# 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. May 9 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. May 9 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. May 9 at 13:31
• Also, all infinite subsets of $\mathbb{R}$ need not be uncountable, $\mathbb{Q}$ or $\mathbb{Z}$ are infinite but countable. May 9 at 13:34
• It means (at most) countable, i.e. it may have finite number of sets or countable number of sets. May 9 at 13:42