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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.

Stefan Hamcke
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  • Related: https://math.stackexchange.com/questions/1789690/, https://math.stackexchange.com/questions/75781/ – Watson Nov 28 '18 at 15:34

1 Answers1

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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 $q_j$.

Tsemo Aristide
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    Ohh... there is a 1-1 correspondence from set of intervals to a subset of rational number. –  May 02 '16 at 18:14
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    And subset of a countable set is countable. Thus we can't find a uncountable disjoint open intervals in R. Thank you...!! –  May 02 '16 at 18:15
  • Is this function well defined? What criteria are we using to pick the rational? – ZSMJ Apr 18 '20 at 15:48
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    @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. – Adam Rubinson Nov 12 '20 at 08:30