As a self-studier, I was reading a proof that any open subset, $U$, of $\mathbb{R}$ is a disjoint union of open intervals. The proof was based on an equivalence relation where $x\sim y$ if $(x,y)$ is contained in $U$.
I have two questions regarding the verification that this is a valid equivalence relation. (Sorry if they are obvious.)
First, for reflexivity, $x\sim x$, is $(x,x)$ the null set?
Second, regarding transitivity, if $x < y < z$ are elements of $U$, with $x\sim y$ and $y\sim z$, then $(x,y)$ and $(y,z)$ are both contained in $U$. My question is how can it be claimed that $x\sim z$, i.e., that $(x,z)$ is in $U$, since $y$ is not in either of the sets (although it was stated that $y$ is an element of $U$).
Thanks.