# Math Help - The Cantor Set

1. ## The Cantor Set

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

2. 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.

3. 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

$x \in [0,1] \mbox{ and } x \in E \mbox{ if and only if } x$
has a ternary expansion whose digits
$b_k \ne 1 \mbox{ }\forall \mbox{ }k$

4. 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

$x \in [0,1] \mbox{ and } x \in E \mbox{ if and only if } x$
has a ternary expansion whose digits
$b_k \ne 1 \mbox{ }\forall \mbox{ }k$
Sorry my mistake

I misread the interior

Sorry.

5. Originally Posted by squarerootof2
how would one show that the interior of the cantor set is the empty set?
If the Cantor set had a non-empty interior, then it would contain an open interval and hence would have positive measure.