aarjaneiro/planetmath huggingface dataset:
topological space

topological space

A

topological space

is a set

X

together with a set

𝒯

whose

elements

are subsets of

X

, such that

•

∅

∈

𝒯

•

X

∈

𝒯

•

If

U

j

∈

𝒯

for all

j

∈

J

, then

⋃

j

∈

J

U

j

∈

𝒯

•

If

U

∈

𝒯

and

V

∈

𝒯

, then

U

∩

V

∈

𝒯

Elements of

𝒯

are called

open sets

of

X

. The set

𝒯

is called a

topology

on

X

. A subset

C

⊂

X

is called a

closed set

if the

complement

X

∖

C

is an open set.

A topology

𝒯

′

is said to be

finer

(respectively,

coarser

) than

𝒯

if

𝒯

′

⊃

𝒯

(respectively,

𝒯

′

⊂

𝒯

).

Examples

•

The

discrete topology

is the topology

𝒯

=

𝒫

⁢

(

X

)

on

X

, where

𝒫

⁢

(

X

)

denotes the

power set

of

X

. This is the largest, or finest, possible topology on

X

.

•

The

indiscrete topology

is the topology

𝒯

=

{

∅

,

X

}

. It is the smallest or coarsest possible topology on

X

.

•

Subspace topology

•

Product topology

•

Metric topology

References

1

J.L. Kelley,

General Topology

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

2

J. Munkres,

Topology

(2nd edition), Prentice Hall, 1999.

Title

topological space

Canonical

name

TopologicalSpace

Date of creation

2013-03-22 11:49:52

Last modified on

2013-03-22 11:49:52

Owner

djao (24)

Last modified by

djao (24)

Numerical id

12

Author

djao (24)

Entry type

Definition

Classification

msc 22-00

Classification

msc 55-00

Classification

msc 54-00

Synonym

topology

Related topic

Neighborhood

Related topic

MetricSpace

Related topic

ExamplesOfCompactSpaces

Related topic

ExamplesOfLocallyCompactAndNotLocallyCompactSpaces

Related topic

Site

Defines

open

Defines

closed

Generated on Fri Feb 9 19:38:49 2018 by

LaTeXML
basic results in topological groups

basic results in topological groups

The purpose of this entry is to list some and useful results concerning the topological of

topological groups

. We will use the following notation whenever

A

,

B

are subsets of a topological group

G

and

r

an element of

G

:

•

A

⁢

r

:=

{

a

⁢

r

:

a

∈

A

}

•

r

⁢

A

:=

{

r

⁢

a

:

a

∈

A

}

•

A

⁢

B

:=

{

a

⁢

b

:

a

∈

A

,

b

∈

B

}

•

A

2

:=

{

a

1

⁢

a

2

:

a

1

,

a

2

∈

A

}

•

A

-

1

:=

{

a

-

1

:

a

∈

A

}

•

A

¯

denotes the

closure

of

A

1 -

Let

G

be a topological group and

r

∈

G

. The left multiplication

s

↦

r

⁢

s

, multiplication

s

↦

s

⁢

r

, and

inversion

s

↦

s

-

1

, are

homeomorphisms

of

G

.

2 -

Let

G

be a topological group and

e

∈

G

the

identity element

. Let

ℬ

be a

neighborhood base

around

e

. Then

{

B

⁢

r

}

B

∈

ℬ

is a neighborhood base around

r

∈

G

and

{

B

⁢

r

:

B

∈

ℬ

⁢

and

⁢

r

∈

G

}

is a basis (

http://planetmath.org/BasisTopologicalSpace

) for the

topology

of

G

.

3 -

Let

G

be a topological group. If

U

⊆

G

is open and

V

is any subset of

G

, then

U

⁢

V

is an

open set

in

G

.

4 -

Let

G

be a topological group and

K

,

L

compact sets

in

G

. Then

K

⁢

L

is also

compact

.

5 -

Let

G

be a topological group and

e

∈

G

the identity element. If

V

is a

neighborhood

of

e

then

V

⊂

V

¯

⊂

V

2

.

6 -

Let

G

be a topological group,

e

∈

G

the identity element and

W

a neighborhood around

e

. Then there exists a neighborhood

U

around

e

such that

U

2

⊂

W

.

7 -

Let

G

be a topological group,

e

∈

G

the identity element and

W

a neighborhood around

e

. Then there exists a symmetric (

http://planetmath.org/SymmetricSet

) neighborhood

U

around

e

such that

U

2

⊆

W

.

8 -

Let

G

be a topological group. If

H

is a

subgroup

of

G

, then so is

H

¯

.

9-

Let

G

be a topological group. If

H

is an

open subgroup

of

G

, then

H

is also closed.

Title

basic results in topological groups

Canonical name

BasicResultsInTopologicalGroups

Date of creation

2013-03-22 17:37:38

Last modified on

2013-03-22 17:37:38

Owner

asteroid (17536)

Last modified by

asteroid (17536)

Numerical id

16

Author

asteroid (17536)

Entry type

Result

Classification

msc 22A05

Related topic

PolishGSpace

Related topic

PolishGroup

Generated on Fri Feb 9 19:38:52 2018 by

LaTeXML
center of a Hausdorff topological group is closed

center of a Hausdorff topological group is closed

Theorem -

Let

G

be a Hausdorff

topological group

. Then the center of

G

is a closed

normal subgroup

.

Proof:

Let

Z

be the center of

G

. We know that

Z

is a normal subgroup of

G

. Let us see that it is closed.

Let

s

∈

Z

¯

, the closure of

Z

. There exists a net

{

s

λ

}

in

Z

converging to

s

. Then, for every

g

∈

G

, we have that

•

g

⁢

s

λ

⟶

g

⁢

s

•

s

λ

⁢

g

⟶

s

⁢

g

But since

Z

is the center of

G

we have that

g

⁢

s

λ

=

s

λ

⁢

g

, and as

G

is Hausdorff one must have

s

⁢

g

=

g

⁢

s

. This implies that

s

∈

Z

, i.e.

Z

is closed.

□

Title

center of a Hausdorff topological group is closed

Canonical name

CenterOfAHausdorffTopologicalGroupIsClosed

Date of creation

2013-03-22 18:01:48

Last modified on

2013-03-22 18:01:48

Owner

asteroid (17536)

Last modified by

asteroid (17536)

Numerical id

4

Author

asteroid (17536)

Entry type

Theorem

Classification

msc 22A05

Generated on Fri Feb 9 19:38:53 2018 by

LaTeXML
component of identity of a topological group is a closed normal subgroup

component of identity of a topological group is a closed normal subgroup

Theorem

-

Let

G

be a

topological group

and

e

its

identity element

. The

connected component

of

e

is a closed

normal subgroup

of

G

.

Proof:

Let

F

be the connected component of

e

. All

components

of a

topological space

are closed, so

F

is closed.

Let

a

∈

F

. Since the

multiplication

and

inversion

functions in

G

are

continuous

, the set

a

⁢

F

-

1

is also

connected

, and since

e

∈

a

⁢

F

-

1

we must have

a

⁢

F

-

1

⊆

F

. Hence, for every

b

∈

F

we have

a

⁢

b

-

1

∈

F

, i.e.

F

is a

subgroup

of

G

.

If

g

is an arbitrary element of

G

, then

g

-

1

⁢

F

⁢

g

is a connected subset containing

e

. Hence

g

-

1

⁢

F

⁢

g

⊂

F

for every

g

∈

G

, i.e.

F

is a normal subgroup.

□

Title

component of

identity

of a topological group is a closed normal subgroup

Canonical

name

ComponentOfIdentityOfATopologicalGroupIsAClosedNormalSubgroup

Date of creation

2013-03-22 18:01:42

Last modified on

2013-03-22 18:01:42

Owner

asteroid (17536)

Last modified by

asteroid (17536)

Numerical id

6

Author

asteroid (17536)

Entry type

Theorem

Classification

msc 22A05

Generated on Fri Feb 9 19:38:58 2018 by

LaTeXML
connected locally compact topological groups are σ-compact

connected locally compact topological groups are

σ

-compact

The main result of this entry is the following theorem (whose proof is given below). The result expressed in the title then follows as a corollary.

Theorem -

Every

locally compact topological group

G

has an open

σ

-

compact

(

http://planetmath.org/SigmaCompact

) subgroup

H

.

Corollary 1 -

Every locally compact topological group is the

topological disjoint union

of

σ

-

compact spaces

.

Corollary 2 -

Every

connected

locally compact topological group is

σ

-compact.

We first outline the proofs of the above corollaries:

Proof (Corollaries 1 and 2) :

Let

G

be a locally compact topological group. The main theorem implies that there is an open

σ

-compact subgroup

H

.

It is known that every

open subgroup

of

G

is also closed (see this entry (

http://planetmath.org/ClosednessOfSubgroupsOfTopologicalGroups

)). Therefore, each

g

⁢

H

is a clopen

σ

-compact subset of

G

, and

G

is the topological disjoint union

⋃

g

∈

G

g

⁢

H

.

Of course, if

G

is connected then

H

must be all of

G

. Hence,

G

is

σ

-compact.

□

Proof (Theorem) :

Let us fix some notation first. If

A

is a subset of

G

we use the notation

A

-

1

:=

{

a

-

1

:

a

∈

A

}

,

A

n

:=

{

a

1

⁢

…

⁢

a

n

:

a

1

,

…

,

a

n

∈

A

}

and

A

¯

denotes the

closure

of

A

.

Pick a

neighborhood

W

of

e

(the identity element of

G

) with compact closure. Then

V

:=

W

∩

W

-

1

is a neighborhood of

e

with compact closure such that

V

=

V

-

1

.

Let

H

:=

⋃

n

=

1

∞

V

n

.

H

is clearly a subgroup of

G

. We now only have to prove that

H

is open and

σ

-compact.

We have that (see this entry (

http://planetmath.org/BasicResultsInTopologicalGroups

) - 3, 4 and 5)

•

V

n

is open

•

V

¯

n

is compact

•

V

¯

n

⊂

V

2

⁢

n

So

H

is open and also

H

=

⋃

n

=

1

∞

V

¯

n

, which implies that

H

is

σ

-compact.

□

Title

connected locally compact topological groups are

σ

-compact

Canonical name

ConnectedLocallyCompactTopologicalGroupsAresigmacompact

Date of creation

2013-03-22 17:37:12

Last modified on

2013-03-22 17:37:12

Owner

asteroid (17536)

Last modified by

asteroid (17536)

Numerical id

9

Author

asteroid (17536)

Entry type

Theorem

Classification

msc 22A05

Classification

msc 22D05

Generated on Fri Feb 9 19:39:00 2018 by

LaTeXML
connected topological group is generated by any neighborhood of identity

connected topological group is generated by any neighborhood of identity

Theorem -

Let

G

be a

connected

topological group

and

e

its

identity element

. If

U

is any open neighborhood of

e

, then

G

is generated by

U

.

Proof:

Let

U

be an open neighborhood of

e

. For each

n

∈

ℕ

we denote by

U

n

the set of elements of the form

u

1

⁢

…

⁢

u

n

, where each

u

i

∈

U

. Let

W

:=

⋃

n

∈

ℕ

U

n

.

Since each

U

n

is open (see this entry (

http://planetmath.org/BasicResultsInTopologicalGroups

) - 3), we have that

W

is an open set. We now see that it is also closed.

Let

g

∈

W

¯

, the closure of

W

. Since

g

⁢

U

-

1

is an open neighborhood of

g

, it must intersect

W

. Thus, let

h

∈

W

∩

g

⁢

U

-

1

.

•

Since

h

∈

g

⁢

U

-

1

, then

h

=

g

⁢

u

-

1

for some element

u

∈

U

.

•

Since

h

∈

W

, then

h

∈

U

n

for some

n

∈

ℕ

, i.e.

h

=

u

1

⁢

…

⁢

u

n

with each

u

i

∈

U

.

We then have

g

=

u

1

⁢

…

⁢

u

n

⁢

u

, i.e.

g

∈

U

n

+

1

⊆

W

. Hence,

W

is closed.

Since

G

is connected and

W

is open and closed, we must have

W

=

G

. This means that

G

is generated by

U

.

□

Title

connected topological group is generated by any neighborhood of identity

Canonical name

ConnectedTopologicalGroupIsGeneratedByAnyNeighborhoodOfIdentity

Date of creation

2013-03-22 18:01:45

Last modified on

2013-03-22 18:01:45

Owner

asteroid (17536)

Last modified by

asteroid (17536)

Numerical id

7

Author

asteroid (17536)

Entry type

Theorem

Classification

msc 22A05

Generated on Fri Feb 9 19:39:03 2018 by

LaTeXML
example of a proof using nets

example of a proof using nets

In this entry we will give a simple example
of how nets can be used to prove topological theorems.
The proof will make use of some of the basic properties of nets
listed in the parent entry (

http://planetmath.org/Net

).

Theorem

.

The centre (

http://planetmath.org/GroupCentre

) of a Hausdorff

topological group

is closed.

Proof.

Let

Z

be the centre of a Hausdorff topological group

G

.
Let

x

∈

Z

¯

.
Then there is a net

(

x

δ

)

in

Z

such that

x

δ

→

x

.
Let

g

∈

G

.
By continuity we have

g

⁢

x

δ

⁢

g

-

1

→

g

⁢

x

⁢

g

-

1

.
But

g

⁢

x

δ

⁢

g

-

1

=

x

δ

, so

g

⁢

x

δ

⁢

g

-

1

→

x

.
As

G

is Hausdorff, these two limits must be the same.
So

g

⁢

x

⁢

g

-

1

=

x

, that is,

g

⁢

x

=

x

⁢

g

.
Thus

x

∈

Z

,
and we have shown that

Z

¯

=

Z

, as required.

Title

example of a proof using nets

Canonical name

ExampleOfAProofUsingNets

Date of creation

2013-03-22 17:11:25

Last modified on

2013-03-22 17:11:25

Owner

yark (2760)

Last modified by

yark (2760)

Numerical id

4

Author

yark (2760)

Entry type

Example

Classification

msc 22A05

Generated on Fri Feb 9 19:39:06 2018 by

LaTeXML
example of an Alexandroff space which cannot be turned into a topological group

example of an Alexandroff space which cannot be turned into a topological group

Let

ℝ

denote the set of real numbers and

τ

=

{

[

a

,

∞

)

|

a

∈

ℝ

}

∪

{

(

b

,

∞

)

|

b

∈

ℝ

}

. One can easily verify that

(

ℝ

,

τ

)

is an Alexandroff space.

Proposition

.

The Alexandroff space

(

ℝ

,

τ

)

cannot be turned into a

topological group

.

Proof.

Assume that

ℝ

=

(

ℝ

,

τ

,

∘

)

is a topological group. It is well known that this implies that there is

H

⊆

ℝ

which is open,

normal subgroup

of

ℝ

. This

subgroup

,,generates” the

topology

(see the parent object for more details). Thus

H

≠

ℝ

because

τ

is not antidiscrete. Let

g

∈

ℝ

such that

g

∉

H

(and thus

g

⁢

H

∩

H

=

∅

). Then

g

⁢

H

is again open (because the

mapping

f

⁢

(

x

)

=

g

∘

x

is a homeomorphism). But since both

H

and

g

⁢

H

are open, then

g

⁢

H

∩

H

≠

∅

. Indeed, every two open subsets in

τ

have nonempty

intersection

.

Contradiction

, because diffrent cosets are

disjoint

.

□

Title

example of an Alexandroff space which cannot be turned into a topological group

Canonical

name

ExampleOfAnAlexandroffSpaceWhichCannotBeTurnedIntoATopologicalGroup

Date of creation

2013-03-22 18:45:46

Last modified on

2013-03-22 18:45:46

Owner

joking (16130)

Last modified by

joking (16130)

Numerical id

4

Author

joking (16130)

Entry type

Example

Classification

msc 22A05

Generated on Fri Feb 9 19:39:08 2018 by

LaTeXML
full subgroup

full subgroup

A subgroup of a

topological group

is

full

if it has a nonempty interior.

Theorem. A

full subgroup

is both open and closed.

References

1

Paul R. Halmos,

Measure Theory

, Springer-Verlag (1974).

Title

full subgroup

Canonical name

FullSubgroup

Date of creation

2013-03-22 17:04:29

Last modified on

2013-03-22 17:04:29

Owner

Mathprof (13753)

Last modified by

Mathprof (13753)

Numerical id

6

Author

Mathprof (13753)

Entry type

Definition

Classification

msc 22A05

Synonym

open subgroup

Synonym

clopen subgroup

Generated on Fri Feb 9 19:39:10 2018 by

LaTeXML
locally closed subgroups of topological groups are closed

locally closed subgroups of topological groups are closed

Let

G

be a

Hausdorff

topological group

and

H

⊆

G

a subgroup (which is a topological group itself under the

subspace topology

).

Theorem -

If

H

is

locally closed

in

G

then

H

is closed.

In particular we see that if

H

is either

•

open, or

•

discrete

(

http://planetmath.org/Discrete

), or

•

locally compact

,

then

H

is closed.

Title

locally closed subgroups of topological groups are closed

Canonical name

LocallyClosedSubgroupsOfTopologicalGroupsAreClosed

Date of creation

2013-03-22 17:36:39

Last modified on

2013-03-22 17:36:39

Owner

asteroid (17536)

Last modified by

asteroid (17536)

Numerical id

11

Author

asteroid (17536)

Entry type

Theorem

Classification

msc 22A05

Generated on Fri Feb 9 19:39:11 2018 by

LaTeXML
subgroup of topological group is either clopen or has empty interior

subgroup of topological group is either clopen or has empty interior

Theorem -

Every subgroup of a

topological group

is either clopen or has empty interior.

Proof:

Let

G

be a topological group and

H

⊆

G

a subgroup. Suppose the interior of

H

is nonempty, i.e. there is a non-empty open set

U

of

G

such that

U

⊆

H

. Translating

U

around

H

we can see that

H

is open: if

u

∈

U

then for every

h

∈

H

the set

h

⁢

u

-

1

⁢

U

is open in

G

, is contained in

H

and contains

h

, which implies that

H

is open in

G

.

Let us now see that

H

is closed. Let

H

¯

denote the closure of

H

and let

H

2

be the set of elements of the form

h

1

⁢

h

2

where

h

1

,

h

2

∈

H

. Of course, since

H

is a subgroup of

G

, we have that

H

2

=

H

. Also, since

H

is open we know that

H

⊆

H

¯

⊆

H

2

(see this entry (

http://planetmath.org/BasicResultsInTopologicalGroups

) - 5). Hence

H

¯

=

H

, i.e.

H

is closed.

We have proven that a subgroup of a topological group must be clopen or it must have empty interior. Since this two topological properties can never be satisfied simultaneously, we have that every subgroup of a topological group is either clopen or it has empty interior.

□

Title

subgroup of topological group is either clopen or has empty interior

Canonical name

SubgroupOfTopologicalGroupIsEitherClopenOrHasEmptyInterior

Date of creation

2013-03-22 18:01:29

Last modified on

2013-03-22 18:01:29

Owner

asteroid (17536)

Last modified by

asteroid (17536)

Numerical id

6

Author

asteroid (17536)

Entry type

Theorem

Classification

msc 22A05

Generated on Fri Feb 9 19:39:15 2018 by

LaTeXML
topological group (obsolete)

topological group (obsolete)

This entry is obsolete, having been superseded by a new entry (

http://planetmath.org/TopologicalGroup2

). It is being retained for a short while because of the attached thread.

A

topological group

is a triple

(

G

,

⋅

,

𝒯

)

where

(

G

,

⋅

)

is a group and

𝒯

is a

topology

on

G

such that under

𝒯

, the

group operation

(

x

,

y

)

↦

x

⋅

y

is

continuous

with respect to the

product topology

on

G

×

G

and the inverse map

x

↦

x

-

1

is continuous on

G

.

Many authors require that the topology be

Hausdorff

.

Title

topological group (obsolete)

Canonical name

TopologicalGroupobsolete

Date of creation

2013-03-22 12:12:54

Last modified on

2013-03-22 12:12:54

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

10

Author

rspuzio (6075)

Entry type

Definition

Classification

msc 22A05

Related topic

Group

Related topic

TopologicalRing

Related topic

BirkhoffKakutaniTheorem

Related topic

CategoryOfPolishGroups

Related topic

AlgebraicTopology

Generated on Fri Feb 9 19:39:19 2018 by

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Borel G-space

Borel G-space

A (standard)

Borel G-space

is defined in

connection

with a standard Borel space which shall be
specified first.

0.1

Basic definitions

•

a. Standard Borel space

Definition 0.1

.

A

standard Borel space

is defined as a

measurable space

, that is, a set

X

equipped with a

σ

-

algebra

𝒮

, such that there exists a Polish

topology

on

X

with

S

its

σ

-algebra of

Borel sets

.

•

b. Borel G-space

Definition 0.2

.

Let

G

be a

Polish group

and

X

a (standard)

Borel space

. An action

a

of

G

on

X

is
defined to be a

Borel action

if

a

:

G

×

X

→

X

is a Borel-measurable map or a
Borel function (

http://planetmath.org/BorelGroupoid

).
In this case, a standard Borel space

X

that is acted upon by a Polish group with a Borel action
is called a

(standard) Borel G-space

.

•

c.

Borel morphisms

Definition 0.3

.

Homomorphisms

,

embeddings

or

isomorphisms

between standard Borel G-spaces
are called

Borel

if they are Borel–measurable.

Remark 0.1

.

Borel G-spaces have the nice

property

that the

product

and sum of a

countable

sequence

of Borel G-spaces

(

X

n

)

n

∈

N

are also Borel G-spaces. Furthermore, the

subspace

of a Borel G-space determined by an

invariant

Borel set is also a Borel G-space.

Title

Borel G-space

Canonical

name

BorelGspace

Date of creation

2013-03-22 18:24:45

Last modified on

2013-03-22 18:24:45

Owner

bci1 (20947)

Last modified by

bci1 (20947)

Numerical id

14

Author

bci1 (20947)

Entry type

Definition

Classification

msc 22A15

Classification

msc 22A25

Classification

msc 22A22

Classification

msc 54H05

Classification

msc 22A05

Classification

msc 22A10

Related topic

BorelSpace

Related topic

BorelMeasure

Related topic

BorelGroupoid

Related topic

CategoryOfBorelSpaces

Defines

Borel action

Defines

Borel-measurable map

Defines

standard Borel space

Generated on Fri Feb 9 19:39:21 2018 by

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topological G-space

topological G-space

0.1

Essential data

Let us recall the

definition

of a

topological group

; this is a group

(

G

,

.

,

e

)

together
with a

topology

on

G

such that

(

x

,

y

)

↦

x

⁢

y

-

1

is

continuous

, i.e., from

G

×

G

into

G

.
Note also that

G

×

G

is regarded as a

topological space

defined by the

product topology

.

Definition 0.1

.

Consider

G

to be a topological group with the above notations, and also let

X

be a topological space, such that an action

a

of

G

on

X

is
continuous if

a

:

G

×

X

→

X

is continuous; with these conditions,

X

is defined to be
a

topological G-space

.

References

1

Howard Becker, Alexander S. Kechris. 1996.

The

Descriptive Set Theory

of

Polish Group

Actions

Cambridge University Press: Cambridge, UK, p.14.

Title

topological G-space

Canonical

name

TopologicalGspace

Date of creation

2013-03-22 18:24:32

Last modified on

2013-03-22 18:24:32

Owner

bci1 (20947)

Last modified by

bci1 (20947)

Numerical id

9

Author

bci1 (20947)

Entry type

Definition

Classification

msc 22A15

Classification

msc 22A25

Classification

msc 22A22

Classification

msc 22A10

Classification

msc 54H05

Classification

msc 22A05

Synonym

G-space

Related topic

PolishSpace

Related topic

PolishGSpace

Related topic

PolishGroup

Generated on Fri Feb 9 19:39:22 2018 by

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Birkhoff-Kakutani theorem

Birkhoff-Kakutani theorem

0.1

Birkhoff-Kakutani theorem

Theorem 0.1

.

A

topological group

(

G

,

*

,

e

)

is

metrizable

if and only if

G

is

Hausdorff

and the

identity

e

of

G

has a

countable

neighborhood

basis. Here

*

is the group

composition

law or

operation

. Furthermore, if G is

metrizable

, then

G

admits a

compatible

metric

d

which is

left-invariant

, that is,

d

⁢

(

g

⁢

x

,

g

⁢

y

)

=

d

⁢

(

x

,

y

)

;

a right-invariant metric

r

also exists under these conditions.

References

1

Howard Becker, Alexander S. Kechris. 1996.

The

Descriptive Set Theory

of

Polish Group

Actions. (

London Mathematical Society

Lecture Note Series)

, Cambridge University Press: Cambridge, UK, p.14.

Title

Birkhoff-Kakutani theorem

Canonical name

BirkhoffKakutaniTheorem

Date of creation

2013-03-22 18:24:34

Last modified on

2013-03-22 18:24:34

Owner

bci1 (20947)

Last modified by

bci1 (20947)

Numerical id

17

Author

bci1 (20947)

Entry type

Theorem

Classification

msc 22A22

Classification

msc 22A10

Classification

msc 22A05

Related topic

TopologicalGroup2

Related topic

T2Space

Related topic

HomotopyDoubleGroupoidOfAHausdorffSpace

Generated on Fri Feb 9 19:39:24 2018 by

LaTeXML
category of representations

category of representations

Definition 0.1

.

The

category

R

⁢

e

⁢

p

⁢

(

G

)

of

representations

has objects the representations of a

groupoid

𝖦

, and as

morphisms

the

intertwiners

i

:

ρ

j

⟶

ρ

k

that are (vector)

bundle morphisms

i

:

E

⟶

E

over the manifold

M

so that

ρ

k

⁢

(

g

)

∘

i

=

i

∘

ρ

j

. Because representations are

functors

ρ

:

𝖦

⟶

𝐕𝐞𝐜𝐭

, an itertwiner

i

is in fact a

natural transformation

between two such functors that are

groupoid representations

of

𝖦

, in this case implemented

via

the

vector bundle

morphisms

i

:

E

⟶

E

.

Title

category of representations

Canonical name

CategoryOfRepresentations

Date of creation

2013-03-22 19:19:26

Last modified on

2013-03-22 19:19:26

Owner

bci1 (20947)

Last modified by

bci1 (20947)

Numerical id

8

Author

bci1 (20947)

Entry type

Definition

Classification

msc 22A22

Classification

msc 20L05

Related topic

GroupoidRepresentation4

Related topic

RepresentationsOfLocallyCompactGroupoids

Related topic

Functor

Defines

morphisms of representations

Defines

representation morphism

Generated on Fri Feb 9 19:39:28 2018 by

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