Algebraic topology: On results of quotient for topological modules

In this paper, we have the principal goal is to study a topology property of important algebraic construction namely the quotient module. We use a new tool with a quotient module which is a tensor product of modules. Therefore all topological submodules in this notion are a tensor product. The meaning of the tensor module introduced in this notion and the important fact of this article is to explain the quotient module when all submodules are tensor. Finally, several results have been obtained about the tenser product of the finite quotient module.


Introduction
Arithmetical geography is one of the significant parts of math and epitomizes the connections between variable based math and geography. The certified beginning of the examination of arithmetical geology during the 1920s through the examination of Topological social occasion. The meaning of the module overall can discover it in [1]. Numerous creators contemplated topography gathering [2]. Consequently, the scientists needed to consider the Topological module. In 1955, Cabaske presented the meaning of the Topological module. A definition and a few properties of the Topological module and Topological submodule can discover it in [1]. To contemplate the remainder module we need to present a simple meaning of shape similarity Topological module: Let be a mapping between two topography modules. µ is called a shape similarity Topological module if is a shape similarity and continuous [3] [4]. A tensor module of Topological R-modules M and N have been defined in and more details about a tensor concept in [5]. A tensor product of Topological R-module is a Topological R-module dented by M together with R-bilinear mapping such that for every R-bilinear mapping . There exists a unique linear mapping such that the diagram:

Quotient of topological module
In this section, we introduce new results about the quotient of the Topological module. We use a tensor product of modules to satisfy our goal.

Definition (2.1):
Let be a Topological submodule of a Topological R-module E. The family F of subsets of E/( ) is a topology on E/( ) and denoted quotient of the Topological module where q-1 is open in E (q: E→ E/( ).

Example (2.2).
The is a tensor module.
Using the Euclidean-ness of , let such that

Proposition (2.3):
Let be a Topological submodule of a Topological R-module E. Then the quotient of the Topological module is a Topological module.

Proof:
We must prove that the mapping from ˟ is continuous.

Proposition (2.8):
Let be a Topological submodule of a Topological module . If is a neighbourhood of zero in , then is a neighbourhood of zero in and if is a system of principle neighbourhood of the zero in E, then of q(F) is a system of a neighbourhood of the zero in .

Proposition (2.9):
Let be a Topological submodule of E and let be another Topological submodule containing . Then the canonical map is homeomorphism.

Proof:
Since 2) The Intersection of two tensor products is also a tensor product.

Proposition (2. 15):
Let are a Topological submodule of Topological module . Then the homeomorphism is continuous.
Proof: Assume that the following shape similarity module is onto shape similarity module and A restriction of this shape similarity on and is a continuous homeomorphism module .

Conclusion
In this paper we study a topological module and study a quotient of topological module ,we use a new tool with quotient which is tensor product ,several result have been obtained about tensor product of finit quotient module was written in the form of new propositions.