Showing a certain map is a homeomorphism

Question

I'm studying point-set topology for the first time, and I'm finding some difficulty in solving problems which ask to verify certain statements in particular examples. I'm ok with the theory, but sometimes I can't quite see how to approach exercises like the following. I'm asked to show that the map $\Bbb{P^1R} \to S^1$, $[x_0,x_1] \mapsto (\frac{x_0^2-x_1^2}{x_0^2+x_1^2},\frac{2x_0x_1}{x_0^2+x_1^2})$ is a homeomorphism. I've thought of using the fact that $\Bbb{P^1R}$ is homeomorphic to $S^1$ with antipodal points identified. This essentially just allows me to say that in the above map $x_0^2+x_1^2=1$, but I can't see how that helps to show that it's a homeomorphism. Is there some geometric picture I'm not seing? Or am I just supposed to do a ton of calculations? More generally, are there any particular things I'm supposed to be thinking when dealing with particular examples in topology, rather than general statements? Thank you in advance, any advice is very much appreciated.

Answer 1

You should be able to do this in a straightforward manner by applying the following theorem:

Theorem: If $f : X \to Y$ is a continuous bijection from a compact space $X$ to a Hausdorff space $Y$, then $f$ is a homeomorphism.

So, just verify that the formula you wrote defines a continuous bijection from $P^1 \mathbb R$ to $S^1$, and verify compactness of $P^1 \mathbb R$ and Hausdorffness of $S^1$.

Let me add that if you are asked to prove a specific function is a homeomorphism, I doubt you could achieve that by using some abstract knowledge of the existence of some other homeomorphism.