Let $\mathcal{I}$ be the family of open subsets of $\mathbb{R}$ comprising of $\mathbb{R}$ and all open intervals with rational endpoints.
Prove that each open interval in $\mathbb{R}$ is a union of countable many of its members.
I am a bit confused by this, it seems to me that each open interval could be a union of three of its members as follow:
Take the interval $(a,b)\in\mathbb{R}$ where $a,b\in\mathbb{Q}$.
Then from the deinsity of the rationals we have that $c\in(a,b)$ where $c\in\mathbb{Q}$
We then have the following two intervals in $\mathcal{I}$ $(a,c)$ and $(c,b)$
Then again from the density of the rationals we have $c_1\in (a,c)$ and $c_2\in(c,b)$ with $c_1,c_2\in\mathbb{Q}$
We then have that $(a,b)=(a,c)\cup(c_1,c_2)\cup(c,b)$
However I'm sure this cant be right, could someone tell me whats wrong with this argument?
Thanks for any help.