Covering Theorem And Related Problems

Covering theorems play a important role in mathematics.This paper mainly in-troduce some covering theorem and it's application.The main content is divided into the following three parts.The first part mainly introduces two covering theorems.Lebesgue covering lem-ma and Heine-Borel theorem,which play a important role in description of compact set;The second part introduces Lindel(?)f space and give a proof of Lindel(?)f theo-rem;The third part introduces two main theorems in metric measure theory,Vitali covering theorem and Besicovitch covering theorem,give the form of Vitali cover-ing theorem with the Lebesgue measure taking placed by Hausdorff measure and doubling measure,and generalize Vitali covering lemma to separable metric space.