Long exact sequence of homotopy of a Serre fibration

Statement

Suppose $p:E\to B$ is a Serre fibration with fiber $F$. Suppose $e\in E$, $b=p(e)$ and $i:F\to E$ is the inclusion of $F$ as the fiber $p^{-1}(b)$. Suppose $f\in F$. We then have the following long exact sequence of homotopy groups induced :

$\!\dots \to \pi _{n}(F,f){\stackrel {\pi _{n}(i)}{\to }}\pi _{n}(E){\stackrel {\pi _{n}(p)}{\to }}\pi _{n}(B)\to \pi _{n-1}(F,f){\stackrel {\pi _{n-1}(i)}{\to }}\dots$

The long exact sequence ends with:

$\!\dots \to \pi _{1}(B,b)\to \pi _{0}(F,f)\to \pi _{0}(E,e)\to \pi _{0}(B,b)$

where the last three arrows are considered as arrows of pointed sets and exactness is interpreted in that fashion.

Note that if $B$ is a paracompact Hasudorff space, and $p$ is a fiber bundle with fiber $F$, then $p$ is also a Serre fibration with fiber $F$.