aarjaneiro/planetmath huggingface dataset:
proof of Riesz’ Lemma

proof of Riesz’ Lemma

Let’s consider

x

∈

E

-

S

and let

r

=

d

⁢

(

x

,

S

)

. Recall that

r

is the

distance

between

x

and

S

:

d

(

x

,

S

)

=

inf

{

d

(

x

,

s

)

such that

s

∈

S

}

. Now,

r

>

0

because

S

is closed. Next, we consider

b

∈

S

such that

∥

x

-

b

∥

<

r

α

This vector

b

exists: as

0

<

α

<

1

then

r

α

>

r

But then the

definition

of infimum implies there is

b

∈

S

such that

∥

x

-

b

∥

<

r

α

Now, define

x

α

=

x

-

b

∥

x

-

b

∥

Trivially,

∥

x

α

∥

=

1

Notice that

x

α

∈

E

-

S

, because if

x

α

∈

S

then

x

-

b

∈

S

, and so

(

x

-

b

)

+

b

=

x

∈

S

, an absurd.
Plus, for every

s

∈

S

we have

∥

s

-

x

α

∥

=

∥

s

-

x

-

b

∥

x

-

b

∥

∥

=

1

∥

x

-

b

∥

⋅

∥

∥

x

-

b

∥

⋅

s

+

b

-

x

∥

≥

r

∥

x

-

b

∥

because

∥

x

-

b

∥

⋅

s

+

b

∈

S

But

r

∥

x

-

b

∥

>

α

r

⋅

r

=

α

QED.

Title

proof of Riesz’ Lemma

Canonical

name

ProofOfRieszLemma

Date of creation

2013-03-22 14:56:14

Last modified on

2013-03-22 14:56:14

Owner

gumau (3545)

Last modified by

gumau (3545)

Numerical id

6

Author

gumau (3545)

Entry type

Proof

Classification

msc 54E35

Classification

msc 15A03

Generated on Sat Feb 10 11:27:35 2018 by

LaTeXML
pseudometric space

pseudometric space

A

pseudometric space

is a set

X

together with a non-negative real-valued function

d

:

X

×

X

⟶

ℝ

(called a

pseudometric

) such that, for every

x

,

y

,

z

∈

X

,

•

d

⁢

(

x

,

x

)

=

0

.

•

d

⁢

(

x

,

y

)

=

d

⁢

(

y

,

x

)

•

d

⁢

(

x

,

z

)

≤

d

⁢

(

x

,

y

)

+

d

⁢

(

y

,

z

)

In other words, a pseudometric space is a

generalization

of a

metric space

in which we allow the possibility that

d

⁢

(

x

,

y

)

=

0

for distinct values of

x

and

y

.

References

1

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

Counterexamples in topology

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

Title

pseudometric space

Canonical

name

PseudometricSpace

Date of creation

2013-03-22 14:40:18

Last modified on

2013-03-22 14:40:18

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

7

Author

mathcam (2727)

Entry type

Definition

Classification

msc 54E35

Synonym

pesudo-metric space

Related topic

MetricSpace

Related topic

QuasimetricSpace

Related topic

NormedVectorSpace

Related topic

Seminorm

Defines

pseudometric

Defines

pseudo-metric

Generated on Sat Feb 10 11:27:37 2018 by

LaTeXML
pseudometric topology

pseudometric topology

Let

(

X

,

d

)

be a

pseudometric space

. As in a

metric space

, we define

B

ε

⁢

(

x

)

=

{

y

∈

X

∣

d

⁢

(

x

,

y

)

<

ε

}

.

for

x

∈

X

,

ε

>

0

.

In the below, we show that the collection of sets

ℬ

=

{

B

ε

⁢

(

x

)

∣

ε

>

0

,

x

∈

X

}

form a base for a

topology

for

X

. We call this topology
the on

X

induced by

d

. Also,
a

topological space

X

is a

pseudometrizable topological space

if there exists a

pseudometric

d

on

X

whose
pseudometric topology coincides with the given topology
for

X

[

1

,

2

]

.

Proposition 1

.

ℬ

is a base for a topology.

Proof.

We shall use the

http://planetmath.org/node/5845

this result
to prove that

ℬ

is a base.

First, as

d

⁢

(

x

,

x

)

=

0

for all

x

∈

X

, it follows
that

ℬ

is a cover.
Second, suppose

B

1

,

B

2

∈

ℬ

and

z

∈

B

1

∩

B

2

.
We claim that there exists a

B

3

∈

ℬ

such that

z

∈

B

3

⊆

B

1

∩

B

2

.

(1)

By definition,

B

1

=

B

ε

1

⁢

(

x

1

)

and

B

2

=

B

ε

2

⁢

(

x

2

)

for some

x

1

,

x

2

∈

X

and

ε

1

,

ε

2

>

0

. Then

d

⁢

(

x

1

,

z

)

<

ε

1

,

d

⁢

(

x

2

,

z

)

<

ε

2

.

Now we can define

δ

=

min

⁡

{

ε

1

-

d

⁢

(

x

1

,

z

)

,

ε

2

-

d

⁢

(

x

2

,

z

)

}

>

0

, and put

B

3

=

B

δ

⁢

(

z

)

.

If

y

∈

B

3

, then for

k

=

1

,

2

, we have by the

triangle inequality

d

⁢

(

x

k

,

y

)

≤

d

⁢

(

x

k

,

z

)

+

d

⁢

(

z

,

y

)

<

d

⁢

(

x

k

,

z

)

+

δ

≤

ε

k

,

so

B

3

⊆

B

k

and condition

1

holds.
∎

Remark

In the proof, we have not used the fact that

d

is

symmetric

. Therefore, we have, in fact, also shown that any

quasimetric

induces a topology.

References

1

J.L. Kelley,

General Topology

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

2

S. Willard,

General Topology

,
Addison-Wesley, Publishing Company, 1970.

Title

pseudometric topology

Canonical name

PseudometricTopology

Date of creation

2013-03-22 14:40:47

Last modified on

2013-03-22 14:40:47

Owner

matte (1858)

Last modified by

matte (1858)

Numerical id

7

Author

matte (1858)

Entry type

Definition

Classification

msc 54E35

Defines

pseudometrizable

Defines

pseudometric topology

Defines

pseudo-metric

Defines

pseudometrizable topological space

Defines

pseudo-metrizable topological space

Generated on Sat Feb 10 11:27:39 2018 by

LaTeXML
quasimetric space

quasimetric space

A

quasimetric space

(

X

,

d

)

is a set

X

together with a non-negative real-valued function

d

:

X

×

X

⟶

ℝ

(called a

quasimetric

) such that, for every

x

,

y

,

z

∈

X

,

•

d

⁢

(

x

,

y

)

≥

0

with equality if and only if

x

=

y

.

•

d

⁢

(

x

,

z

)

≤

d

⁢

(

x

,

y

)

+

d

⁢

(

y

,

z

)

In other words, a quasimetric space is a

generalization

of a

metric space

in which we drop the requirement that, for two points

x

and

y

, the “distance” between

x

and

y

is the same as the “distance” between

y

and

x

(i.e. the

symmetry

axiom of metric spaces).

Some

properties

:

•

If

(

X

,

d

)

is a quasimetric space, we can form a metric space

(

X

,

d

′

)

where

d

′

is defined for all

x

,

y

∈

X

by

d

′

⁢

(

x

,

y

)

=

1

2

⁢

(

d

⁢

(

x

,

y

)

+

d

⁢

(

y

,

x

)

)

.

•

Every metric space is trivially a quasimetric space.

•

A quasimetric that is (i.e.

d

⁢

(

x

,

y

)

=

d

⁢

(

y

,

x

)

for all

x

,

y

∈

X

is a metric.

References

1

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

Counterexamples in topology

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

2

Z. Shen,

Lectures of

Finsler geometry

, World Sientific, 2001.

Title

quasimetric space

Canonical

name

QuasimetricSpace

Date of creation

2013-03-22 14:40:21

Last modified on

2013-03-22 14:40:21

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

8

Author

mathcam (2727)

Entry type

Definition

Classification

msc 54E35

Synonym

quasi-metric space

Related topic

PseudometricSpace

Related topic

MetricSpace

Related topic

GeneralizationOfAPseudometric

Defines

quasimetric

Defines

quasi-metric

Generated on Sat Feb 10 11:27:41 2018 by

LaTeXML
Smirnov metrization theorem

Smirnov metrization theorem

The

Smirnov metrization theorem

establishes necessary and sufficient conditions for a

topological space

to be

metrizable

. The

theorem

reduces questions of metrizability to

paracompactness

and a local metrizability condition.

Definition

:

A space

X

is

locally metrizable

if every point

x

∈

X

has a

neighborhood

that is metrizable in the

subspace topology

.

Theorem (Smirnov metrization theorem):

A space is metrizable if and only if it is paracompact and locally metrizable.

Title

Smirnov metrization theorem

Canonical

name

SmirnovMetrizationTheorem

Date of creation

2013-03-22 18:01:00

Last modified on

2013-03-22 18:01:00

Owner

rm50 (10146)

Last modified by

rm50 (10146)

Numerical id

7

Author

rm50 (10146)

Entry type

Theorem

Classification

msc 54E35

Defines

locally metrizable

Generated on Sat Feb 10 11:27:43 2018 by

LaTeXML
some structures on ℝ^n

some structures on

ℝ

n

Let

n

∈

{

1

,

2

,

…

}

. Then, as a set,

ℝ

n

is the

n

-fold

Cartesian
product

of the

real numbers

.

0.0.1

Vector space structure of

ℝ

n

If

u

=

(

u

1

,

…

,

u

n

)

and

v

=

(

v

1

,

…

,

v

n

)

are points in

ℝ

n

, we define their sum
as

u

+

v

=

(

u

1

+

v

1

,

…

,

u

n

+

v

n

)

.

Also, if

λ

is a scalar (real number), then

scalar multiplication

is defined as

λ

⋅

u

=

(

λ

⁢

u

1

,

…

,

λ

⁢

u

n

)

.

With these

operations

,

ℝ

n

becomes a

vector space

(over

ℝ

) with

dimension

n

.
In other words, with this

structure

, we can talk about, vectors, lines,

subspaces

of different dimension.

0.0.2

Inner product for

ℝ

n

For

u

and

v

as above, we define the

inner product

as

⟨

u

,

v

⟩

=

u

1

⁢

v

1

+

⋯

+

u

n

⁢

v

n

.

With this

product

,

ℝ

n

is called an

Euclidean space

.

We have also an induced norm

∥

u

∥

=

⟨

u

,

u

⟩

, which gives

ℝ

n

the structure of a

normed space

(and thus

metric space

).
This inner product let us talk about length, angle between vectors,

orthogonal vectors

.

0.0.3

Topology for

ℝ

n

The

usual topology

for

ℝ

n

is the

topology

induced by the metric

d

⁢

(

x

,

y

)

=

∥

x

-

y

∥

.

As a basis for the topology induced by the above norm, one can take
open balls

B

⁢

(

x

,

r

)

=

{

y

∈

ℝ

n

∣

∥

x

-

y

∥

<

r

}

where

r

>

0

and

x

∈

ℝ

n

.

Properties

of the

topological space

ℝ

n

are:

1.

ℝ

n

is

second countable

, i.e.,

ℝ

n

has a

countable basis

.

2.

(

Heine-Borel theorem

)
A set in

ℝ

n

is

compact

if and only if it is closed and

bounded

.

3.

Since

ℝ

n

is a metric space,

ℝ

n

is a

Hausdorff space

.

Title

some structures on

ℝ

n

Canonical

name

SomeStructuresOnmathbbRn

Date of creation

2013-03-22 14:03:47

Last modified on

2013-03-22 14:03:47

Owner

drini (3)

Last modified by

drini (3)

Numerical id

10

Author

drini (3)

Entry type

Definition

Classification

msc 54E35

Classification

msc 53A99

Generated on Sat Feb 10 11:27:47 2018 by

LaTeXML
sphere (metric space)

sphere (metric space)

The set

{

x

∣

d

⁢

(

x

,

c

)

=

r

}

is called the

sphere

of radius

r

with centre

c

. This generalizes the notion of spheres to

metric spaces

.

Note that the sphere in a metric space need not look like a sphere in

Euclidean space

. For

instance

, if we impose the metric

d

⁢

(

x

,

y

)

=

m

⁢

a

⁢

x

⁢

{

|

x

1

-

y

1

|

,

|

x

2

-

y

2

|

,

|

x

3

-

y

3

|

}

on

ℝ

3

instead of the

Euclidean metric

, spheres according to this metric are actually cubes! Even more bizarre situations can

occur in

general — a sphere might be

disconnected

, or it may be

discrete

, or it may even be an

empty set

.

Title

sphere (metric space)

Canonical name

SpheremetricSpace

Date of creation

2013-03-22 14:47:38

Last modified on

2013-03-22 14:47:38

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

6

Author

rspuzio (6075)

Entry type

Definition

Classification

msc 54E35

Synonym

sphere

Generated on Sat Feb 10 11:27:49 2018 by

LaTeXML
topology of the complex plane

topology of the complex plane

The

usual topology

for the

complex plane

ℂ

is the

topology

induced by the metric

d

⁢

(

x

,

y

)

:=

|

x

-

y

|

for

x

,

y

∈

ℂ

.
Here,

|

⋅

|

is the

complex modulus

(

http://planetmath.org/ModulusOfComplexNumber

).

If we identify

ℝ

2

and

ℂ

, it is clear that the above
topology coincides with topology induced by the

Euclidean metric

on

ℝ

2

.

Some basic topological concepts for

ℂ

:

1.

The

open balls

B

r

(

ζ

)

=

{

z

∈

ℂ

⋮

|

z

-

ζ

|

<

r

}

are often called

open disks

.

2.

A point

ζ

is an

accumulation point

of a subset

A

of

ℂ

, if any open disk

B

r

⁢

(

ζ

)

contains at least one point of

A

distinct from

ζ

.

3.

A point

ζ

is an

interior point

of the set

A

, if there exists an open disk

B

r

⁢

(

ζ

)

which is contained in

A

.

4.

A set

A

is

open

, if each of its points is an interior point of

A

.

5.

A set

A

is

closed

, if all its accumulation points belong to

A

.

6.

A set

A

is

bounded

, if there is an open disk

B

r

⁢

(

ζ

)

containing

A

.

7.

A set

A

is

compact

, if it is closed and bounded.

Title

topology of the complex plane

Canonical name

TopologyOfTheComplexPlane

Date of creation

2013-03-22 13:38:40

Last modified on

2013-03-22 13:38:40

Owner

matte (1858)

Last modified by

matte (1858)

Numerical id

8

Author

matte (1858)

Entry type

Definition

Classification

msc 54E35

Classification

msc 30-00

Related topic

IdentityTheorem

Related topic

PlacesOfHolomorphicFunction

Defines

open disk

Defines

accumulation point

Defines

interior point

Defines

open

Defines

closed

Defines

bounded

Defines

compact

Generated on Sat Feb 10 11:27:50 2018 by

LaTeXML
totally bounded

totally bounded

Let

A

be a subset of a

topological vector space

X

.

A

is called

totally bounded

if , for each

neighborhood

G

of 0,
there exists a finite subset

S

of

A

with

A

contained in the sumset

S

+

G

.

The

definition

can be restated in the following form when

X

is a

metric space

:

A set

A

⊆

X

is said to be

totally bounded

if for every

ϵ

>

0

, there exists a finite subset

{

s

1

,

s

2

,

…

,

s

n

}

of

A

such that

A

⊆

⋃

k

=

1

n

B

⁢

(

s

k

,

ϵ

)

, where

B

⁢

(

s

k

,

ϵ

)

denotes the open ball around

s

k

with radius

ϵ

.

References

1

G. Bachman, L. Narici,

Functional analysis

, Academic Press, 1966.

2

A. Wilansky,

Functional Analysis

, Blaisdell Publishing Co., 1964

3

W. Rudin,

Functional Analysis

, 2nd ed. McGraw-Hill , 1973

Title

totally bounded

Canonical

name

TotallyBounded

Date of creation

2013-03-22 13:09:54

Last modified on

2013-03-22 13:09:54

Owner

Mathprof (13753)

Last modified by

Mathprof (13753)

Numerical id

12

Author

Mathprof (13753)

Entry type

Definition

Classification

msc 54E35

Related topic

MetricSpace

Related topic

Bounded

Related topic

Subset

Generated on Sat Feb 10 11:27:52 2018 by

LaTeXML
totally bounded subset of a metric space is bounded

totally bounded subset of a metric space is bounded

Theorem 1

.

Every

totally bounded

subset of a

metric space

is

bounded

.

Proof.

Let

K

be a totally bounded subset of a metric space. Suppose

x

,

y

∈

K

. We will show that there exists

M

>

0

such that for any x,y we have

d

⁢

(

x

,

y

)

<

M

. From the definition of totally bounded,
we can find an

ε

>

0

and a finite subset

{

x

1

,

x

2

⁢

…

,

x

n

}

of

K

such that

K

⊆

⋃

k

=

1

n

B

⁢

(

x

k

,

ε

)

, so

x

∈

B

⁢

(

x

i

,

ε

)

,

y

∈

B

⁢

(

x

l

,

ε

)

,

i

,

l

∈

{

1

,

2

,

…

⁢

n

}

. So we have that

d

⁢

(

x

,

y

)

≤

d

⁢

(

x

,

x

i

)

+

d

⁢

(

x

i

,

x

l

)

+

d

⁢

(

x

l

,

y

)

<

ε

+

max

1

≤

s

,

t

≤

n

⁡

d

⁢

(

x

s

,

x

t

)

+

ε

=

M

∎

Title

totally bounded subset of a metric space is bounded

Canonical name

TotallyBoundedSubsetOfAMetricSpaceIsBounded

Date of creation

2013-03-22 15:25:27

Last modified on

2013-03-22 15:25:27

Owner

georgiosl (7242)

Last modified by

georgiosl (7242)

Numerical id

12

Author

georgiosl (7242)

Entry type

Theorem

Classification

msc 54E35

Generated on Sat Feb 10 11:27:54 2018 by

LaTeXML
totally bounded uniform space

totally bounded uniform space

A

uniform space

X

with uniformity

𝒰

is called

totally bounded

if for every entourage

U

∈

𝒰

, there is a

finite cover

C

1

,

…

,

C

n

of

X

, such that

C

i

×

C

i

∈

U

for every

i

=

1

,

…

,

n

.

𝒰

is called a

totally bounded uniformity

.

Remark

. A uniform space is

compact

(under the

uniform topology

) iff it is

complete

and totally bounded.

References

1

S. Willard,

General Topology

,
Addison-Wesley, Publishing Company, 1970.

Title

totally bounded uniform space

Canonical name

TotallyBoundedUniformSpace

Date of creation

2013-03-22 16:44:09

Last modified on

2013-03-22 16:44:09

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

5

Author

CWoo (3771)

Entry type

Definition

Classification

msc 54E35

Defines

totally bounded

Defines

totally bounded uniformity

Generated on Sat Feb 10 11:28:02 2018 by

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ultrametric

ultrametric

Any metric

d

:

X

×

X

→

ℝ

on a set

X

must

satisfy

the

triangle inequality

:

(

∀

x

,

y

,

z

)

d

⁢

(

x

,

z

)

≤

d

⁢

(

x

,

y

)

+

d

⁢

(

y

,

z

)

An

ultrametric

must additionally satisfy a

stronger

version of the triangle inequality:

(

∀

x

,

y

,

z

)

d

⁢

(

x

,

z

)

≤

max

⁡

{

d

⁢

(

x

,

y

)

,

d

⁢

(

y

,

z

)

}

Here is an example of an ultrametric on a space with 5 points, labelled

a

,

b

,

c

,

d

,

e

:

a

b

c

d

e

a

0

12

4

6

12

b

0

12

12

5

c

0

6

12

d

0

12

e

0

In the table above, an entry

n

in the for

element

x

and the for element

y

indicates that

d

⁢

(

x

,

y

)

=

n

, where

d

is the ultrametric. By

symmetry

of the ultrametric (

d

⁢

(

x

,

y

)

=

d

⁢

(

y

,

x

)

), the above table yields all values of

d

⁢

(

x

,

y

)

for all

x

,

y

∈

{

a

,

b

,

c

,

d

,

e

}

.

The ultrametric condition is

equivalent

to the ultrametric three point condition:

(

∀

x

,

y

,

z

)

x

,

y

,

z

⁢

can be renamed such that

⁢

d

⁢

(

x

,

z

)

≤

d

⁢

(

x

,

y

)

=

d

⁢

(

y

,

z

)

Ultrametrics can be used to model bifurcating hierarchical systems.  The

distance

between nodes in a weight-balanced binary tree is an ultrametric. Similarly, an ultrametric can be modelled by a weight-balanced binary tree, although the choice of tree is not necessarily unique.  Tree models of ultrametrics are sometimes called

ultrametric trees

.

The metrics

induced

by

non-Archimedean valuations

are ultrametrics.

Title

ultrametric

Canonical

name

Ultrametric

Date of creation

2013-03-22 13:28:28

Last modified on

2013-03-22 13:28:28

Owner

Koro (127)

Last modified by

Koro (127)

Numerical id

21

Author

Koro (127)

Entry type

Definition

Classification

msc 54E35

Related topic

MetricSpace

Related topic

Valuation

Related topic

UltrametricSpace

Generated on Sat Feb 10 11:28:04 2018 by

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ultrametric space

ultrametric space

The

metric space

(

X

,

d

)

is called an

ultrametric space

, if its metric

d

is an

ultrametric

, i.e. if

d

⁢

(

x

,

z

)

≦

max

⁡

{

d

⁢

(

x

,

y

)

,

d

⁢

(

y

,

z

)

}

∀

x

,

y

,

z

∈

X

.

Example.

The field

ℚ

together with any of its

p

-adic metrics

d

p

⁢

(

x

,

y

)

=

|

x

-

y

|

p

,

where

|

⋅

|

p

is the

p

-adic valuation (

http://planetmath.org/PAdicValuation

) of

ℚ

,  forms an ultrametric space.

Title

ultrametric space

Canonical

name

UltrametricSpace

Date of creation

2013-03-22 14:55:28

Last modified on

2013-03-22 14:55:28

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

6

Author

pahio (2872)

Entry type

Definition

Classification

msc 54E35

Related topic

UltrametricTriangleInequality

Related topic

Ultrametric

Generated on Sat Feb 10 11:28:06 2018 by

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uniform base

uniform base

Let

X

be a

Hausdorff topological space

. A basis for

X

is said to be a

uniform base

if for all

x

∈

X

and every

neighborhood

U

of

x

, only a finite number of the basis sets containing

x

intersect the complement of

U

.

For example, in any

metric space

, the open balls of radius

1

n

form a uniform base of

X

.

Any uniform base of

X

is a

point countable base

.

References

1

Steen, Lynn Arthur and Seebach, J. Arthur,

Counterexamples in Topology

, Dover Books, 1995.

Title

uniform base

Canonical

name

UniformBase

Date of creation

2013-03-22 14:49:56

Last modified on

2013-03-22 14:49:56

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 11:28:07 2018 by

LaTeXML
Urysohn metrization theorem

Urysohn metrization theorem

Let

X

be a

topological space

which is

regular

and

second countable

and in which singleton sets are closed. Then

X

is

metrizable

.

Title

Urysohn metrization theorem

Canonical name

UrysohnMetrizationTheorem

Date of creation

2013-03-22 12:12:36

Last modified on

2013-03-22 12:12:36

Owner

Evandar (27)

Last modified by

Evandar (27)

Numerical id

7

Author

Evandar (27)

Entry type

Theorem

Classification

msc 54E35

Related topic

SecondCountable

Related topic

Metrizable

Generated on Sat Feb 10 11:28:09 2018 by

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ε-net

ε

-net

Definition

Suppose

X

is a metric space with a metric

d

, and suppose

S

is a subset of

X

. Let

ε

be a positive real number.
A subset

N

⊂

S

is an

ε

-net
for

S

if, for all

x

∈

S

, there is an

y

∈

N

,
such that

d

⁢

(

x

,

y

)

<

ε

.

For any

ε

>

0

and

S

⊂

X

, the set

S

is trivially an

ε

-net for itself.

Theorem

Suppose

X

is a metric space with a metric

d

, and suppose

S

is a subset of

X

. Let

ε

be a positive real number.
Then

N

is an

ε

-net for

S

, if and only if

{

B

ε

⁢

(

y

)

∣

y

∈

N

}

is a cover for

S

. (Here

B

ε

⁢

(

x

)

is
the open ball with center

x

and radius

ε

.)

Proof.

Suppose

N

is an

ε

-net for

S

.
If

x

∈

S

, there is an

y

∈

N

such that

x

∈

B

ε

⁢

(

y

)

.
Thus,

x

is covered by some set in

{

B

ε

⁢

(

x

)

∣

x

∈

N

}

.

Conversely

, suppose

{

B

ε

⁢

(

y

)

∣

y

∈

N

}

is
a cover for

S

, and suppose

x

∈

S

. By

assumption

,
there is an

y

∈

N

, such that

x

∈

B

ε

⁢

(

y

)

.
Hence

d

⁢

(

x

,

y

)

<

ε

with

y

∈

N

.

□

Example

In

X

=

ℝ

2

with the usual
Cartesian metric, the set

N

=

{

(

a

,

b

)

∣

a

,

b

∈

ℤ

⁢

Ã

⁢



⁢

Â

⁢

}

is an

ε

-net for

X

assuming that

ε

>

2

/

2

.

□

The above definition and example can be found in

[

1

]

, page 64-65.

References

1

G. Bachman, L. Narici,

Functional analysis

,
Academic Press, 1966.

Title

ε

-net

Canonical

name

varepsilonnet

Date of creation

2013-03-22 13:37:54

Last modified on

2013-03-22 13:37:54

Owner

Koro (127)

Last modified by

Koro (127)

Numerical id

4

Author

Koro (127)

Entry type

Definition

Classification

msc 54E35

Related topic

Cover

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

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