aarjaneiro/planetmath huggingface dataset:
Krull dimension

Krull dimension

If

R

is a ring, the

Krull dimension

(or simply

dimension

) of

R

,

dim

⁡

R

is the

supremum

of all

integers

n

such that there is an

increasing sequence

of

prime ideals

𝔭

0

⊊

⋯

⊊

𝔭

n

of length

n

in

R

.

If

X

is a

topological space

, the Krull dimension (or simply dimension) of

X

,

dim

⁡

X

is the supremum of all integers

n

such that there is a

decreasing sequence

of

irreducible

closed subsets

F

0

⊋

⋯

⊋

F

n

of

X

.

Title

Krull dimension

Canonical name

KrullDimension

Date of creation

2013-03-22 12:03:27

Last modified on

2013-03-22 12:03:27

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

8

Author

mathcam (2727)

Entry type

Definition

Classification

msc 54-00

Synonym

dimension (Krull)

Related topic

HeightOfAPrimeIdeal

Related topic

Dimension3

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

LaTeXML
Kuratowski’s embedding theorem

Kuratowski’s embedding theorem

Let

X

be a set and

Bou

⁡

(

X

,

ℝ

)

be the set of

bounded functions

f

:

X

→

ℝ

with norm

||

f

||

=

sup

⁡

{

|

f

⁢

(

x

)

|

:

x

∈

X

}

. Kuratowski’s

embedding

theorem

states that every

metric space

(

X

,

d

)

can be embedded isometrically into the

Banach space

E

=

Bou

⁡

(

X

,

ℝ

)

.

Proof.

One can assume that

X

≠

∅

. Fix a point

a

0

∈

X

and for every

a

∈

X

define a

function

f

a

:

X

→

ℝ

by

f

a

⁢

(

x

)

=

d

⁢

(

x

,

a

)

-

d

⁢

(

x

,

a

0

)

.

Then

|

f

a

⁢

(

x

)

|

≤

d

⁢

(

a

,

a

0

)

for every

x

∈

X

so

f

a

is

bounded

. By setting

φ

:

X

→

E

,

φ

⁢

(

a

)

=

f

a

, we have the

mapping

φ

:

X

→

E

. It requires to prove that

φ

is an

isometry

.

Let

a

,

b

∈

X

. As

x

∈

X

we have that

|

f

a

⁢

(

x

)

-

f

b

⁢

(

x

)

|

=

|

d

⁢

(

x

,

a

)

-

d

⁢

(

x

,

b

)

|

≤

d

⁢

(

a

,

b

)

.

Therefore

||

f

a

-

f

b

||

≤

d

⁢

(

a

,

b

)

. On the other hand

|

f

a

⁢

(

a

)

-

f

b

⁢

(

a

)

|

=

|

d

⁢

(

a

,

a

)

-

d

⁢

(

a

,

a

0

)

-

d

⁢

(

a

,

b

)

+

d

⁢

(

a

,

a

0

)

|

=

d

⁢

(

a

,

b

)

.

Therefore

||

φ

⁢

(

a

)

-

φ

⁢

(

b

)

||

=

||

f

a

-

f

b

||

=

d

⁢

(

a

,

b

)

.

References

1

J. VÃÂisÃÂlÃÂ:

Topologia II

.  2nd corrected issue, Limes ry., Helsinki, Finland (2005), ISBN 951-745-209-8

Title

Kuratowski’s embedding theorem

Canonical name

KuratowskisEmbeddingTheorem

Date of creation

2013-03-22 18:24:48

Last modified on

2013-03-22 18:24:48

Owner

puuhikki (9774)

Last modified by

puuhikki (9774)

Numerical id

10

Author

puuhikki (9774)

Entry type

Theorem

Classification

msc 54-00

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

LaTeXML
ladder connected

ladder connected

Definition

Suppose

X

is a

topological space

. Then

X

is called

ladder connected

provided that any

open cover

𝒰

of

X

has the following property:
If

p

,

q

∈

X

, then there exists a finite number of

open sets

U

1

,

…

,

U

N

from

𝒰

such that

p

∈

U

1

,

U

1

∩

U

2

≠

∅

,

…

,

U

N

-

1

∩

U

N

≠

∅

,
and

q

∈

U

N

.

Title

ladder connected

Canonical name

LadderConnected

Date of creation

2013-03-22 14:00:29

Last modified on

2013-03-22 14:00:29

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

6

Author

mathcam (2727)

Entry type

Definition

Classification

msc 54-00

Related topic

CharacterizationOfConnectedCompactMetricSpaces

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

LaTeXML
list of common topologies

list of common topologies

The aim of this entry is to give an overview of

topologies

commonly used in mathematics with links to the
entries at Planetmath where these are discussed in greater detail.

Topologies on arbitrary sets

The below topologies show that any set can be endowed with a topology.

1.

discrete topology

2.

indiscrete topology

3.

cofinite topology

4.

cocountable topology

New topologies from old ones

1.

subspace topology

2.

quotient topology

3.

box topology

4.

product topology

5.

inductive limit

topology

Topologies on sets with more structure

1.

order topology

(

standard topologies

on

ℤ

,

ℕ

,

ℝ

)

2.

metric topology

3.

seminorm

topology (

topological vector spaces

)

4.

graph topology

(

graph theory

)

5.

Zariski topology

(

algebraic geometry

)

Topologies for functions spaces and mappings

An extensive list of can be
found here (

http://planetmath.org/FunctionSpaces

).

1.

Whitney topology

(differential geometry)

2.

Compact-open topology

Title

list of common topologies

Canonical name

ListOfCommonTopologies

Date of creation

2013-03-22 14:38:27

Last modified on

2013-03-22 14:38:27

Owner

matte (1858)

Last modified by

matte (1858)

Numerical id

5

Author

matte (1858)

Entry type

Topic

Classification

msc 54-00

Classification

msc 55-00

Classification

msc 22-00

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

LaTeXML
local base

local base

Let

x

be a point in a

topological space

X

. A

local base

around

x

, is a family of

neighborhoods

such that every neighborhood of

x

contains a member of the family. A local base is also known as a

neighborhood base

.

Title

local base

Canonical name

LocalBase

Date of creation

2013-03-22 12:22:16

Last modified on

2013-03-22 12:22:16

Owner

yark (2760)

Last modified by

yark (2760)

Numerical id

9

Author

yark (2760)

Entry type

Definition

Classification

msc 54-00

Synonym

base of a

neighborhood system

Synonym

neighborhood base

Synonym

base of a neighbourhood system

Synonym

neighbourhood base

Related topic

FirstAxiomOfCountability

Related topic

Neighborhood

Related topic

NeighborhoodSystem

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

LaTeXML
locally homeomorphic

locally homeomorphic

Let

X

and

Y

be

topological spaces

. Then

X

is

locally
homeomorphic

to

Y

, if for every

x

∈

X

there is a

neighbourhood

U

⊆

X

of

x

and an

http://planetmath.org/node/380

open set

V

⊆

Y

, such that

U

and

V

with their respective

subspace topology

are

homeomorphic

.

Examples

•

Let

X

=

{

1

}

and

Y

=

{

2

,

3

}

be

discrete spaces

with one resp. two elements. Since

X

and

Y

have different cardinalities,
they cannot be homeomorphic. They are, however, locally homeomorphic
to each other.

•

Again, let

X

=

{

1

}

be a discrete space with one element, but
now let

Y

=

{

2

,

3

}

the space with

topology

{

∅

,

{

2

}

,

Y

}

. Then

X

is still locally homeomorphic to

Y

, but

Y

is not locally homeomorphic to

X

, since the smallest
neighbourhood of

3

already has more elements than

X

.

•

Now, let

X

be as in the previous examples, and

Y

=

{

2

,

3

}

be

http://planetmath.org/node/3120

indiscrete. Then neither

X

is locally homeomorphic to

Y

nor
the other way round.

•

Non-trivial examples arise with

locally Euclidean spaces

,
especially

manifolds

.

Title

locally homeomorphic

Canonical

name

LocallyHomeomorphic

Date of creation

2013-03-22 15:14:34

Last modified on

2013-03-22 15:14:34

Owner

GrafZahl (9234)

Last modified by

GrafZahl (9234)

Numerical id

4

Author

GrafZahl (9234)

Entry type

Definition

Classification

msc 54-00

Synonym

local homeomorphy

Related topic

LocallyEuclidean

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

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loop

loop

A

loop

based at

x

0

in a

topological space

X

is simply a

continuous map

f

:

[

0

,

1

]

→

X

with

f

⁢

(

0

)

=

f

⁢

(

1

)

=

x

0

.

The collection of all such loops, modulo

homotopy equivalence

, forms a group known as the

fundamental group

.

More generally, the space of loops in

X

based at

x

0

with the

compact-open topology

, represented by

Ω

x

0

, is known as the

loop space

of

X

. And one has the

homotopy groups

π

n

⁢

(

X

,

x

0

)

=

π

n

-

1

⁢

(

Ω

x

0

,

ι

)

, where

π

n

represents

the higher homotopy groups, and

ι

is the basepoint in

Ω

x

0

consisting of the constant loop at

x

0

.

Title

loop

Canonical name

Loop1

Date of creation

2013-03-22 12:16:21

Last modified on

2013-03-22 12:16:21

Owner

nerdy2 (62)

Last modified by

nerdy2 (62)

Numerical id

5

Author

nerdy2 (62)

Entry type

Definition

Classification

msc 54-00

Generated on Sat Feb 10 11:11:24 2018 by

LaTeXML
loop space

loop space

Let

X

be a

topological space

, and give the space of

continuous maps

[

0

,

1

]

→

X

, the

compact-open topology

, that is a

subbasis

for the

topology

is the collection of sets

{

σ

:

σ

⁢

(

K

)

⊂

U

}

for

K

⊂

[

0

,

1

]

compact

and

U

⊂

X

open.

Then for

x

∈

X

, let

Ω

x

⁢

X

be the subset of loops based at

x

(that is

σ

such that

σ

⁢

(

0

)

=

σ

⁢

(

1

)

=

x

), with the

relative topology

.

Ω

x

⁢

X

is called the

loop space

of

X

at

x

.

Title

loop space

Canonical name

LoopSpace

Date of creation

2013-03-22 12:15:26

Last modified on

2013-03-22 12:15:26

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

8

Author

mathcam (2727)

Entry type

Definition

Classification

msc 54-00

Related topic

Suspension

Related topic

EilenbergMacLaneSpace

Generated on Sat Feb 10 11:11:26 2018 by

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metacompact

metacompact

A

topological space

is said to be

metacompact

if every

open cover

has an open

point finite

refinement

.

Title

metacompact

Canonical name

Metacompact

Date of creation

2013-03-22 16:06:21

Last modified on

2013-03-22 16:06:21

Owner

Mathprof (13753)

Last modified by

Mathprof (13753)

Numerical id

5

Author

Mathprof (13753)

Entry type

Definition

Classification

msc 54-00

Synonym

pointwise paracompact

Synonym

weakly paracompact

Related topic

PointFinite

Defines

metacompactness

Generated on Sat Feb 10 11:11:28 2018 by

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metrizable

metrizable

A

topological space

(

X

,

𝒯

)

is said to be

metrizable

if there is a metric

d

:

X

→

[

0

,

∞

)

such that the

topology

induced by

d

is

𝒯

.

Title

metrizable

Canonical name

Metrizable

Date of creation

2013-03-22 12:12:44

Last modified on

2013-03-22 12:12:44

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

6

Author

mathcam (2727)

Entry type

Definition

Classification

msc 54-00

Classification

msc 55-00

Classification

msc 22-00

Synonym

metrization

Synonym

metrizable space

Related topic

Metric

Related topic

UrysohnMetrizationTheorem

Related topic

CategoryOfPolishGroups

Generated on Sat Feb 10 11:11:29 2018 by

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neighborhood system

neighborhood system

Let

(

X

,

τ

)

be a

topological space

and

x

∈

X

. The

neighborhood system

around

x

is the family of all the

neighborhoods

of

x

.

Title

neighborhood system

Canonical name

NeighborhoodSystem

Date of creation

2013-03-22 12:22:18

Last modified on

2013-03-22 12:22:18

Owner

Mathprof (13753)

Last modified by

Mathprof (13753)

Numerical id

6

Author

Mathprof (13753)

Entry type

Definition

Classification

msc 54-00

Synonym

fundamental system of neighborhoods

Related topic

Neighborhood

Related topic

LocalBase

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

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Niemytzki plane

Niemytzki plane

Let

Γ

be the

Euclidean

half plane

Γ

=

{

(

x

,

y

)

∣

y

≥

0

}

⊆

ℝ

2

,
with the usual

subspace topology

. We enrich the

topology

on

Γ

by throwing
in

open sets

of the form

{

(

x

,

0

)

}

∪

B

r

⁢

(

x

,

r

)

, that is an

open ball

of radius

r

around

(

x

,

r

)

together with its point

tangent

to

ℝ

×

{

0

}

(Fig.

1

).

Figure 1:

A new open set in

Γ

. It consists of an open
disc and the point tangent to

y

=

0

.

The space

Γ

endowed with the enriched topology is called the

Niemytzki plane

.

Some miscellaneous properties of the Niemytzki plane are

•

the

subspace

ℝ

×

{

0

}

of

Γ

is

discrete

, hence
the only

convergent sequences

in this subspace are constant ones;

•

it is

Hausdorff

;

•

it is

completely regular

;

•

it is not normal.

Title

Niemytzki plane

Canonical name

NiemytzkiPlane

Date of creation

2013-03-22 13:36:53

Last modified on

2013-03-22 13:36:53

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

7

Author

PrimeFan (13766)

Entry type

Example

Classification

msc 54-00

Classification

msc 54G99

Synonym

Niemytzki space

Generated on Sat Feb 10 11:11:45 2018 by

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paracompact topological space

paracompact topological space

A

topological space

X

is said to be

paracompact

if every

open cover

of

X

has a

locally finite

open refinement.

In more detail, if

(

U

i

)

i

∈

I

is any family of

open subsets

of

X

such that

∪

i

∈

I

U

i

=

X

,

then there exists another family

(

V

i

)

i

∈

I

of open sets such that

∪

i

∈

I

V

i

=

X

V

i

⊂

U

i

⁢

for all

⁢

i

∈

I

and any specific

x

∈

X

is in

V

i

for only finitely many

i

.

Some properties:

•

Any metric or

metrizable space

is paracompact (A. H. Stone).

•

Given an open cover of a

paracompact space

X

, there exists a (

continuous

)

partition of unity

on

X

subordinate to that cover.

•

A paracompact ,

Hausdorff space

is

regular

.

•

A

compact

or

pseudometric space

is paracompact.

Title

paracompact topological space

Canonical name

ParacompactTopologicalSpace

Date of creation

2013-03-22 12:12:47

Last modified on

2013-03-22 12:12:47

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

9

Author

mathcam (2727)

Entry type

Definition

Classification

msc 54-00

Classification

msc 55-00

Synonym

paracompact space

Related topic

ExampleOfParacompactTopologicalSpaces

Defines

paracompact

Defines

paracompactness

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

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pointed topological space

pointed topological space

Definition

Suppose

X

is a non-empty

topological space

and

x

0

is an
element of

X

. Then the pair

(

X

,

x

0

)

is called a

pointed topological space

, or a

based topological space

.

The idea with pointed topological spaces is simply that one fixes a base point
in the space. This is necessary, for instance, when defining the

fundamental
group

of a topological space.

Title

pointed topological space

Canonical name

PointedTopologicalSpace

Date of creation

2013-03-22 14:01:15

Last modified on

2013-03-22 14:01:15

Owner

bwebste (988)

Last modified by

bwebste (988)

Numerical id

7

Author

bwebste (988)

Entry type

Definition

Classification

msc 54-00

Classification

msc 54E99

Synonym

based topological space

Related topic

CategoryOfPointedTopologicalSpaces

Related topic

PolishGSpace

Related topic

OmegaSpectrum

Generated on Sat Feb 10 11:11:50 2018 by

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quasi-compact

quasi-compact

A

topological space

is called

quasi-compact

if every

open cover

of it has a finite subcover. The distinction

compact spaces

and quasi-compact spaces exists because some people insist that a space that is

compact

must be a

Hausdorff space

. This distinction occurs for example in Robin Hartshorne’s book

Algebraic Geometry

.

Title

quasi-compact

Canonical

name

Quasicompact

Date of creation

2013-03-22 12:15:31

Last modified on

2013-03-22 12:15:31

Owner

Wkbj79 (1863)

Last modified by

Wkbj79 (1863)

Numerical id

7

Author

Wkbj79 (1863)

Entry type

Definition

Classification

msc 54-00

Synonym

quasicompact

Related topic

Compact

Generated on Sat Feb 10 11:11:51 2018 by

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regularly open

regularly open

Given a

topological space

(

X

,

τ

)

, a

regularly open

set

is an

open set

A

∈

τ

such that

int

⁢

A

¯

=

A

(the

interior

of the

closure

is the set itself).

An example of non regularly open set on the

standard topology

for

ℝ

is

A

=

(

0

,

1

)

∪

(

1

,

2

)

since

int

⁢

A

¯

=

(

0

,

2

)

.

Title

regularly open

Canonical name

RegularlyOpen

Date of creation

2013-03-22 12:19:43

Last modified on

2013-03-22 12:19:43

Owner

drini (3)

Last modified by

drini (3)

Numerical id

5

Author

drini (3)

Entry type

Definition

Classification

msc 54-00

Generated on Sat Feb 10 11:11:53 2018 by

LaTeXML
separated

separated

Definition

Suppose

A

and

B

are subsets of a

topological space

X

. Then

A

and

B

are

separated

provided that

A

¯

∩

B

=

∅

,

A

∩

B

¯

=

∅

,

where

A

¯

is the

closure operator

(

http://planetmath.org/

Closure

) in

X

.

Properties

1.

If

A

,

B

are separated in

X

, and

f

:

X

→

Y

is a

homeomorphism

,
then

f

⁢

(

A

)

and

f

⁢

(

B

)

are separated in

Y

.

Examples

1.

On

ℝ

, the

intervals

(

0

,

1

)

and

(

1

,

2

)

are separated.

2.

If

d

⁢

(

x

,

y

)

≥

r

+

s

, then the

open balls

B

r

⁢

(

x

)

and

B

s

⁢

(

y

)

are
separated (proof.) (

http://planetmath.org/WhenAreBallsSeparated

).

3.

If

A

is a

clopen set

, then

A

and

A

∁

are separated.
This follows since

S

¯

=

S

when

S

is a

closed set

.

Remarks

The above definition follows

[

1

]

. In

[

2

]

, separated sets are called

strongly

disjoint

sets.

References

1

J.L. Kelley,

General Topology

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

2

G.J. Jameson,

Topology

and

Normed Spaces

,
Chapman and Hall, 1974.

Title

separated

Canonical name

Separated

Date of creation

2013-03-22 15:16:34

Last modified on

2013-03-22 15:16:34

Owner

matte (1858)

Last modified by

matte (1858)

Numerical id

15

Author

matte (1858)

Entry type

Definition

Classification

msc 54-00

Classification

msc 54D05

Generated on Sat Feb 10 11:11:55 2018 by

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spherical metric

spherical metric

Suppose that

ℂ

^

:=

ℂ

∪

{

∞

}

is the

extended complex plane

(the Riemann sphere).

Definition

.

Suppose

γ

:

[

0

,

1

]

→

ℂ

^

is a path in

ℂ

^

.
The

spherical length

of

γ

is defined as

ℓ

⁢

(

γ

)

:=

2

⁢

∫

γ

|

d

⁢

z

|

1

+

|

z

|

2

=

2

⁢

∫

0

1

|

γ

′

⁢

(

t

)

|

1

+

|

γ

⁢

(

t

)

|

2

⁢

𝑑

t

.

Definition

.

Let

z

1

,

z

2

∈

ℂ

^

, and let

Γ

be the set of all paths
in

ℂ

^

from

z

1

to

z

2

, then the

distance

from

z

1

to

z

2

in the

spherical metric

is defined as

σ

⁢

(

z

1

,

z

2

)

:=

inf

γ

∈

Γ

⁡

ℓ

⁢

(

γ

)

.

More intuitivelly this is the shortest distance to travel from

z

1

to

z

2

if we think of these points as being on the Riemann sphere, and we can
only travel on the Riemann sphere itself (we cannot “drill” a

straight line

from

z

1

to

z

2

).

References

1

Theodore B. Gamelin.
.
Springer-Verlag, New York, New York, 2001.

Title

spherical metric

Canonical name

SphericalMetric

Date of creation

2013-03-22 14:18:41

Last modified on

2013-03-22 14:18:41

Owner

jirka (4157)

Last modified by

jirka (4157)

Numerical id

6

Author

jirka (4157)

Entry type

Definition

Classification

msc 54-00

Classification

msc 30A99

Defines

spherical length

Generated on Sat Feb 10 11:11:57 2018 by

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support of function

support of function

Definition

Suppose

X

is a

topological space

, and

f

:

X

→

ℂ

is a function.
Then the

support

of

f

(written as

supp

⁡

f

), is the set

supp

⁡

f

=

{

x

∈

X

∣

f

⁢

(

x

)

≠

0

}

¯

.

In other words,

supp

⁡

f

is the

closure

of the set where

f

does not vanish.

Properties

Let

f

:

X

→

ℂ

be a function.

1.

supp

⁡

f

is closed.

2.

If

x

∉

supp

⁡

f

, then

f

⁢

(

x

)

=

0

.

3.

If

supp

⁡

f

=

∅

, then

f

=

0

.

4.

If

χ

:

X

→

ℂ

is such that

χ

=

1

on

supp

⁡

f

, then

f

=

χ

⁢

f

.

5.

If

f

,

g

:

X

→

ℂ

are functions, then we have

supp

⁡

(

f

⁢

g

)

⊂

supp

⁡

f

∩

supp

⁡

g

,

supp

⁡

(

f

+

g

)

⊂

supp

⁡

f

∪

supp

⁡

g

.

6.

If

Y

is another topological space, and

Ψ

:

Y

→

X

is a

homeomorphism

, then

supp

⁡

(

f

∘

Ψ

)

=

Ψ

-

1

⁢

(

supp

⁡

f

)

.

Title

support of function

Canonical name

SupportOfFunction

Date of creation

2013-03-22 13:46:10

Last modified on

2013-03-22 13:46:10

Owner

matte (1858)

Last modified by

matte (1858)

Numerical id

16

Author

matte (1858)

Entry type

Definition

Classification

msc 54-00

Synonym

support

Synonym

carrier

Related topic

ZeroOfAFunction

Related topic

ApplicationsOfUrysohnsLemmaToLocallyCompactHausdorffSpaces

Generated on Sat Feb 10 11:12:00 2018 by

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topological invariant

topological invariant

A

topological invariant

of a space

X

is a

property

that depends only on the

topology

of the space, i.e. it is shared by any

topological space

homeomorphic

to

X

. Common examples include

compactness

(

http://planetmath.org/

Compact

),

connectedness

(

http://planetmath.org/ConnectedSpace

), Hausdorffness (

http://planetmath.org/T2Space

),

Euler characteristic

, orientability (

http://planetmath.org/Orientation2

),

dimension

(

http://planetmath.org/InvarianceOfDimension

), and like

homology

,

homotopy groups

, and

K-theory

.

Properties of a space depending on an extra

structure

such as a metric (i.e. volume, curvature, symplectic

invariants

) typically are not topological invariants, though sometimes there are useful

interpretations

of topological invariants which seem to depend on extra information like a metric (for example, the Gauss-Bonnet theorem).

Title

topological invariant

Canonical

name

TopologicalInvariant

Date of creation

2013-03-22 13:42:07

Last modified on

2013-03-22 13:42:07

Owner

bwebste (988)

Last modified by

bwebste (988)

Numerical id

5

Author

bwebste (988)

Entry type

Definition

Classification

msc 54-00

Generated on Sat Feb 10 11:12:01 2018 by

LaTeXML