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The Cantor set , sometimes also called the Cantor comb or no middle third set (Cullen 1968, pp. 78-81), is given by taking the interval (set ), removing the open middle third (), removing the middle third of each of the two remaining pieces (), and continuing this procedure ad infinitum. It is therefore the set of points in the interval whose ternary expansions do not contain 1, illustrated above.

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With the discrete topology on $X$, and the product topology on $X^\mathbb{N}$. –  1015 Apr 4 '13 at 13:53
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Here again, you should not have changed the question... It does not make sense anymore. –  1015 Apr 4 '13 at 18:20
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@user68507: your edit has changed a question into a statement. –  Grumpy Parsnip Apr 4 '13 at 18:20

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Hint: Any element of the Cantor set has the form $\displaystyle \sum\limits_{n \geq 1} \frac{2x_n}{3^n}$ for some $(x_n) \in \{0,1\}^{\mathbb{N}}$.

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