## 5.2. Compact and Perfect Sets

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

The definition of the Cantor set is as follows: let*A*_{0}= [0, 1]

*n*, the sets

*A*recursively as

_{n}*A*_{n}= A_{n-1}\

**C**= A_{n}

*A*can be written as a finite union of

_{n}*2*closed intervals, each of which has a length of

^{n}*1 / 3*, as follows:

^{n}*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]- ...

**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

*(a, b)*can not be contained in the set

*A*, because that set is comprised of intervals of length

_{N}*1 / 3*. But if that interval is not contained in

^{N}*A*it can not be contained in

_{N}**C**. Hence, no open set can be contained in the Cantor set

**C**.