TITLE: Milnor's proof that a smooth manifold has the homotopy type of a CW complex
QUESTION [5 upvotes]: I have some questions about the proof of Theorem 3.5 of Milnor’s “Morse Theory”:

At the end of the proof of this theorem, Milnor addresses the case when $f$ has infinitely many critical points:

Questions:

Is the limit map $g$ the colimit of the graphs of the maps $g_i : M^{a_i} \to K_i$?, i.e. $\textrm{Graph}(g_i) = \{(g_i(m), m) : m \in M^{a_i}\}$ (as each map extends the previous one, the colimit of the graphs (i.e. a colimit of sets) makes sense.)
Why does $g$ induce isomorphisms of homotopy groups in all dimensions?
Why does $M$ being a retract of its tubular neighbourhood in a euclidean space mean it is dominated by a CW complex? (Hatcher, Proposition A.11 p. 528. has an elementary proof that "A space dominated by a CW complex is homotopy equivalent to a CW complex.", so if it is obvious that $M$ is dominated by a CW complex, why was Morse theory even needed to prove Theorem 3.5?)

Google books has the section available to view (p23-24) https://books.google.com.au/books?id=A9QZZ3S_QxwC&lpg=PA25&pg=PA23#v=onepage&q&f=false
Thank you.

REPLY [2 votes]: Ad 3) $M$ can be embedded in $\mathbb{R}^N$ for $N \in \mathbb{N}$ by Whitney. Then, a tubular neighbourhood of $M$ in $\mathbb{R}^N$ is an open subset of $\mathbb{R}^N$. And open subsets of $\mathbb{R}^N$ are CW-complexes (choose a grid of edge length $1/n$ to fill the open set up). Therefore $M$ is dominated by a CW-complex per definitionem. Finally, as you mentioned A.11 in Hatcher and Whitehead's theorem complete the proof.