Topological Immersion of Peanian Continua in a Spherical Surface
Topological Immersion of Peanian Continua in a Spherical Surface
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球面中皮亚连续体的拓扑浸入
DOI：
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发表时间：
1934
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影响因子：
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通讯作者：
Schieffelin Claytor
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文献类型：
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作者：
Schieffelin Claytor
在1930年Kuratowski 2建立了以下结果：定理A：一个只包含有限个简单闭曲线的peanian连续统，与平面的一个子集同胚，只要它不包含复形Cs和Ad中任何一个的拓扑像，其中（s）由两组各有三个顶点和九个1-胞组成，以一个组的每个顶点与另一个组的每个顶点一起界定在1-胞腔中的方式; D由5个顶点和10个1-胞元组成，以每对顶点限定一个1-胞元的方式。4这个定理提出了本文中处理的更一般的问题，即，与球面的子集同胚的Peanian连续统的特征。我们发现它方便，首先，指定作为一个原始的斜曲线任何点集同胚与任何一个复杂的（s和A”;然后，区分所有的花生连续集的一个类，Y，包括那些未能包含一个原始的斜曲线。定义I：M中的简单闭曲线T称为M的边界曲线，只要在M-T中不存在不同的分量E和F，使得
In 1930 Kuratowski2 established the following result: THEOREM A: A peanian continuum,3 containing but a finite number of simple closed curves, is homneomorphic with a subset of the plane, provided that it does not contain a topological image of either of the complexes Cs and Ad where (s consists of two groups of three vertices each and nine 1-cells, in a fashion that each vertex of one group together with each vertex of the other group bounds in a 1-cell; D consists of five vertices and ten 1-cells, in a fashion that each pair of vertices bounds a 1-cell.4 This theorem suggests the more general problem treated in this paper, namely, the characterization of the peanian continua which are homeomorphic with a subset of the surface of a sphere. We find it convenient, first, to designate as a primitive skew curve any pointset homeomorphic with either of the complexes (s and A"; and then, to distinguish among the set of all peanian continua a class, Y, consisting of those which fail to contain a primitive skew curve. Certain boundary sets are defined in an arbitrary continuum, M, of class Y, as follows: DEFINITION I: A simple closed curve, T, in M is called a boundary curve of M provided that there do not exist in M-T distinct components E and F such that