Show that the quotient space is not Hausdorff.

Question

Let $\mathcal{R}$ be the equivalence relation on $\mathbb{R}$ consisting of those pairs $(r, s)$ of real numbers with the property that there exist two integers $m$ and $n$ such that $r − s = m + n\sqrt{2}$. Show that the resulting quotient space is not Hausdorff.

Answer 1

SKETCH: Let $D=\left\{m+n\sqrt2:m,n\in\Bbb Z\right\}$; for each $r\in\Bbb R$, the $\mathcal{R}$-equivalence class $[r]$ of $r$ is $r+D=\{r+d:d\in D\}$. Use this result to show that $D$ is dense in $\Bbb R$ and hence that $[r]$ is dense in $\Bbb R$ for each $r\in\Bbb R$, and use that fact to show that if $U$ and $V$ are non-empty open sets in the quotient space, then $U\cap V\ne\varnothing$. Conclude that the quotient space is not Hausdorff.