how would one show that the interior of the cantor set is the empty set?

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- Apr 13th 2008, 01:57 PMsquarerootof2The Cantor Set
how would one show that the interior of the cantor set is the empty set?

- Apr 13th 2008, 02:40 PMPlato
- Apr 13th 2008, 02:44 PMTheEmptySet
You can't because the cantor set is nonempty compact set of measure zero.

I will not show this but... let E be the cantor set then

$\displaystyle x \in [0,1] \mbox{ and } x \in E \mbox{ if and only if } x $

has a ternary expansion whose digits

$\displaystyle b_k \ne 1 \mbox{ }\forall \mbox{ }k $ - Apr 13th 2008, 02:46 PMTheEmptySet
- Apr 13th 2008, 04:24 PMawkward