Let $X$ be a Čech-complete space, and $Y$ a paracompact space. Suppose $f\colon X\to Y$ is a continuous and open surjection.
Since $Y$ is completely regular we have that $\beta(Y)$ is homeomorphic to $Y$ as a dense subset of $\beta Y$ (the Stone-Čech compactification).
We can, if so, take $\hat f\colon X\to\beta Y$ defined as $\beta\circ f$, as a continuous function from $X$ into a compact Hausdorff space.
By the universal property of $\beta X$ we can uniquely extend $\hat f$ to a continuous $\tilde f\colon\beta X\to\beta Y$ such that $\tilde f|_{\beta(X)} = \hat f\circ\beta$. In particular $\tilde f$ is onto $\beta Y$ due to two reasons:
- $\tilde f$ is continuous from a compact domain, therefore its image is closed; and
- $\tilde f$ is onto a dense subset of $\beta Y$.
Therefore it is onto its closure which is $\beta Y$.
My question is whether or not the fact $Y$ is paracompact lets us extend the map such that $\tilde f$ is also an open surjection.
(The motivation is to write a proof for the theorem mentioned in my previous question, and a result as above would give a quick solution to the problem. Regardless, this question is interesting on its own accord)
Edit (Oct. 3rd): If in a few more days there won't be an answer, I'll try cross-posting this on MathOverflow as well.