Interactions between Homotopy and Topological Groups in Covering ( C , R ) Space Embeddings

: The interactions between topological covering spaces, homotopy and group structures in a fibered space exhibit an array of interesting properties. This paper proposes the formulation of finite covering space components of compact Lindelof variety in topological ( C , R ) spaces. The covering spaces form a Noetherian structure under topological injective embeddings. The locally path ‐ connected components of covering spaces establish a set of finite topological groups, main ‐ taining group homomorphism. The homeomorphic topological embedding of covering spaces and base space into a fibered non ‐ compact topological ( C , R ) space generates two classes of fibers based on the location of identity elements of homomorphic groups. A compact general fiber gives rise to the discrete variety of fundamental groups in the embedded covering subspace. The path ‐ homotopy equivalence is admitted by multiple identity fibers if, and only if, the group ho ‐ momorphism is preserved in homeomorphic topological embeddings. A single identity fiber maintains the path ‐ homotopy equivalence in the discrete fundamental group. If the fiber is an identity ‐ rigid variety, then the fiber ‐ restricted finite and symmetric translations within the em ‐ bedded covering space successfully admits path ‐ homotopy equivalence involving kernel. The topological projections on a component and formation of 2 ‐ simplex in fibered compact covering space embeddings generate a prime order cyclic group. Interestingly, the finite translations of the 2 ‐ simplexes in a dense covering subspace assist in determining the simple connectedness of the covering space components, and preserves cyclic group structure.


Introduction
The structures and properties of topological covering space o C of any arbitrary base space X have been studied in detail by Lubkin [1]. The construction of covering space of an arbitrary base space is formulated without any requirement of additional local as well as global topological properties, such as path connectedness. In general, if a continuous surjection between two spaces is given by is a locally trivial sheaf [1]. If the local connectedness of the topological space is further re-  X is Hausdorff and contractible (i.e., compact CW-complex with finitely many components) [2]. The characterizations of Hausdorff topological spaces can be made by employing the concept of compact-covering maps, which is analogous to the definition of sequence-covering maps [3]. Note that such covering projective spaces can successfully admit fibrations and injective embeddings. The preservation of fibrations in a covering space and projection requires specific topological properties. For example, if a topological space is locally compact then the preservation of covering fibration is maintained by the local homotopy of the respective topological space [4]. A topological space can admit a class of coverable topological groups by generalizing the concept of cover if the corresponding topological space is of a metrizable and connected variety [5].
This paper investigates the nature of interactions between homeomorphically embedded covering ) , ( R C spaces of Lindelof as well as Noetherian variety (under topological injective embeddings) and the finite topological groups within the covering spaces under fibration. The topological properties of corresponding structures and interactions are presented in detail, in view of algebraic as well as geometric topology. First we present the associated concepts and the resulting motivation as well as the summary of contributions made in this paper to address the wider audience. Note that in this paper  denotes an index set and ) , hom (

Motivation and Contributions
It is known that the Poincare group displays a set of interactive properties in the topological covering spaces. It is noted that every filtered Galois group is isomorphic to the corresponding Poincare filtered group of regular covering space where the topological space is a connected variety [1]. Moreover in a covering projection if a connected topological group G exists in the simply connected base space then there is a connected covering space group preserves respective covering projection properties while incorporating group homomorphism [6]. Interestingly, the varieties of Hausdorff and contractible topological spaces of continuous functions admit fibrations and topological embeddings under injective inclusions [2]. Recall that the covering spaces are generally considered to be path-connected topological spaces. There is a natural way to establish fibrations in a path-connected space X . If ] 1 , 0 [ X denotes the space of every path in a path-connected topological space X then there is a natural fibration given by [7]. However a discretely fibered covering space can also be formulated in a covering projection [8]. These observations suggest the importance of investigating the properties of interactions of topological groups and covering spaces under embeddings in a fibered topological ) , ( R C space which admits Noetherian convex P-separations [9,10]. Hence, the motivating questions can be summa-rized as: (1) what are the properties of interactions between the topological groups in homeomorphically embedded compact Lindelof-Noetherian planar covering spaces and the fibers in a path connected ) , ( R C subspace, (2) is it possible to categorize the fibers in such covering spaces to establish a topological structure in path-connected embeddings in a ) , ( R C space, and (3) what are the properties of generated planar simplicial complexes in view of geometric topology within the respective embedded covering ) , ( R C spaces under fibration? These questions are addressed in this paper in relative detail.
The main contributions made in this paper can be summarized as follows. The concepts related to compact Lindelof as well as finite Noetherian (under topological injective embeddings) covering space components are introduced and the formulation of homomorphic topological groups within such covering spaces is presented. Next, the topological properties of embeddings of such covering spaces in a fibered topological ) , ( R C space are investigated in detail, involving finite as well as symmetric translations restricted on the identity-rigid fibers. It is shown that the embeddings give rise to two varieties of fibers and the path-homotopy equivalence is preserved by different topological structures within the embedded subspace. Moreover a discrete-loop variety of fundamental groups is generated within the embedded subspace under fibration.
The rest of the paper is organized as follows. The preliminary concepts are presented in Section 2 as a set of existing definitions and theorems. The definitions related to proposed topological structures are presented in Section 3. The main results are presented in Section 4. Finally, Section 5 concludes the paper.

Preliminary Concepts
In algebraic topology, the structure and properties of topological spaces are investigated by studying the behaviors of open as well as closed continuous functions within the space along with their continuous deformations. In this section, a set of classical as well as contemporary results are presented in relation to the covering spaces, fibrations, covering homotopy varieties and associated isomorphism of fundamental groups. Let two topological spaces be denoted as 2 1 The concept of covering space is central to the algebraic topology. An elementary neighborhood of a topological space ) , ( X X  is a subspace of X which can be surjectively mapped under additional conditions leading to the concept of covering space. The definition of covering space and projection is presented as follows.

Definition: Covering Spaces
Let a continuous surjective function be given as where the topological space X is called a base space. If it is true that X B   is called a covering projection or covering map.
It is important to note that base subspace B is considered to be path-connected topological subspace. If we consider C S  1 on a complex plane then it results in the formulation of covering path theorem which is presented as follows [11,12].
The above theorem can be further generalized for any R r  . Suppose a continuous function is defined as 1 : then the corresponding unique covering path (.)  can also be formulated. The respective covering homotopy is a related concept which can be presented in the following theorem [11,13,14].

Theorem 2. If the function
then the covering homotopy in a covering space can be established as presented in the following theorem [11].
there is a covering homotopy in the covering space given by The fibration in a path-connected topological space can be defined in the respective covering spaces and in terms of homotopy lifting. Moreover, there exists a special variety of coverings called compact-covering in a metrizable space. Let us consider two continuous functions between two topological spaces Y X , which are denoted as . The definitions of fibration in covering space, compact-covering and point-countable cover are presented as follows.

Definitions: Fibration and Covering Varieties
is a covering projection [2]. First, we present the definition of covering fibration and its discrete variety in a covering space. The definitions of two different notions related to coverings are presented next.
Fibration [8,13]: A fibering of a topological covering space is a structure given by G-covering space contains discrete fibers in the covering projection X Y u  : (i.e., the covering space is a discretely fibered space).
Compact-covering [15]: A function It is important to note that the covering maps or covering projections preserve complete metrizability under certain conditions. Point-countable cover [16]: Suppose P is a cover of space X and P F  is a finite subcover. The space X is defined as having a point-countable cover P if The equivalence between multiple covering spaces and the corresponding multiple covering projections can be established in the presence of covering varieties. If two different covering projections are given as then they are called equivalent if, and only if, there is a homeomorphism . This leads to the conjugacy theorem of fundamental groups in the covering spaces, as presented in the following theorem [11,13].
is given then any two universal covering spaces of a base space are isomorphic to each other. Furthermore the path con- x X  in the space.

Topological Structures and Definitions
Let a second countable Hausdorff as well as compact normal topological and an open set and each compact cover is a path-connected component. In this paper the i-th path-connected component of the covering spaces is denoted as Note that the compactness of spaces X and cov X enables the formulation of locally homeomorphic embeddings into a non-compact In this paper, the topological embeddings are employed to construct the embedded subspaces, preserving local homeomorphism. It is important to note that the topological injective embeddings forming a Noetherian structure in covering spaces do not consider the group algebraic standpoints and such embeddings are completely topological in nature. The reason is that the set of finite groups in covering components in a topological , ( R C spaces through the topological embeddings and the construction of a set of homeomorphic finite groups within such spaces. Note that in this paper all topological spaces are considered to be second-countable Hausdorff spaces in nature and the topological groups are compactible as well as connected.

Definition: LN Covering of (C, R) Space
, ( R C space and the corresponding embedding within covering spaces generated by Note that the LN covering path components are finite and countable maintaining the property that  and the space Y is not compact then a set of suitable injective embeddings can be formulated maintaining local homeomorphisms.

Definition: Covering
are homeomorphic topological embeddings. The corresponding embeddings are called covering . This property assists to establish a set of finite group algebraic structures in cov X which are embeddable, retaining the respective group homomorphism in Y . First, we define the topological group structures and group homomorphism in the covering spaces under topological embeddings.

Definition: Finite Covering
be a countable set of finite (locally compactible as well as locally connected) topological groups such that is also a group in Y . In addition, the locally homeomorphic embeddings maintain that

Definition: Identity Fiber and Rigidity
Let the two path components in covering . This results in the formation of a discrete variety of fundamental group in Y under homotopic path products, as presented in the following theorem. ,

Proof. Let
we consider a set of continuous functions given by If we consider the existence of topological groups in the path-connected LN covering spaces (i.e., the covering ) , ( R C spaces are in a dense topological subspace under embeddings), then the fiber-connected distributed groups exhibit interesting homotopy properties if two such topological groups maintain group homomorphism under the identity fiber. The following theorem presents this interesting observation.
Hence, a path-homotopy is formulated in cov Y which is given by . We show in the following theorem that such translation establishes a path-homotopy equivalence on the identity-rigid fiber  Interestingly, if a group in the LN covering of ) , ( R C space is a trivial group then an equivalence relation involving the finite linear translation on an identity-rigid fiber can be established. This observation is presented in the following lemma.
is a trivial group then Hence we can conclude that and this results in the property given by The formulation of path-homotopy equivalences on a single fiber requires the specific condition about the position of identity elements in a fibered topological , (  , ( R C spaces generate a cyclic group structure under certain conditions. First, the projections of fibers on real subspace need to be finite, and second the resulting planar subspace forms a 2-simplex. This interesting property is presented in the following theorem.  The proof of aforesaid corollary is straightforward and it shows that a fibered topological ) , ( R C space containing embedded LN covering spaces and base space suc-cessfully preserves the classification of covering spaces in terms of fundamental groups. In other words the interplay of homotopy and topological groups in a fibered topological ) , ( R C covering space of LN variety with embeddings does not interfere with the classical results related to the covering space classifications based on fundamental groups.

Conclusions
The topological covering (C, R) spaces enable suitable incorporation of additional structures enhancing the richness of their properties. The compact Lindelof variety of path-connected components of covering (C, R) spaces enables the formulation of finite group algebraic structures within the spaces. The groups can be equipped with homomorphism and the corresponding finite Noetherian covering spaces formed by homeomorphic embeddings allowtable the formulation of various path-homotopy equivalences in the fibered topological (C, R) space. The interplay of finite homomorphic groups in the path-connected components of covering spaces and the topological fibers generates a discrete variety of fundamental group structure within the embedded dense subspaces. The topological fibers get classified into several varieties depending on the position of identity elements of the embedded homomorphic groups. As a result a wide array of path-homotopy equivalences is formulated within the embedded LN covering spaces including the base space. The rigidity of fibers based on identity and the multiplicity of fibers support path-homotopy equivalences considering that the path connected covering components are simply connected in view of nullhomotopy. Interestingly, the resulting 2-simplex structures and finite as well as symmetric translations within the fibered covering space assists in determining the simple connectedness of the path-connected covering components under topological embeddings.