Abstract
This chapter discusses the Cantor set problems. A Cantor set is characterized as a topological space that is totally disconnected, perfect, compact, and metric. Any two such spaces C1 and C2 are homeomorphic, but if C1 and C2 are subspaces of ℝn, n ≥3, there may not be a homeomorphism of ℝn to itself taking C1 to C2. In this case, C1 and C2 are said to be inequivalent embeddings of the Cantor set. Cantor set C in ℝn is tame, if it is equivalent to the standard middle thirds Cantor set. If it is not tame, it is wild. A Cantor set C is strongly homogeneously embedded in ℝn if every self-homeomorphism of C extends to a self-homeomorphism of ℝn. At the opposite extreme, a Cantor set C in ℝn is rigidly embedded if the identity homeomorphism is the only self-homeomorphism of C that extends to a homeomorphism of ℝn.
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