This is not a homework problem, just a question from a discussion with my classmates about the Cantor set. The original goal is to prove Cantor set is closed. My earlier attempt is to show the complement of the Cantor set is open. Since when construct the Cantor set each time the sets removed are open, I thought the complement of the Cantor set contains uncountably infinite many open sets, and the union of all of these open sets is still open. So the Cantor set is closed. But first, one of my classmate disagree with this proof. He thought the complement of the Cantor set only contains countably infinite many open sets. Second, he thought only countably infinite union of open sets is open, uncountably infinite union of open sets may not be open. But he could not give me convincing proof. So I posted my question here. Any suggestions are highly appreciated!