Irreducible components of fiber bundle Suppose $\pi:X \rightarrow Y$ is a (locally trivial) fiber bundle $F$, where all spaces are Noetherian. Suppose $F$ and $Y$ are irreducible; show that $X$ is also irreducible. 
Here a similar question is discussed; but this question should be simpler, and should have an easier proof. 
Thanks in advance.

Since $\pi$ is locally trivial, it is an open map.  Thus the result follows by http://stacks.math.columbia.edu/tag/004Z which states:
Let $f : X \to Y$ be a continuous map of topological spaces.
Assume that
(a) $Y$ is irreducible,
(b) $f$ is open, and
(c) there exists a dense collection of points $y \in Y$ such
that $f^{-1}(y)$ is irreducible.
Then $X$ is irreducible.
