Is it possible to find an uncountable number of disjoint open intervals in $R$?
Several times I saw the sentence
every open set in $\mathbb{R}$ can be expressed as a countable number of open intervals (Because $\mathbb{R}$ is second countable)
Suppose we are able to find an uncountable number of disjoint open intervals in $\mathbb{R}$, then union of these intervals is an open set (say $G$) in $\mathbb{R}$. But $G$ cannot be expressed as a countable number of open intervals.
Thus my answer is there is no such a collection exist.
Is my think is correct? Give more hints and clarify it..!!
Thanks in advance.
Best Answer
No, in a disjoint union of open intervals $(I_j)_{j\in J}$ each interval $I_j$ contains a rational number $q_j$ which enables to define an injection $J\rightarrow Q$ which sends $j$ to $q_j$.