I have a question concerning the proof of theorem 3.5 in Milnor's Morse Theory.

This theorem states that if $f$ is a differentiable function on a Manifold M with no critical points, and if each $M^a = \{x\in M | f(x)\leq a \} $ is compact then $M$ has the homotopy type of a CW-complex with one cell of dimension $\lambda$ for each critical point of index $\lambda$

Basically the proof shows via induction and appealing to previous theorems that $M^a$ has the homotopy type of a CW complex satisfying the conclusion above. But then Milnor says "If $M$ is compact this completes the proof".

I don't understand how this just follows directly? Is there some theorem involving Compact manifolds or CW complexes that I am missing? Does $M^a$ have to be homotopic to $M$ for some $a$ if $M$ is compact?

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    $\begingroup$ Presumably in your 2nd paragraph you mean to say $f$ is a Morse function, not that $f$ has no critical points? $\endgroup$ – Ryan Budney Feb 3 '11 at 5:08
  • $\begingroup$ I merged your two unregistered accounts. Please try to remember to log in using the same credentials so you can at least post comments as comments, and not as answers. $\endgroup$ – Willie Wong Feb 3 '11 at 11:48

If $M$ is compact, one of the $M^a$ is equal to $M$!

  • $\begingroup$ Basically this is because you can Just take a= max f which is achieved since $M$ is compact. Then $M^a = M$ and since $M^a$ has the correct form for all $a$,thus so does $M$? $\endgroup$ – user6562 Feb 3 '11 at 11:46
  • $\begingroup$ @anonymous: exactly. $\endgroup$ – Mariano Suárez-Álvarez Feb 3 '11 at 15:24

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