On a difference in topological nature between Cantor middle-third set and Sierpiński carpet

2014/02/15 by Akihiko Kitada, Kitada, Akihiko, Tomoyuki Yamamoto +5
Mathematics · #FOS: Mathematics #General Topology (math.GN) #math.GN

paper · pdf · doi:10.48550/arxiv.1402.7290

7 pages

arxiv created 2014/03/22 · arxiv updated 2014/03/25

Abstract

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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