Let $\Phi$ be the diffeomorphism $\mathbb{R}^2/\{0\}\to \mathbb{R}^2/\{0\}$, given by $$\Phi(x,y)=(2x,\frac{1}{2}y).$$ Let $\text{~}$ be equivalence relation generated by $p \text{~} \Phi(p)$. Let $M$ be the quotient space, with quotient map $\pi:\mathbb{R}^2/\{0\}\to M$. Every open subset $V\subseteq \mathbb{R}^2/\{0\}\to $ for which the restriction $\pi|_V: V\to M$ is injective defines a chart $(U,\phi)$, where $$ U=\pi(V), \phi\circ \pi|_V=id_V. $$
I am wondering if we can use counterexamples such as $p_1=\pi((1,1)),p_2=\pi(1,0)$. Such that every chart containing $p_2$ would also contain $p_1$, thus this is non-Hausdorff. However, I became kind of skeptical that whether these points are nonseparable, since it seems that we can always take nbhd infinitesimally small such that it can actually separates the two points.
Can anyone help me with this?