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## Example 5.2.13(d): Properties of the Cantor Set

The Cantor set does not contain any open set.
The definition of the Cantor set is as follows: let
• A 0 = [0, 1]
and define, for each n, the sets A n recursively as
• A n = A n-1 \
Then the Cantor set is given as:
• C = A n
Another way to write the Cantor set is to note that each of the sets A n can be written as a finite union of 2 n closed intervals, each of which has a length of 1 / 3 n, as follows:
• A 0 = [0, 1]
• A 1 = [0, 1/3] [2/3, 1]
• A 2 = [0, 1/9] [2/9, 3/9] [6/9, 7/9] [8/9, 1]
• ...
Now suppose that there is an open set U contained in C. Then there must be an open interval (a, b) contained in C. Now pick an integer N such that
• 1 / 3 N < b - a
Then the interval (a, b) can not be contained in the set AN, because that set is comprised of intervals of length 1 / 3N. But if that interval is not contained in AN it can not be contained in C. Hence, no open set can be contained in the Cantor set C.
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