TITLE: Existence of continuous sections of a continuous open surjection.
QUESTION [1 upvotes]: Let $f : X \to Y$ be a continuous open surjection between Hausdorff spaces.
Does there always exists a continuous section $s: Y \to X$ of $f$? 
If not, under what conditions does one exist?
For the problem I'm considering I have a few additional assuptions available, but they may be red herrings:

$X$ and $Y$ are compact/locally compact, and
$\sup_{x \in X} \# f^{-1}(x)<\infty$.

REPLY [3 votes]: Consider the map $f(z)=z^2$ of the complex plane and try to construct a local section near zero.
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