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I am looking at the complement of the cantor set in $[0,1]$ as the union of open intervals of decreasing length $\displaystyle\bigcup_{i=1}^{\infty}A_i$ where $A_1 = (\frac{1}{3},\frac{2}{3}), A_2 = (\frac{1}{9},\frac{2}{9})\cup(\frac{7}{9},\frac{8}{9})$ and so on.

I am trying to prove that $\overline{\displaystyle\bigcup_{i=1}^{\infty}A_i} = \displaystyle\bigcup_{i=1}^{\infty}\overline{A_i}$.

I know that this is not true in general and is probably not true in this case, but can someone give a proof or a counterexample

Thank you very much

1 Answers 1

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Here are some hints for one method of answering this:

  • The complement of the Cantor set is dense in $[0,1]$.
  • The closure of each individual $A_n$ only has finitely many extra points.
  • The Cantor set is uncountable.
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    @Henning: Oh, right. For some reason it seemed to me that equality was hinted somehow.. :-)2011-09-09