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I consider a bounded subset $A \subset \mathbb{R}$ and want to know if $A$ can be written as a countable union of disjoint intervals (open or half open)?

In case that is true: Does it mean that a bounded subset $A \subset \mathbb{R}$ can have no more than countable many points of discontinuity (holes)?

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The Cantor set is a bounded set of $\Bbb R$ which can not be expressed as a countable disjoint union of open intervals.

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