Concerning upper semi-continuous collections of continua
Concerning upper semi-continuous collections of continua
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DOI：
10.1090/s0002-9947-1925-1501320-8
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发表时间：
1925-04
影响因子：
1.3
通讯作者：
Robert L. Moore
中科院分区：
文献类型：
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作者：
Robert L. Moore
连续统的集合i称为上半连续集合，如果对于集合G的每个元素y和每个正数e，存在一个正数d，使得如果x是G中距离g小于d的任何元素，则x到g的上距离小于e。这样一个集合G的元素p被称为G的子集合K的极限元素，如果对于每个正数e，存在K的某个元素不同于p，并且其到p的上距离小于e。在这一节中，将证明，如果在平面S中，G是互斥有界连续统的任何上连续集合，使得S的每个点都属于集合G的某个连续统，并且没有G的连续统将S分开，那么如果G的每个连续统都被认为是一个点，并且适当地定义了术语区域，如果把作者的文章《论平面分析位置的基础》中的空间S解释为元素G的集合，则该文章中的公理1-8全部成立。因此，就该论文中定义的极限点的概念而言，元素集G拓扑等价于平面S中的普通点集。此外，这样定义的极限点的概念，对于上半连续集合的情况，与上面给出的极限元素的自然解释一致。从这里开始，在这一节中，可以理解，已经选择了有界连续统的某些确定的上半连续集合。
A collectioni of continua is said to be an upper semi-continuous collection if for each element y of the collection G and each positive number e there exists a positive number d such that if x is any element of G at a lower distancet from g less than d then the upper distance of x from g is less than e. The element p of such a collection G is said to be a limit element of the subcollection K of G if for every positive number e there exists some element of K which is distinct from p and whose upper distance from p is less than e. In this section it will be shown that if, in a plane S, G is any upper semicontinuous collection of mutually exclusive bounded continua such that every point of S belongs to some continuum of the collection G and no continuum of G separates S, then if each continuum of G is considered as a point, and the term region is suitably defined, all the Axioms 1-8 of the author's articlet On the foundations of plane analysis situs hold true, if the space S of that article is interpreted to mean the collection of elements G. Thus the set of elements G is, with respect to the notion of limit point defined in that paper, topologically equivalent to the set of ordinary points in a plane S. Furthermore the notion of linmit point so defined coincides, for the case of an upper semi-continuous collection, with the natural interpretation of limit element given above. From here on, in this section, it is understood that there has been selected some definite upper semi-continuous collection of bounded continua