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We prove that every compact metric space is the continuous image of the Cantor set. We characterize the Cantor set as the unique 0-dimensional compact metric space without isolated points. We show that every 0-dimensional compact metric space can be embedded in the Cantor set. We describe two important mappings on the Cantor set: the Adding Machine and the Shift mapping.

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The Cantor Set

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  • 发表日期：

    2025年

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  • Alejandro Illanes

摘要

We prove that every compact metric space is the continuous image of the Cantor set. We characterize the Cantor set as the unique 0-dimensional compact metric space without isolated points. We show that every 0-dimensional compact metric space can be embedded in the Cantor set. We describe two important mappings on the Cantor set: the Adding Machine and the Shift mapping.