# The Cantor Set

• Apr 13th 2008, 01:57 PM
squarerootof2
The Cantor Set
how would one show that the interior of the cantor set is the empty set?
• Apr 13th 2008, 02:40 PM
Plato
Quote:

Originally Posted by squarerootof2
how would one show that the interior of the cantor set is the empty set?

In my view the fastest way to prove that is to show that each point of the Cantor Set is a boundary point.
• Apr 13th 2008, 02:44 PM
TheEmptySet
Quote:

Originally Posted by squarerootof2
how would one show that the interior of the cantor set is the empty set?

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 PM
TheEmptySet
Quote:

Originally Posted by TheEmptySet
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
$\displaystyle b_k \ne 1 \mbox{ }\forall \mbox{ }k$

Sorry my mistake