# Examples of open sets of $\mathbb{R}$ that are nontrivial countable unions of mutually disjoint open intervals

Could anyone please give me an example of a open set of $$\mathbb{R}$$ that can be written as a countable union of mutually disjoint open intervals?

I could not understand why the union of disjoint countable open sets could be an open subset of $$\mathbb{R}$$.

I mean, how could a union of countable sets equal an uncountable sets? As far as I understand, all infinite subsets of $$\mathbb{R}$$ are uncountable. I am not sure whether or not I misunderstood the theorem somehow.

Thank you so much, Riemann, Podiki, Soumik Mukherjee by saying countable union of sets, does it mean the union is a countable set, i.e. with the same cardinality of $$\mathbb{N}$$ (the set of natural numbers), or it just means a finite number of sets in a union?

• It's not "countable open sets", but "countable family of open sets". There are countable number of sets in a family or collection, and each set in the family or collection is an open set. Commented May 9, 2023 at 13:29
• The union is countable (meaning it is the union of countably many sets), but each interval you consider will have uncountably many elements in it. Commented May 9, 2023 at 13:30
• It is not union of countable sets, it is countable union of sets. That is, total number of sets is countable, the individual sets are open sets. Commented May 9, 2023 at 13:31
• Also, all infinite subsets of $\mathbb{R}$ need not be uncountable, $\mathbb{Q}$ or $\mathbb{Z}$ are infinite but countable. Commented May 9, 2023 at 13:34
• It means (at most) countable, i.e. it may have finite number of sets or countable number of sets. Commented May 9, 2023 at 13:42