Algebraic Topology - Computer Graphics and Geometric Modeling - page 390

Note that the relative homeomorphism f need not be a homeomorphism because
it might not be one-to-one on A .

Definition. Let A be a closed subspace of X . We shall say that X is obtained from A
by adjoining k n-cells c i n , n ≥ 0 and 0 £ i < k (we allow k =•), if the following holds:

(1) Each c i n is a subspace of X and X = A » c 0 » c 1 »....
(2) If

. i n = c i n « A ,
then c i n - . i n and c j n - . j n are disjoint for i π j.
(3) X has the weak topology with respect to the sets A and c i n .
(4) For each n-cell c i n , there exists a relative homeomorphism

.

( ) Æ

nn - 1

i n i n

f i : DS , ( cc , )

that maps S n-1 onto the set . i . The map f i is called a characteristic map for
the n-cell c i n and g i = f i ΩS n-1 is called the attaching map for the n-cell c i n .

Condition (4) justifies us calling c i n an n-cell or cell or closed n-cell (one can show that

c i n is a closed subset of X ). Note however that, although c i n - . i is an open n-cell since
it is homeomorphic to R n , c i n may not be homeomorphic to the closed disk D n because
f is not required to be one-to-one on S n-1 .

Like in Section 5.3, one can think of these attaching maps as specifying a way to
glue an n-disk D n to a space along its boundary S n-1 . This continues the cut-and-paste
paradigm from the last chapter except that we are not doing any “cutting” right now.
An alternate description of X is that

n n

XA =» D » D » ....

g g

0 1

7.2.4.1. Example. The n-sphere S n can be thought of as a space obtained from a
point by attaching an n-cell using an attaching map that collapses the boundary of
the n-cell to a point. For example, consider S 1 . A natural characteristic map is

1

[ ] Æ

1

() = ( )

f : D =- 11 , S , f t cos t pp ,sin t ,

which shows that S 1 can be thought of as the point (-1,0) with a 1-cell attached.

Definition. A CW complex C is a Hausdorff space X together with a sequence of
closed subspaces X n of X , n =-1, 0, 1,..., satisfying

(1) f = X -1 Ã X 0 Ã X 1 Ã ...

•

U

n

(2) XX = .

n0 =

(3) Each X n is obtained from X n-1 by adjoining n-cells c i n via characteristic

- . i = c i n maps f n,i . The n-cells c i n are called the ( closed ) n-cells of C and c i n

-

X n-1 , the open n-cells .
(4) X has the weak topology with respect to the subspaces X n .