I'm having trouble understanding the Cantor set. The idea of a nowhere dense uncountable set makes no sense to me, because of the following argument I thought of:(adsbygoogle = window.adsbygoogle || []).push({});

Let x be any element of the Cantor set. Since the Cantor set is nowhere dense, there exists some open interval I_x containing x but containing no other elements of Cantor set. Moreover, if x and y are distinct elements of the Cantor set then I_x and I_y are disjoint. Since the Cantor set is uncountable, the set {I_x: x is in the Cantor set} is also uncountable. But it is impossible to have an infinite number of pairwise disjoint open intervals, because each interval contains a rational number, and there are only countably many rational numbers.

Where is the error in my reasoning? It's probably something obvious, but I don't see any errors in reasoning.

Any help would be greatly appreciated.

Thank You in Advance.

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# Trouble with the Cantor Set

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