Hopf fibration homeomorphism injectivity

Question

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?