笛卡尔型集的上(下)极限的计算及应用  

Calculation and Application of the Superior(Inferior) Limit of Cartesian Set

作　　者：叶明露[1] 陈家欣 YE Minglu;CHEN Jiaxin(School of Mathematics and Information,China West Normal University,Nanchong Sichuan 637009,China)

机构地区：[1]西华师范大学数学与信息学院

出　　处：《西华师范大学学报（自然科学版）》2019年第3期251-255,共5页Journal of China West Normal University(Natural Sciences)

基　　金：国家自然科学基金面上项目(11871059);国家自然科学基金青年项目(11801455);2018年校级大学生创新创业训练计划项目(cxcy2018204)

摘　　要：集列的上(下)极限的计算是实变函数教学中的一个重点和难点,但其计算公式对复杂的集列并不适用。本文主要证明了当集列{Bn}n=1^∞和{Cn}n=1^∞均单调时,集列{Bn×Cn}]n=1^∞的上(下)极限集分别等于{Bn}n=1^∞和{Cn}n=1^∞的上(下)极限集的笛卡尔集。当笛卡尔型集列的每个坐标集列均单调时,我们还证明了上述等式关系可以推广到有限个集合中去,但不能推广到可数个集合中去。通过例子,我们还展示了上述等式关系的运用。Calculation of the superior (inferior) limit of set sequence is a key and difficult point in the teaching of real variable function.Although the calculation formula about the superior (inferior) limit of set sequence is given,it does not apply to the case when the set sequence is complex.This paper mainly proves that the superior (inferior) limit of set sequence {Bn×Cn}n=1^∞ equals to the Cartesian set of superior (inferior) limit of set sequence {Bn}n=1^∞ and {Cn}n=1^∞ when both of them are monotone.Moreover,if each coordinate set sequence of the Cartesian set is monotone,the above equality relationship holds for the case when the Cartesian set is constructed by finite number of set but not for infinite number of set.Examples are also given to show the application of the above equality relationship.

关 键 词：集列 上极限集 下极限集 笛卡尔集 

分 类 号：O29[理学—应用数学]