Topological Projective Covers for Topological Groups

Abstract

       The primary objective of this paper is to evaluate the tensor product of topological projective cover for topological groups. After we explain that not every topological group has topological group cover. We depend on tensor product properties of topological groups for algebra and we inputed topological properties that suitable of algebra construction as topology from the definition of topological groups and projective on topological projective groups , such that; a topological group q is called topological projective group  if for all topological group epimorphism

 g : A ¾® B and for all topological group morphism f : q ¾® B, there exists a topological group morphism f ' :q ¾® A, for which the following diagram commutes'':

Difinition:- Amorphism of topologicl group   is a continuous                  homomorphism between topological groups.

Difinition:- epimorphism in the category of all topological groups are easily seen to be surjective if G is a   topological group and any subgroup there exist agroup  which endow with the indiscrete topology and two homomorphism from G into  which agree only on it.                                                                                                                      

Furthermore, new theorems are given at the end of the paper.