Example: The Cantor set does not contain any open set.

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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 A_N, because that set is comprised of intervals of length 1 / 3^N. But if that interval is not contained in A_N it can not be contained in C. Hence, no open set can be contained in the Cantor set C.

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(bgw)