Some Three Space Properties In Topological Groups

A topological property P is said to be a three space property if for every topological group G and a closed invariant subgroup H of G,the fact that both spaces H and G/H have P implies that G also has P.The three space property of some topological properties was discussed and some three space properties under spacial conditions were studied in this article.This article is divided into three chapters.Some basic concepts and basic theorems of topology space,group and topological group were introduced in the first chapter.The three space property and inverse fiber property of axioms of separation and axioms of countability were introduced in the second chapter.Some three space properties under spacial conditions were studied in the third chapter.The following results are proved:（1）Satisfying₁T axiom is a three space property;satisfying₂T axiom is a three space property under the condition that the invariant subgroup is open and close;satisfying₄T axiom is not a inverse fiber property.（2）The concepts of three space property for open compact sets and open inverse fiber property for compact sets were given,and the result that every open inverse fiber property for compact sets is a three space property for open compact sets was proved;the concepts of three space property for Lindel?f sets and inverse fiber property for Lindel?f sets were given,and the result that every inverse fiber property for Lindel?f sets is a three space property for Lindel?f sets was proved.（3）Countability and finiteness are inverse fiber properties for Lindel?f sets;separabilityisanopeninversefiberpropertyforcompactsets;Lindel?f—compactness is an inverse fiber property.Further countability and finiteness are three space properties for Lindel?f sets;separability is three space property for open compact sets;Lindel?f—compactness is a three space property.