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..."

intentionwas that it be completely different to that. I know lots of good properties of CW-complexes. I don't really need telling of those. That people have interpreted it that way is slightly unfortunate, but some have answered my real question which is great. $\endgroup$ – Loop Space Sep 15 '11 at 20:32