1、 河南理工大学 本科毕业设计（论文） 外文文献资料翻译 院（系部） 数学与信息科学学院 专业名称 数学与应用数学专业 年级班级 0902 班 学生姓名 陈勇 学生学号 310911010214 2013 年 6 月 1 日 共 54 页 河南理工大学本科 毕业论文外文文献资料翻译 第 1 页 指导教师：赵勇 学生：陈勇 Compact Spaces The notion of component is not nearly so natural as that of connectedness.From the beginnings of topology,it was clear that
2、the closed interval ba, of the real line had a certain proerty that was crucial for proving such thorrems as the maximum value theorem and the uniform continuity theorem .But for a long time,it was not clear how this property should be formulated for an arbitrary topological space.It used to be thou
3、gh that the crucial property of ba, was the fact that every infinte subset of ba, has a limit point,and this property was the one dignified with the name of compactness.Later ,mathematicians realized that this formulation does not lie at the heart of the matter,but rather that a stronger formulation
4、,in terms of open coverings of the space,is morecentral.The latter formulation is what we now call compactness.It is not as natural or intuitive as the former,some familiarity with it is needed before its usefulness becomes apparent. Definition. A collection of subsets of a space X is said to cover
5、X ,or to be a covering of X ,if the union of the elements of is equal to X .It is called an open covering of X if its elements are open subsets of X . Definition. A space X is said to be compact if every open covering of X contains a finite subcollection that also covers X . EXAMPLE1. The real line
6、R is not compact,for the covering of R by open intervals Znnn 2. contains no finite subcollection that covers R . EXAMPLE2. The following subspace of R is compact: ZnnX 10 . Given an open covering of X ,there is an element U of containing 0.The set U contains all but finitely many of the points n1 ；
7、choose,for each points of X not in U ,an element of containing it.The collection consisting of these elements of ,along with the element U ,is a finite subcollection of that covers X . EXAMPLE3. Any space X containing only finitely many points is necessarily 共 54 页 河南理工大学本科 毕业论文外文文献资料翻译 第 2 页 指导教师：赵
8、勇 学生：陈勇 compacts,because in this case every open covering of X is finite. EXAMPLE4 The interal 1,0 is not compact；the open covering Znn 1,1 contains no finite subcollection covering 1,0 .Nor is the interal 1,0 compact；the same argument applies.On the other hand,interal 1,0 is copact；you are probably
9、 already familiar with this fact from analysis.In any case,we shall prove it shortly. In general,it takes some effort to decide whether a given space is compact or not.First we shall prove some general theorems that show us how to construct new compact spaces out of existing ones. Then in the next s
10、ection we shall show certain specific spaces are compact.The spaces include all closed interals in the real line,and all closed and bounded subsets of nR Let us first prove some facts about subspaces.If Y is a subspace of X ,a collection of subsets of X is said to cover Y if the union of its element
11、s contaons Y . Lemma 1. Let Y be a subspace of X .Then Y is compact if and only if every covering of Y by sets open in X contains a finite subcollection covering Y . Proof . Suppose that Y is compact and JA is a covering of Y by sets open in X .Then the collection JYA is a covering of Y by sets of Y
12、 ；hence a finite subcollection YY,A 1 nA covers Y .Then ,1 nAA is a subcollection of that covers Y . Coversely,suppose the given condition holds:we wish to prove Y compact.Let A be a covering of Y by sets open in Y .For each ,choose a set A open In X such that YAA The collection A is a covering of Y by sets open in X .By
