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Let $f:X\to Y$ be an open, surjective, continuous map between two topological spaces $X$ and $Y$ with compact fibers. Let $X$ be a Hausdorff topological space. Then $Y$ is Hausdorff.

I wanted to show that by taking two distinct points $a_1$ and $a_2$ in $Y$. If I take their compact fibers $B_1=f^{-1}(a_1)$ and $B_2=f^{-1}(a_2)$, then I can separate them by disjoint open subsets $U_1$ and $U_2$ where $B_i\subset U_i$ as $B_i$ is compact. However, I struggle to show that $f(U_1)$ and $f(U_2)$ are going to be disjoint. Am I doing something wrong, or is there another way to show that?

eightc
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    In the title you have the assertion of compact fibers but not in the body of the question. – mathcounterexamples.net Oct 26 '21 at 06:58
  • Updated, but I mentioned compactness when I defined $B_i$. – eightc Oct 26 '21 at 07:01
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    You can show from these assumptions that $f$ is in fact "proper": $f^{-1}[K]$ is compact when $K \subseteq Y$ is compact. That might help some. – Henno Brandsma Oct 26 '21 at 08:22
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    There is another question that has a Lemma that may be helpfull: https://math.stackexchange.com/questions/1385996/if-every-image-under-a-continuous-open-map-of-a-hausdorff-space-is-hausdorff-sh – Elma Oct 26 '21 at 16:34
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    What is the source of your question? Are you sure that $f$ is required to be open and not to be closed? – Paul Frost Oct 27 '21 at 22:19
  • @PaulFrost, I basically want to use that statement to show that $\mathbb{P}^1(\mathbb{C})$ is Hausdorff by considering the quotient map $\pi: S^3 \to \mathbb{P}^1(\mathbb{C})$. I am not sure if that quotient map is closed. – eightc Oct 28 '21 at 04:32
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    I recommend to reformulate your question to "Does a continuous open surjective map $f$ with compact fibers preserve Hausdorffness?" I doubt that it is true in general, so perhaps somebody can give a counterexample. It is well-known that a continuous closed surjective map $f$ with compact fibers preserves Hausdorffness. Now look at https://math.stackexchange.com/q/3998245. – Paul Frost Oct 28 '21 at 08:39
  • @PaulFrost, awesome! Thank you! Yeah, I suspected that as I did not see how I could proceed to show that. – eightc Oct 28 '21 at 19:05

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