In Milnor's Morse Theory the Theorem 3.1 is given as follows:
Let f be a smooth real valued function on a manifold $M$. Let $a < b$ and suppose that the set $f^{−1}[a,b]$, consisting of all $p \in M$ with $a \le f(p) \le b$, is compact, and contains no critical points of $f$. Then $M^a$ is diffeomorphic to $M^b$. Furthermore, $M^a$ is a deformation retract of $M^b$, so that the inclusion map $M^a \to M^b$ is a homotopy equivalence.
Edit: The set $M^a$ is defined as $M^a:= f^{-1}(-\infty,a]$.
After the proof, he states that
The condition that $f^{−1}[a,b]$ is compact cannot be omitted. For example Figure 3.2 indicates a situation in which this set is not compact. The manifold $M$ does not contain the point $p$. Clearly $M^a$ is not a deformation retract of $M^b$.
Figure 3.2 is the following:
But i dont understand why $f^{−1}[a,b]$ is supposedly not compact and how is it possible that $p \not\in M$ ? What am i missing? Why does $p \not\in M$ imply that $f^{−1}[a,b]$ is not compact and why is it clear that $M^a$ is not a deformation retract of $M^b$?
Could someone elaborate?
Thanks for any help!