## 5.2. Compact and Perfect Sets

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

The definition of the Cantor set is as follows: letand define, for eachA_{0}= [0, 1]

*n*, the sets

*recursively as*

**A**_{n}Then the Cantor set is given as:A_{n}=A_{n-1}\

To be more specific, we have:=CA_{n}

That is, at theA_{0}= [0, 1]

A_{1}= [0, 1] \ (1/3, 2/3)

A_{2}=A_{1}\ [(1/9, 2/9) (7/9, 8/9)] =

[0,1] \ (1/3, 2/3) ) \ (1/9, 2/9) \ (7/9, 8/9)

...

*n*-th stage (

*n > 0*) we remove

*2*intervals from each previous set, each having length

^{n-1}*1 / 3*. Therefore, we will remove a total length of

^{n}from the unit interval

*[0, 1]*. Since we remove a set of total length 1 from the unit interval, the length of the remaining Cantor set must be 0.

The Cantor set contains uncountably many points because it is a perfect set.