A
cell complex
satisfying the following conditions: (C) for any
the complex
is finite, that is, consists of a finite number of cells. (For any subset
of a cell complex
,
is the notation for the intersection of all subcomplexes of
containing
.)
(W) If
is some subset of
and if for any cell
in
the intersection
is closed in
(and therefore in
as well), then
is a closed subset of
.
In this connection, each point
belongs to a definite set
of
,
and
.
The notation CW comes from the initial letters of the (English)
names for the above two conditions — (C)
for closure finiteness and (W) for weak topology.
A finite cell complex
satisfies both conditions (C) and (W). More generally, a cell complex
each point
of which is contained in some finite subcomplex
is a CW-complex. Let
be a subset of
such that
is closed in
for each cell
in
.
Then for any
the intersection
is closed in
.
If the point
does not belong to
,
then the open set
contains
and does not intersect
.
The set
is open and
is closed.
The class of CW-complexes (or the class of spaces of
the same homotopy type as a CW-complex) is the most
suitable class of topological spaces in relation to homotopy theory. Thus, if a subset
of a CW-complex
is closed, then a mapping
from the topological space
into a topological space
is continuous if and only if the restrictions of
to the closures of the cells of
are continuous. If
is a compact subset of a CW-complex
,
then the complex
is finite. There exists for every cell
of a CW-complex
a set
that is open in
and has
as a
deformation retract.
In practice, CW-complexes are constructed by an inductive procedure:
Each stage consists in glueing cells of given dimension to the
result of the previous stage. The cellular structure of such a
complex is directly related to its homotopy properties. Even for such
"good"
spaces as polyhedra it is helpful to consider
their representation as CW-complexes: There are usually fewer in
such a representation than in a simplicial triangulation. If
is obtained by attaching
-dimensional
cells to the space
,
then the subset
,
where
,
is a
strong deformation retract
of
.
A
relative CW-complex
is a pair
consisting of a topological space
and a closed subset
,
together with a sequence of closed subspaces
,
,
satisfying the following conditions: a) the space
is obtained from
by adjoining
-cells; b) for
,
is obtained from
by adjoining
-dimensional
cells; c)
;
d) the topology of
is compatible with the family
.
The space
is called the
-dimensional skeleton
of
relative to
.
When
,
the relative CW-complex
is a CW-complex in the previous sense and its
-dimensional
skeleton is
.
Examples.
1) The pair
of simplicial complexes
,
with
,
defines a relative CW-complex
,
where
.
2) The ball
is a CW-complex:
for
,
and
for
.
The sphere
is a subcomplex of the CW-complex
.
3) If the pair
is a relative CW-complex, then so is
,
and
(when
,
is, by definition,
).
4) If
is a relative CW-complex, then
is a CW-complex and
,
where
is the quotient space of
obtained by identifying all points of
with a single point.