Definition 0.1
A
complex, denoted as

, is a special
type
of
topological
space (

) which is the
union of an expanding sequence of subspaces

, such that, inductively, the first member of this expansion sequence is

- a discrete set of points called the
vertices of

, and

is the
pushout obtained from

by attaching disks

along “attaching maps”

. Each resulting map

is called a
cell. (The subscript “

” in

, stands for the fact that this (CW) type of topological space

is called
cellular, or “made of cells”). The subspace

is called the “

-skeleton” of

.
Pushouts, expanding sequence and unions are here understood in the topological sense, with the compactly generated
topologies (
viz. p.71 in P. J. May, 1999 [
1]).
Remark 0.1
An earlier, alternative definition of CW complex is also in use that may have
advantages in certain applications where the concept of pushout might not be apparent; on the other hand
as pointed out in [
1] the
Definition 0.1 presented here has advantages in proving
results, including generalized, or extended
theorems
in
Algebraic Topology,
(as for example in [
1]).