How would I prove that every open set in $\mathbb{R}$ is the countable union of open intervals with rational endpoints?

Context: I am trying to see whether the set of rational open subsets of $\mathbb{R}$ is countable.

  • 4
    $\begingroup$ How would you write $(e,\pi)$ in that way ? $\endgroup$ – Nitrogen Oct 29 '15 at 0:04
  • $\begingroup$ Delete "disjoint" and try again. $\endgroup$ – B. S. Thomson Oct 29 '15 at 0:12
  • $\begingroup$ I've edited the question. :) $\endgroup$ – Evan Oct 29 '15 at 0:17
  • $\begingroup$ What do you mean by a rational open set? $\endgroup$ – DanielWainfleet May 12 '16 at 2:26
  • $\begingroup$ @Nitrogen you would union subintervals whose endpoints are $\omega$-sequences of rationals converging on the endpoints. For example, decimal expansions that converge down to $e$ and up to $\pi$. $e$ and $\pi$ are actually easy because they are computable. $\endgroup$ – Spencer Jun 1 '16 at 4:37

Let $U$ be an open set in $\mathbb R$ we have to prove there is a countable family of open intervals $(a_i,b_i)$ with $a_i,b_i \in \mathbb R$ so that $U=\bigcup\limits_{i\in\mathbb N}(a_i,b_i)$.

To do this for each $x\in U$ we have $(a_x,b_x)$ so that $x\in(a_x,b_x) \subseteq U$. This is because $U$ is open. Of course, there is a rational number $a'_x$ between $a_x$ and $x$ and a rational number $b'_x$ between $b_x$ and $x$. This is because the rational numbers are dense in $\mathbb R$.

So we can consider the familiy of open intervals with rational endpoints. $(a'_x,b'_x)$ Why is this family countable? Because the number of intervals with rational coordinates is countable, since its cardinality does not exceed $|\mathbb Q\times \mathbb Q |=|\mathbb N|$.

Moreover $\bigcup\limits_{x\in U}(a'_x,b'_x)$ is clearly a subset of $U$ since every interval in the union is contained in $U$. But this union also contains $U$ since every element $x\in U$ is contained in the interval $(a'_x,b'_x)$ and hence is contained in the union. Hence $U$ is a countable union of rational open intervals.

  • $\begingroup$ why we can consider the familiy of open intervals with rational endpoints? $\endgroup$ – DuFong May 12 '16 at 0:54

The following link shows that opne set can is countable disjoint union of open intervals:

Any open subset of $\Bbb R$ is a at most countable union of disjoint open intervals. [Collecting Proofs]

Let's asume $\mathcal{O}=\bigcup_{i=1}^\infty(a_i,b_i)$,where $a_i,b_i\in \mathbb{R}$, then for each interval $(a_i,b_i)$, since rational number is dense in real number, we can always find rational sequence $\{b_{i,n}\}$ increasingly approaching $b_i$ and $\{a_{i,n}\}$ decreasingly approaching $a_i$, then $(a_i,b_i)=\bigcup_{n=1}^\infty (a_{i,n},b_{i,n})$, then $\mathcal{O}=\bigcup_{i=1}^\infty(a_i,b_i)=\bigcup_{i=1}^\infty\bigcup_{n=1}^\infty(a_{i,n},b_{i,n})$, and countable union of countable union is countable union too.

As for can we find disjoint countable union of rational intervals, I didn't prove it.


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