aarjaneiro/planetmath huggingface dataset:
equicontinuous

equicontinuous

1

Definition

Let

X

be a

topological space

,

(

Y

,

d

)

a

metric space

and

C

⁢

(

X

,

Y

)

the set of

continuous functions

X

→

Y

.

Let

ℱ

be a subset of

C

⁢

(

X

,

Y

)

. A

function

f

∈

ℱ

is continuous at a point

x

0

when given

ϵ

>

0

there is a

neighbourhood

U

of

x

0

such that

d

⁢

(

f

⁢

(

x

)

,

f

⁢

(

x

0

)

)

<

ϵ

for every

x

∈

U

. When the same neighbourhood

U

can be chosen for all functions

f

∈

ℱ

, the family

ℱ

is said to be

equicontinuous

. More precisely:

Definition -

Let

ℱ

be a subset of

C

⁢

(

X

,

Y

)

. The set of functions

ℱ

is said to be

equicontinuous at

x

0

∈

X

if for every

ϵ

>

0

there is a neighbourhood

U

of

x

0

such that for every

x

∈

U

and every

f

∈

ℱ

we have

d

⁢

(

f

⁢

(

x

)

,

f

⁢

(

x

0

)

)

<

ϵ

The set

ℱ

is said to be

equicontinuous

if it is equicontinuous at every point

x

∈

X

.

2

Examples

•

A

finite set

of functions in

C

⁢

(

X

,

Y

)

is always equicontinuous.

•

When

X

is also a metric space, a family of functions in

C

⁢

(

X

,

Y

)

that share the same

Lipschitz constant

is equicontinuous.

•

The family of functions

{

f

n

}

n

∈

ℕ

, where

f

n

:

ℝ

→

ℝ

is given by

f

n

⁢

(

x

)

:=

arctan

⁡

(

n

⁢

x

)

is not equicontinuous at

0

.

3

Properties

•

If a subset

ℱ

⊆

C

⁢

(

X

,

Y

)

is

totally bounded

under the uniform metric, then

ℱ

is equicontinuous.

•

Suppose

X

is

compact

. If a

sequence

of functions

{

f

n

}

in

C

⁢

(

X

,

ℝ

k

)

is

equibounded

and equicontinuous, then the sequence

{

f

n

}

has a

uniformly convergent

subsequence. (ArzelÃÂ ’s

theorem

(

http://planetmath.org/AscoliArzelaTheorem

))

•

Let

{

f

n

}

be a sequence of functions in

C

⁢

(

X

,

Y

)

. If

{

f

n

}

is equicontinuous and

converges

pointwise

to a function

f

:

X

→

Y

, then

f

is

continuous

and

{

f

n

}

converges to

f

in the

compact-open topology

.

References

1

J. Munkres,

Topology

(2nd edition), Prentice Hall, 1999.

Title

equicontinuous

Canonical name

Equicontinuous

Date of creation

2013-03-22 18:38:10

Last modified on

2013-03-22 18:38:10

Owner

asteroid (17536)

Last modified by

asteroid (17536)

Numerical id

6

Author

asteroid (17536)

Entry type

Definition

Classification

msc 54E35

Classification

msc 54C35

Synonym

equicontinuity

Related topic

Equicontinuous2

Generated on Sat Feb 10 11:26:54 2018 by

LaTeXML
example of pseudometric space

example of pseudometric space

Let

X

=

ℝ

2

and consider the function

d

:

X

×

X

to the non-negative

real numbers

given by

d

⁢

(

(

x

1

,

x

2

)

,

(

y

1

,

y

2

)

)

=

|

x

1

-

y

1

|

.

Then

d

⁢

(

x

,

x

)

=

|

x

1

-

x

1

|

=

0

,

d

⁢

(

x

,

y

)

=

|

x

1

-

y

1

|

=

|

y

1

-

x

1

|

=

d

⁢

(

y

,

z

)

and the

triangle inequality

follows from the triangle inequality on

ℝ

1

, so

(

X

,

d

)

satisfies the defining conditions of a

pseudometric space

.

Note, however, that this is not an example of a

metric space

, since we can have two distinct points that are

distance

0 from each other, e.g.

d

⁢

(

(

2

,

3

)

,

(

2

,

5

)

)

=

|

2

-

2

|

=

0

.

Other examples:

•

Let

X

be a set,

x

0

∈

X

, and let

F

⁢

(

X

)

be functions

X

→

R

.
Then

d

⁢

(

f

,

g

)

=

|

f

⁢

(

x

0

)

-

g

⁢

(

x

0

)

|

is a

pseudometric

on

F

⁢

(

X

)

[

1

]

.

•

If

X

is a vector space and

p

is a seminorm over

X

, then

d

⁢

(

x

,

y

)

=

p

⁢

(

x

-

y

)

is a pseudometric on

X

.

•

The

trivial pseudometric

d

⁢

(

x

,

y

)

=

0

for all

x

,

y

∈

X

is a
pseudometric.

References

1

S. Willard,

General Topology

,
Addison-Wesley, Publishing Company, 1970.

Title

example of pseudometric space

Canonical name

ExampleOfPseudometricSpace

Date of creation

2013-03-22 14:40:24

Last modified on

2013-03-22 14:40:24

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

6

Author

mathcam (2727)

Entry type

Example

Classification

msc 54E35

Related topic

Seminorm

Related topic

VectorSpace

Related topic

MetricSpace

Related topic

Metric

Defines

trivial pseudometric

Generated on Sat Feb 10 11:26:56 2018 by

LaTeXML
examples of metric spaces

examples of metric spaces

The aim of this entry is to list

metric spaces

that have a

standard metric

.

Distance on set

•

Euclidean metric on

ℝ

n

and on

open subsets

of

ℝ

n

,

•

Normed vector spaces

,

•

Hausdorff metric

on sets

•

Valuated field, i.e. a field and its valuation

Function spaces

•

L

p

functions

(

http://planetmath.org/LpSpace

)

•

𝒪

⁢

(

G

)

;

holomorphic functions

(

http://planetmath.org/SpaceOfAnalyticFunctions

)

•

𝒮

; the

Schwartz space

of rapidly

decreasing functions

(

http://planetmath.org/SpaceOfRapidlyDecreasingFunctions

)

Title

examples of metric spaces

Canonical name

ExamplesOfMetricSpaces

Date of creation

2013-03-22 15:06:13

Last modified on

2013-03-22 15:06:13

Owner

matte (1858)

Last modified by

matte (1858)

Numerical id

9

Author

matte (1858)

Entry type

Topic

Classification

msc 54E35

Generated on Sat Feb 10 11:26:58 2018 by

LaTeXML
generalization of a pseudometric

generalization of a pseudometric

Let

X

be a set. Let

d

:

X

×

X

→

ℝ

be a

function

with the

property

that

d

⁢

(

x

,

y

)

≥

0

for all

x

,

y

∈

X

. Then

d

is a

1.

semi-pseudometric

if

d

⁢

(

x

,

y

)

=

d

⁢

(

y

,

x

)

for all

x

,

y

∈

X

,

2.

quasi-pseudometric

if

d

⁢

(

x

,

z

)

≤

d

⁢

(

x

,

y

)

+

d

⁢

(

y

,

z

)

for all

x

,

y

,

z

∈

X

.

X

equipped with a function

d

described above is called a

semi-pseudometric space

or a

quasi-pseudometric

space, depending on whether

d

is a semi-pseudometric or a quasi-pseudometric. A

pseudometric

is the same as a semi-pseudometric that is a quasi-pseudometric at the same time.

If

d

satisfies

the property that

d

⁢

(

x

,

y

)

=

0

implies

x

=

y

, then

d

is called a

semi-metric

if

d

is a semi-pseudometric, or a

quasi-metric

if

d

is a quasi-pseudometric.

Title

generalization

of a pseudometric

Canonical

name

GeneralizationOfAPseudometric

Date of creation

2013-03-22 16:43:06

Last modified on

2013-03-22 16:43:06

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

6

Author

CWoo (3771)

Entry type

Definition

Classification

msc 54E35

Synonym

semipseudometric

Synonym

quasipseudometric

Synonym

semipseudometric space

Synonym

quasipseudometric space

Related topic

semimetric

Related topic

quasimetric

Related topic

GeneralizationOfAUniformity

Defines

semi-pseudometric space

Defines

quasi-pseudometric space

Defines

semi-pseudometric

Defines

quasi-pseudometric

Generated on Sat Feb 10 11:27:06 2018 by

LaTeXML
Hausdorff metric

Hausdorff metric

Let

(

X

,

d

)

be a

metric space

, and let

ℱ

X

be the family of all
closed and

bounded

subsets of

X

. Given

A

∈

ℱ

X

, we will denote by

N

r

⁢

(

A

)

the

neighborhood

of

A

of radius

r

, i.e. the set

∪

x

∈

A

B

⁢

(

x

,

r

)

.

The

upper

Hausdorff hemimetric

is defined by

δ

*

⁢

(

A

,

B

)

=

inf

⁡

{

r

>

0

:

B

⊂

N

r

⁢

(

A

)

}

.

Analogously, the

lower Hausdorff hemimetric

is

δ

*

⁢

(

A

,

B

)

=

inf

⁡

{

r

>

0

:

A

⊂

N

r

⁢

(

B

)

}

.

Finally, the

Hausdorff metric

is given by

δ

⁢

(

A

,

B

)

=

max

⁡

{

δ

*

⁢

(

A

,

B

)

,

δ

*

⁢

(

A

,

B

)

}

.

for

A

,

B

∈

ℱ

X

.

The following

properties

follow straight from the

definitions

:

1.

δ

*

⁢

(

A

,

B

)

=

δ

*

⁢

(

B

,

A

)

;

2.

δ

*

⁢

(

A

,

B

)

=

0

if and only if

B

⊂

A

;

3.

δ

*

⁢

(

A

,

B

)

=

0

if and only if

A

⊂

B

;

4.

δ

*

⁢

(

A

,

C

)

≤

δ

*

⁢

(

A

,

B

)

+

δ

*

⁢

(

B

,

C

)

, and similarly for

δ

*

.

From this it is clear that

δ

is a metric: the

triangle inequality

follows from that of

δ

*

and

δ

*

; symmetry follows from

δ

*

⁢

(

A

,

B

)

=

δ

*

⁢

(

A

,

B

)

; and

δ

⁢

(

A

,

B

)

=

0

iff both

δ

*

⁢

(

A

,

B

)

and

δ

*

⁢

(

A

,

B

)

are zero iff

A

⊂

B

and

B

⊂

A

iff

A

=

B

.

Hausdorff metric inherits completeness

; i.e. if

(

X

,

d

)

is

complete

, then so is

(

ℱ

X

,

δ

)

. Also, if

(

X

,

d

)

is

totally bounded

, then so is

(

ℱ

X

,

δ

)

.

Intuitively, the Hausdorff hemimetric

δ

*

(resp.

δ

*

) measure how much bigger (resp. smaller) is a set compared to another. This allows us to define hemicontinuity of correspondences.

Title

Hausdorff metric

Canonical

name

HausdorffMetric

Date of creation

2013-03-22 13:28:34

Last modified on

2013-03-22 13:28:34

Owner

Koro (127)

Last modified by

Koro (127)

Numerical id

11

Author

Koro (127)

Entry type

Definition

Classification

msc 54E35

Synonym

Hausdorff distance

Defines

Hausdorff hemimetric

Generated on Sat Feb 10 11:27:08 2018 by

LaTeXML
Hausdorff metric inherits completeness

Hausdorff metric inherits completeness

Theorem 1

.

If

(

X

,

d

)

is a complete metric space, then the

Hausdorff metric

induced by

d

is also

complete

.

Proof.

Suppose

(

A

n

)

is a

Cauchy sequence

with respect to the Hausdorff metric. By selecting a

subsequence

if necessary, we may assume that

A

n

and

A

n

+

1

are within

2

-

n

of each other, that is, that

A

n

⊂

K

⁢

(

A

n

+

1

,

2

-

n

)

and

A

n

+

1

⊂

K

⁢

(

A

n

,

2

-

n

)

.
Now for any

natural number

N

, there is a

sequence

(

x

n

)

n

≥

N

in

X

such that

x

n

∈

A

n

and

d

⁢

(

x

n

,

x

n

+

1

)

<

2

-

n

. Any such sequence is Cauchy
with respect to

d

and thus

converges

to some

x

∈

X

. By applying the

triangle inequality

, we see that for any

n

≥

N

,

d

⁢

(

x

n

,

x

)

<

2

-

n

+

1

.

Define

A

to be the set of all

x

such that

x

is the limit of a sequence

(

x

n

)

n

≥

0

with

x

n

∈

A

n

and

d

⁢

(

x

n

,

x

n

+

1

)

<

2

-

n

.
Then

A

is nonempty.
Furthermore, for any

n

, if

x

∈

A

,
then there is some

x

n

∈

A

n

such that

d

⁢

(

x

n

,

x

)

<

2

-

n

+

1

, and so

A

⊂

K

⁢

(

A

n

,

2

-

n

+

1

)

. Consequently, the set

A

¯

is
nonempty, closed and

bounded

.

Suppose

ϵ

>

0

. Thus

ϵ

>

2

-

N

>

0

for some

N

. Let

n

≥

N

+

1

.
Then by applying the claim in the first paragraph, we have that for any

x

n

∈

A

n

, there is some

x

∈

X

with

d

⁢

(

x

n

,

x

)

<

2

-

n

+

1

. Hence

A

n

⊂

K

⁢

(

A

¯

,

2

-

n

+

1

)

.
Hence the sequence

(

A

n

)

converges to

A

in the Hausdorff metric.
∎

This proof is based on a sketch given in an exercise in

[

1

]

.
An exercise for the reader: is the set

A

constructed above closed?

References

1

J. Munkres,

Topology

(2nd edition), Prentice Hall, 1999.

Title

Hausdorff metric inherits completeness

Canonical

name

HausdorffMetricInheritsCompleteness

Date of creation

2013-03-22 14:08:51

Last modified on

2013-03-22 14:08:51

Owner

mps (409)

Last modified by

mps (409)

Numerical id

8

Author

mps (409)

Entry type

Theorem

Classification

msc 54E35

Related topic

Complete

Generated on Sat Feb 10 11:27:11 2018 by

LaTeXML
isometry

isometry

Let

(

X

1

,

d

1

)

and

(

X

2

,

d

2

)

be

metric spaces

.
A

function

f

:

X

1

→

X

2

is said to be an

isometric mapping

(or

isometric embedding

) if

d

1

⁢

(

x

,

y

)

=

d

2

⁢

(

f

⁢

(

x

)

,

f

⁢

(

y

)

)

for all

x

,

y

∈

X

1

.

Every isometric mapping is

injective

,
for if

x

,

y

∈

X

1

with

x

≠

y

then

d

1

⁢

(

x

,

y

)

>

0

,
and so

d

2

⁢

(

f

⁢

(

x

)

,

f

⁢

(

y

)

)

>

0

, and then

f

⁢

(

x

)

≠

f

⁢

(

y

)

.
One can also easily show that every isometric mapping is

continuous

.

An isometric mapping that is

surjective

(and therefore

bijective

)
is called an

isometry

.
(Readers are warned, however,
that some authors do not require isometries to be surjective;
that is, they use the term

isometry

for what we have called an isometric mapping.)
Every isometry is a

homeomorphism

.

If there is an isometry between the metric spaces

(

X

1

,

d

1

)

and

(

X

2

,

d

2

)

,
then they are said to be

isometric

.
Isometric spaces are essentially identical as metric spaces,
and in particular they are homeomorphic.

Given any metric space

(

X

,

d

)

,
the set of all isometries

X

→

X

forms a group under

composition

.
This group is called the

isometry group

(or

group of isometries

) of

X

,
and may be denoted by

Iso

⁡

(

X

)

or

Isom

⁡

(

X

)

.
In general,

an

(as opposed to

the

) isometry group
(or group of isometries) of

X

is any subgroup of

Iso

⁡

(

X

)

.

Title

isometry

Canonical

name

Isometry

Date of creation

2013-03-22 12:19:08

Last modified on

2013-03-22 12:19:08

Owner

yark (2760)

Last modified by

yark (2760)

Numerical id

13

Author

yark (2760)

Entry type

Definition

Classification

msc 54E35

Related topic

RealTree

Related topic

IsometricIsomorphism

Defines

isometric

Defines

isometric mapping

Defines

isometric embedding

Defines

isometry group

Defines

group of isometries

Generated on Sat Feb 10 11:27:12 2018 by

LaTeXML
metric equivalence

metric equivalence

Let

X

be a set equipped with two metrics

ρ

and

σ

. We say that

ρ

is

equivalent

to

σ

(on

X

) if the

identity map

on

X

, is a

homeomorphism

between the

metric topology

on

X

induced by

ρ

and the metric topology on

X

induced by

σ

.

For example, if

(

X

,

ρ

)

is a metric space, then the

function

σ

:

X

→

ℝ

defined by

σ

⁢

(

x

,

y

)

:=

ρ

⁢

(

x

,

y

)

1

+

ρ

⁢

(

x

,

y

)

is a metric on

X

that is equivalent to

ρ

. This shows that every metric is equivalent to a

bounded

metric.

Title

metric equivalence

Canonical

name

MetricEquivalence

Date of creation

2013-03-22 19:23:11

Last modified on

2013-03-22 19:23:11

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

6

Author

CWoo (3771)

Entry type

Definition

Classification

msc 54E35

Defines

equivalent

Generated on Sat Feb 10 11:27:21 2018 by

LaTeXML
metric space

metric space

A

metric space

is a set

X

together with a real valued

function

d

:

X

×

X

⟶

ℝ

(called a

metric

, or sometimes a

distance

function) such that, for every

x

,

y

,

z

∈

X

,

•

d

⁢

(

x

,

y

)

≥

0

, with equality

1

1

This condition can be replaced with the weaker statement

d

⁢

(

x

,

y

)

=

0

⇔

x

=

y

without affecting the definition.

if and only if

x

=

y

•

d

⁢

(

x

,

y

)

=

d

⁢

(

y

,

x

)

•

d

⁢

(

x

,

z

)

≤

d

⁢

(

x

,

y

)

+

d

⁢

(

y

,

z

)

For

x

∈

X

and

ε

∈

ℝ

with

ε

>

0

, the

open ball

around

x

of radius

ε

is the set

B

ε

⁢

(

x

)

:=

{

y

∈

X

∣

d

⁢

(

x

,

y

)

<

ε

}

. An

open set

in

X

is a set which equals an arbitrary (possibly empty) union of open balls in

X

, and

X

together with these open sets forms a

Hausdorff topological space

. The

topology

on

X

formed by these open sets is called the

metric topology

, and in fact the open sets form a basis for this topology (proof (

http://planetmath.org/PseudometricTopology

)).

Similarly, the set

B

¯

ε

⁢

(

x

)

:=

{

y

∈

X

∣

d

⁢

(

x

,

y

)

≤

ε

}

is called a

closed ball

around

x

of radius

ε

. Every closed ball is a

closed subset

of

X

in the metric topology.

The prototype example of a metric space is

ℝ

itself, with the metric defined by

d

⁢

(

x

,

y

)

:=

|

x

-

y

|

. More generally, any

normed vector space

has an underlying metric space

structure

; when the

vector space

is finite dimensional, the resulting metric space is

isomorphic

to

Euclidean space

.

References

1

J.L. Kelley,

General Topology

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

Title

metric space

Canonical name

MetricSpace

Date of creation

2013-03-22 11:53:19

Last modified on

2013-03-22 11:53:19

Owner

djao (24)

Last modified by

djao (24)

Numerical id

15

Author

djao (24)

Entry type

Definition

Classification

msc 54E35

Classification

msc 82-00

Classification

msc 83-00

Classification

msc 81-00

Related topic

Neighborhood

Related topic

VectorNorm

Related topic

T2Space

Related topic

Ultrametric

Related topic

QuasimetricSpace

Related topic

NormedVectorSpace

Related topic

PseudometricSpace

Defines

distance metric

Defines

metric

Defines

distance

Defines

metric topology

Defines

open ball

Defines

closed ball

Generated on Sat Feb 10 11:27:23 2018 by

LaTeXML
nested sphere theorem

nested sphere theorem

In a

metric space

X

, let

B

¯

r

⁢

(

x

)

be the closed
ball centered at

x

∈

X

with radius

r

>

0

.

Theorem 1

(Nested sphere theorem

[

KF

]

).

A metric space

X

is

complete

if and only if every

sequence

{

B

¯

r

n

⁢

(

x

n

)

}

n

such that

B

¯

r

i

+

1

⁢

(

x

i

+

1

)

⊆

B

¯

r

i

⁢

(

x

i

)

and

r

n

→

0

when

n

→

∞

has a nonempty

intersection

(i.e.

⋂

n

=

1

∞

B

¯

r

n

⁢

(

x

n

)

≠

∅

).

References

KF

Kolmogorov, A.N. & Fomin, S.V.: Introductory Real Analysis, Translated & Edited by Richard A. Silverman. Dover Publications, Inc. New York, 1970.

Title

nested sphere theorem

Canonical name

NestedSphereTheorem

Date of creation

2013-03-22 14:57:12

Last modified on

2013-03-22 14:57:12

Owner

Daume (40)

Last modified by

Daume (40)

Numerical id

5

Author

Daume (40)

Entry type

Theorem

Classification

msc 54E35

Classification

msc 54E50

Generated on Sat Feb 10 11:27:25 2018 by

LaTeXML
open set in ℝ^n contains an open rectangle

open set in

ℝ

n

contains an open rectangle

Theorem

Suppose

ℝ

n

is
equipped with the

usual topology

induced by the

open balls

of the
Euclidean metric.
Then, if

U

is a non-empty

open set

in

ℝ

n

, there
exist

real numbers

a

i

,

b

i

for

i

=

1

,

…

,

n

such that

a

i

<

b

i

and

[

a

1

,

b

1

]

×

⋯

×

[

a

n

,

b

n

]

is a subset of

U

.

Proof.

Since

U

is non-empty, there exists some point

x

in

U

. Further, since

U

is a

topological space

,

x

is contained in
some open set. Since the

topology

has a basis consisting of
open balls, there exists a

y

∈

U

and

ε

>

0

such that

x

is contained in the open ball

B

⁢

(

y

,

ε

)

.
Let us now set

a

i

=

y

i

-

ε

2

⁢

n

and

b

i

=

y

i

+

ε

2

⁢

n

for all

i

=

1

,

…

,

n

.
Then

D

=

[

a

1

,

b

1

]

×

⋯

×

[

a

n

,

b

n

]

can be
parametrized as

D

=

{

y

+

(

λ

1

,

…

,

λ

n

)

⁢

ε

2

⁢

n

∣

λ

i

∈

[

-

1

,

1

]

⁢

for all

⁢

i

=

1

,

…

,

n

}

.

For an arbitrary point in

D

, we have

|

y

+

(

λ

1

,

…

,

λ

n

)

⁢

ε

2

⁢

n

-

y

|

=

|

(

λ

1

,

…

,

λ

n

)

⁢

ε

2

⁢

n

|

=

ε

2

⁢

n

⁢

λ

1

2

+

⋯

+

λ

n

2

≤

ε

2

<

ε

,

so

D

⊂

B

⁢

(

y

,

ϵ

)

⊂

U

, and the claim follows.

□

Title

open set in

ℝ

n

contains an open

rectangle

Canonical name

OpenSetInmathbbRnContainsAnOpenRectangle

Date of creation

2013-03-22 14:07:46

Last modified on

2013-03-22 14:07:46

Owner

matte (1858)

Last modified by

matte (1858)

Numerical id

5

Author

matte (1858)

Entry type

Theorem

Classification

msc 54E35

Related topic

Interval

Generated on Sat Feb 10 11:27:27 2018 by

LaTeXML
point countable base

point countable base

Let

X

be a

topological space

. A basis

ℬ

of

X

is a

point countable base

if every point of

X

is contained in at most countably many sets of

ℬ

.

Any

uniform base

is a point countable base, and a theorem of R. W. Heath states that every

semimetric

space with a point countable base is

developable

.

References

1

Steen, Lynn Arthur and Seebach, J. Arthur,

Counterexamples in Topology

, Dover Books, 1995.

Title

point countable base

Canonical name

PointCountableBase

Date of creation

2013-03-22 14:49:59

Last modified on

2013-03-22 14:49:59

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

4

Author

mathcam (2727)

Entry type

Definition

Classification

msc 54E35

Generated on Sat Feb 10 11:27:29 2018 by

LaTeXML
product of metric spaces

product of metric spaces

Theorem 1

.

Let

(

X

i

,

ϱ

i

)

be a

metric space

for each

i

=

1

,

2

,

…

,

where
the

diameter

of

X

i

using

ϱ

i

is less than

1

/

i

. Then the

product topology

for the space

∏

i

=

1

∞

X

i

is given by the metric

ϱ

⁢

(

x

,

y

)

=

∑

i

=

1

∞

1

2

i

⁢

ϱ

i

⁢

(

x

i

,

y

i

)

.

Hence, a countable product of

metrizable

topological spaces

is again metrizable.

Since the

convergence

in the product topology is the

pointwise convergence

, the same is true for the metric space with the above metric.

Title

product of metric spaces

Canonical name

ProductOfMetricSpaces

Date of creation

2013-03-22 16:11:44

Last modified on

2013-03-22 16:11:44

Owner

kompik (10588)

Last modified by

kompik (10588)

Numerical id

6

Author

kompik (10588)

Entry type

Theorem

Classification

msc 54E35

Generated on Sat Feb 10 11:27:31 2018 by

LaTeXML
proof of Cauchy condition for limit of function

proof of Cauchy condition for limit of function

The forward direction is . Assume that

lim

x

→

x

0

⁡

f

⁢

(

x

)

=

L

. Then given

ϵ

there is a

δ

such that

|

f

⁢

(

u

)

-

L

|

<

ϵ

/

2

⁢

when

⁢

0

<

|

u

-

x

0

|

<

δ

.

Now for

0

<

|

u

-

x

0

|

<

δ

and

0

<

|

v

-

x

0

|

<

δ

we have

|

f

⁢

(

u

)

-

L

|

<

ϵ

/

2

⁢

and

⁢

|

f

⁢

(

v

)

-

L

|

<

ϵ

/

2

and so

|

f

⁢

(

u

)

-

f

⁢

(

v

)

|

=

|

f

⁢

(

u

)

-

L

-

(

f

⁢

(

v

)

-

L

)

|

≤

|

f

⁢

(

u

)

-

L

|

+

|

f

⁢

(

v

)

-

L

|

<

ϵ

/

2

+

ϵ

/

2

=

ϵ

.

We prove the reverse by

contradiction

.
Assume that the condition holds.
Now suppose that

lim

x

→

x

0

⁡

f

⁢

(

x

)

does not exist. This means that for
any

l

and any

ϵ

sufficiently small then for any

δ

>

0

there is

x

l

such that

0

<

|

x

⁢

l

-

x

0

|

<

δ

⁢

and

⁢

|

f

⁢

(

x

l

)

-

l

|

≥

ϵ

.
For any such

ϵ

choose

u

such that

0

<

|

u

-

x

0

|

<

δ

and
put

l

=

f

⁢

(

v

)

then substituting in the condition with

u

=

x

l

we get

|

f

⁢

(

x

l

)

-

l

|

<

ϵ

. A contradiction.

Title

proof of Cauchy condition for limit of function

Canonical name

ProofOfCauchyConditionForLimitOfFunction

Date of creation

2013-03-22 18:59:08

Last modified on

2013-03-22 18:59:08

Owner

puff (4175)

Last modified by

puff (4175)

Numerical id

8

Author

puff (4175)

Entry type

Proof

Classification

msc 54E35

Classification

msc 26A06

Classification

msc 26B12

Generated on Sat Feb 10 11:27:33 2018 by

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