关于Cantor中三分之一集与Sierpienski地毯拓扑性质的区别
On a difference in topological nature between Cantor middle-third set and Sierpiński carpet
摘要：
Cantor中三分之一集和Sierpienski地毯都是自相似的、完备的、紧的度量空间。尽管相似的数学程序的建设，有它们之间存在一个根本的区别在拓扑性质，这种差异影响的方法建设一个有趣的非平凡的商空间。完全不连通性（或者更一般地说，零维）使得康托中三分之一集具有自相似的非平凡商空间。另一方面，对于Sierpienski地毯，由于其结构的连通性，没有一个自相似的非平凡商空间可以像Cantor中三分之一集那样用一个优雅的过程构造.康托中三分之一集合所特有的各种拓扑意义的性质主要归因于该集合的完全不连通性。
英文摘要：
Both Cantor middle-third set and Sierpiński carpet are self-similar, perfect, compact metric spaces. In spite of the similarity of the mathematical procedure of construction, there exists between them a fundamental difference in topological nature, and this difference affects the methods of construction of an interesting non-trivial quotient space of them. The totally disconnectedness (or, more generally, zero-dimensional) enables Cantor middle-third set to have a non-trivial quotient space which is self-similar. On the other hand, concerning Sierpiński carpet, because of the connectedness of its structure, no non-trivial quotient space which is self-similar can be constructed by such an elegant procedure as that for Cantor middle-third set. Various topologically significant nature specific to Cantor middle-third set owe mainly to the totally disconnectedness of the set.
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