Injective Objects And Projective Objects In The Category Of Topological Groups

It is very meaningful to study injective objects and projective objects in a given category.In this thesis,we consider the injective objects and projective objects in the category TopGp.We proved the following results:1.The injective objects in the category TopGp are exactly the trivial group {e}.2.Let M be the class of all dense embeddings on topological groups,then the injective objects according to M are exactly all Raikov com-plete groups.3.In precompact topological group category PCGp,the injective objects according to dense emdeddings are exactly all compact groups.4.In Abelian topological group category TopAb,the injective objects according to open emdeddings are exactly all divisible topological groups.5.The projective objects according to quotient homomorphisms in the category of topological groups are exactly all retracts of some free topological group.6.We define a special class of epimorphisms,prove that the projective objects according to this class of epimorphisms are also retracts of some free topological group.