In the appendix of Allen Hatcher's book "Algebraic Topology", a CW complex is defined to be a space iteratively constructed by attaching $n$-cells onto an $(n-1)$ skeleton. There is a more general notion where a space can be built iteratively by attaching cells, however we pose no restriction on the order of attachment. For instance the endpoints of a $1$-cell could be glued onto the interior of a $2$-cell. This notion is discussed in Chris Schommer-Pries' answer to another MO question:

What does actually being a CW-complex provide in algebraic topology?

Does this kind of complex have a name?

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    $\begingroup$ Cofibrant objects in the model category of topological spaces, Serre fibrations, and weak homotopy equivalences are precisely those spaces that can be obtained by a transfinite gluing of cells and closing under retracts. Thus if you allow retracts, you can refer to such spaces as cofibrant topological spaces (in the Quillen model structure). $\endgroup$ – Dmitri Pavlov Dec 17 '11 at 13:49
  • $\begingroup$ Thank you Dmitri, I am aware of this. I think that I'm for a more friendly notion for a broader audience. $\endgroup$ – James Griffin Dec 19 '11 at 10:48

A cell complex.

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    $\begingroup$ This answer, a cell complex, not only has the merit of brevity, but also it makes sense and applies prior to any knowledge of model category. This is standard, and helpful while proving the model axioms. More substantially, it applies in many general situations where one may or may not have a model structure. $\endgroup$ – Peter May Dec 17 '11 at 15:07
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    $\begingroup$ Can you persuade me that this is standard language? For instance J.H.C. Whitehead in Combinatorial Homotopy I, describes a cell complex as a space with a CW-like decomposition but without the topological conditions. In particular he asks that the boundary of an n-cell is in the (n-1)-skeleton. $\endgroup$ – James Griffin Dec 19 '11 at 10:46
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    $\begingroup$ @James: the language is very common, if not quite standard. There probably is no standard language. $\endgroup$ – John Klein Apr 22 '13 at 12:15

Whitehead's definition of CW-complex had a long gestation, and was initially formulated by him as a "membrane complex", see his very original 1941 Annals paper on "On incidence matrices, nuclei and homotopy types", which includes the initial research on what he later called simple homotopy theory. The idea for such complexes was partially to have something coarser than a simplicial complex. Later he developed the notions of adjunction space, and also the general topology; Ioan James said it took him a year to prove his product theorem for CW-complexes.

Now we can see more clearly that the great advantage of CW-complexes is the inductive definition of the structure, since this allows, as shown by Whitehead, proofs by induction. So the inductive definition is often taken as a starting point.

The skeletal filtration of a CW-complex is also usefully seen as an example of a filtered space, i.e. a space $X$ with an increasing sequence of subspaces $X_0 \subseteq X_1 \subseteq X_2 \subseteq \cdots$. The book advertised here develops algebraic topology at the border between homotopy and homology on the basis of this structure, without singular homology or simplicial approximation, and relating it to Whitehead's key notion of free crossed module (developed 1941-1949), an example of a universal property in $2$-dimensional homotopy theory.

The notion of a space with structure seems to me to be related to Grothendieck's remarks in "Esquisse d'un programme" Section 5, where he claims the notion of topological space is derived from analysis and is inadequate for geometry. For his purposes, notions of stratifications are crucial. In any case, to specify a topological space one needs some kind of data, and it is not unreasonable to suggest that the invariants we seek should have some structure reflecting that of the data defining the space.

In particular, there are van Kampen type theorems for filtered spaces (as in the above mentioned book) and also for $n$-cubes of spaces, as in work with J.-L. Loday. The latter work is related to classical work on the homotopy theory of $n$-ads, their connectivity theorems, and the determinations of the critical (i.e. first non zero) groups (Blakers-Massey, Barratt-Whitehead). See also the paper by Ellis and Steiner referenced there.


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