Understanding how map between spaces induces map between quotient spaces

Question

Recently I was trying to prove that $\mathbb{C}P^1$ (complex projective line) is homeomorphic to $\mathbb{C}\cup\{\infty\}$ (Riemann sphere). My question is partly based on a comment made at the end of this question.

Suppose that I have an open continuous map $\phi: \mathbb{C}^2 \setminus \{0\} \to \mathbb{C} \cup \{\infty\}$ given by $\phi(u,v) = \frac{u}{v}$ which is clearly a surjection but not injective. How can I derive an "induced" map $ \phi' :\mathbb{C}P^1 \to \mathbb{C}\cup \{ \infty\}$ that is a homeomorphism.

One idea is that since we have the quotient map $q: \mathbb{C}^2 \setminus \{0\} \to \mathbb{C}P^1$, I try the composition $\phi \circ q^{-1}: \mathbb{C}P^1 \to \mathbb{C} \cup \{ \infty\}$. However, I don't think it is clear that $q^{-1}$ is continuous (I know $q$ is by definition of quotient topology).

In general, if I am given two spaces $X,Y$ with corresponding quotients $X /\sim $ and $Y / \sim$. Under what conditions does a continuous map $X \to Y$ "descend" to a continuous map between $X /\sim \to Y / \sim$ and how can I construct this derived map.

Sorry if this sounds like nonsense. I am quite confused about how one should go about proving homeomorphism between quotient spaces.

Answer 1

However, I don't think it is clear that $q^{-1}$ is continuous (I know $q$ is by definition of quotient topology).

$q^{-1}$ is not even well defined. $q$ is not injective and therefore not invertible.

The crucial observation is the following:

Lemma. Let $f:X\to Y$ and $g:X\to Z$ be two quotient maps. Assume that $f(x)=f(y)$ if and only if $g(x)=g(y)$. Then $Y$ is homeomorphic to $Z$.

which I leave as an exercise.

Now note that $\phi(u,v)=\phi(u',v')$ if and only if $u/v=u'/v'$ which is if and only if $u'=\lambda u$ and $v'=\lambda v$ where $\lambda=v'/v$ or $\lambda=u'/u$ depending on whether $v$ is nonzero or $u$ (obviously you have to consider the special case of $\infty$ here, which I leave to you as well). Therefore $\phi(u,v)=\phi(u',v')$ if and only if $(u,v)\sim (u',v')$ which is if and only if $q(u,v)=q(u',v')$. The conclusion thus follows from the lemma.

In general, if I am given two spaces $X,Y$ with corresponding quotients $X /\sim $ and $Y / \sim$. Under what conditions does a continuous map $X \to Y$ "descend" to a continuous map between $X /\sim \to Y / \sim$ and how can I construct this derived map.

As long as the derived map is well defined then it is continuous. If $f:X\to Y$ is continuous and $\sim$ is such that "if $x\sim y$ then $f(x)=f(y)$" then $f$ induces $F:X/\sim\to Y$ such that $F([x])=f(x)$. It is well defined and continuous. Now applying another relationship to the right side is a simple matter of composing $F$ with the quotient map. Although I don't see how all of that is related to the original problem.