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(2) If C is a CW complex, then its skeletons C
n
are subcomplexes.
(3) Arbitrary unions and intersections of subcomplexes are subcomplexes.
(4) Every compact subset of a CW complex lies in a finite subcomplex.
(5) A CW complex is compact if and only if it is finite.
(6) If
X
is a locally finite CW complex, then any union of closed cells is a closed
subset of
X
.
(7) If
X
is a CW complex and
Y
is an arbitrary topological space, then a map f :
X
Æ
Y
is continuous if and only if fΩs is continuous for every closed cell s in
X
.
(8) Let
X
and
Y
be CW complexes and let
A
be a subcomplex of
X
. Any continu-
ous map f :
X
Æ
Y
that is cellular on
A
is homotopic to a cellular map g :
X
Æ
Y
rela-
tive to
A
, that is, there is a homotopy h :
X
¥
I
Æ
Y
so that h(
x
,0) = f(
x
), h(
x
,1) = g(
x
),
and h(
a
,t) = f(
a
), for all
a
Œ
A
and t Œ
I
.
(9) If the spaces
X
and
Y
are finite CW complexes, then so is
X
¥
Y
.
(10) If the subspace
A
is a subcomplex of a CW complex
X
, then the cells in
X
-
A
and one 0-cell corresponding to
A
define a cell decomposition for
X
/
A
that make it
into a CW complex.
(11) Attaching cells to a CW complex produces a CW complex.
(12) Let
X
and
Y
be CW complexes and let
A
be a subcomplex of
X
. If f :
A
Æ
Y
is a cellular map, then
Y
»
f
X
is a CW complex.
(13) A CW complex is a paracompact space and hence also a normal space.
(14) A connected CW complex is metrizable if and only if it is locally finite.
The cell decompositions of CW complexes are usually more natural decomposi-
tions of a space than triangulations and there are typically substantially fewer cells in
a cell decomposition than simplices in a triangulation, but there are a number of other
advantages to using CW complexes, making them the spaces of choice for topologists.
For example, properties (9)-(11) are easy for CW complexes and would be more com-
plicated for simplicial complexes because it would involve subdivisions.
It is possible to define a homology theory for a finite regular normal CW complex
C (actually for any CW complex, but this gets more involved) merely by copying what
was done in the case of simplicial complexes. For details see [CooF67]. In other words,
one can define the notion of an oriented cell and then the group of q-chains, C
q
(C),
of C is obtained by taking formal linear combinations of oriented q-cells in C. There
is also a natural boundary map
()
Æ
()
∂
qq
:
CC
C
C
q
-1
and well-defined group
ker ∂
∂
q
()
=
HC
q
im
q
+
1
called the qth homology group of C.
Using the fact that any continuous map f : ΩCΩÆΩC¢Ω between the underlying
spaces of two CW complexes C and C¢ is homotopic to a cellular map, there is an
induced natural homomorphism
()
Æ
()
f
:
HCHC
¢
.
*
q
q
q
