It would also be nice if someone can explain this comment appearing on the Wikipedia page on CW-complexes:


"The homotopy category of CW complexes is, in the opinion of some experts, the best if not the only candidate for the homotopy category."



What is an example of a topological space that is not homotopy equivalent to a CW-complex?
My favorite example of a space which is not homotopy equivalent to a CW complex is the Long Line. All it's homotopy groups vanish (exercise 1) but the long line is not contractible (exercise 2). It's too long!


