Concerning upper semi-continuous collections of continua
Concerning upper semi-continuous collections of continua
复制标题
关于 Continua 的上半连续集合
DOI：
10.1090/s0002-9947-1949-0033533-5
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发表时间：
1949
期刊：
影响因子：
--
通讯作者：
R. D. Anderson
中科院分区：
文献类型：
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作者：
OF Continua;R. D. Anderson
Hurewicz [1]（2）和Mazurkiewicz [2]分别证明了：若M是紧度量连续统，则在三维欧氏空间中存在一个一维连续统K和一个充满K的互斥连续统的上半连续集[3 ]，该连续统关于其元素为点与M拓扑等价。在每一个这些解决方案的情况下，所使用的证明方法并不容易适用于本文的主要结果的解决方案，这是证明，如果M是任何紧凑的连续曲线，存在三维欧氏空间中的一维连续曲线K和上半连续曲线K，填充K的互斥连续体的连续集合，K的元素作为点在拓扑上等价于M。已知在Hurewicz和Mazurkiewicz问题中，如果M不是连续曲线，则K不能是连续曲线。本文的主要结果，即定理II，是R。L.摩尔。我谨向摩尔教授表示衷心的感谢，感谢他耐心而富有启发性的教学，感谢他对数学研究的热情。定义.连续曲线的集合Q被称为具有X性质，如果Q的任何子集的连续体的公共部分只有有限个分量，每个分量都是连续曲线。
Hurewicz [1](2) and Mazurkiewicz [2] showed independently that if M is any compact metric continuum, there exist a one-dimensional continuum K in three-dimensional Euclidean space and an upper semi-continuous collection [3 ] of mutually exclusive continua filling up K which with respect to its elements as points is topologically equivalent to M. In the case of each of these solutions the method of proof used does not lend itself readily to the solution of the principal result of this paper which is the demonstration that if M is any compact continuous curve, there exist a one-dimensional continuous curve K in three-dimensional Euclidean space and an upper semi-continuous collection of mutually exclusive continua filling up K which with respect to its elements as points is topologically equivalent to M. It is known that, in the problem of Hurewicz and Mazurkiewicz, if M is not a continuous curve then K cannot be a continuous curve. The principal result in this paper, Theorem II, was proposed to me as a problem by Professor R. L. Moore. I wish to express my sincere appreciation to Professor Moore for his patient and stimulating teaching and for his contagious enthusiasm for mathematical research. DEFINITION. A collection Q of continuous curves will be said to have the X property if the common part of the continua of any subcollection of Q has only a finite number of components, each a continuous curve.