拓扑学中可数集的教学探讨  

Research on Teaching Countable Set in Topology

作　　者：张德燕[1] 安佰玲[1] ZHANG Deyan;AN Bailing(School of Mathmatical Sciences,Huaibei Normal University,235000,Huaibei,Anhui,China)

机构地区：[1]淮北师范大学数学科学学院

出　　处：《淮北师范大学学报（自然科学版）》2019年第3期86-89,共4页Journal of Huaibei Normal University：Natural Sciences

基　　金：国家自然科学基金项目(11671298,11561020);安徽省自然科学基金项目(1908085MA05);省级教学团队项目(2017jxtd146);淮北师范大学教研项目(JY180185)

摘　　要：可数集是拓扑学中一个非常基本且重要的概念,很多重要的拓扑概念和性质都依赖于此概念,因此,对可数集的进一步学习和研究是必要的.论文中利用可数集的定义,给出命题“有理数集是可数集”的一个严谨而完整的证明,并给出命题“可数多个可数集的并是可数集”的一种新证明方法.同时,引入2个集族等价的定义,结合集合的可数性证明两个结论,分别为:如果两个等价集族中有一个是有限集族,则这两个集族的交不空;若一个满足第二可数性的拓扑空间有最小基,则最小基是可数集.The countable set is a basic and important concept in topology,and many topological concepts and properties depend on this concept.Hence,it is necessary to study the countable set.In this paper,based on the definition of the countable set,a complete and strict proof of the proposition“Rational number set is a countable set”is given,and a new proof of the proposition“Union of denumerable countable sets is a count?able set”is obtained too.In addition,the equivalence of two collections is introduced,which together with the definition of the countable set gives two results:if one of two equivalent collections is finite,then their in?tersection is not empty,and if a topological space satisfying the sencond countable property has the minimal basis,then its minimal basis is countable.

关 键 词：可数集 有理数集 集族的等价 

分 类 号：O189.11[理学—数学]