Cantor set
See the link, I am referring to cantor set on the real line. I wish to show that it is compact. I am doing this by pointing following arguments. I am not sure if this is enough.
- Cantor set is bounded by definition in the region $[0,1]$
- Cantor set is the union of closed intervals, and hence it is a closed set.
- Since the Cantor set is both bounded and closed it is compact by Heine-Borel Theorem.
Best Answer
Cantor set is defined as $C=\cap_n C_n$ where $C_{n+1}$ is obtained from $C_n$ by dropping 'middle third' of each closed interval in $C_n$
As you have noted, Cantor set is bounded.
Since each $C_n$ is closed and $C$ is an intersection of such sets, $C$ is closed (arbitrary intersection of closed sets is a closed set).
As $C$ is closed and bounded, it is compact by Heine-Borel theorem.
PS: You cannot say that Cantor set is a union of closed intervals. Rudin is giving Cantor set as an example for a perfect set that contains no open interval!