TITLE: Why is the Riemann sphere the compactified complex plane?
QUESTION [8 upvotes]: Physicist here, although math inclined but I certainly won't brag about it on a math forum, I am encountering a confusion with the compactification of the complex plane. I am learning conformal field theory in 2d and, as we use conformal map mapping some points to infinity and vice versa, we work on the Riemann sphere defined as $$\hat{\mathbb{C}} = \mathbb{C} \cup \{\infty\}$$
I have a doubt on having only "one infinity" and not seeing it as a full circle whose radius tends to $\infty$. For example compactifying $\mathbb{R}$ means adding the two infinities $\pm \infty$.
Is it because all those infinities could "topologically" connected when one needs to cross all the real axis to go from $+\infty$ to $-\infty$?

REPLY [6 votes]: It might help to think first about compactifying $\mathbb R$ by adding one point rather than two.  Here is one way: identify $(-\infty,\infty)$ with $(-\pi,\pi)$ in the usual way, say using $x \mapsto 2\arctan x$, and then instead of
compactifying $(-\pi,\pi)$ to the closed interval $[-\pi,\pi],$ instead compactify it to a circle by wrapping around the unit circle (in terms of formulas, you can use $\theta \mapsto e^{i \theta}$): this adds just one point instead of two.
(If you don't want to make all these changes of variables, you just think about taking $[-\infty,\infty]$ and then identifying the endpoints to get a circle, just as identifying the endpoints of any closed interval gives a circle.)
So you can form a one-point compactification of $\mathbb R$ to get a space that is topologically a circle.
Note that one nice thing about this is that the one-point compactification is topologically homogeneous (no point looks different to any other; it is a one-dimensional manifold); this is different to the two-point compactification, where the endpoints have different topology to all the other points.

Now we can form the one-point compaticification of $\mathbb C$ to get the Riemann sphere. Note that if we close up $\mathbb R$ inside, we get the one-point (circle) compatification of $\mathbb R$ discussed above.  
The Riemann sphere is also topoogically homogeneous.
We could also compactify $\mathbb C$ by adding a circle at infinity instead of just a point, but this isn't as useful when studying complex analysis and related questions (for example because holomorphicity involves considering derivatives along multiple directions --- think of the Cauchy--Riemann equations --- and so adding points at infinity for each direction from the origin isn't so helpful when studying complex analysis).  And it wouldn't be topologically homogeneous.
There are situations where you can see a "circle at infinity" in the context of the Riemman sphere.  For example, the one-point compactification of $\mathbb R$ cuts the Riemann sphere into two halves (the upper and lower halfplanes), and we can think of the one-point compatification of $\mathbb R$ as being a "circle at infinity" for each of these half-planes.