TITLE: The fiber of a Serre fibration
QUESTION [6 upvotes]: If $p:E\to B$ is a Serre fibration (assume it is surjective), then for each 
$b\in B$  we get a comparison map $p^{-1}(b) \to F_b$, where $F_b$ is the homotopy
fiber of $p$ over $b$.  
It is easy to see that these maps induce isomorphisms on $\pi_n$ for $n\geq 1$, but I
wonder about $\pi_0$.  
Question:  Is it true that $p^{-1}(b) \to F_b$ is a weak homotopy 
equivalence?

REPLY [2 votes]: Dear Jeff,
The map you describe is a homotopy equivalence. This is proved as Proposition 1.1 in the paper
Varadarajan, K. On fibrations and category.  Math. Z.  88  1965 267–273.