[Math] Can every closed subset of real numbers be written as the countable (or finite) union of disjoint closed intervals

Question

I want to show that if $C\subseteq \mathbb{R}$ is closed, then $C$ can be written as the union of countably many (or finitely many) disjoint closed intervals.

Note: If $C$ is itself a closed interval then this is trivially true, a bunch of people I have asked say $[0,1]$ is a counterexample, but it is not because $[0,1]=\cup\{[0,1]\}$ which is a finite union of disjoint closed intervals.

I know a similar theorem is true for open intervals.

Answer 1

The Cantor set is a union of uncountably many points, but contains no closed intervals. So, it cannot be written was the union of countably many disjoint closed intervals.