [20]. Interestingly, the holomorphic and principal fiber bundle of a compact complex manifold supports a complex Lie group structure [22]. The topological decomposition indicates that Minkowski space can be completely decomposed into two components, such as 3D real Euclidean subspace and 1D real Euclidean subspace [21]. On the contrary, the topological ) , ( R C space is a quasinormed space supporting non-uniform scaling, and it can be topologically decomposed into two components, such as 2D complex subspace and 1D real subspace.

Conclusions
A fiber space is a topological space where it locally behaves as a product space, but globally the space has a different topological structure. The analytical properties of fibrations and resulting fiber spaces vary depending upon the constructions as well as the structures of topological spaces. The , ( R C space, Minkowski space, and Seifert fiber space illustrates that the supporting group algebraic structures are different for each space. Moreover, the topologically decomposed subspaces have different dimensions as well as properties in each space. Finally, this is to note that the proposed constructions of topological fibrations may find applications in mathematical sciences (i.e., manifold immersion and surface classifications) and in physical sciences (i.e., supersymmetry, topological string theory, cosmology, and analyzing symmetry fibration in biological networks).
Author Contributions: The author (S.B., Department of Aerospace and Software Engineering (Informatics), Gyeongsang National University, Jinju, ROK) is the sole and single author of this paper, and the paper contains his own contributions. The author has read and agreed to the published version of the manuscript.
Funding: The research is funded by Gyeongsang National University, Jinju, Korea.