Complex Analysis/Complex Numbers/Topology

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We have already seen that \mathbb{C} is an algebraically closed field. Here we consider some of its other analytic and topological properties.

[edit] Metric property

Define the metric d:\mathbb{C}^2\to\mathbb{R} as

d(z1,z2) = | z1 − z2 |

It can easily be seen that d satisfies positive definiteness, symmetry and the triangle inequality, implying that \mathbb{C} is a metric space.

[edit] Completeness

Recall that a metric space is said to be complete if every Cauchy sequence converges to a limit.


For any point z_0\in \mathbb{C}, we call the open ball Bδ(z0), consisting of all the points z such that | z − z0 | < δ, a neighborhood of z0. Similarly, a set consisting of points z such that | z | > δ for a positive δ will be called a neighborhood of infinity. Given a set \mathfrak{G}\subset\mathbb{C}, we call the set open if every point in \mathfrak{G} has a neighborhood completely contained in \mathfrak{G}. Similarly, we call a set closed if its complement is open. A point z is called an accumulation point of \mathfrak{G} if every neighborhood of z contains a point in \mathfrak{G} other than z itself. It can be shown that a set is closed if and only if it contains all of its accumulation points: see proof.

[edit] The Riemann Sphere

The Riemann Sphere

An interesting idea related to the extension of the complex numbers is the construction of the Riemann Sphere. The Riemann Sphere, essentially a stereographic projection, is constructed by projecting the Complex plane onto the unit sphere about the point (0,0,1).

Formally, the rectangular coördinates of the projection (ξ,η,ζ) can be given by the transformations

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

Or equivalently, the reverse transformation,

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

The Riemann sphere is this transformation, together with the point (0,0,1) labeled as \infty

It can also be shown 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\bar a} \sqrt{1+b\bar 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 z0 is the set of all points z satisfying

χ(z,z0) < δ,

where z0 is allowed to be infinity.

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