A Related Study Of M-decomposable Topological Groups

In this thesis,we mainly study M-factorizable topological groups,which origi-nated from the ancient problem of the factorization of mappings.Pontryagin proved that every compact topological group G has the following property(see[21,Example37]):For every continuous real-valued function f on G,there exists a continuous ho-momorphism?of G onto a second-countable topological group H such that f=g°?,for some continuous real-valued function g on H.Motivated by this fact,Tkachenko introduced in[21]R-factorizable groups as the groups that have the above property.In the definition of R-factorizable topological groups,the property H is second-countable topological group is so strong that the R-factorizable topological groups does not contain a very important kind of topological groups:measurable topological group.Hence,how to modify the definition of R-factorizable topological groups to include measurable topological groups is a very meaningful question.We introduce and discuss M-factorizable groups,which both generalizes metrizable topological groups and R-factorizable topological groups.The first part of this thesis studies the properties of M-factorizable topological groups.We give a sufficient and necessary condition of M-factorizable,which shows that all the continuous real-valued functions on it are?-continuous.We prove that if G is an M-factorizabl topological group,then G is either?-precompact or?-fine,where?is a uncountable cardinal.The concept of?-precompact describes the trans-verse“width”of topological groups and?-fine characterizes the longitudinal“depth”of topological groups.The above theorem shows that the“width”and the“depth”influence each other on M-factorizable topological groups.We also generalize many important properties of R-factorizable topological groups.For example,we prove z-embedded subgroups of M-factorizable M-factorizable groups inherit this property.It is shown that continuous d-open homomorphism preserve M-factorizability.We also study the products of M-factorizable groups.In the second part of the thesis,we continue to study the M-factorizability in topological groups with a special emphasis on feathered groups.It is shown that a feathered group G is M-factorizable if and only if G is either metrizable or R-factorizable.According to this result,we obtain that every subgroup of an M-factorizable feathered group is M-factorizable.We show that the product G=?_n??G_n of countably many M-factorizable feathered groups is M-factorizable if and only if all factors are first countable or R-factorizable.At the same time,we find some other interesting properties:(1)the C-embedded subgroups of an M-factorizable feathered topological groups is exactly a closed subgroup;(2)M-factorizable (?)ech-complete subgroup H of a topological group G is C-embedded in G;(3)For an M-factorizable group G,the product G×K is M-factorizable for every locally compact separable metrizable topological group K.In the third part of the thesis,we continue the study of M-factorizability in topo-logical groups with a special emphasis on P-groups.We give different characteriza-tions of M-factorizable P-groups,which show that R-factorizability,fine,pseudo-p(G)-compactness are equivalent.This result is applied to prove the validity of the conjecture of Comfort and Hager formulated in 1975:A topological group G is fine iff it is pseudo-p(G)-compact.In particular,a P-group is M-factorizable iff it is fine.We verify that a subgroup of a M-factorizable P-group G is M-factorizable iff it is C-embedded in G.We also study products of M-factorizable P-groups and their powers.In particular,it is proved that all finite and countable powers of an M-factorizable P-group are M-factorizable.