Take the tour ×  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?  share|improve this question 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). –  Dmitri Pavlov Dec 17 '11 at 1