摘要
借助于[0,1]区间中的两两不相交的开集的无穷序列的重新排列,证明了[0,1]区间中的两个康托集之间存在着保序的同胚。分析了Hausdorff空间X中的任意一条路f:[0,1]→X的结构。通过回归时段常值化,将f改造为一条不含有回归时段的路h:[0,1]→X。特别是,通过在严格单调时段中增添无穷多个停滞时段,通过将[0,1]区间中的一个康托集更换为一个勒贝格测度处处大于0的康托集,通过将无穷多个停滞时段切除,进一步将只含有停滞时段而不含有回归时段的路h:[0,1]→X改造为一个连续的单射,从而证明了每一个路连通的Hausdorff空间都是一个弧连通空间。
By means of rearrangement of infinite sequence of pairwise disjoint open intervals in[0,1],it is proved that there exists an order-preserving homeomorphism between any two Cantor sets in[0,1].For any Hausdorff space X,the structure of any path f:[0,1]→X is investigated.By changing f on some recurrent intervals to be constants,we obtain a path h:[0,1]→X which has no recurrent intervals.Further,by putting infinitely many stagnant intervals into those strictly monotonic intervals,by replacing a Cantor set in[0,1]to be another Cantor set in[0,1]which has positive Lebesgue measure everywhere,and then by cutting infinitely many stagnant intervals,we rechange the path h:[0,1]→X to be a continuous injection.Therefore,it is proved that every path connected Hausdorff space is an arcwise connected space.
作者
麦结华
MAI Jie-hua(College of Information and Statistics,Guangxi University of Finance and Economics,Nanning 530003,China)
出处
《广西大学学报（自然科学版）》
CAS
北大核心
2020年第3期674-680,共7页
Journal of Guangxi University（Natural Science Edition）
基金
国家自然科学基金资助项目(11761012)。
关键词
路连通
弧连通
返回点
回归时段
停滞时段
康托集
勒贝格测度
HAUSDORFF空间
path connectedness
arcwise connectedness
return point
recurrent interval
stagnant interval
Cantor set
Lebesgue measure
Hausdorff space
作者简介
通讯作者:麦结华(1942-),男,广西桂平人,广西财经学院特聘教授,博士研究生导师,E-mail:jiehuamai@163.com。
