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Mathematics-Online course: Basic Mathematics - Real Numbers

Cantor Set

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We divide the unit interval $ [0,1]$ into three equals parts and remove the middle subinterval $ (1/3,2/3)$. Then, we repeat the procedure with the two remaining subintervals, i.e., we split each of them into three parts and remove the middle subintervals $ (1/9,2/9)$ and $ (7/9,8/9)$, respectively.

\includegraphics[width=.8\linewidth]{a_cantormenge_bild.eps}

Continuing the subdivision process indefinitely, we arrive in the limit at an uncountable set with measure

$\displaystyle 1-\frac{1}{3} - \frac{1}{3}\cdot\frac{2}{3} - \frac{1}{3}\cdot \left(\frac{2}{3} \right)^2 - \ldots = 0\,. $

It is referred to as Cantor Set.

(Authors: Höllig/Kopf/Abele)
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  automatically generated 10/31/2008