aarjaneiro/planetmath huggingface dataset:
totally disconnected space

totally disconnected space

A

topological space

X

is called

totally disconnected

if all

connected components

of

X

are

singletons

.

Examples of

totally disconnected spaces

include:

•

Every

discrete space

,

•

Cantor set,

•

the set of rational numbers

ℚ

endowed with the

subspace topology

induced

by

ℚ

↪

ℝ

and,

•

the set of

irrational numbers

ℝ

∖

ℚ

again with the subspace topology.

Title

totally disconnected space

Canonical

name

TotallyDisconnectedSpace

Date of creation

2013-03-22 13:25:31

Last modified on

2013-03-22 13:25:31

Owner

Dr_Absentius (537)

Last modified by

Dr_Absentius (537)

Numerical id

8

Author

Dr_Absentius (537)

Entry type

Definition

Classification

msc 54D99

Synonym

totally disconnected

Defines

totally disconnected space

Generated on Sat Feb 10 11:24:15 2018 by

LaTeXML
Moore space

Moore space

A

regular

developable

topological space

X

with a nested development

{

F

i

}

is called a

Moore space

, named after topologist R. L. Moore.

Moore spaces form a fertile bridge between point-set topology, metrizability theory, and

Zermelo-Fraenkel set theory

.

For example,

Jones’

Theorem

states that if

2

ℵ

0

<

2

ℵ

1

, then every

separable

normal Moore space is

metrizable

.

References

1

Steen, Lynn Arthur and Seebach, J. Arthur,

Counterexamples in Topology

, Dover Books, 1995.

Title

Moore space

Canonical name

MooreSpace

Date of creation

2013-03-22 14:49:51

Last modified on

2013-03-22 14:49:51

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

5

Author

mathcam (2727)

Entry type

Definition

Classification

msc 54E05

Classification

msc 54E30

Defines

Jones’ theorem

Generated on Sat Feb 10 11:24:17 2018 by

LaTeXML
complete uniform space

complete uniform space

Let

X

be a

uniform space

with uniformity

𝒰

. A filter

ℱ

on

X

is said to be a

Cauchy filter

if for each entourage

V

in

𝒰

, there is an

F

∈

ℱ

such that

F

×

F

⊆

V

.

We say that

X

is

complete

if every Cauchy filter is a

convergent filter

in the

topology

T

𝒰

induced

(

http://planetmath.org/TopologyInducedByUniformStructure

) by

𝒰

.

𝒰

in this case is called a

complete uniformity

.

A

Cauchy sequence

{

x

i

}

in a uniform space

X

is a

sequence

in

X

whose

section filter

is a Cauchy filter. A Cauchy sequence is said to be convergent if its section filter is convergent.

X

is said to be

sequentially complete

if every Cauchy sequence

converges

(every section filter of it converges).

Remark

. This is a

generalization

of the

concept

of completeness in a

metric space

, as a metric space is a uniform space. As we see above, in the course of this generalization, two notions of completeness emerge: that of completeness and sequentially completeness. Clearly, completeness always imply sequentially completeness. In the context of a metric space, or a

metrizable

uniform space, the two notions are

indistinguishable

: sequentially completeness also implies completeness.

Title

complete uniform space

Canonical

name

CompleteUniformSpace

Date of creation

2013-03-22 16:41:40

Last modified on

2013-03-22 16:41:40

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

7

Author

CWoo (3771)

Entry type

Definition

Classification

msc 54E15

Synonym

semicomplete

Synonym

semi-complete

Related topic

Complete

Defines

Cauchy filter

Defines

Cauchy sequence

Defines

sequentially complete

Defines

complete uniformity

Generated on Sat Feb 10 11:24:47 2018 by

LaTeXML
equivalent condition for being a fundamental system of entourages

equivalent condition for being a fundamental system of entourages

Lemma

.

Let

X

be a set and let

B

be a nonempty family of subsets of

X

×

X

. Then

B

is a

fundamental system of entourages

of a

uniformity

on

X

if and only if it

satisfies

the following axioms.

•

(B1) If

S

,

T

∈

ℬ

, then

S

∩

T

contains an element of

ℬ

.

•

(B2) Each element of

ℬ

contains the diagonal

Δ

⁢

(

X

)

.

•

(B3) For any

S

∈

ℬ

, the

inverse relation

of

S

contains an element of

ℬ

.

•

(B4) For any

S

∈

ℬ

, there is an

element

T

∈

ℬ

such that the

relational composition

T

∘

T

is

contained

in

S

.

Proof.

Suppose

ℬ

is a fundamental system of entourages for a uniformity

𝒰

. Verification of axiom (B2) is immediate, since

ℬ

⊆

𝒰

and each entourage is already required to contain the diagonal of

X

. We will prove that

ℬ

satisfies (B1); the proofs that (B3) and (B4) hold are analogous.

Let

S

,

T

be entourages in

ℬ

⊆

𝒰

. Since

𝒰

is

closed under

binary

intersections

,

S

∩

T

∈

𝒰

. By the definition of fundamental system of entourages, since

S

∩

T

∈

𝒰

, there exists an entourage

B

∈

ℬ

such that

B

⊆

S

∩

T

. Thus

ℬ

satisfies axioms (B1) through (B4).

To prove the

converse

, define a family of subsets of

X

×

X

by

𝒰

=

{

S

⊆

X

×

X

:

B

⊆

S

⁢

for some

B

∈

ℬ

}

.

By construction, each element of

𝒰

contains an element of

ℬ

, so all that remains is to show that

𝒰

is a uniformity. Suppose

T

is a subset of

X

×

X

that contains an element

S

∈

𝒰

. By the definition of

𝒰

, there exists some

B

∈

ℬ

such that

B

⊆

S

. Since

S

⊆

T

, it follows that

B

⊆

T

, so

T

satisfies the requirement for membership in

𝒰

. Thus

𝒰

is closed under taking

supersets

. The remaining axioms for a uniformity follow directly from the appropriate axioms for the fundamental system of entourages by applying the axiom we have just checked. Hence

ℬ

is a fundamental system of entourages for a uniformity on

X

.
∎

Title

equivalent

condition for being a fundamental system of entourages

Canonical name

EquivalentConditionForBeingAFundamentalSystemOfEntourages

Date of creation

2013-03-22 16:30:02

Last modified on

2013-03-22 16:30:02

Owner

mps (409)

Last modified by

mps (409)

Numerical id

5

Author

mps (409)

Entry type

Derivation

Classification

msc 54E15

Generated on Sat Feb 10 11:24:49 2018 by

LaTeXML
fundamental system of entourages

fundamental system of entourages

Let

(

X

,

𝒰

)

be a

uniform space

. A subset

ℬ

⊆

𝒰

is a

fundamental system of entourages

for

𝒰

provided that each entourage in

𝒰

contains an element of

ℬ

.

To see that each uniform space

(

X

,

𝒰

)

has a fundamental system of entourages, define

ℬ

=

{

U

∩

U

-

1

:

U

∈

𝒰

}

,

where

U

-

1

denotes the

inverse relation

of

U

. Since

𝒰

is

closed under

taking relational

inverses

and binary

intersections

,

ℬ

⊆

𝒰

. By construction, each

U

∈

𝒰

contains the element of

U

∩

U

-

1

∈

ℬ

.

There is a useful

equivalent

condition for being a fundamental system of entourages. Let

ℬ

be a nonempty family of subsets of

X

×

X

. Then

ℬ

is a fundamental system of entourages of a uniformity on

X

if and only if it the following axioms.

•

(B1) If

S

,

T

∈

ℬ

, then

S

∩

T

contains an element of

ℬ

.

•

(B2) Each element of

ℬ

contains the diagonal

Δ

⁢

(

X

)

.

•

(B3) For any

S

∈

ℬ

, the inverse relation of

S

contains an element of

ℬ

.

•

(B4) For any

S

∈

ℬ

, there is an

element

T

∈

ℬ

such that the

relational composition

T

∘

T

is

contained

in

S

.

Suppose

ℬ

is a fundamental system of entourages for uniformities

𝒰

and

𝒱

. Then

𝒰

⊂

𝒱

. To see this, suppose

S

∈

𝒰

. Since

ℬ

is a fundamental system of entourages for

𝒰

, there is some element

B

∈

ℬ

such that

B

⊂

S

. But

ℬ

⊂

𝒱

, so

B

∈

𝒱

. Hence by applying the fact that

𝒱

is closed under taking

supersets

we may conclude that

S

∈

𝒱

. So if

ℬ

is a fundamental system of entourages, it is a fundamental system for a unique uniformity

𝒰

. Thus it makes sense to call

𝒰

the

uniformity generated by the fundamental system

ℬ

.

References

1

Nicolas Bourbaki,

Elements of Mathematics: General Topology: Part 1

, Hermann, 1966.

Title

fundamental system of entourages

Canonical name

FundamentalSystemOfEntourages

Date of creation

2013-03-22 16:29:55

Last modified on

2013-03-22 16:29:55

Owner

mps (409)

Last modified by

mps (409)

Numerical id

5

Author

mps (409)

Entry type

Definition

Classification

msc 54E15

Defines

uniformity generated by

Generated on Sat Feb 10 11:24:51 2018 by

LaTeXML
generalization of a uniformity

generalization of a uniformity

Let

X

be a set. Let

𝒰

be a family of subsets of

X

×

X

such that

𝒰

is a filter, and that every element of

𝒰

contains the

diagonal relation

Δ

(

reflexive

). Consider the following possible “axioms”:

1.

for every

U

∈

𝒰

,

U

-

1

∈

𝒰

2.

for every

U

∈

𝒰

, there is

V

∈

𝒰

such that

V

∘

V

∈

U

,

where

U

-

1

is defined as the inverse relation (

http://planetmath.org/OperationsOnRelations

) of

U

, and

∘

is the

composition of relations

(

http://planetmath.org/OperationsOnRelations

). If

𝒰

satisfies

Axiom 1, then

𝒰

is called a

semi-uniformity

. If

𝒰

satisfies Axiom 2, then

𝒰

is called a

quasi-uniformity

. The underlying set

X

equipped with

𝒰

is called a

semi-uniform space

or a

quasi-uniform space

according to whether

𝒰

is a semi-uniformity or a quasi-uniformity.

A

semi-pseudometric space

is a semi-uniform space. A quasi-pseudometric space is a quasi-uniform space.

A

uniformity

is one that satisfies both axioms, which is

equivalent

to saying that it is both a semi-uniformity and a quasi-uniformity.

References

1

W. Page,

Topological Uniform Structures

, Wiley, New York 1978.

Title

generalization

of a uniformity

Canonical name

GeneralizationOfAUniformity

Date of creation

2013-03-22 16:43:09

Last modified on

2013-03-22 16:43:09

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

5

Author

CWoo (3771)

Entry type

Definition

Classification

msc 54E15

Synonym

semiuniformity

Synonym

quasiuniformity

Synonym

semiuniform space

Synonym

quasiuniform space

Synonym

semi-uniform

Synonym

quasi-uniform

Synonym

semiuniform

Synonym

quasiuniform

Related topic

GeneralizationOfAPseudometric

Defines

semi-uniformity

Defines

quasi-uniformity

Defines

semi-uniform space

Defines

quasi-uniform space

Generated on Sat Feb 10 11:25:01 2018 by

LaTeXML
product of uniform spaces

product of uniform spaces

Definition

.

Let

{

X

α

}

α

∈

I

be a nonempty family of

uniform spaces

. The

is the weakest uniformity (

http://planetmath.org/UniformitiesOnASetFormACompleteLattice

) on the

Cartesian product

X

=

∏

α

∈

I

X

α

making all the

projection maps

π

α

:

X

→

X

α

uniformly continuous

.

Title

product of uniform spaces

Canonical name

ProductOfUniformSpaces

Date of creation

2013-03-22 16:30:49

Last modified on

2013-03-22 16:30:49

Owner

mps (409)

Last modified by

mps (409)

Numerical id

4

Author

mps (409)

Entry type

Definition

Classification

msc 54E15

Generated on Sat Feb 10 11:25:03 2018 by

LaTeXML
proof that uniformly continuous is proximity continuous

proof that uniformly continuous is proximity continuous

Let

f

:

X

→

Y

be a

uniformly continuous function

from

uniform spaces

X

to

Y

with uniformities

𝒰

and

𝒱

respectively. Let

δ

and

ϵ

be the

proximities

generated by (

http://planetmath.org/UniformProximity

)

𝒰

and

𝒱

respectively. It is known that

X

and

Y

are proximity spaces with proximities

δ

and

ϵ

respectively. Furthermore, we have the following:

Theorem 1

.

f

:

X

→

Y

is

proximity continuous

.

Proof.

Let

A

,

B

be any subsets of

X

with

A

⁢

δ

⁢

B

. We want to show that

f

⁢

(

A

)

⁢

ϵ

⁢

f

⁢

(

B

)

, or equivalently,

V

⁢

[

f

⁢

(

A

)

]

∩

V

⁢

[

f

⁢

(

B

)

]

≠

∅

,

for any

V

∈

𝒱

. Pick any

V

∈

𝒱

. Since

f

is

uniformly continuous

, there is

U

∈

𝒰

such that

U

⁢

[

x

]

⊆

f

-

1

⁢

(

V

⁢

[

f

⁢

(

x

)

]

)

,

for any

x

∈

X

. As a result,

U

⁢

[

A

]

⊆

f

-

1

⁢

(

V

⁢

[

f

⁢

(

A

)

]

)

,

which implies that

f

⁢

(

U

⁢

[

A

]

)

⊆

V

⁢

[

f

⁢

(

A

)

]

.

Similarly

f

⁢

(

U

⁢

[

B

]

)

⊆

V

⁢

[

f

⁢

(

B

)

]

. Now,

A

⁢

δ

⁢

B

is

equivalent

to

U

⁢

[

A

]

∩

U

⁢

[

B

]

≠

∅

, so we can pick

z

∈

U

⁢

[

A

]

∩

U

⁢

[

B

]

.

Then

f

⁢

(

z

)

∈

f

⁢

(

U

⁢

[

A

]

)

∩

f

⁢

(

U

⁢

[

B

]

)

⊆

V

⁢

[

f

⁢

(

A

)

]

∩

V

⁢

[

f

⁢

(

B

)

]

,

and therefore

V

⁢

[

f

⁢

(

A

)

]

∩

V

⁢

[

f

⁢

(

B

)

]

≠

∅

.

This shows that

f

is proximity continuous.
∎

Title

proof that uniformly continuous is proximity continuous

Canonical name

ProofThatUniformlyContinuousIsProximityContinuous

Date of creation

2013-03-22 18:07:50

Last modified on

2013-03-22 18:07:50

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

6

Author

CWoo (3771)

Entry type

Proof

Classification

msc 54E15

Classification

msc 54C08

Classification

msc 54C05

Classification

msc 54E17

Classification

msc 54E05

Generated on Sat Feb 10 11:25:14 2018 by

LaTeXML
separated uniform space

separated uniform space

Let

X

be a

uniform space

with uniformity

𝒰

.

X

is said to be

separated

or

Hausdorff

if it

satisfies

the following

separation axiom

:

⋂

𝒰

=

Δ

,

where

Δ

is the

diagonal relation

on

X

and

⋂

𝒰

is the

intersection

of all

elements

(entourages) in

𝒰

. Since

Δ

⊆

⋂

𝒰

, the separation axiom says that the only elements that belong to every entourage of

𝒰

are precisely the diagonal elements

(

x

,

x

)

. Equivalently, if

x

≠

y

, then there is an entourage

U

such that

(

x

,

y

)

∉

U

.

The reason for calling

X

separated has to do with the following assertion:

X

is separated iff

X

is a Hausdorff space under the

topology

T

𝒰

induced

by (

http://planetmath.org/TopologyInducedByAUniformStructure

)

𝒰

.

Recall that

T

𝒰

=

{

A

⊆

X

∣

for each

⁢

x

∈

A

⁢

, there is

⁢

U

∈

𝒰

⁢

, such that

⁢

U

⁢

[

x

]

⊆

A

}

, where

U

⁢

[

x

]

is some

uniform neighborhood

of

x

where, under

T

𝒰

,

U

⁢

[

x

]

is also a

neighborhood

of

x

. To say that

X

is Hausdorff under

T

𝒰

is the same as saying every pair of distinct points in

X

have

disjoint

uniform neighborhoods.

Proof.

(

⇒

)

. Suppose

X

is separated and

x

,

y

∈

X

are distinct. Then

(

x

,

y

)

∉

U

for some

U

∈

𝒰

. Pick

V

∈

𝒰

with

V

∘

V

⊆

U

. Set

W

=

V

∩

V

-

1

, then

W

is

symmetric

and

W

⊆

V

. Furthermore,

W

∘

W

⊆

V

∘

V

⊆

U

. If

z

∈

W

⁢

[

x

]

∩

W

⁢

[

y

]

, then

(

x

,

z

)

,

(

y

,

z

)

∈

W

. Since

W

is symmetric,

(

z

,

y

)

∈

W

, so

(

x

,

y

)

=

(

x

,

z

)

∘

(

z

,

y

)

∈

W

∘

W

⊆

U

, which is a

contradiction

.

(

⇐

)

. Suppose

X

is Hausdorff under

T

𝒰

and

(

x

,

y

)

∈

U

for every

U

∈

𝒰

for some

x

,

y

∈

X

. If

x

≠

y

, then there are

V

⁢

[

x

]

∩

W

⁢

[

y

]

=

∅

for some

V

,

W

∈

𝒰

. Since

(

x

,

y

)

∈

V

by

assumption

,

y

∈

V

⁢

[

x

]

. But

y

∈

W

⁢

[

y

]

, contradicting the disjointness of

V

⁢

[

x

]

and

W

⁢

[

y

]

. Therefore

x

=

y

.
∎

Title

separated uniform space

Canonical name

SeparatedUniformSpace

Date of creation

2013-03-22 16:42:34

Last modified on

2013-03-22 16:42:34

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

5

Author

CWoo (3771)

Entry type

Definition

Classification

msc 54E15

Synonym

separating

Synonym

Hausdorff uniform space

Generated on Sat Feb 10 11:25:22 2018 by

LaTeXML
topology induced by uniform structure

topology induced by uniform structure

Let

𝒰

be a

uniform structure

on a set

X

. We define a subset

A

to be open if and only if for each

x

∈

A

there exists an entourage

U

∈

𝒰

such that whenever

(

x

,

y

)

∈

U

, then

y

∈

A

.

Let us verify that this defines a

topology

on

X

.

Clearly, the subsets

∅

and

X

are open. If

A

and

B

are two

open sets

, then for each

x

∈

A

∩

B

, there exist an entourage

U

such that, whenever

(

x

,

y

)

∈

U

, then

y

∈

A

, and an entourage

V

such that, whenever

(

x

,

y

)

∈

V

, then

y

∈

B

. Consider the entourage

U

∩

V

: whenever

(

x

,

y

)

∈

U

∩

V

, then

y

∈

A

∩

B

, hence

A

∩

B

is open.

Suppose

ℱ

is an arbitrary family of open subsets. For each

x

∈

⋃

ℱ

, there exists

A

∈

ℱ

such that

x

∈

A

. Let

U

be the entourage whose existence is granted by the

definition

of open set. We have that whenever

(

x

,

y

)

∈

U

, then

y

∈

A

; hence

y

∈

⋃

ℱ

, which concludes the proof.

Title

topology induced by uniform structure

Canonical

name

TopologyInducedByUniformStructure

Date of creation

2013-03-22 12:46:44

Last modified on

2013-03-22 12:46:44

Owner

Mathprof (13753)

Last modified by

Mathprof (13753)

Numerical id

7

Author

Mathprof (13753)

Entry type

Derivation

Classification

msc 54E15

Related topic

UniformNeighborhood

Defines

uniform topology

Generated on Sat Feb 10 11:25:24 2018 by

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uniform continuity

uniform continuity

In this entry, we extend the usual definition of a

uniformly continuous function

between

metric spaces

to arbitrary

uniform spaces

.

Let

(

X

,

𝒰

)

,

(

Y

,

𝒱

)

be uniform spaces (the second

component

is the uniformity on the first component). A

function

f

:

X

→

Y

is said to be

uniformly continuous

if for any

V

∈

𝒱

there is a

U

∈

𝒰

such that for all

x

∈

X

,

U

⁢

[

x

]

⊆

f

-

1

⁢

(

V

⁢

[

f

⁢

(

x

)

]

)

.

Sometimes it is useful to use an alternative but

equivalent

version of uniform continuity of a function:

Proposition 1

.

Suppose

f

:

X

→

Y

is a function and

g

:

X

×

X

→

Y

×

Y

is defined by

g

⁢

(

x

1

,

x

2

)

=

(

f

⁢

(

x

1

)

,

f

⁢

(

x

2

)

)

. Then

f

is uniformly continuous iff for any

V

∈

V

, there is a

U

∈

U

such that

U

⊆

g

-

1

⁢

(

V

)

.

Proof.

Suppose

f

is uniformly continuous. Pick any

V

∈

𝒱

. Then

U

∈

𝒰

exists with

U

⁢

[

x

]

⊆

f

-

1

⁢

(

V

⁢

[

f

⁢

(

x

)

]

)

for all

x

∈

X

. If

(

a

,

b

)

∈

U

, then

b

∈

U

⁢

[

a

]

⊆

f

-

1

⁢

(

V

⁢

[

f

⁢

(

a

)

]

)

, or

f

⁢

(

b

)

⊆

V

⁢

[

f

⁢

(

a

)

]

, or

g

⁢

(

a

,

b

)

=

(

f

⁢

(

a

)

,

f

⁢

(

b

)

)

∈

V

. The

converse

is straightforward.
∎

Remark

. Note that we could have picked

U

so the

inclusion

becomes an equality.

Proposition 2

.

. If

f

:

X

→

Y

is uniformly continuous, then it is

continuous

under the

uniform topologies

of

X

and

Y

.

Proof.

Let

A

be open in

Y

and set

B

=

f

-

1

⁢

(

A

)

. Pick any

x

∈

B

. Then

y

=

f

⁢

(

x

)

has a

uniform neighborhood

V

⁢

[

y

]

⊆

A

. By the uniform continuity of

f

, there is an entourage

U

∈

𝒰

with

x

∈

U

⁢

[

x

]

⊆

f

-

1

⁢

(

V

⁢

[

y

]

)

⊆

f

-

1

⁢

(

A

)

=

B

.
∎

Remark

. The converse is not true, even in metric spaces.

Title

uniform continuity

Canonical name

UniformContinuity

Date of creation

2013-03-22 16:43:15

Last modified on

2013-03-22 16:43:15

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

7

Author

CWoo (3771)

Entry type

Definition

Classification

msc 54E15

Related topic

UniformlyContinuous

Related topic

UniformContinuityOverLocallyCompactQuantumGroupoids

Defines

uniformly continuous

Generated on Sat Feb 10 11:25:33 2018 by

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uniform space

uniform space

A

uniform structure

(or

uniformity

) on a set

X

is a non

empty set

𝒰

of subsets of

X

×

X

which

satisfies

the following axioms:

1.

Every subset of

X

×

X

which contains a set of

𝒰

belongs to

𝒰

.

2.

Every finite

intersection of sets

of

𝒰

belongs to

𝒰

.

3.

Every set of

𝒰

is a

reflexive relation

on

X

(i.e. contains the diagonal).

4.

If

V

belongs to

𝒰

, then

V

′

=

{

(

y

,

x

)

:

(

x

,

y

)

∈

V

}

belongs to

𝒰

.

5.

If

V

belongs to

𝒰

, then exists

V

′

in

𝒰

such that, whenever

(

x

,

y

)

,

(

y

,

z

)

∈

V

′

, then

(

x

,

z

)

∈

V

(i.e.

V

′

∘

V

′

⊆

V

).

The sets of

𝒰

are called

entourages

or

vicinities

. The set

X

together with the uniform structure

𝒰

is called a

uniform space

.

If

V

is an entourage, then for any

(

x

,

y

)

∈

V

we say that

x

and

y

are

V

-close

.

Every uniform space can be considered a

topological space

with a natural topology induced by uniform structure. The uniformity, however, provides in general a richer

structure

, which formalize the

concept

of relative closeness: in a uniform space we can say that

x

is close to

y

as

z

is to

w

, which makes no sense in a topological space. It follows that uniform spaces are the most natural environment for

uniformly continuous functions

and

Cauchy sequences

, in which these concepts are naturally involved.

Examples of uniform spaces are

metric spaces

,

topological groups

, and

topological vector spaces

.

Title

uniform space

Canonical

name

UniformSpace

Date of creation

2013-03-22 12:46:26

Last modified on

2013-03-22 12:46:26

Owner

mps (409)

Last modified by

mps (409)

Numerical id

12

Author

mps (409)

Entry type

Definition

Classification

msc 54E15

Defines

uniform structure

Defines

uniformity

Defines

entourage

Defines

V

-close

Defines

vicinity

Generated on Sat Feb 10 11:25:46 2018 by

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uniform structure of a metric space

uniform structure of a metric space

Let

(

X

,

d

)

be a

metric space

. There is a natural

uniform structure

on

X

, which induces the same

topology

as the metric. We define a subset

V

of the

Cartesian product

X

×

X

to be an entourage if and only if it contains a subset of the form

V

ε

=

{

(

x

,

y

)

∈

X

×

X

:

d

⁢

(

x

,

y

)

<

ε

}

for some

ε

.

Title

uniform structure of a metric space

Canonical name

UniformStructureOfAMetricSpace

Date of creation

2013-03-22 12:47:18

Last modified on

2013-03-22 12:47:18

Owner

n3o (216)

Last modified by

n3o (216)

Numerical id

6

Author

n3o (216)

Entry type

Derivation

Classification

msc 54E15

Generated on Sat Feb 10 11:25:47 2018 by

LaTeXML
uniform structure of a topological group

uniform structure of a topological group

Let

G

be a

topological group

. There is a natural

uniform structure

on

G

which induces its

topology

. We define a subset

V

of the

Cartesian product

G

×

G

to be an entourage if and only if it contains a subset of the form

V

N

=

{

(

x

,

y

)

∈

G

×

G

:

x

⁢

y

-

1

∈

N

}

for some

N

neighborhood

of the

identity element

. This is called the

right uniformity

of the topological group, with which

multiplication

becomes a uniformly continuous map.
The

left uniformity

is defined in a fashion, but in general they don’t coincide, although they both induce the same topology on

G

.

Title

uniform structure of a topological group

Canonical name

UniformStructureOfATopologicalGroup

Date of creation

2013-03-22 12:47:21

Last modified on

2013-03-22 12:47:21

Owner

mps (409)

Last modified by

mps (409)

Numerical id

10

Author

mps (409)

Entry type

Derivation

Classification

msc 54E15

Defines

right uniformity

Defines

left uniformity

Generated on Sat Feb 10 11:25:49 2018 by

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