While reading "Morse Theory" by Milnor, I noticed that certain arguments would not work, if the considered manifolds have nonempty boundary. Example: Proof of 3.5
I could not find the definition inside the book:
What is Milnors definition of a manifold in "Morse Theory" (by Milnor)? -> more specific: are they allowed to have boundary? do the have to be closed (compact and without boundary)?
Thus: Which manifolds are homotopy equivalent to CW-complexes?
My second question is related to the first one:
In 6.7 of "Morse Theory": Why does $M^a$ have to be compact for all a?
Thanks for your help.