product topology

product topology

Definition

Let

(

(

X

α

,

𝒯

α

)

)

α

∈

A

be a family of

topological spaces

, and let

Y

be the

Cartesian product

(

http://planetmath.org/GeneralizedCartesianProduct

)
of the sets

X

α

, that is

Y

=

∏

α

∈

A

X

α

.

Recall that an

element

y

∈

Y

is a

function

y

:

A

→

⋃

α

∈

A

X

α

such that

y

⁢

(

α

)

∈

X

α

for each

α

∈

A

,
and that for each

α

∈

A

the

projection map

π

α

:

Y

→

X

α

is defined by

π

α

⁢

(

y

)

=

y

⁢

(

α

)

for each

y

∈

Y

.

The (

Tychonoff

)

product topology

𝒯

for

Y

is defined to be the

initial topology

with respect to the projection maps;
that is,

𝒯

is the smallest

topology

such that each

π

α

is

continuous

(

http://planetmath.org/Continuous

).

Subbase

If

U

⊆

X

α

is open,
then

π

α

-

1

⁢

(

U

)

is an

open set

in

Y

.
Note that this is the set of all elements of

Y

in which the

α

component is restricted to

U

and all other components are unrestricted.
The open sets of

Y

are the unions of finite

intersections

of such sets.
That is,

{

π

α

-

1

⁢

(

U

)

∣

α

∈

A

⁢

and

⁢

U

∈

𝒯

α

}

is a

subbase

for the topology on

Y

.

Theorems

The following

theorems

assume the product topology on

∏

α

∈

A

X

α

.
Notation is as above.

Theorem 1

Let

Z

be a topological space
and let

f

:

Z

→

∏

α

∈

A

X

α

be a function.
Then

f

is continuous if and only if

π

α

∘

f

is continuous
for each

α

∈

A

.

Theorem 2

The product topology on

∏

α

∈

A

X

α

is the topology induced by the subbase

{

π

α

-

1

⁢

(

U

)

∣

α

∈

A

⁢

and

⁢

U

∈

𝒯

α

}

.

Theorem 3

The product topology on

∏

α

∈

A

X

α

is the topology induced by the base

{

⋂

α

∈

F

π

α

-

1

⁢

(

U

α

)

|

F

⁢

is a finite subset of

⁢

A

⁢

and

⁢

U

α

∈

𝒯

α

⁢

for each

⁢

α

∈

F

}

.

Theorem 4

A net

(

x

i

)

i

∈

I

in

∏

α

∈

A

X

α

converges

to

x

if and only if each coordinate

(

x

i

α

)

i

∈

I

converges to

x

α

in

X

α

.

Theorem 5

Each projection map

π

α

:

∏

α

∈

A

X

α

→

X

α

is continuous and open (

http://planetmath.org/OpenMapping

).

Theorem 6

For each

α

∈

A

, let

A

α

⊆

X

α

.
Then

∏

α

∈

A

A

α

¯

=

∏

α

∈

A

A

α

¯

.

In particular, any

product

of

closed sets

is closed.

Theorem 7

(Tychonoff’s Theorem)
If each

X

α

is

compact

, then

∏

α

∈

A

X

α

is compact.

Comparison with box topology

There is another well-known way to topologize

Y

, namely the

box topology

.
The product topology is a subset of the box topology;
if

A

is finite, then the two topologies are the same.

The product topology is generally more useful than the box topology.
The main reason for this can be expressed in terms of

category theory

:
the product topology is the topology of the
direct categorical product (

http://planetmath.org/CategoricalDirectProduct

)
in the

category

Top

(see Theorem 1 above).

References

1

J. L. Kelley,

General Topology

,
D. van Nostrand Company, Inc., 1955.

2

J. Munkres,

Topology

(2nd edition),
Prentice Hall, 1999.

Title

product topology

Canonical name

ProductTopology

Date of creation

2013-03-22 12:47:09

Last modified on

2013-03-22 12:47:09

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

38

Author

CWoo (3771)

Entry type

Definition

Classification

msc 54B10

Synonym

Tychonoff product topology

Related topic

BoxTopology

Related topic

GeneralizedCartesianProduct

Related topic

ASpaceMathnormalXIsHausdorffIfAndOnlyIfDeltaXIsClosed

Related topic

InitialTopology

Defines

product

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