# Is it possible to find an uncountable number of disjoint open intervals in $R$?

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.

• Nov 28, 2018 at 15:34

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 $q_j$.
• @Akash - You can pick any rational in each interval $I_j$. If the amount of intervals, $|J|$ was uncountable then since each interval $I_j$ contains at least one rational number, certainly the set of rationals would be uncountable which is not true. Nov 12, 2020 at 8:30