# Show that the quotient space is not Hausdorff.

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.

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.