Existence of continuous sections of a continuous open surjection.

Question

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
• $\sup_{x \in X} \# f^{-1}(x)<\infty$.

Answer 1

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