Every open set in R is the countable union of open intervals: open?

Question

I am trying to prove this and there is something that I am missing.

Let $E\subseteq\mathbb{R}$ and let us define an equivalence relation on E as follows. If $a\in E$ and $b\in E$ we say that $a\sim b$, if the entire open interval $(a,b)$ is contained in $E$.

Now, if I accept that this equivalence relationship partitions E into a disjoint union of classes, we can easily prove that:

• an equivalence class is an interval and it's open

• since $\mathbb{Q}$ is dense in $\mathbb{R}$, and $\mathbb{Q}$ is countable, we can pick a rational to associate with each $(a,b)$

And we are done.

My issue is: since we specify that $(a,b)$ are open disjoint intervals, how can they partition $E$? In particular, what happens to the boundaries $a$ and $b$? We may pick them to be in $E$, sure, but the collection of open intervals bounded by $a_i$ and $b_i$, $i<\infty$, will exclude $a_i$ and $b_i$, thus not covering $E$. So say $E=(1,3)$. The element $2\notin(1,2)\cup (2,3)$. What am I missing?

Answer 1

Let In$[a,b]$ denote the closed interval from $a$ to $b.$ That is In$[a,b]=[a,b]\cup [b,a].$ This removes the need to distinguish the cases $a<b,a=b, a>b.$

For open $ E\subset \mathbb R $ and $ a,b\in E $ let $ a\sim b $ iff In$[a,b]\subset E.$ Obviously $ a\sim b$ iff $b\sim a, $ and also $ a\sim a, $ for any $ a,b\in E.$

Exercise. For any real $a,b,c$ we have $$In[a,b]\cup In[b,c]=In [\min (a,b,c),\max (a,b,c)]=[\min (a,b,c),\max (a,b,c)].$$ Corollary : $\sim$ is transitive.

So $\sim$ is an equivalence relation on $E.$

Exercise.The set of equivalence classes of any equivalence relation on any set comprise a partition of that set.

Note that in your title we implicitly include open half-lines, the whole real line, and the empty set among the open intervals. The members of the partition $E_{/\sim}$ are convex open sets (open because if $x\in E$ then $(x-d,x+d)\subset E$ for some $d>0,$ so $(x-d,x+d)\subset [x]_{\sim}$ ),... but not necessarily bounded open intervals. For example if $E=(1,\infty)$ then $E_{/\sim}=\{E\}.$

If $x,y\in \mathbb R$ with $x<y$ and $(x,y)\in E_{\sim}$ then neither $x$ nor $y$ belongs to $E$.

Another way of describing $E_{/\sim}$ is : For $a\in E, $ let $a^*$ be the set of convex open real set(s) $T$ such that $a\in T\subset E.$ Let $a^{**}=\cup a^*.$ Then $a^{**} \in a^*,$ so $a^{**}$ is the $\subset$-maximum member of $a^*.$ And $a^{**}=[a]_{\sim},$ the $\sim$-equivalence class containing $a.$