Some Topics On Paratopological Groups And Semitopological Groups

Paratopological groups and semitopological groups are two important branches of topological algebra.The theory of topological algebra originated from Lie group founded by Sophus Lie,established from the nineteenth Century to the twentieth Cen-tury approximately.Functional analysis and differential geometry have promoted the development of this discipline.It is a comprehensive discipline which aims to study the topological structure and group structure in a certain way.In modern mathemat-ics,it is closely linked with analysis,algebra,geometry and topology.and becomes an indispensable part in the study and research of modern mathematics.In this thesis,we mainly discuss a number of issues of paratopological groups and semitopological groups,which mainly include the following three parts.In Chapter 1,we prove that every saturated paratopological group with countable weak extent is w-narrow.This answers partly the problem posed by Arhangel’skii et al in[3,Problem 5].This result generalizes the theorem which Sanchez had showed on topological groups in[37,Proposition 2.22].Applying this property,we prove that for each saturated Hausdorff paratopological group G with ωe(G)πx({G)≤ ω,G condenses onto a Hausdorff space with a countable base.At the same time,we construct a ω-balanced paratopological group with no countable weak extent.This gives a positive answer to the problem posed by Sanchez in[36,Problem 2.13].In Chapter 2,we prove that if some dense subgroup of paratopological group is ω-narrow,then the paratopological group is ω-narrow.This answers positively the problem posed by Arhangel’skii et al in[1,Problem 5.21].At the same time,we show the property-Let G be a T3 semitopological group and H a dense subgroup of G.If H is a P-space and ω-narrow,then G is ω-narrow as well.Furthermore,we also study the topological properties of dense subgroups of paratopological groups and semitopological groups,such as first countable property,2-pseudocompactness,precompactness,period and topological period and so on.We obtain some interesting results.In addition,we give the topological space endowed with any algebraic structure,which can not be a semitopological group.In Chapter 3,we mainly prove that for T3 paratopological group 2-pseudocompactness has three space property.This answers partly the problem posed by Tkachenko in[40,Problem 5.7].At the same time,we prove that for each paratopological group,ω-narrowness has the three space property.Applying this property,we can show that for each topological group,second countable property has the three space property.This gives a characterization to second countable topological group.Furthermore,we sum-marize other topological properties of three space properties in paratopological groups,such as,ω-balancedness,countable π-character and so on.