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?

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    $\begingroup$ 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. $\endgroup$ – user98602 Jul 11 '16 at 19:50
  • $\begingroup$ 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 $\endgroup$ – Salvatore Jul 11 '16 at 19:57
  • $\begingroup$ Out of curiosity, what is your definition of $H$? $\endgroup$ – Ted Shifrin Jul 12 '16 at 0:38
  • $\begingroup$ @TedShifrin H(z,w):=(2zw*,|z|^2 - |w|^2) $\endgroup$ – Salvatore Jul 12 '16 at 4:25
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    $\begingroup$ So, this is just the map to $\Bbb CP^1$ followed by the inverse of stereographic projection to the sphere. $\endgroup$ – Ted Shifrin Jul 12 '16 at 15:36

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