边界点和聚点定义的另类含义

Implications for Definitions of Boundary Point and Accumulation Point

距离空间中边界点和聚点是重点,也是难点;与它们关联的还有内点、外点和孤立点,对这些定义的深度理解,有助于学生对开集合、闭集合等概念的掌握。在一般教材中都是直接给出相应定义,只解决了“是这样”的问题,但对于“为什么这样”的问题,基本少有提及。本文基于分类理论,就R~n的子集E,对R~n中的点P_(0)做分类,针对P_(0)的领域与集合E和E~C的包含关系,给出边界点定义的含义;针对P_(0)的领域中含集合E的元素个数,给出聚点的意义所在。挖掘了距离空间中基本定义的内涵所在,清楚地阐述了五点(边界点、聚点、内点、外点、孤立点)之间的联系和区别。

The boundary point and accumulation point are the key and difficult points in the distance space,which are also associat‐ed with the interior point,exterior point,and isolated point.A deep understanding of these definitions is helpful to master the con‐cepts of open sets and closed sets.In general teaching textbooks,definitions are directly given,and only the problem of“It is this”is solved,but the problem of“Why is this”is rarely mentioned.This paper classified the point P_(0) in Rfor the subset E of R,based on the classification theory,and gave the meaning of the definition of the boundary point in one new way according to the inclusion rela‐tionship between the neighbourhood of P_(0) and the sets E and E.For the element number of sets E in the neighbourhood of P_(0),the definition of accumulation point in some new way was given as well.The connotation of the basic definition in the distance space was excavated,and the relationship and difference between the boundary point,accumulation point,interior point,exterior point,and isolated point were clearly explained.