Question:
On $\mathbb{R}^2 \setminus\{0\}$, introduce an equivalence relation by declaring that $(x,y)\sim (x^{\prime}, y^{\prime})$ if and only if there exists $t>0$ and that $x^{\prime} = tx$ and $y^{\prime} = y/t$. Let $M$ be the set of equivalence classes.
Show that $M$ has a natural $ 1-$ dimensional atlas, which is NOT Hausdorff. (Remark: Here, "Natural" means that if you sketch the equivalence classes, you will see a way to choose little transverse arcs that suggest a $1-$dimensional atlas.)
I thought of this as a projective space $\mathbb{R}P^n$, but I do not know how to attack this question. I am not sure I can sketch the equivalence classes, either. Any help would be appreciated.