aarjaneiro/planetmath huggingface dataset:
uniformities on a set form a complete lattice

uniformities on a set form a complete lattice

Theorem

.

The

collection

of

uniformities

on a given set ordered by

set inclusion

forms a

complete lattice

.

Proof.

Let

X

be a set. Let

𝔘

⁢

(

X

)

denote the collection of uniformities on

X

. The coarsest uniformity on

X

is

{

X

×

X

}

, and the finest is the

discrete uniformity

:

{

S

⊂

X

×

X

:

Δ

⁢

(

X

)

⊆

S

}

.

Hence

𝔘

⁢

(

X

)

is

bounded

. To show that

𝔘

⁢

(

X

)

is

complete

, we must prove that it has the least

upper bound

property

.

Suppose

{

𝒰

α

}

α

∈

I

is a nonempty family of uniformities on

X

. Let

ℬ

consist of all finite

intersections

of elements of the

𝒰

α

. Let us check that

ℬ

is a

fundamental system of entourages

for a uniformity on

X

.

(B1) Let

S

,

T

∈

ℬ

. Each of

S

and

T

is a finite intersection of elements of the

𝒰

α

, so their intersection is as well. Hence

S

∩

T

∈

ℬ

.

(B2) Every element of

ℬ

is a finite intersection of subsets of

X

×

X

containing

Δ

⁢

(

X

)

. So every element of

ℬ

contains the diagonal.

(B3) Let

S

∈

ℬ

. Without loss of generality,

S

=

S

α

∩

S

β

, where

S

α

∈

𝒰

α

and

S

β

∈

𝒰

β

. Since

S

α

∈

𝒰

α

,

S

α

-

1

∈

𝒰

α

. Similarly,

S

β

-

1

∈

𝒰

β

. Since the process of taking the

inverse

of a

relation

commutes with taking finite intersections,

(

S

α

∩

S

β

)

-

1

∈

ℬ

.

(B4) Let

S

∈

ℬ

. Again suppose

S

=

S

α

∩

S

β

with

S

α

∈

𝒰

α

and

S

β

∈

𝒰

β

. Then there exist

T

α

∈

𝒰

α

and

T

β

∈

𝒰

β

such that

T

α

∘

T

α

⊆

S

α

and

T

β

∘

T

β

⊆

S

β

. The set

T

=

T

α

∩

T

β

is in

𝒰

, and since

T

∘

T

is a subset of both

S

α

and

S

β

, it is a subset of

S

.

The fundamental system

ℬ

generates a uniformity

𝒰

. By construction,

𝒰

is an upper bound of the

𝒰

α

. But any upper bound of the

𝒰

α

would have to contain all finite intersections of elements of the

𝒰

α

. So

𝒰

=

sup

α

∈

I

⁡

𝒰

α

.
∎

This

theorem

is useful because it allows us to assert the

existence

of the coarsest uniform space satisfying a particular property.

Corollary

.

Let

X

be a set and let

{

Y

α

}

α

∈

I

be a family of uniform spaces. Then for any family of

functions

{

f

α

:

X

→

Y

α

}

, there is a coarsest uniformity on

X

making all the

f

α

uniformly continous.

The coarsest uniformity making a family of functions

uniformly continuous

is called the

initial uniformity

or

weak uniformity

.

References

1

Nicolas Bourbaki,

Elements of Mathematics: General Topology: Part 1

, Hermann, 1966.

Title

uniformities on a set form a complete lattice

Canonical

name

UniformitiesOnASetFormACompleteLattice

Date of creation

2013-03-22 16:30:46

Last modified on

2013-03-22 16:30:46

Owner

mps (409)

Last modified by

mps (409)

Numerical id

4

Author

mps (409)

Entry type

Derivation

Classification

msc 54E15

Classification

msc 06B23

Defines

discrete uniformity

Defines

initial uniformity

Defines

weak uniformity

Generated on Sat Feb 10 11:25:51 2018 by

LaTeXML
uniformizable space

uniformizable space

Let

X

be a

topological space

with

𝒯

the

topology

defined on it.

X

is said to be

uniformizable

1.

there is a

uniformity

𝒰

defined on

X

, and

2.

𝒯

=

T

𝒰

, the

uniform topology

induced by

𝒰

.

It can be shown that a topological space is uniformizable iff it is

completely regular

.

Clearly, every

pseudometric space

is uniformizable. The converse is true if the space has a countable basis. Pushing this idea further, one can show that a uniformizable space is

metrizable

iff it is

separating

(or

Hausdorff

) and has a countable basis.

Let

X

,

𝒯

, and

𝒰

be defined as above. Then

X

is said to be

completely uniformizable

if

𝒰

is a

complete uniformity

.

Every

paracompact space

is completely uniformizable. Every completely uniformizable space is completely regular, and hence uniformizable.

Title

uniformizable space

Canonical name

UniformizableSpace

Date of creation

2013-03-22 16:49:05

Last modified on

2013-03-22 16:49:05

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

5

Author

CWoo (3771)

Entry type

Definition

Classification

msc 54E15

Defines

uniformizable

Defines

completely uniformizable

Generated on Sat Feb 10 11:25:59 2018 by

LaTeXML
proximity generated by uniformity

proximity generated by uniformity

Definition

. Let

X

be a

uniform space

with uniformity

𝒰

. The

uniform proximity

, or

proximity

generated by

U

, is a

binary relation

𝛿

on the subsets of

X

, given by the

formula

A

𝛿

B

⇔

∀

U

∈

𝒰

:

U

∩

(

A

×

B

)

≠

∅

.

Correctness of the above proximity is given by the below

theorem

:

Theorem

. Proximity generated by a uniformity is a proximity. In other words,

X

is a proximity space with proximity

𝛿

.

Title

proximity generated by uniformity

Canonical name

ProximityGeneratedByUniformity

Date of creation

2013-03-22 16:56:20

Last modified on

2013-03-22 16:56:20

Owner

porton (9363)

Last modified by

porton (9363)

Numerical id

9

Author

porton (9363)

Entry type

Definition

Classification

msc 54E17

Classification

msc 54E15

Classification

msc 54E05

Defines

uniform proximity

Generated on Sat Feb 10 11:26:01 2018 by

LaTeXML
P-space

P-space

Suppose

X

is a

completely regular

topological space

. Then

X

is said to be a

P-space

if every

prime ideal

in

C

⁢

(

X

)

, the

ring of continuous functions

on

X

, is

maximal

.

For example, every space with the

discrete topology

is a P-space.

Algebraically, a

commutative

reduced ring

R

with

1

such that every prime ideal is maximal is

equivalent

to any of the following statements:

•

R

is von-Neumann

regular

,

•

every ideal in

R

is the

intersection

of prime ideals,

•

every ideal in

R

is the intersection of

maximal ideals

,

•

every

principal ideal

is generated by an

idempotent

.

When

R

=

C

⁢

(

X

)

, then

R

is commutative

reduced

with

1

. In

addition

to the

algebraic

characterizations

of

R

above,

X

being a P-space is equivalent to any of the following statements:

•

every

zero set

is open

•

if

f

,

g

∈

C

⁢

(

X

)

, then

(

f

,

g

)

=

(

f

2

+

g

2

)

.

Some properties of P-spaces:

1.

Every

subspace

of a P-space is a P-space,

2.

Every

quotient space

of a P-space is a P-space,

3.

Every finite

product

of P-spaces is a P-space,

4.

Every P-space has a base of

clopen sets

.

For more properties of P-spaces, please see the reference below. For proofs of the above properties and equivalent characterizations, see

http://planetmath.org/?op=getobj&from=books&id=46

here.

References

1

L. Gillman, M. Jerison:

Rings of Continuous Functions

, Van Nostrand, (1960).

Title

P-space

Canonical name

Pspace

Date of creation

2013-03-22 18:53:13

Last modified on

2013-03-22 18:53:13

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

6

Author

CWoo (3771)

Entry type

Definition

Classification

msc 54E18

Classification

msc 16S60

Generated on Sat Feb 10 11:26:19 2018 by

LaTeXML
hemimetric

hemimetric

A

hemimetric

on a set

X

is a function

d

:

X

×

X

→

ℝ

such that

1.

d

⁢

(

x

,

y

)

≥

0

;

2.

d

⁢

(

x

,

z

)

≤

d

⁢

(

x

,

y

)

+

d

⁢

(

y

,

z

)

;

3.

d

⁢

(

x

,

x

)

=

0

;

for all

x

,

y

,

z

∈

X

.

Hence, essentially

d

is a metric which fails to satisfy symmetry and the

property

that distinct points have

positive

distance

.
A hemimetric induces a

topology

on

X

in the same way that a metric does, a basis of open sets being

{

B

⁢

(

x

,

r

)

:

x

∈

X

,

r

>

0

}

,

where

B

⁢

(

x

,

r

)

=

{

y

∈

X

:

d

⁢

(

x

,

y

)

<

r

}

is the

r

-ball centered at

x

.

Title

hemimetric

Canonical

name

Hemimetric

Date of creation

2013-03-22 14:24:12

Last modified on

2013-03-22 14:24:12

Owner

Koro (127)

Last modified by

Koro (127)

Numerical id

5

Author

Koro (127)

Entry type

Definition

Classification

msc 54E25

Generated on Sat Feb 10 11:26:21 2018 by

LaTeXML
semimetric

semimetric

A

semimetric

on a set

X

is a

function

d

:

X

×

X

→

ℝ

which satisfies:

1.

d

⁢

(

x

,

y

)

≥

0

2.

d

⁢

(

x

,

y

)

=

0

if and only if

x

=

y

;

3.

d

⁢

(

x

,

y

)

=

d

⁢

(

y

,

x

)

.

A semimetric differs from a metric in that the

triangle inequality

is not required to hold.

Title

semimetric

Canonical name

Semimetric

Date of creation

2013-03-22 14:24:15

Last modified on

2013-03-22 14:24:15

Owner

Koro (127)

Last modified by

Koro (127)

Numerical id

8

Author

Koro (127)

Entry type

Definition

Classification

msc 54E25

Related topic

GeneralizationOfAPseudometric

Generated on Sat Feb 10 11:26:23 2018 by

LaTeXML
relatively prime integer topology

relatively prime integer topology

Let

X

be the set of

strictly

positive integers

. The

relatively

prime integer topology

on

X

is the

topology

determined by a basis consisting of the sets

U

⁢

(

a

,

b

)

=

{

a

⁢

x

+

b

∣

x

∈

X

}

for any

a

and

b

are relatively

prime integers

. That this does indeed form a basis is found in this entry. (

http://planetmath.org/HausdorffSpaceNotCompletelyHausdorff

)

Equipped with this topology,

X

is

T

0

(

http://planetmath.org/T0Space

),

T

1

(

http://planetmath.org/T1Space

),and

T

2

(

http://planetmath.org/T2Space

), but

satisfies

none of the higher

separation axioms

(and hence meet very few

compactness

criteria).

We can define a

coarser

topology on

X

by considering the

subbasis

of the above basis consisting of all

U

⁢

(

a

,

b

)

with

a

being a prime. This is called the

prime integer topology

on

ℤ

+

.

References

1

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

Counterexamples in topology

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

Title

relatively prime integer topology

Canonical

name

RelativelyPrimeIntegerTopology

Date of creation

2013-03-22 14:42:07

Last modified on

2013-03-22 14:42:07

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

7

Author

mathcam (2727)

Entry type

Definition

Classification

msc 54E30

Defines

prime integer topology

Generated on Sat Feb 10 11:26:24 2018 by

LaTeXML
alternative definition of metric space

alternative definition of metric space

1

Definition

The notion of

metric space

may be defined in a way which makes use only of

rational numbers

as opposed to

real numbers

. To avoid mention of real numbers we take a three place

relation

as our

primitive

term instead of the distance function. Thus, one can use this

definition

to define the set of real numbers as the

completion

of the set of rational numbers as a metric space. Let

ℚ

+

denote the set of

strictly

positive rational numbers

Definition 1

.

A

metric space

consists of a set

X

and a relation

D

⊂

X

×

X

×

Q

+

which

satisfies

the following

properties

:

1.

For all

x

,

y

∈

X

, it is the case that

D

⁢

(

x

,

y

,

r

)

for all

r

∈

ℚ

+

if and only if

x

=

y

.

2.

For all

x

,

y

∈

X

and all

r

∈

ℚ

+

, it is the case that

D

⁢

(

x

,

y

,

r

)

if and only if

D

⁢

(

y

,

x

,

r

)

.

3.

For all

x

,

y

∈

X

and all

r

,

s

∈

ℚ

+

, if

r

≤

s

and

D

⁢

(

x

,

y

,

r

)

then

D

⁢

(

x

,

y

,

s

)

.

4.

For all

x

,

y

,

z

∈

X

and all

r

,

s

∈

ℚ

+

, if

D

⁢

(

x

,

y

,

r

)

and

D

⁢

(

y

,

z

,

s

)

, then

D

⁢

(

x

,

z

,

r

+

s

)

.

2

Equivalence with usual definition

The definition presented above is

equivalent

to the definition presented in
metric space (

http://

planetmath

.org/MetricSpace

).
This equivalence is
given by the following equations which relate the primitive term “

D

” used in this definition with the primitive
term “

d

” of the previous definition:

d

⁢

(

x

,

y

)

=

inf

⁡

{

r

∣

D

⁢

(

x

,

y

,

r

)

}

D

⁢

(

x

,

y

,

r

)

↔

d

⁢

(

x

,

y

)

≤

r

Title

alternative definition of metric space

Canonical

name

AlternativeDefinitionOfMetricSpace

Date of creation

2013-03-22 15:09:19

Last modified on

2013-03-22 15:09:19

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

8

Author

rspuzio (6075)

Entry type

Definition

Classification

msc 54E35

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

LaTeXML
bounded

bounded

Given a metric space

(

X

,

d

)

, a subset

A

⊆

X

is said to be

bounded

if there is some positive real number

M

such that

d

⁢

(

x

,

y

)

≤

M

whenever

x

,

y

∈

A

.

A function

f

:

X

→

Y

from a set

X

to a metric space

Y

is said to be

bounded

if its range is bounded in

Y

.

Title

bounded

Canonical

name

Bounded

Date of creation

2013-03-22 12:05:11

Last modified on

2013-03-22 12:05:11

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

12

Author

PrimeFan (13766)

Entry type

Definition

Classification

msc 54E35

Related topic

TotallyBounded

Related topic

AlternateStatementOfBolzanoWeierstrassTheorem

Generated on Sat Feb 10 11:26:36 2018 by

LaTeXML
bounded

bounded

Let

X

be a subset of

ℝ

. We say that

X

is

bounded

when there exists a

real number

M

such that

|

x

|

<

M

for all

x

∈

X

. When

X

is an

interval

, we speak of a

bounded interval

.

This can be generalized first to

ℝ

n

. We say that

X

⊆

ℝ

n

is bounded if there is a real number

M

such that

∥

x

∥

<

M

for all

x

∈

X

and

∥

⋅

∥

is the

Euclidean distance

between

x

and

y

.

This condition is

equivalent

to the statement: There is a real number

T

such that

∥

x

-

y

∥

<

T

for all

x

,

y

∈

X

.

A further

generalization

to any

metric space

V

says that

X

⊆

V

is bounded when there is a real number

M

such that

d

⁢

(

x

,

y

)

<

M

for all

x

,

y

∈

X

, where

d

is the metric on

V

.

Title

bounded

Canonical name

Bounded1

Date of creation

2013-03-22 14:00:00

Last modified on

2013-03-22 14:00:00

Owner

yark (2760)

Last modified by

yark (2760)

Numerical id

11

Author

yark (2760)

Entry type

Definition

Classification

msc 54E35

Related topic

EuclideanDistance

Related topic

MetricSpace

Defines

bounded interval

Generated on Sat Feb 10 11:26:38 2018 by

LaTeXML
Cauchy sequence

Cauchy sequence

A

sequence

x

0

,

x

1

,

x

2

,

…

in a

metric space

(

X

,

d

)

is a

Cauchy sequence

if, for every

real number

ϵ

>

0

, there exists a

natural number

N

such that

d

⁢

(

x

n

,

x

m

)

<

ϵ

whenever

n

,

m

>

N

.

Likewise, a sequence

v

0

,

v

1

,

v

2

,

…

in a

topological vector space

V

is a

Cauchy sequence

if and only if for every

neighborhood

U

of

𝟎

, there exists a natural number

N

such that

v

n

-

v

m

∈

U

for all

n

,

m

>

N

. These two

definitions

are

equivalent

when the

topology

of

V

is induced by a metric.

Title

Cauchy sequence

Canonical

name

CauchySequence

Date of creation

2013-03-22 11:55:04

Last modified on

2013-03-22 11:55:04

Owner

djao (24)

Last modified by

djao (24)

Numerical id

10

Author

djao (24)

Entry type

Definition

Classification

msc 54E35

Classification

msc 26A03

Synonym

fundamental sequence

Related topic

MetricSpace

Generated on Sat Feb 10 11:26:40 2018 by

LaTeXML
city-block metric

city-block metric

The

city-block metric

, defined on

ℝ

n

, is

d

⁢

(

a

,

b

)

=

∑

i

=

1

n

|

b

i

-

a

i

|

where

a

and

b

are vectors in

ℝ

n

with

a

=

(

a

1

,

…

,

a

n

)

and

b

=

(

b

1

,

…

,

b

n

)

.

In two

dimensions

and with discrete-valued vectors, when we can picture the set of points in

ℤ

×

ℤ

as a grid, this is simply the number of edges between points that must be traversed to get from

a

to

b

within the grid. This is the same problem as getting from corner

a

to

b

in a rectilinear downtown area, hence the name “city-block metric.”

Title

city-block metric

Canonical

name

CityblockMetric

Date of creation

2013-03-22 12:12:57

Last modified on

2013-03-22 12:12:57

Owner

akrowne (2)

Last modified by

akrowne (2)

Numerical id

9

Author

akrowne (2)

Entry type

Definition

Classification

msc 54E35

Synonym

city-block distance

Synonym

taxicab metric

Generated on Sat Feb 10 11:26:42 2018 by

LaTeXML
closed complex plane

closed complex plane

The

complex plane

ℂ

, i.e. the set of the

complex numbers

z

satisfying

|

z

|

<

∞

,

is open but not closed, since it doesn’t contain the

accumulation points

of all sets of complex numbers, for example of the set

{

1

,

2

,

3

,

…

}

.  One can

ℂ

to the

closed complex plane

ℂ

∪

{

∞

}

by adding to

ℂ

the

infinite

point

∞

which the lacking accumulation points. One settles that

|

∞

|

=

∞

,  where the latter

∞

means the real infinity.

The resulting space is the

one-point compactification

of

ℂ

. The

open sets

are the open sets in

ℂ

together with sets containing

∞

whose complement is

compact

in

ℂ

. Conceptually, one thinks of the additional open sets as those open sets “around

∞

”.

The one-point compactification of

ℂ

is also the complex projective line

ℂ

⁢

ℙ

1

, as well as the Riemann sphere.

Title

closed complex plane

Canonical name

ClosedComplexPlane

Date of creation

2013-03-22 17:37:48

Last modified on

2013-03-22 17:37:48

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

5

Author

pahio (2872)

Entry type

Definition

Classification

msc 54E35

Classification

msc 30-00

Synonym

extended complex plane

Related topic

RiemannSphere

Related topic

StereographicProjection

Related topic

RegularAtInfinity

Generated on Sat Feb 10 11:26:44 2018 by

LaTeXML
completely metrizable

completely metrizable

Let

X

be a

topological space

.

X

is said to be

completely metrizable

if there is a metric

d

on

X

under which

X

is

complete

.

In particular, a completely metrizable space is

metrizable

.

\PMlinkescapephrase

completely metrizable

Title

completely metrizable

Canonical name

CompletelyMetrizable

Date of creation

2013-03-22 12:13:56

Last modified on

2013-03-22 12:13:56

Owner

Evandar (27)

Last modified by

Evandar (27)

Numerical id

5

Author

Evandar (27)

Entry type

Definition

Classification

msc 54E35

Synonym

complete metrizability

Synonym

topologically complete

Synonym

topological completeness

Generated on Sat Feb 10 11:26:45 2018 by

LaTeXML
convergent sequence

convergent sequence

A

sequence

x

0

,

x

1

,

x

2

,

…

in a

metric space

(

X

,

d

)

is a

convergent sequence

if there exists a point

x

∈

X

such that, for every

real number

ϵ

>

0

, there exists a

natural number

N

such that

d

⁢

(

x

,

x

n

)

<

ϵ

for all

n

>

N

.

The point

x

, if it exists, is unique, and is called the

limit point

or

limit

of the sequence. One can also say that the sequence

x

0

,

x

1

,

x

2

,

…

converges

to

x

.

A sequence is said to be

divergent

if it does not converge.

Title

convergent sequence

Canonical

name

ConvergentSequence

Date of creation

2013-03-22 11:55:07

Last modified on

2013-03-22 11:55:07

Owner

djao (24)

Last modified by

djao (24)

Numerical id

10

Author

djao (24)

Entry type

Definition

Classification

msc 54E35

Classification

msc 40A05

Related topic

AxiomOfAnalysis

Related topic

BolzanoWeierstrassTheorem

Related topic

Sequence

Defines

limit point

Defines

limit

Defines

converge

Defines

diverge

Defines

divergent sequence

Generated on Sat Feb 10 11:26:47 2018 by

LaTeXML
disjoint disks property

disjoint disks property

A

metric space

(

X

,

d

)

is said to have the

disjoint disks

property

(DDP)
if for every pair of

continuous maps

f

,

g

:

B

2

→

X

of the closed
unit 2-ball (

http://planetmath.org/StandardNBall

)

B

2

into

X

and every

ϵ

>

0

there exist continuous maps

f

′

,

g

′

:

B

2

→

X

such that

d

⁢

(

f

,

f

′

)

<

ϵ

,

d

⁢

(

g

,

g

′

)

<

ϵ

and

f

′

⁢

(

B

2

)

∩

g

′

⁢

(

B

2

)

=

∅

.

Title

disjoint disks property

Canonical

name

DisjointDisksProperty

Date of creation

2013-03-22 16:50:45

Last modified on

2013-03-22 16:50:45

Owner

Mathprof (13753)

Last modified by

Mathprof (13753)

Numerical id

8

Author

Mathprof (13753)

Entry type

Definition

Classification

msc 54E35

Generated on Sat Feb 10 11:26:48 2018 by

LaTeXML
distance to a set

distance to a set

Let

X

be a

metric space

with a metric

d

. If

A

is a non-empty
subset of

X

and

x

∈

X

, then the

distance from

x

to

A

[

1

]

is defined as

d

⁢

(

x

,

A

)

:=

inf

a

∈

A

⁡

d

⁢

(

x

,

a

)

.

We also write

d

⁢

(

x

,

A

)

=

d

⁢

(

A

,

x

)

.

Suppose that

x

,

y

are points in

X

, and

A

⊂

X

is non-empty.
Then we have the following

triangle inequality

d

⁢

(

x

,

A

)

=

inf

a

∈

A

⁡

d

⁢

(

x

,

a

)

≤

d

⁢

(

x

,

y

)

+

inf

a

∈

A

⁡

d

⁢

(

y

,

a

)

=

d

⁢

(

x

,

y

)

+

d

⁢

(

y

,

A

)

.

If

X

is only a pseudo-metric space, then the above

definition

and triangle-inequality also hold.

References

1

J.L. Kelley,

General Topology

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

Title

distance to a set

Canonical

name

DistanceToASet

Date of creation

2013-03-22 13:38:37

Last modified on

2013-03-22 13:38:37

Owner

bbukh (348)

Last modified by

bbukh (348)

Numerical id

4

Author

bbukh (348)

Entry type

Definition

Classification

msc 54E35

Generated on Sat Feb 10 11:26:50 2018 by

LaTeXML
equibounded

equibounded

Let

X

and

Y

be

metric spaces

. A family

F

of functions from

X

to

Y

is said to be

equibounded

if there exists a

bounded

subset

B

of

Y

such that for all

f

∈

F

and all

x

∈

X

it holds

f

⁢

(

x

)

∈

B

.

Notice that if

F

⊂

𝒞

b

⁢

(

X

,

Y

)

(

continuous

bounded functions) then

F

is equibounded if and only if

F

is bounded (with respect to the metric of

uniform convergence

).

Title

equibounded

Canonical name

Equibounded

Date of creation

2013-03-22 13:16:55

Last modified on

2013-03-22 13:16:55

Owner

paolini (1187)

Last modified by

paolini (1187)

Numerical id

5

Author

paolini (1187)

Entry type

Definition

Classification

msc 54E35

Synonym

equi bounded

Synonym

equi-bounded

Generated on Sat Feb 10 11:26:52 2018 by

LaTeXML