*(Cross posted from this math.SE question)*

Let $X$ be a Cech-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-Cech 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 implies that the extend is also an open surjection?