Complex Analysis/Complex Numbers/Topology

The triangle inequality is a useful inequality that holds for complex numbers:

$|a+b|\le \left|a\right|+\left|b\right|$.

This is obvious geometrically, but a formal proof can be found here.

We now define some basic topological constructs that we need to define the notion of continuous functions and derivatives.

The natural metric on the complex numbers $d(x,y)=|x-y|$ gives rise to a Euclidean space. For any point $z_0\in ℂ$, we call the open ball $B_{\delta }(z_0)$, consisting of all the points $z$ s.t. $|z-z_0|<\delta$, a neighborhood of $z_0$. Similarly, a set consisting of points z such that $|z|>\delta$ for a positive δ will be called a neighborhood of infinity. Given a set $𝔊\subset ℂ$, we call the set open if every point in $𝔊$ has a neighborhood completely contained in $𝔊$. Similarly, we call a set closed if its complement is open. A point $z$ is called an accumulation point of $𝔊$ if every neighborhood of z contains a point in $𝔊$ other than z itself. It can be shown that a set is closed if and only if it contains all of its limit points: see proof.

We would also like to define infinity for complex numbers. This can be achieved conveniently through a construct called a stereographic projection. Unlike the projection for the two-dimensional real plane, we extend the complex plane by only one point at infinity. The resulting surface, which we call the extended complex plane, consists of the complex plane and a single point at infinity. Because of similar triangles, the transformations are relatively simple:

$z=\frac{\xi +\eta i}{1+\zeta },$

and the reverse transformation,

$\xi =\frac{z+\overline{z}}{1+z\overline{z}},\eta =\frac{1}{i}\frac{z-\overline{z}}{1+z\overline{z}},\zeta =\frac{1-z\overline{z}}{1+z\overline{z}}.$

We can also show that the stereographic projection preserves angles, and that circles and lines in the plane correspond to circles on the sphere: see proof.

In the metric |a-b| used earlier, the point z=∞ causes problems. However, using the stereographic projection, we can define another metric where the distance between two points a and b is the chordal distance

$\chi (a,b)=\frac{2|a-b|}{\sqrt{1+a\overline{a}}\sqrt{1+b\overline{b}}}$,

which has a well-defined meaning even when one of the points is ∞. We will only employ this metric when dealing with infinite values. For example, using this metric, neighborhoods of infinity do not require special treatment; we say that a neighborhood of a point $z_0$ is the set of all points z satisfying

$\chi (z,z_0)<\delta$,

where $z_0$ is allowed to be infinity.

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