Obtaining a Quotient Space of $\mathbb{R}^2$ Using stereographic projection of a sphere, $S^2$, we can obtain the one point compactification of $\mathbb{R}^2$ is sphere, i.e. $S^2$ can be thought of as $\mathbb{R}^2 \cup \{ \infty \}$. Now I am wondering how can $S^2$ be obtained by quotienting $\mathbb{R^2}$.
I have an idea (maybe it's vague),that can we identify the boundary of $\mathbb{R}^2$ to one point and other points as singletons. But I have thought for a while that the boundary of $\mathbb{R}^2$ is not in $\mathbb{R}^2$, so we may not get sphere by quotienting the space $\mathbb{R}^2$ only.
So I think we have to take $\mathbb{R}^2$ union its boundary at first then proceed in the above way I said. But I have no confidence in my intuition. Please, someone help me to clear my doubts. (But I know it can be done by any bounded closed subset of $\mathbb{R}^2$ and identifying its boundary to one point.)

$\mathbb R^2$ has no boundary. But that doesn't matter, we can still get $S^2$ by quotienting $\mathbb R^2$: take the open unit disc and collapse every point which is not in the disc into a single point. This point is essentially the pole needed for the one-point compactification of the unit disc.
