The book I am reading contains the following two definitions:
Two sets $A$ and $B$ have the same cardinality if there exists $f: A \rightarrow B$ that is one to one and onto. In this case, we write $A \sim B$.
A set $A$ is countable if $\mathbb{N} \sim A$. An infinite set that is not countable is called an uncountable set.
Following on, I read the following statement:
Every open set is either a finite or countable union of open intervals.
Here, what does countable union mean? Clearly it can't mean that the resultant set formed by the union of open intervals is countable (since open intervals are uncountable). But I am not sure how the use of "countable union" connects with the definition provided earlier.