# 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.

• – Watson Nov 28 '18 at 15:34

## 1 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 $q_j$.

• Ohh... there is a 1-1 correspondence from set of intervals to a subset of rational number. – user322806 May 2 '16 at 18:14
• And subset of a countable set is countable. Thus we can't find a uncountable disjoint open intervals in R. Thank you...!! – user322806 May 2 '16 at 18:15