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I've been thinking about the Cantor set lately and I'm struggling with the proof of the obvious fact that the Cantor set is self-similar. This question Show that the Cantor set is self-similar and the accepted answer were very useful. However, I lack the other direction of the proof. How to show that if $c\in C$, then $c\in \frac{1}{3}C$ or $c\in(\frac{2}{3}+\frac{1}{3}C)$?

Definitions and the other direction of the proof can be read from the link. Btw. there isn't a tag for self-similarity.

Edit: this book seems to have a complete proof. It requires the result which says that the Cantor set contains only numbers which can be expressed in ternary with only zeros and twos, tough. I would still like to see a set-theoretic proof if there is any.

Edit 2: this book has the approach I was looking for.

jvkloc
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  • What is your definition of the Cantor set? How do you construct it? It appears that you are are defining the Cantor set as the set of real numbers possessing ternary expansions consisting of only 0 and 2. Is this your definition? Or something else? – Xander Henderson Jul 15 '20 at 21:53
  • @XanderHenderson "Definitions and the other direction of the proof can be read from the link" – jvkloc Jul 16 '20 at 08:12
  • Questions on Math SE should be self-contained. Please edit your question to include the relevant definitions here. – Xander Henderson Jul 16 '20 at 16:09

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