= Riemann sphere
{c}
{title2=$\widehat{\mathbb C}=\mathbb C\cup\{\infty\}$}
{wiki}
The Riemann sphere is $\mathbb C\cup\{\infty\}$ with two charts related by reciprocal coordinates.
The extra point completes the <complex plane> into a <compact Riemann surface>. It is one point shared by every direction of escape: $n$, $-n$ and $in$ all tend to $\infty$ as $n\to\infty$. A basis of <neighbourhoods> at $\infty$ is
$$
U_R=\{\infty\}\cup\{z\in\mathbb C:|z|>R\},\qquad R>0.
$$
This is the <one-point compactification> of the <complex plane>; convergence to $\infty$ means that the <modulus> tends to infinity.<Complex Analysis Lecture Notes by Dan Romik>[Complex Analysis Lecture Notes, section 9, item 84, printed p. 33, topological definition]{ref}
For a concrete picture, use the <unit sphere>
$$
S^2=\{(X,Y,Z)\in\mathbb R^3:X^2+Y^2+Z^2=1\}
$$
and identify $z=x+iy$ with $(x,y,0)$ in its equatorial plane. The north pole $N=(0,0,1)$ represents $\infty$ and the south pole $S=(0,0,-1)$ represents $0$. The <stereographic projection> below gives the identification. This convention fixes the scale: the equator corresponds to $|z|=1$.<Complex Variables with Applications, MIT 18.04, Spring 2018/Topic 2 Notes>[Complex Variables with Applications, MIT 18.04, Spring 2018; Topic 2 Notes, section 2.5.2, printed p. 7, equatorial-plane convention]{ref}
Why add this point? The <Möbius transformation> $z\mapsto1/z$ then exchanges $0$ and $\infty$ and becomes a <holomorphic map> everywhere on the <Riemann sphere>. Its apparent blow-up at zero disappears when its output is expressed in the <reciprocal chart at infinity>: $w=1/(1/z)=z$. This is a concrete instance of a <meromorphic function as a holomorphic map to the Riemann sphere>. The additional point makes the <complex plane> <compact>, while its second <manifold chart> lets us do ordinary <complex analysis> near infinity.
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