# Hopf fibration homeomorphism injectivity

I was wondering if there is a possibility to proof the homeomorphism between the 1-dimensional complex projective space $\mathbb{C}P^1$ and the 2-sphere $S^2$ via the Hopf fibration (I know already that there is an explicit homeomorphism between these two spaces). Here is my idea: let $\pi:S^3 \rightarrow \mathbb{C}P^1$ be the natural projection to the quotient and let $H:S^3 \rightarrow S^2$. It is easy to see that these two maps pass to the quotient, then because of the universal property of the quotient topology there is an unique continuous map $f:\mathbb{C}P^1 \rightarrow S^2$ such that $f \circ \pi=H$. f is surjective because $\pi$ and $H$ are surjective maps and $f$ is also closed because is a map which satisfies the hypothesis of the closed map lemma. The problem I have is to show that $f$ is injective and I think it is not trivial to show. Could somebody help me please?

• On a note that doesn't answer your question, you'll find that your continuous map $f$ is the standard identification between $\Bbb{CP}^1$ and $S^2$ you know exists but don't want to use. – user98602 Jul 11 '16 at 19:50
• Thank you, yes exactly I know there is an explicit homeomorphism, I was just wondering if without that one could deduct the injectivity of f – Salvatore Jul 11 '16 at 19:57
• Out of curiosity, what is your definition of $H$? – Ted Shifrin Jul 12 '16 at 0:38
• @TedShifrin H(z,w):=(2zw*,|z|^2 - |w|^2) – Salvatore Jul 12 '16 at 4:25
• So, this is just the map to $\Bbb CP^1$ followed by the inverse of stereographic projection to the sphere. – Ted Shifrin Jul 12 '16 at 15:36