Let $X,Y$ be separable and metrizable, with $Y$ Polish, and suppose there is a topological quotient map $f:X\to Y$ with compact fibers. Is $X$ Polish?


I have a specific space in mind, so if the answer is no and some extra criterion comes to mind ($Y$ is not locally compact, and $f$ is not a fiber bundle), I would appreciate it. 



Polish by compact is Polish?
I think the answer is no. Consider the following subsets of the Euclidean plane. 


Let $X\_0 = \{(p,q)\in (\mathbb R\setminus \mathbb Q)\times \mathbb R: 
 0\le p \le 1, 0 \le q \le 1 \}$, $X\_1 = \{(p,q)\in ( \mathbb Q)\times \mathbb R: 
 0\le p \le 1, -1 \le q \le 0 \}$, 


Let $X=X\_0\cup X\_1$, $Y=[0,1]$, and let $f$ be the projection. 


(ADDED: $X$ is not Polish, as $X$ contains a closed subset homeomorphic to $\mathbb Q$. Closed subsets of Polish spaces are Polish, but $\mathbb Q$ is not.) 


