Some Topics On Topological Algebras

Topological algebra is a comprehensive subject whose aim is to study the problems of compatibilities of topological structures and algebraic structures. And it has a wealth of applications in abstract harmonic analysis, operator theory, algebraic number theory, Lie group theory, topological dynamical system and so on. In this thesis, we discuss a number of issues of topological algebras which are mainly divided into the following three parts.In Chapter 1, we prove that every locally K topological group has a nonzero outer regular invariant Borel measure when K is an admissible invariant family which is separated by NG.In this case, every open set and every member of S(K0)are K-inner regular. This extends the existence theorem of Haar measure on locally compact Hausdorff groups.In Chapter 2, we give a useful criterion for the existence of reflections. Base on this criterion, we obtain a number of results about topological free objects, completions, compact reflections and (ω-)totally bounded reflections.In Chapter 3, we define the operator topologies on the Hom-sets, and discuss the topologically left exact properties of the Hom functors. Moreover, we give dual theorems for the topological modules over admissible rings. In addition, we give the structures of the topological tensor products in the category of semitopological modules, and discuss the topologically right exact properties of the tensor product functors in detail.