TITLE: What does actually being a CW-complex provide in algebraic topology?
QUESTION [50 upvotes]: From time to time, I pretend to be an algebraic topologist.  But I'm not really hard-core and some of the deeper mysteries of the subject are still ... mysterious.  One that came up recently is the exact role of CW-complexes.  I'm very happy with the mantra "CW-complexes Good, really horrible pathological spaces Bad." but there's a range in the middle there where I'm not sure if the classification is "Good" or just "Pretty Good".  These are the spaces with the homotopy type of a CW-complex.
In the algebraic topology that I tend to do then I treat CW-complexes in the same way that I treat Riemannian metrics when doing differential topology.  I know that there's always a CW-complex close to hand if I really need it, but what I'm actually interested in doesn't seem to depend on the space actually being a CW-complex.  But, as I said, I'm only a part-time algebraic topologist and so there may be whole swathes of this subject that I'm completely unaware of where actually having a CW-complex is of extreme importance.
Thus, my question:
In algebraic topology, if I have a space that actually is a CW-complex, what can I do with it that I couldn't do with a space that merely had the homotopy type of a CW-complex?

REPLY [5 votes]: There is a fundamental reason why CW-complexes $X$  say  are useful, namely for the purpose of constructing continuous functions $f: X \to Y$ by induction on the skeletal filtration $X^n, n \geqslant 0$,  and for going from $X^{n-1} $ to $X^{n}$ knowing that you have to construct the extension only on each $n$-cell $e^n$. The topology on $X$ is arranged precisely for this purpose. So a complicated space is put together from simple ones. 
It is this possibility of constructing continuous functions, and also homotopies, which allows for the proof of the Theorem of Whitehead mentioned above. 
It interesting to trace the development of this concept from his earlier papers for example
Whitehead, J H.C. On incidence matrices, nuclei and homotopy types. Ann. of Math. (2) 42 (1941) 1197--1239, 
which introduced the term "membrane complexes, which are so to speak more elastic than simplicial complexes". The idea was to avoid the subdivision into simplices by amalgamating them.  This paper, and others, were rewritten after the war, in terms of the CW-complexes we know and love. 
A crucial point was to be able to construct not only continuous functions but also continuous homotopies of these. For these a result on the product of CW-complexes was needed,  which I was told took him a year to prove. He also early on formulated basic results on adjunction spaces, which are now clear in terms of pushouts.  
I feel the emphasis on filtrations is crucial, and allows for other methods in algebraic topology. 
22 Nov 2013: Grothendieck in his "Esquisse d'un programme" Section 5 (English version available here, see p.258) discussed what he feels is the inadequate nature of topological spaces for certain geometric considerations. 
When foundations are under consideration, one should take a "no holds barred" attitude. This agrees with the advice of Einstein: 
"What is essential and what is based only on the accidents of development?... Concepts which have proved useful for ordering things easily assume so great an authority over us, that we forget their terrestrial origin and accept them as unalterable facts. They then become labelled as "conceptual necessities", "a priori situations", etc. The road of scientific progress is frequently blocked for long periods by such errors. It is therefore not just an idle game to exercise our ability to analyse familiar concepts, and to demonstrate the conditions on which their justification and usefulness depend, and the way in which these developed, little by little..."