On The Study Of Weaker Forms Of Compactness In Topological Groups

Compactness is one of the most important properties and the central concept in topology.It has wide applications in other mathematical disciples such as geometry,analysis,dynamical system and so on.As the natural generalizations of compact-ness,minimality,precompactness,pseudocompactness and countable compactness al-so paly significant roles and status in topology.At the same time,we can see that these concepts have been also spread widely into the number theory,geometry,analysis,and also topology.On the other side,topological group theory has a close connection with analysis,topological dynamic system,Lie group theory,representation the-ory and number theory,and supplies them with the solid foundation and wide space for their development.Therefore it’s meaningful to discuss the applications of these weaker forms of compactness in topological groups.This thesis is concentrated on the applications of these concepts in topological groups,we generalize some classical results hold for compact and Bohr compactification cases in topological groups,also pose some open questions which deserve to be done further research.The contents are as follows:In chapter 1,we give three different descriptions of the τ-precompact Hausdorff group reflection of topological groups.In particular,we describe the ω-narrow Haus-dorff reflection of a given topological group and precompact Hausdorff reflection of a given topological group,therefore we give the description of the compact Hausdorff group reflection of a given topological group(i.e.Bohr compactification).We also prove that the τ-precompact Hausdorff reflection functor preserves perfect surjective homomorphisms,quotient homomorphisms and arbitrary products.As a direct appli-cation,we deduce that the compact Hausdorff reflection functor preserves arbitrary products.In chapter 2,according to a well known theorem of Prodanov every subgroup of an infinite compact abelian group K is minimal if and only if K is isomorphic to the group Zp of p-adic integers for some prime p.We find a remarkable connection of local minimality to Lie groups and p-adic numbers by means of the following results extending Prodanov’s theorem to the locally compact case:every subgroup of a locally compact abelian group K is locally minimal if and only if K is either a Lie group or K has an open subgroup isomorphic to Zp for some prime p.In the non-abelian case we prove that all subgroups of a connected locally compact group are locally minimal if and only if K is a Lie group,resolving in the positive Problem 7.49 from[23].In chapter 3,we study the properties of the class CSS of topological groups in which all closed subgroups are separable.One of the main problems here is whether the class CSS is finitely productive or not.We solve the problem in the negative and present three examples with different combinations of additional properties:(1)there exist in ZFC precompact Abelian groups H and K in CSS such that H×K is not in CSS;(2)under the assumption of 2ω1=2ω,there exists a pseudocompact Abelian group G ∈ CSS such that G x G is not in CSS;(3)under the assumption of MA&(?)CH,there exist countably compact Abelian groups G and H in CSS such that G×H is not in CSS.Our example in(2)im-proves upon an example presented in a recent article[56]by A.Leiderman and M.Tkachenko,while the example in(3)answers a question raised in[56].