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?