aarjaneiro/planetmath huggingface dataset:
scattered space

scattered space

A

topological space

X

is said to be

scattered

if for every

closed subset

C

of

X

, the set of

isolated points

of

C

is dense in

C

. Equivalently,

X

is a scattered space if no non-empty closed subset of

X

is dense in itself: for every closed subset

C

of

X

, the

closure

of the

interior

of

C

is not

C

.

A subset of a topological space is called

scattered

if it is a scattered space with the

subspace topology

.

Every

discrete space

is scattered, since every

singleton

is open, hence isolated.

Scattered line

. Let

ℝ

be the real line equipped with the

usual topology

T

(formed by the

open intervals

). Let’s define a new

topology

S

on

ℝ

as follows: a subset

A

is open under

S

(

A

∈

S

) if

A

=

B

∪

C

, where

B

is open under

T

(

B

∈

T

) and

C

⊆

ℝ

-

ℚ

, a subset of the

irrational numbers

. We make the following observations:

1.

S

is a topology on

ℝ

which is finer than

T

2.

ℝ

is a

Hausdorff space

under

S

,

3.

a singleton in

ℝ

is clopen iff it contains an irrational number

4.

any subset of irrationals is scattered under the subspace topology of

ℝ

under

S

Proof.

1.

First note that every element of

T

is an element of

S

, so

∅

,

ℝ

∈

S

in particular. Suppose

A

1

,

A

2

∈

S

with

A

1

=

B

1

∪

C

1

and

A

2

=

B

2

∪

C

2

, where

B

i

,

C

i

are defined as in the setup above. Then

A

1

∩

A

2

=

B

∪

C

, where

B

=

B

1

∩

B

2

∈

T

and

C

=

(

C

1

∩

B

2

)

∪

(

(

B

1

∪

C

1

)

∩

C

2

)

is a subset of the irrationals. So

A

1

∩

A

2

∈

S

. If

A

i

∈

S

with

A

i

=

B

i

∪

C

i

, then

⋃

A

i

=

⋃

B

i

∪

⋃

C

i

∈

S

. So

S

is a topology which is finer than

T

2.

ℝ

is

Hausdorff

under

S

is clear, the topological property is inherited from

T

.

3.

First, any singleton is closed since

X

is Hausdorff under

S

. If

x

is irrational, then

{

x

}

is open (under

S

) as well. So

{

x

}

is clopen. If

x

is

rational

and

{

x

}

∈

S

, then it is the union of a

T

-

open set

B

and a subset

C

of the irrationals. The only

T

-open subset of

{

x

}

is the

empty set

, so

{

x

}

is a subset of the irrationals, a

contradiction

.

4.

Let

C

is a subset of the irrational numbers. and considered the subspace topology under

S

. Then every point

r

of

C

is isolated, since

{

r

}

is the open subset of

C

separating

it from the rest. The closure of the

collection

of these points is clearly

C

itself, so

C

is scattered.

∎

The real line under the topology

S

is called a

scattered line

.

Remark

. Every topological space is a

disjoint union

of a

perfect set

and a scattered set.

Title

scattered space

Canonical

name

ScatteredSpace

Date of creation

2013-03-22 16:42:59

Last modified on

2013-03-22 16:42:59

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

6

Author

CWoo (3771)

Entry type

Definition

Classification

msc 54G12

Related topic

DenseInItself

Defines

scattered

Defines

scattered set

Defines

scattered line

Generated on Sat Feb 10 11:31:10 2018 by

LaTeXML
hedgehog space

hedgehog space

For any

cardinal number

K

, we can form a

topological space

, called the

K

-

hedgehog space

, consisting of the

disjoint union

of

K

real unit intervals identified at the origin. Each unit interval is referred to as one of the hedgehog’s “spines.”

The hedgehog space admits a somewhat surprising metric, by defining

d

⁢

(

x

,

y

)

=

|

x

-

y

|

if

x

and

y

lie in the same spine, and by

d

⁢

(

x

,

y

)

=

x

+

y

if

x

and

y

lie in different spines.

The hedgehog space is an example of a

Moore space

, and

satisfies

many strong

normality

and

compactness

properties.

References

1

L.A. Steen, J.A.Seebach, Jr.,

Counterexamples in topology

,
Holt, Rinehart and Winston, Inc., 1970.

Title

hedgehog space

Canonical name

HedgehogSpace

Date of creation

2013-03-22 14:50:02

Last modified on

2013-03-22 14:50:02

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

10

Author

mathcam (2727)

Entry type

Definition

Classification

msc 54G20

Generated on Sat Feb 10 11:31:12 2018 by

LaTeXML
long line

long line

The

long line

is a non-paracompact

Hausdorff

1

-dimensional

manifold

constructed as
follows. Let

Ω

be the first uncountable ordinal (viewed as an

ordinal space

) and consider the set

L

:=

Ω

×

[

0

,

1

)

endowed with the

order topology

induced by the
lexicographical order, that is the order defined by

(

α

1

,

t

1

)

<

(

α

2

,

t

2

)

⇔

α

1

<

α

2

or

(

α

1

=

α

2

and

t

1

<

t

2

)

.

Intuitively

L

is obtained by “filling the gaps” between consecutive

ordinals

in

Ω

with

intervals

, much the same way that
nonnegative reals are
obtained by filling the gaps between consecutive

natural numbers

with intervals.

Some of the properties of the long line:

•

L

is a chain.

•

L

is not

compact

; in fact

L

is not Lindelöf.

Indeed

{

[

0

,

α

)

:

α

<

Ω

}

is an

open cover

of

L

that has no

countable

subcovering. To see this notice that

⋃

{

[

0

,

α

x

)

:

x

∈

X

}

=

[

0

,

sup

⁡

{

α

x

:

x

∈

X

}

)

and since the

supremum

of a countable

collection

of countable ordinals is a countable ordinal such a union can
never be

[

0

,

Ω

)

.

•

However,

L

is

sequentially compact

.

Indeed every

sequence

has a convergent subsequence. To see this notice that
given a sequence

a

:=

(

a

n

)

of elements of

L

there is an ordinal

α

such
that all the terms of

a

are in the subset

[

0

,

α

]

. Such a subset is
compact since it is

homeomorphic

to

[

0

,

1

]

.

•

L

therefore is not

metrizable

.

•

L

is a

1

–dimensional locally Euclidean

•

L

therefore is not

paracompact

.

•

L

is

first countable

.

•

L

is not

separable

.

•

All

homotopy groups

of

L

are

trivial

.

•

However,

L

is not

contractible

.

Variants

There are several variations of the above construction.

•

Instead of

[

0

,

Ω

)

one can use

(

0

,

Ω

)

or

[

0

,

Ω

]

. The latter (obtained by adding a single point to

L

) is compact.

•

One can consider the “double” of the above construction. That is the
space obtained by gluing two copies of

L

along

0

. The resulting

open
manifold

is not homeomorphic to

L

∖

{

0

}

.

Title

long line

Canonical name

LongLine

Date of creation

2013-03-22 13:29:40

Last modified on

2013-03-22 13:29:40

Owner

Dr_Absentius (537)

Last modified by

Dr_Absentius (537)

Numerical id

17

Author

Dr_Absentius (537)

Entry type

Definition

Classification

msc 54G20

Generated on Sat Feb 10 11:31:14 2018 by

LaTeXML
Sierpinski space

Sierpinski space

Sierpinski space

is the

topological space

X

=

{

x

,

y

}

with the

topology

given by

{

X

,

{

x

}

,

∅

}

.

Sierpinski space is

T

0

(

http://planetmath.org/T0

) but not

T

1

(

http://planetmath.org/T1

). It is

T

0

because

{

x

}

is the

open set

containing

x

but not

y

. It is not

T

1

because every open set

U

containing

y

(namely

X

) contains

x

(in other words, there is no open set containing

y

but not containing

x

).

Remark

. From the Sierpinski space, one can construct many non-

T

1

T

0

spaces, simply by taking any set

X

with at least two

elements

, and take any non-empty

proper subset

U

⊂

X

, and set the topology

𝒯

on

X

by

𝒯

=

P

⁢

(

U

)

∪

{

X

}

.

Title

Sierpinski space

Canonical name

SierpinskiSpace

Date of creation

2013-03-22 12:06:26

Last modified on

2013-03-22 12:06:26

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

9

Author

CWoo (3771)

Entry type

Definition

Classification

msc 54G20

Synonym

Sierpiński space

Related topic

T1Space

Related topic

T2Space

Related topic

SeparationAxioms

Generated on Sat Feb 10 11:31:15 2018 by

LaTeXML
Borel groupoid

Borel groupoid

0.1

Definitions

•

a.

Borel function

Definition 0.1

.

A

function

f

B

:

(

X

;

ℬ

)

→

(

X

;

𝒞

) of
Borel spaces (

http://planetmath.org/BorelSpace

) is defined to be a

Borel function

if the

inverse image

of every Borel set under

f

B

-

1

is also a Borel set.

•

b. Borel groupoid

Definition 0.2

.

Let

𝔾

be a

groupoid

and

𝔾

(

2

)

a subset of

𝔾

×

𝔾

– the set of its

composable pairs

.
A

Borel groupoid

is defined as a groupoid

𝔾

B

such that

𝔾

B

(

2

)

is a Borel set in the product structure on

𝔾

B

×

𝔾

B

, and also with functions

(

x

,

y

)

↦

x

⁢

y

from

𝔾

B

(

2

)

to

𝔾

B

, and

x

↦

x

-

1

from

𝔾

B

to

𝔾

B

defined such that they are all
(measurable) Borel functions (

http://planetmath.org/MeasurableFunctions

) (ref.

[

1

]

).

0.1.1

Analytic Borel space

𝔾

B

becomes an analytic groupoid (

http://planetmath.org/LocallyCompactGroupoids

) if its Borel structure is

analytic

(

http://planetmath.org/Analytic

).

A Borel space

(

X

;

ℬ

)

is called

analytic

if it is
countably

separated

, and also if it is the image of a Borel function from a

standard
Borel space

.

References

1

M.R. Buneci. 2006.,

http://www.utgjiu.ro/math/mbuneci/preprint/p0024.pdf

Groupoid C*-

Algebras

.,

Surveys in Mathematics and its Applications

, Volume 1, p.75 .

Title

Borel groupoid

Canonical

name

BorelGroupoid

Date of creation

2013-03-22 18:23:30

Last modified on

2013-03-22 18:23:30

Owner

bci1 (20947)

Last modified by

bci1 (20947)

Numerical id

39

Author

bci1 (20947)

Entry type

Definition

Classification

msc 54H05

Classification

msc 28A05

Classification

msc 18B40

Classification

msc 28A12

Classification

msc 22A22

Classification

msc 28C15

Synonym

Borel space

Synonym

measure groupoid

Related topic

BorelSpace

Related topic

BorelMeasure

Related topic

MeasurableFunctions

Related topic

Groupoid

Related topic

Groupoids

Related topic

GroupoidRepresentationsInducedByMeasure

Related topic

LocallyCompactGroupoids

Related topic

BorelGSpace

Related topic

CategoryOfBorelGroupoids

Defines

Borel function

Defines

analytic groupoid

Defines

set of composable pairs

Defines

G

⁢

r

⁢

p

⁢

(

2

)

Defines

analytic (Borel) groupoid

Defines

analytic Borel space

Defines

product structure

Generated on Sat Feb 10 11:31:17 2018 by

LaTeXML
Borel isomorphism

Borel isomorphism

Definition 0.1

.

A

Borel isomorphism

between two

Borel spaces

(

X

;

𝔹

⁢

(

X

)

)

and

(

Y

;

𝔹

⁢

(

Y

)

)

is defined as the

bijection

ι

:

(

X

;

𝔹

⁢

(

X

)

)

≅

(

Y

;

𝔹

⁢

(

Y

)

)

.

References

1

M.R. Buneci. 2006.,

http://www.utgjiu.ro/math/mbuneci/preprint/p0024.pdf

Groupoid

C*-

Algebras

.,

Surveys in Mathematics and its Applications

, Volume 1: 71–98.

2

A. Connes.1979. Sur la théorie

noncommutative

de l’ integration,

Lecture Notes in
Math.

, Springer-Verlag, Berlin,

725

: 19-14.

Title

Borel isomorphism

Canonical name

BorelIsomorphism

Date of creation

2013-03-22 18:23:05

Last modified on

2013-03-22 18:23:05

Owner

bci1 (20947)

Last modified by

bci1 (20947)

Numerical id

5

Author

bci1 (20947)

Entry type

Definition

Classification

msc 54H05

Classification

msc 28A05

Synonym

one-to-one and onto correspondence

Related topic

Bijection

Generated on Sat Feb 10 11:31:19 2018 by

LaTeXML
Cantor-Bendixson derivative

Cantor-Bendixson derivative

Let

A

be a subset of a

topological space

X

. Its

Cantor-Bendixson
derivative

A

′

is defined as the set of

accumulation points

of

A

. In
other words

A

′

=

{

x

∈

X

∣

x

∈

A

∖

{

x

}

¯

}

.

Through

transfinite induction

, the Cantor-Bendixson derivative can be
defined to any order

α

, where

α

is an arbitrary

ordinal

.
Let

A

(

0

)

=

A

. If

α

is a

successor ordinal

, then

A

(

α

)

=

(

A

(

α

-

1

)

)

′

. If

λ

is a

limit
ordinal

, then

A

(

λ

)

=

⋂

α

<

λ

A

(

α

)

.
The

Cantor-Bendixson rank

of the set

A

is the least ordinal

α

such that

A

(

α

)

=

A

(

α

+

1

)

. Note that

A

′

=

A

implies that

A

is a

perfect set

.

Some basic properties of the Cantor-Bendixson derivative include

1.

(

A

∪

B

)

′

=

A

′

∪

B

′

,

2.

(

⋃

i

∈

I

A

i

)

′

⊇

⋃

i

∈

I

A

i

′

,

3.

(

⋂

i

∈

I

A

i

)

′

⊆

⋂

i

∈

I

A

i

′

,

4.

(

A

∖

B

)

′

⊇

A

′

∖

B

′

,

5.

A

⊆

B

⇒

A

′

⊆

B

′

,

6.

A

¯

=

A

∪

A

′

,

7.

A

′

¯

=

A

′

.

The last property requires some justification. Obviously,

A

′

⊆

A

′

¯

. Suppose

a

∈

A

′

¯

, then every

neighborhood

of

a

contains some points of

A

′

distinct from

a

. But by definition of

A

′

, each such neighborhood must also contain some points of

A

. This
implies that

a

is an accumulation point of

A

, that is

a

∈

A

′

.
Therefore

A

′

¯

⊆

A

′

and we have

A

′

¯

=

A

′

.

Finally, from the definition of the Cantor-Bendixson rank and the above
properties, if

A

has Cantor-Bendixson rank

α

, the sets

A

(

1

)

⊃

A

(

2

)

⊃

⋯

⊃

A

(

α

)

form a

strictly decreasing

chain of

closed sets

.

Title

Cantor-Bendixson derivative

Canonical name

CantorBendixsonDerivative

Date of creation

2013-03-22 15:01:37

Last modified on

2013-03-22 15:01:37

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

9

Author

CWoo (3771)

Entry type

Definition

Classification

msc 54H05

Classification

msc 03E15

Synonym

set derivative

Related topic

DerivedSet

Defines

Cantor-Bendixson rank

Generated on Sat Feb 10 11:31:21 2018 by

LaTeXML
categories of Polish groups and Polish spaces

categories of Polish groups and Polish spaces

0.1

Introduction

Definition 0.1

.

Let us recall that a

Polish space

is a

separable

,

completely metrizable

topological space

, and
that

Polish groups

G

P

are

metrizable

(topological) groups whose

topology

is Polish, and thus they admit a

compatible

metric

d

which is

left-invariant

; (a

topological group

G

T

is

metrizable

iff

G

T

is

Hausdorff

, and the

identity

e

of

G

T

has a

countable

neighborhood

basis).

Remark 0.1

.

Polish spaces can be classified up to a (Borel)

isomorphism

according to the following provable
results (

http://planetmath.org/PolishSpacesUpToBorelIsomorphism

):

•

“

All

uncountable

Polish spaces are Borel isomorphic to

R

equipped with the

standard topology

;

”

This also implies that all uncountable Polish space have the

cardinality

of the

continuum

.

•

“

Two Polish spaces are Borel isomorphic if and only if they have the same cardinality.

”

Furthermore, the

subcategory

of Polish spaces that are Borel isomorphic is, in fact, a

Borel groupoid

.

0.2

Category of Polish groups

Definition 0.2

.

The

category

of Polish groups

𝒫

has, as its objects, all Polish groups

G

P

and, as its

morphisms

the

group homomorphisms

g

P

between Polish groups, compatible with the

Polish topology

Π

on

G

P

.

Remark 0.2

.

𝒫

is obviously a subcategory of

𝒯

g

⁢

r

⁢

p

the category of topological groups; moreover,

𝒯

g

⁢

r

⁢

p

is a subcategory of

𝒯

𝔾

-the category of

topological groupoids

and topological groupoid

homomorphisms

.

Title

categories of Polish groups and Polish spaces

Canonical name

CategoriesOfPolishGroupsAndPolishSpaces

Date of creation

2013-03-22 18:25:04

Last modified on

2013-03-22 18:25:04

Owner

bci1 (20947)

Last modified by

bci1 (20947)

Numerical id

19

Author

bci1 (20947)

Entry type

Topic

Classification

msc 54H05

Classification

msc 28A05

Classification

msc 28A12

Classification

msc 28C15

Synonym

subcategory of topological groupoid category

Related topic

Category

Related topic

Metrizable

Related topic

CategoryOfBorelSpaces

Related topic

PolishSpacesUpToBorelIsomorphism

Related topic

TopologicalGroup2

Related topic

MeasureSpace

Related topic

PolishSpace

Defines

Polish group homomorphism

Defines

metrizable topological groups

Generated on Sat Feb 10 11:31:23 2018 by

LaTeXML
category of Borel spaces

category of Borel spaces

Definition 0.1

.

The

category of Borel spaces

𝔹

has, as its objects, all

Borel spaces

(

X

b

;

ℬ

⁢

(

X

b

)

)

, and as its

morphisms

the

Borel morphisms

f

b

between Borel spaces; the Borel morphism

composition

is defined so that it preserves the

Borel structure

determined by the

σ

-

algebra

of

Borel sets

.

Remark 0.1

.

The

category of (standard)

Borel G-spaces

𝔹

G

is defined in a

similar

manner to

𝔹

, with the additional condition that Borel G-space morphisms commute with
the

Borel actions

a

:

G

×

X

→

X

defined as Borel functions (

http://planetmath.org/BorelGroupoid

)
(or Borel-measurable maps). Thus,

𝔹

G

is a subcategory of

𝔹

; in its turn,

𝔹

is a subcategory of

𝕋

⁢

o

⁢

p

–the category of topological spaces and

continuous
functions

.

The

category of rigid Borel spaces

can be defined as above with the additional condition that the
only

automorphism

f

:

X

b

→

X

b

(

bijection

) is the

identity

1

(

X

b

;

ℬ

⁢

(

X

b

)

)

.

Title

category of Borel spaces

Canonical name

CategoryOfBorelSpaces

Date of creation

2013-03-22 18:25:01

Last modified on

2013-03-22 18:25:01

Owner

bci1 (20947)

Last modified by

bci1 (20947)

Numerical id

16

Author

bci1 (20947)

Entry type

Definition

Classification

msc 54H05

Classification

msc 28A05

Classification

msc 28A12

Classification

msc 28C15

Synonym

category of measure spaces

Related topic

Category

Related topic

BorelSpace

Related topic

BorelGSpace

Related topic

BorelMorphism

Related topic

CategoryOfPointedTopologicalSpaces

Related topic

CategoryOfSets

Related topic

CategoryOfPolishGroups

Related topic

IndexOfCategories

Defines

composition of Borel morphisms

Defines

category of rigid Borel spaces

Generated on Sat Feb 10 11:31:25 2018 by

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topological transformation group

topological transformation group

Let

G

be a

topological group

and

X

any

topological space

. We say that

G

is a

topological transformation group

of

X

if

G

acts

on

X

continuously, in the following sense:

1.

there is a

continuous function

α

:

G

×

X

→

X

, where

G

×

X

is given the

product topology

2.

α

⁢

(

1

,

x

)

=

x

, and

3.

α

⁢

(

g

1

⁢

g

2

,

x

)

=

α

⁢

(

g

1

,

α

⁢

(

g

2

,

x

)

)

.

The

function

α

is called the

(left) action

of

G

on

X

. When there is no confusion,

α

⁢

(

g

,

x

)

is simply written

g

⁢

x

, so that the two conditions above read

1

⁢

x

=

x

and

(

g

1

⁢

g

2

)

⁢

x

=

g

1

⁢

(

g

2

⁢

x

)

.

If a topological transformation group

G

on

X

is

effective

, then

G

can be viewed as a group of

homeomorphisms

on

X

: simply define

h

g

:

X

→

X

by

h

g

⁢

(

x

)

=

g

⁢

x

for each

g

∈

G

so that

h

g

is the

identity function

precisely when

g

=

1

.

Some Examples.

1.

Let

X

=

ℝ

n

, and

G

be the group of

n

×

n

matrices over

ℝ

. Clearly

X

and

G

are both topological spaces with the

usual topology

. Furthermore,

G

is a topological group.

G

acts on

X

continuous

if we view elements of

X

as

column vectors

and take the action to be the

matrix multiplication

on the left.

2.

If

G

is a topological group,

G

can be considered a topological transformation group on itself. There are many continuous actions that can be defined on

G

. For example,

α

:

G

×

G

→

G

given by

α

⁢

(

g

,

x

)

=

g

⁢

x

is one such action. It is continuous, and

satisfies

the two action axioms.

G

is also effective with respect to

α

.

Title

topological transformation group

Canonical name

TopologicalTransformationGroup

Date of creation

2013-03-22 16:43:58

Last modified on

2013-03-22 16:43:58

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

5

Author

CWoo (3771)

Entry type

Definition

Classification

msc 54H15

Classification

msc 22F05

Defines

effective topological transformation group

Generated on Sat Feb 10 11:31:33 2018 by

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asymptotically stable

asymptotically stable

Let

(

X

,

d

)

be a

metric space

and

f

:

X

→

X

a

continuous function

. A point

x

∈

X

is said to be

Lyapunov stable

if for each

ϵ

>

0

there is

δ

>

0

such that for all

n

∈

ℕ

and all

y

∈

X

such that

d

⁢

(

x

,

y

)

<

δ

, we have

d

⁢

(

f

n

⁢

(

x

)

,

f

n

⁢

(

y

)

)

<

ϵ

.

We say that

x

is

asymptotically stable

if it belongs to the

interior

of its

stable set

, i.e. if there is

δ

>

0

such that

lim

n

→

∞

⁡

d

⁢

(

f

n

⁢

(

x

)

,

f

n

⁢

(

y

)

)

=

0

whenever

d

⁢

(

x

,

y

)

<

δ

.

In a similar way, if

φ

:

X

×

ℝ

→

X

is a flow, a point

x

∈

X

is said to be Lyapunov stable if for each

ϵ

>

0

there is

δ

>

0

such that, whenever

d

⁢

(

x

,

y

)

<

δ

, we have

d

⁢

(

φ

⁢

(

x

,

t

)

,

φ

⁢

(

y

,

t

)

)

<

ϵ

for each

t

≥

0

; and

x

is called asymptotically stable if there is a

neighborhood

U

of

x

such that

lim

t

→

∞

⁡

d

⁢

(

φ

⁢

(

x

,

t

)

,

φ

⁢

(

y

,

t

)

)

=

0

for each

y

∈

U

.

Title

asymptotically stable

Canonical name

AsymptoticallyStable

Date of creation

2013-03-22 13:55:19

Last modified on

2013-03-22 13:55:19

Owner

Koro (127)

Last modified by

Koro (127)

Numerical id

10

Author

Koro (127)

Entry type

Definition

Classification

msc 54H20

Classification

msc 37B99

Related topic

UnstableFixedPoint

Related topic

LiapunovStable

Defines

Lyapunov stable

Generated on Sat Feb 10 11:31:35 2018 by

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Markov Kakutani fixed point theorem

Markov Kakutani fixed point theorem

Theorem

[Markov - Kakutani]

-

Let

X

be a

topological vector space

and

𝒯

a commuting family of continuous linear operators in

X

. Suppose

K

is a

compact

convex subset of

X

such that

T

⁢

(

K

)

⊆

K

for every

T

∈

𝒯

. Then there is a point

x

0

∈

K

such that

T

⁢

x

0

=

x

0

for all

T

∈

𝒯

.

Title

Markov Kakutani fixed point theorem

Canonical name

MarkovKakutaniFixedPointTheorem

Date of creation

2013-03-22 17:33:25

Last modified on

2013-03-22 17:33:25

Owner

asteroid (17536)

Last modified by

asteroid (17536)

Numerical id

6

Author

asteroid (17536)

Entry type

Theorem

Classification

msc 54H25

Classification

msc 46A99

Classification

msc 46A50

Generated on Sat Feb 10 11:31:41 2018 by

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Schauder fixed point theorem

Schauder fixed point theorem

Let

X

be a

normed vector space

, and let

K

⊂

X

be a non-empty,

compact

, and convex set.
Then given any

continuous mapping

f

:

K

→

K

there exists

x

∈

K

such that

f

⁢

(

x

)

=

x

.

Notice that the

unit disc

of a finite dimensional vector space is always convex and compact hence this theorem extends

Brouwer Fixed Point Theorem

.

Notice that the space

X

is not required to be

complete

, however the subset

K

being compact,
is complete with respect to the metric

induced

by

X

.

References

1

Rudin,

Functional Analysis

, Chapter 5.

Title

Schauder fixed point theorem

Canonical name

SchauderFixedPointTheorem

Date of creation

2013-03-22 13:45:17

Last modified on

2013-03-22 13:45:17

Owner

paolini (1187)

Last modified by

paolini (1187)

Numerical id

12

Author

paolini (1187)

Entry type

Theorem

Classification

msc 54H25

Classification

msc 47H10

Classification

msc 46B50

Related topic

BrouwerFixedPointTheorem

Related topic

FixedPoint

Related topic

TychonoffFixedPointTheorem

Generated on Sat Feb 10 11:31:43 2018 by

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