# Prove that spherical distance on $\bar{\mathbb{C}}$ is a metric which induces the standard topology on $\bar{\mathbb{C}}$.

Prove that spherical distance on $\bar{\mathbb{C}}$ is a metric which induces the standard topology on $\bar{\mathbb{C}}$.

Just starting complex analysis and having a bit of trouble understanding the question. Part of the exercise was to derive a formula for the spherical distance on $\bar{\mathbb{C}}$. With $\bar{\mathbb{C}}$ being the Riemann sphere. But the second part is rather confusing to me. Any help to guide me in the right direction would be much appreciated.

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Metric defined as $p(z_{1}, z_{2}) = \frac{2|z_2-z_1|}{\sqrt[]{1+|z_2|^2}\sqrt[]{1+|z_1|^2}}$

Extend to $\infty$

$p(z, \infty) = \frac{2}{\sqrt[]{1+|z^|2}}$

I hope that is not too small, if so how do I make it larger?

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Does it mean I have to define the notions of neighborhood and openness using the distance metric I had previously defined? –  John Thomas Sep 9 '12 at 20:52
How was the topology introduced? As one-point compactification of $\bf C$? –  tomasz Sep 9 '12 at 21:20
So what was the formula for spherical distance that you derived? –  Jesse Madnick Sep 9 '12 at 23:16
@JohnThomas: It means you have to prove that: (1) the "spherical distance" is in fact a metric, and (2) a subset of $\bar{\mathbb{C}}$ is open with respect to the spherical metric if and only if it is open with respect to whatever is meant by "standard topology" on $\bar{\mathbb{C}}$. (This brings us to tomasz' question above: how was the topology on $\bar{\mathbb{C}}$ introduced?) –  Jesse Madnick Sep 9 '12 at 23:21