CW complex

CW complex

A

Hausdorff topological space

X

is said to be a

CW complex

if it the following conditions:

1.

There exists a

filtration

by

subspaces

X

(

-

1

)

⊆

X

(

0

)

⊆

X

(

1

)

⊆

X

(

2

)

⊆

⋯

with

X

=

⋃

n

≥

-

1

X

(

n

)

.

2.

X

(

-

1

)

is empty, and, for

n

≥

0

,

X

(

n

)

is obtained
from

X

(

n

-

1

)

by attachment of a

collection

{

e

ι

n

:

ι

∈

I

n

}

of

n

-cells.

3.

(“

closure-finite

”)

Every

closed cell

is

contained

in a finite union of open cells.

4.

(“weak topology”)

X

has the weak topology with respect to the collection of all cells.
That is,

A

⊂

X

is closed in

X

if and only if the

intersection

of

A

with every closed cell

e

is closed in

e

with respect to the subspace topology.

The letters ‘C’ and ‘W’ stand for “closure-finite” and “weak topology,” respectively. In particular, this means that one shouldn’t look too closely at
the initials of J.H.C. Whitehead, who invented CW complexes.

The subspace

X

(

n

)

is called the

n

-

skeleton

of

X

.

Note that there normally are many possible choices of a filtration by skeleta for a given CW complex. A particular choice of skeleta and attaching maps for the cells is called a

CW

structure

on the space.

Intuitively,

X

is a CW complex if it can be constructed, starting from a

discrete space

, by first attaching one-cells, then two-cells, and so on. Note that the definition above does not allow one to attach

k

-cells before

h

-cells if

k

>

h

.

While some authors allow this in the definition, it seems to be common usage to restrict CW complexes to the definition given here, and to call a space constructed by cell attachment with unrestricted order of

dimensions

a

cell complex.

This is not essential for

homotopy

purposes, since any cell complex is

homotopy equivalent

to a CW complex.

CW complexes are a

generalization

of

simplicial complexes

, and have some of the same advantages. In particular, they allow inductive reasoning on the of skeleta. However, CW complexes are far more flexible than simplicial complexes. For a space

X

drawn from “everyday”

topological spaces

, it is a good bet that it is homotopy equivalent, or even

homeomorphic

, to a CW complex. This includes, for

instance

, smooth finite-dimensional manifolds, algebraic varieties, certain smooth infinite-dimensional manifolds (such as Hilbert manifolds), and

loop spaces

of CW complexes. This makes the

category

of spaces homotopy equivalent to a CW complex a very popular category for doing

homotopy theory

.

Remark 1

.

There is potential for confusion in the way words like “open” and “

interior

” are used for cell complexes. If

e

k

is a closed

k

-cell in CW complex

X

it does

not

follow that the corresponding open cell

e

∘

k

is an

open set

of

X

.

It is, however, an open set of the

k

-skeleton.
Also, while

e

∘

k

is often referred to as the “interior” of

e

k

,

it is not necessarily the case that it is the interior of

e

k

in the sense of pointset

topology

. In particular, any closed

0

-cell is its own corresponding open

0

-cell, even though it has empty interior in most cases.

Title

CW complex

Canonical name

CWComplex

Date of creation

2013-03-22 13:26:02

Last modified on

2013-03-22 13:26:02

Owner

antonio (1116)

Last modified by

antonio (1116)

Numerical id

10

Author

antonio (1116)

Entry type

Definition

Classification

msc 57-XX

Classification

msc 55-XX

Synonym

CW-complex

Related topic

SimplicialComplex

Related topic

CellAttachment

Related topic

ApproximationTheoremForAnArbitrarySpace

Related topic

SpinNetworksAndSpinFoams

Related topic

CWComplexDefinitionRelatedToSpinNetworksAndSpinFoams

Related topic

GeneralizedHurewiczFundamentalTheorem

Related topic

VariableTopology3

Related topic

QuantumAlgebraicTopologyOfCWComplexRepres

Defines

skeleton

Defines

skeleta

Defines

closure-finite

Defines

cell complex

Defines

CW structure

Defines

CW-structure

Generated on Sat Feb 10 11:47:00 2018 by

LaTeXML