## 5.2. Compact and Perfect Sets

### Example 5.2.13(a): 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*is closed, the sets

_{0}*A*are all closed as well, which can be shown by induction. Also, each set

_{n}*A*is a subset of

_{n}*A*, so that all sets

_{0}*A*are bounded.

_{n}
Hence, **C** is the intersection of closed, bounded sets, and
therefore **C** is also closed and bounded. But then
**C** is compact.