limit points at infinity

Question

State why the limit points at the infinity in a extended complex plane are unique.

I know that the obvious answer to this is to use equivalent definitions and then assume that it has two different limit points say L and L' and reach upto a contradiction, like how we do it for real numbers. But I dont think that fulfils the questions requirement as the question asks "state why.." and not "show that..". Any alternative answers?

Answer 1

As Sangchul Lee said, there are many compactifications of the complex plane - i.e., ways of adding extra points to the plane to serve as unbounded limit points.

• The Riemann Sphere, which you - and perhaps no one else - call the "extended complex plane", is the one-point compactification, where you add a single point $\infty$ to $\Bbb C$. As Jean Marie describes, you can use stereographic projection to map $\Bbb C$ onto the unit sphere except for one point, which then acts as $\infty$. This turns out to be very useful in complex analysis, since linear fractional transformations are convolutions of the Riemann Sphere. You can do a similar thing in the real numbers: instead of adding both $-\infty$ and $+\infty$ to each end of the reals, add a single $\infty$ that ties the two ends of the real line together into a circle. However, as this breaks the order relation on the real line, which is one of its most useful features, we generally prefer not to do this.
• In $\Bbb C$, you could instead add a separate infinity for each ray eminating from $0$. That is, you add $\{\infty_\theta \mid \theta \in [0, 2\pi)\}$. If you denote $\infty := \infty_0$, then you can define multiplication so that $\infty_\theta = \infty e^{i\theta}$. However, addition is not well-defined between these infinities. This is similar to the extended real numbers, where $\infty \cdot -\infty$ is well-defined, but $\infty + -\infty$ is not. This extension of the complex numbers forms a closed disk instead of a sphere.
• Instead of rays, we can add infinities for each line through $0$. Effectively identifying $\infty_\theta$ with $\infty_{\theta+\pi}$ in the previous extension. This extension of the complex numbers gives the projective plane.

There are in fact infinitely many variants. However, by far the most useful is the Riemann Sphere.