topological group

topological group

\PMlinkescapephrase

completely regular

Definitions

A

topological group

is a group

G

endowed with a

topology

such that the

multiplication

and

inverse

operations

of

G

are

continuous

(

http://planetmath.org/Continuous

).
That is, the map

G

×

G

→

G

defined by

(

x

,

y

)

↦

x

⁢

y

is continuous,
where the topology on

G

×

G

is the

product topology

,
and the map

G

→

G

defined by

x

↦

x

-

1

is also continuous.

Many authors require the topology on

G

to be

Hausdorff

,
which is

equivalent

to requiring that the

trivial subgroup

be a

closed set

.

A topology on a group

G

that makes

G

into a topology group
is called a

group topology

for

G

.

Examples

Any group becomes a topological group if it is given the

discrete topology

.

Any group becomes a topological group if it is given the

indiscrete topology

.

The

real numbers

with the

standard topology

form a topological group.
More generally, an

ordered group

with its
is a topological group.

Lie groups

are topological groups with additional

structure

.

Profinite groups

are another important class of topological groups;
they arise, for example, in infinite Galois theory.

Subgroups, quotients and products

Every

subgroup

(

http://planetmath.org/Subgroup

) of a topological group
either has empty

interior

or is clopen.
In particular, all

proper subgroups

of a

connected

topological group
have empty interior.
The

closure

of any subgroup is also a subgroup,
and the closure of a

normal subgroup

is normal
(for proofs, see the entry
“closure of sets closed under a finitary operation (

http://planetmath.org/ClosureOfSetsClosedUnderAFinitaryOperation

)”).
A subgroup of a topological group is itself a topological group,
with the

subspace topology

.

If

G

is a topological group and

N

is a normal subgroup of

G

,
then the

quotient group

G

/

N

is also a topological group,
with the

quotient topology

.
This quotient

G

/

N

is Hausdorff if and only if

N

is a closed subset of

G

.

If

(

G

i

)

i

∈

I

is a family of topological groups,
then the

unrestricted direct product

∏

i

∈

I

G

i

is also a topological group, with the product topology.

Morphisms

Let

G

and

H

be topological groups, and let

f

:

G

→

H

be a

function

.

The function

f

is said to be a

homomorphism of topological groups

if it is a

group homomorphism

and is also continuous.
It is said to be an

isomorphism of topological groups

if it is both a group isomorphism and a

homeomorphism

.

Note that it is possible for

f

to be a

continuous group

isomorphism

(that is, a

bijective

homomorphism of topological groups)
and yet not be an isomorphism of topological groups.
This occurs, for example, if

G

is

ℝ

with the discrete topology,
and

H

is

ℝ

with its usual topology,
and

f

is the

identity map

on

ℝ

.

Topological properties

While every group can be made into a topological group,
the same cannot be said of every

topological space

.
In this

section

we mention some of the properties
that the underlying topological space must have.

Every topological group is

bihomogeneous

and completely regular (

http://planetmath.org/Tychonoff

).
Note that our earlier claim that a topological group
is Hausdorff if and only if its trivial subgroup is closed
follows from this:
if the trivial subgroup is closed,
then homogeneity ensures that all singletons are closed,
and so the space is

T

1

(

http://planetmath.org/T1Space

),
and being completely regular is therefore Hausdorff.
A topological group is not necessarily

http://planetmath.org/node/1530

normal, however,
a counterexample being the unrestricted direct product
of uncountably many copies of the

discrete group

ℤ

.

Every topological group is obviously an

H-space

.
Consequently, the

fundamental group

of a topological group is

abelian

.
Note that because topological groups are

homogeneous

,
the fundamental group does not depend (up to isomorphism)
on the choice of

basepoint

.

Every

locally compact topological group

is

http://planetmath.org/node/1530

normal and

strongly paracompact

.

Every connected locally compact topological group is

σ

-

compact

.

Other notes

Every topological group possesses a natural

uniformity

,
which induces the topology.
See the entry about the uniformity of a topological group (

http://planetmath.org/UniformStructureOfATopologicalGroup

).

A locally compact topological group
possesses a natural measure, called the Haar measure.

Title

topological group

Canonical name

TopologicalGroup

Date of creation

2013-03-22 15:47:09

Last modified on

2013-03-22 15:47:09

Owner

yark (2760)

Last modified by

yark (2760)

Numerical id

29

Author

yark (2760)

Entry type

Definition

Classification

msc 22A05

Related topic

Group

Related topic

TopologicalSpace

Defines

homomorphism of topological groups

Defines

isomorphism of topological groups

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