aarjaneiro/planetmath huggingface dataset:
examples of locally compact and not locally compact spaces

examples of locally compact and not locally compact spaces

Examples of locally compact spaces include:

•

The

Euclidean spaces

ℝ

n

with the

standard topology

: their

local compactness

follows from the

Heine-Borel theorem

. The complex plane (

http://planetmath.org/Complex

)

ℂ

carries the same

topology

as

ℝ

2

and is therefore also locally compact.

•

All

topological manifolds

are locally compact since locally they look like Euclidean space.

•

Any closed or

open subset

of a locally compact space is locally compact. In fact, a subset of a

locally compact Hausdorff space

X

is locally compact if and only if it is the difference (

http://planetmath.org/SetDifference

) of two

closed subsets

of

X

(equivalently: the

intersection

of an open and a closed subset of

X

).

•

The space of

p

-adic rationals (

http://planetmath.org/PAdicIntegers

) is

homeomorphic

to the Cantor set minus one point, and since the Cantor set is

compact

as a closed

bounded

subset of

ℝ

, we see that the

p

-adic rationals are locally compact.

•

Any

discrete space

is locally compact, since the singletons can serve as compact

neighborhoods

.

•

The

long line

is a locally compact topological space.

•

If you take any unbounded

totally ordered set

and equip it with the left

order topology

(or right order topology), you get a locally compact space. This space, unlike all the others we have looked at, is not

Hausdorff

.

Examples of spaces which are

not

locally compact include:

•

The rational numbers

ℚ

with the standard topology inherited from

ℝ

: each of its compact subsets has empty

interior

.

•

All infinite-dimensional

normed vector spaces

: a normed vector space is finite-dimensional if and only if its closed

unit ball

is compact.

•

The subset

X

=

{

(

0

,

0

)

}

∪

{

(

x

,

y

)

⁢

∣

x

>

⁢

0

}

of

ℝ

2

: no compact
subset of

X

contains a neighborhood of

(

0

,

0

)

.

Title

examples of locally compact and not locally compact spaces

Canonical name

ExamplesOfLocallyCompactAndNotLocallyCompactSpaces

Date of creation

2013-03-22 12:48:36

Last modified on

2013-03-22 12:48:36

Owner

AxelBoldt (56)

Last modified by

AxelBoldt (56)

Numerical id

18

Author

AxelBoldt (56)

Entry type

Example

Classification

msc 54D45

Related topic

TopologicalSpace

Generated on Sat Feb 10 11:23:28 2018 by

LaTeXML
local compactness is hereditary for locally closed subspaces

local compactness is hereditary for locally closed subspaces

Theorem

-

Let

X

be a locally compact space and

Y

⊆

X

a

subspace

. If

Y

is

locally closed

in

X

then

Y

is also

locally compact

.

The

converse

of this theorem is also true with the additional

assumption

that

X

is

Hausdorff

.

Theorem 2 -

Let

X

be a

locally compact Hausdorff space

(

http://planetmath.org/LocallyCompactHausdorffSpace

) and

Y

⊆

X

a subspace. If

Y

is locally compact then

Y

is locally closed in

X

.

Title

local compactness is hereditary for locally closed subspaces

Canonical name

LocalCompactnessIsHereditaryForLocallyClosedSubspaces

Date of creation

2013-03-22 17:36:33

Last modified on

2013-03-22 17:36:33

Owner

asteroid (17536)

Last modified by

asteroid (17536)

Numerical id

5

Author

asteroid (17536)

Entry type

Theorem

Classification

msc 54D45

Related topic

LocallyCompactHausdorffSpace

Generated on Sat Feb 10 11:23:29 2018 by

LaTeXML
locally compact

locally compact

A

topological space

X

is

locally compact

at a point

x

∈

X

if there exists a

compact set

K

which contains a nonempty

neighborhood

U

of

x

. The space

X

is

locally compact

if it is locally compact at every point

x

∈

X

.

Note that local compactness at

x

does not require that

x

have a neighborhood which is actually

compact

, since compact

open sets

are fairly rare and the more relaxed condition turns out to be more useful in practice. However, it is true that a space is locally compact at

x

if and only if

x

has a

precompact

neighborhood.

Title

locally compact

Canonical name

LocallyCompact

Date of creation

2013-03-22 12:38:24

Last modified on

2013-03-22 12:38:24

Owner

djao (24)

Last modified by

djao (24)

Numerical id

6

Author

djao (24)

Entry type

Definition

Classification

msc 54D45

Related topic

Compact

Defines

local compactness

Generated on Sat Feb 10 11:23:31 2018 by

LaTeXML
precompact set

precompact set

Definition 1

.

A subset in a

topological space

is

precompact

if its

closure

is

compact

[

1

]

.

For

metric spaces

, we have the following theorem due to

Hausdorff

[

2

]

.

Theorem

Suppose

K

is a set in a

complete metric space

X

.
Then

K

relatively compact if and only if for any

ε

>

0

there is a finite

ε

-net (

http://planetmath.org/VarepsilonNet

) for

K

.

Examples

1.

In

ℝ

n

every point has a precompact

neighborhood

.

2.

On a manifold, every point has a precompact neighborhood.
This follows from the previous example, since a

homeomorphism

commutes with the closure operator, and since the

continuous

image
of a compact set is compact.

Notes

A synonym is

relatively compact

[

2

,

3

]

.

Some authors (notably Bourbaki see

[

4

]

) use precompact differently - as a synonym for

totally bounded

(

http://planetmath.org/TotallyBounded

) (in the generality of

topological groups

). “Relatively compact” is then used to mean “precompact ”as it is defined here

References

1

J.M. Lee,

Introduction to Smooth Manifolds

,
Graduate Texts in Mathematics series, 218,
Springer-Verlag, 2002.

2

R. Cristescu,

Topological vector spaces

,
Noordhoff International Publishing, 1977.

3

E. Kreyszig,

Introductory

Functional Analysis

With Applications

,
John Wiley & Sons, 1978.

4

N. Bourbaki,

Topological Vector Spaces

Springer-Verlag, 1981

Title

precompact set

Canonical name

PrecompactSet

Date of creation

2013-03-22 14:39:59

Last modified on

2013-03-22 14:39:59

Owner

matte (1858)

Last modified by

matte (1858)

Numerical id

12

Author

matte (1858)

Entry type

Definition

Classification

msc 54D45

Synonym

precompact

Synonym

relatively compact

Generated on Sat Feb 10 11:23:33 2018 by

LaTeXML
vanish at infinity

vanish at infinity

Let

X

be a locally compact space. A

function

f

:

X

⟶

ℂ

is said to

vanish at infinity

if, for every

ϵ

>

0

, there is a

compact set

K

⊆

X

such that

|

f

⁢

(

x

)

|

<

ϵ

for every

x

∈

X

-

K

, where

∥

⋅

∥

denotes the standard norm (

http://planetmath.org/Norm2

) on

ℂ

.

If

X

is non-compact, let

X

∪

{

∞

}

be the

one-point compactification

of

X

. The above definition can be rephrased as: The

extension

of

f

to

X

∪

{

∞

}

satisfying

f

⁢

(

∞

)

=

0

is

continuous at

the point

∞

.

The set of

continuous functions

X

⟶

ℂ

that vanish at infinity is an algebra over the complex field and is usually denoted by

C

0

⁢

(

X

)

.

0.0.1

Remarks

•

When

X

is

compact

, all functions

X

⟶

ℂ

vanish at infinity. Hence,

C

0

⁢

(

X

)

=

C

⁢

(

X

)

.

Title

vanish at infinity

Canonical name

VanishAtInfinity

Date of creation

2013-03-22 17:50:57

Last modified on

2013-03-22 17:50:57

Owner

asteroid (17536)

Last modified by

asteroid (17536)

Numerical id

6

Author

asteroid (17536)

Entry type

Definition

Classification

msc 54D45

Classification

msc 54C35

Synonym

zero at infinity

Synonym

vanishes at infinity

Related topic

RegularAtInfinity

Related topic

ApplicationsOfUrysohnsLemmaToLocallyCompactHausdorffSpaces

Defines

C

0

Generated on Sat Feb 10 11:23:35 2018 by

LaTeXML
σ-compact

σ

-compact

A

topological space

is

σ

-

compact

if it is a countable union of compact sets.

Title

σ

-compact

Canonical name

sigmacompact

Date of creation

2013-03-22 14:06:10

Last modified on

2013-03-22 14:06:10

Owner

Koro (127)

Last modified by

Koro (127)

Numerical id

5

Author

Koro (127)

Entry type

Definition

Classification

msc 54D45

Synonym

sigma-compact

Synonym

sigma compact

Synonym

σ

compact

Generated on Sat Feb 10 11:23:36 2018 by

LaTeXML
Hewitt-Marczewski-Pondiczery theorem

Hewitt-Marczewski-Pondiczery theorem

The

Hewitt–Marczewski–Pondiczery

Theorem

is a result on the

density

of

products

of

topological spaces

.
This theorem was arrived at independently by
Hewitt

[

1

]

, Marczewski

[

2

]

and Pondiczery

[

3

]

in the 1940s.

Theorem

.

Let

κ

be an

infinite

cardinal number

and

S

an

index set

of

cardinality

at most

2

κ

.
If

X

s

(

s

∈

S

)

are topological spaces with

d

⁢

(

X

s

)

≤

κ

then

d

⁢

(

∏

s

∈

S

X

s

)

≤

κ

.

The special case

κ

=

ℵ

0

says that the product of at most

continuum many

separable spaces

is

separable

.

References

1

Edwin Hewitt,

A remark on density characters

,
Bull. Amer. Math. Soc. 52 (1946), 641–643.
(This paper is available as a PDF file from the AMS website:

http://www.ams.org/journals/bull/1946-52-08/home.html

Bull. Amer. Math. Soc., Volume 52, Number 8.)

2

Edward Marczewski,

Séparabilité et

multiplication

cartésienne des espaces topologiques

,
Fund. Math. 34 (1947), 127–143.
(This paper is available as a PDF file from the Polish Virtual Library of Science:

http://matwbn.icm.edu.pl/tresc.php?wyd=1&tom=34&jez=en

Fundamenta Mathematicae, Volume 34.)

3

E. S. Pondiczery,

Power problems in abstract spaces

,
Duke Math. J. 11 (1944), 835–837.

Title

Hewitt-Marczewski-Pondiczery theorem

Canonical

name

HewittMarczewskiPondiczeryTheorem

Date of creation

2013-03-22 17:16:56

Last modified on

2013-03-22 17:16:56

Owner

yark (2760)

Last modified by

yark (2760)

Numerical id

6

Author

yark (2760)

Entry type

Theorem

Classification

msc 54D65

Classification

msc 54A25

Related topic

Dense

Related topic

Separable

Generated on Sat Feb 10 11:23:38 2018 by

LaTeXML
separable space

separable space

Definition

A

topological space

is said to be

separable

if it has a

countable

dense subset

.

Properties

All second-countable spaces are separable.
A

metric space

is separable if and only if it is

second-countable

.

A

continuous

image of a

separable space

is separable.

An

open subset

of a separable space is separable (in the

subspace topology

).

A

product

(

http://planetmath.org/ProductTopology

) of

2

ℵ

0

or fewer separable spaces
is separable. This is a special case of the

Hewitt-Marczewski-Pondiczery Theorem

.

A

Hilbert space

is separable if and only if it has a countable

orthonormal basis

.

Title

separable space

Canonical name

SeparableSpace

Date of creation

2013-03-22 12:05:45

Last modified on

2013-03-22 12:05:45

Owner

yark (2760)

Last modified by

yark (2760)

Numerical id

13

Author

yark (2760)

Entry type

Definition

Classification

msc 54D65

Synonym

separable topological space

Related topic

SecondCountable

Related topic

Lindelof

Related topic

EverySecondCountableSpaceIsSeparable

Related topic

HewittMarczewskiPondiczeryTheorem

Defines

separable

Generated on Sat Feb 10 11:23:39 2018 by

LaTeXML
a compact metric space is second countable

a compact metric space is second countable

Proposition

.

Every

compact metric space is second countable

.

Proof.

Let

(

X

,

d

)

be a

compact

metric space

, and for each

n

∈

ℤ

+

define

𝒜

n

=

{

B

⁢

(

x

,

1

/

n

)

:

x

∈

X

}

, where

B

⁢

(

x

,

1

/

n

)

denotes the open ball centered about

x

of

http://planetmath.org/node/1296

radius

1

/

n

. Each such

collection

is an

open cover

of the

compact space

X

, so for each

n

∈

ℤ

+

there exists a finite collection

ℬ

n

⊆

𝒜

n

that

X

. Put

ℬ

=

⋃

n

=

1

∞

ℬ

n

. Being a

countable

union of

finite sets

, it follows that

ℬ

is countable; we assert that it forms a basis for the metric topology on

X

. The first property of a basis is satisfied trivially, as each set

ℬ

n

is an open cover of

X

. For the second property, let

x

,

x

1

,

x

2

∈

X

,

n

1

,

n

2

∈

ℤ

+

, and suppose

x

∈

B

⁢

(

x

1

,

1

/

n

1

)

∩

B

⁢

(

x

2

,

1

/

n

2

)

. Because the sets

B

⁢

(

x

1

,

1

/

n

1

)

and

B

⁢

(

x

2

,

1

/

n

2

)

are open in the metric topology on

X

, their

intersection

is also open, so there exists

ϵ

>

0

such that

B

⁢

(

x

,

ϵ

)

⊆

B

⁢

(

x

1

,

1

/

n

1

)

∩

B

⁢

(

x

2

,

1

/

n

2

)

. Select

N

∈

ℤ

+

such that

1

/

N

<

ϵ

. There must exist

x

3

∈

X

such that

x

∈

B

⁢

(

x

3

,

1

/

2

⁢

N

)

(since

ℬ

2

⁢

N

is an open cover of

X

). To see that

B

⁢

(

x

3

,

1

/

2

⁢

N

)

⊆

B

⁢

(

x

1

,

1

/

n

1

)

∩

B

⁢

(

x

2

,

1

/

n

2

)

, let

y

∈

B

⁢

(

x

3

,

1

/

2

⁢

N

)

. Then we have

d

⁢

(

x

,

y

)

≤

d

⁢

(

x

,

x

3

)

+

d

⁢

(

x

3

,

y

)

<

1

2

⁢

N

+

1

2

⁢

N

=

1

N

<

ϵ

⁢

,

(1)

so that

y

∈

B

⁢

(

x

,

ϵ

)

, from which it follows that

y

∈

B

⁢

(

x

1

,

1

/

n

1

)

∩

B

⁢

(

x

2

,

1

/

n

2

)

, hence that

B

⁢

(

x

3

,

1

/

2

⁢

N

)

⊆

B

⁢

(

x

1

,

1

/

n

1

)

∩

B

⁢

(

x

2

,

1

/

n

2

)

. Thus the countable collection

ℬ

forms a basis for a

topology

on

X

; the verification that the topology by

ℬ

is in fact the metric topology follows by an to that used to verify the second property of a basis, and

completes

the proof that

X

is

second countable

.
∎

It is worth nothing that, because a countable

union of countable sets

is countable, it would have been

sufficient

to assume that

(

X

,

d

)

was a Lindelöf space.

Title

a compact metric space is second countable

Canonical name

ACompactMetricSpaceIsSecondCountable

Date of creation

2013-03-22 17:00:49

Last modified on

2013-03-22 17:00:49

Owner

azdbacks4234 (14155)

Last modified by

azdbacks4234 (14155)

Numerical id

17

Author

azdbacks4234 (14155)

Entry type

Theorem

Classification

msc 54D70

Related topic

MetricSpace

Related topic

Compact

Related topic

Lindelof

Related topic

Ball

Related topic

basisTopologicalSpace

Related topic

Cover

Related topic

BasisTopologicalSpace

Generated on Sat Feb 10 11:23:42 2018 by

LaTeXML
second countability is hereditary

second countability is hereditary

Suppose

X

is a

second countable

topological space

and

A

is a
subset in

X

. Then

A

(when equipped with the

subspace topology

)
is also second countable.

Title

second countability is hereditary

Canonical name

SecondCountabilityIsHereditary

Date of creation

2013-03-22 15:22:45

Last modified on

2013-03-22 15:22:45

Owner

matte (1858)

Last modified by

matte (1858)

Numerical id

4

Author

matte (1858)

Entry type

Theorem

Classification

msc 54D70

Generated on Sat Feb 10 11:23:43 2018 by

LaTeXML
second countable

second countable

A

topological space

is said to be

second

if it has a countable basis (

http://planetmath.org/BasisTopologicalSpace

).
It can be shown that a space is both Lindelöf and

separable

, although the converses fail. For instance, the

lower limit topology

on the real line is both Lindelöf and separable, but not

second countable

.

Title

second countable

Canonical name

SecondCountable

Date of creation

2013-03-22 12:05:06

Last modified on

2013-03-22 12:05:06

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

17

Author

rspuzio (6075)

Entry type

Definition

Classification

msc 54D70

Synonym

second axiom of countability

Synonym

completely separable

Synonym

perfectly separable

Synonym

second-countable

Related topic

Separable

Related topic

Lindelof

Related topic

EverySecondCountableSpaceIsSeparable

Related topic

LindelofTheorem

Related topic

UrysohnMetrizationTheorem

Related topic

FirstAxiomOfCountability

Related topic

LocallyCompactGroupoids

Related topic

FirstCountable

Generated on Sat Feb 10 11:23:45 2018 by

LaTeXML
first countable

first countable

Let

X

be a

topological space

and let

x

∈

X

.

X

is said to be

at

x

if there is a sequence

(

B

n

)

n

∈

ℕ

of

open sets

such that whenever

U

is an open set containing

x

, there is

n

∈

ℕ

such that

x

∈

B

n

⊆

U

.

The space

X

is said to be

if for every

x

∈

X

,

X

is

first countable

at

x

.

Remark

. Equivalently, one can take each

B

n

in the sequence to be open neighborhood of

x

.

Title

first countable

Canonical name

FirstCountable

Date of creation

2013-03-22 12:23:33

Last modified on

2013-03-22 12:23:33

Owner

Evandar (27)

Last modified by

Evandar (27)

Numerical id

5

Author

Evandar (27)

Entry type

Definition

Classification

msc 54D99

Synonym

first axiom of countability

Related topic

SecondCountable

Related topic

TestingForContinuityViaNets

Generated on Sat Feb 10 11:23:55 2018 by

LaTeXML
homogeneous topological space

homogeneous topological space

\PMlinkescapephrase

initial point

Definitions

A

topological space

X

is said to be

homogeneous

if for all

a

,

b

∈

X

there is a

homeomorphism

ϕ

:

X

→

X

such that

ϕ

⁢

(

a

)

=

b

.

A topological space

X

is said to be

bihomogeneous

if for all

a

,

b

∈

X

there is a homeomorphism

ϕ

:

X

→

X

such that

ϕ

⁢

(

a

)

=

b

and

ϕ

⁢

(

b

)

=

a

.

Examples

The

long line

(without initial point) is homogeneous,
but it is not bihomogeneous
as its self-homeomorphisms are all order-preserving.
This can be considered a pathological example,
as most homogeneous topological spaces encountered in practice
are also bihomogeneous.

Every

topological group

is bihomogeneous.
To see this, note that if

G

is a topological group and

a

,

b

∈

G

,
then

x

↦

a

⁢

x

-

1

⁢

b

defines a homeomorphism interchanging

a

and

b

.

Every

connected

topological manifold

without

boundary

is homogeneous.
This is true even if we do not require our

manifolds

to be

paracompact

,
as any two points share a

Euclidean

neighbourhood

,
and a suitable homeomorphism for this neighbourhood
can be extended to the whole manifold.
In fact, except for the long line (as mentioned above),
every connected topological manifold without boundary is bihomogeneous.
This is for essentially the same reason,
except that the argument breaks down for

1

-manifolds.

Title

homogeneous topological space

Canonical name

HomogeneousTopologicalSpace

Date of creation

2013-03-22 16:18:21

Last modified on

2013-03-22 16:18:21

Owner

yark (2760)

Last modified by

yark (2760)

Numerical id

5

Author

yark (2760)

Entry type

Definition

Classification

msc 54D99

Synonym

homogeneous space

Defines

homogeneous

Defines

bihomogeneous

Defines

bihomogeneous space

Defines

bihomogeneous topological space

Generated on Sat Feb 10 11:23:57 2018 by

LaTeXML
Lindelöf theorem

Lindelöf theorem

If a

topological space

(

X

,

τ

)

satisfies the

second axiom of countability

, and if

A

is any subset of

X

, then any

open cover

for

A

has a countable subcover.

In particular, we have that

(

X

,

τ

)

is a Lindelöf space (

http://planetmath.org/lindelofspace

).

Title

Lindelöf theorem

Canonical name

LindelofTheorem

Date of creation

2014-11-06 13:45:28

Last modified on

2014-11-06 13:45:28

Owner

drini (3)

Last modified by

pahio (2872)

Numerical id

7

Author

drini (2872)

Entry type

Theorem

Classification

msc 54D99

Related topic

SecondCountable

Related topic

Lindelof

Generated on Sat Feb 10 11:23:58 2018 by

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locally closed

locally closed

-

A subset

Y

of a

topological space

X

is said to be

locally closed

if it is the

intersection

of an open and a

closed subset

.

The following result provides some

definitions

:

-

The following are

equivalent

:

1.

Y

is locally closed in

X

.

2.

Each point in

Y

has an open neighborhood

U

⊆

X

such that

U

∩

Y

is closed in

U

(with the

subspace topology

).

3.

Y

is open in its

closure

Y

¯

(with the subspace topology).

Title

locally closed

Canonical

name

LocallyClosed

Date of creation

2013-03-22 17:36:12

Last modified on

2013-03-22 17:36:12

Owner

asteroid (17536)

Last modified by

asteroid (17536)

Numerical id

5

Author

asteroid (17536)

Entry type

Definition

Classification

msc 54D99

Generated on Sat Feb 10 11:24:00 2018 by

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M. H. Stone’s representation theorem

M. H. Stone’s representation theorem

Theorem 1

.

Given a

Boolean algebra

B

there exists a

totally disconnected

compact

Hausdorff space

X

such that

B

is

isomorphic

to the Boolean algebra
of

clopen subsets

of

X

.

Proof.

Let

X

=

B

*

, the

dual space

(

http://planetmath.org/DualSpaceOfABooleanAlgebra

) of

B

, which is composed of all

maximal ideals

of

B

. According to this entry (

http://planetmath.org/DualSpaceOfABooleanAlgebra

),

X

is a

Boolean space

(totally disconnected compact

Hausdorff

) whose

topology

is generated by the basis

ℬ

:=

{

M

⁢

(

a

)

∣

a

∈

B

}

,

where

M

⁢

(

a

)

=

{

M

∈

B

*

∣

a

∉

M

}

.

Next, we show a general fact about the dual space

B

*

:

Lemma 2

.

ℬ

is the set of

all

clopen sets in

X

.

Proof.

Clearly, every element of

ℬ

is clopen, by definition.

Conversely

, suppose

U

is clopen. Then

U

=

⋃

{

M

⁢

(

a

i

)

∣

i

∈

I

}

for some

index set

I

, since

U

is open. But

U

is closed, so

B

*

-

U

=

⋃

{

M

⁢

(

a

j

)

∣

j

∈

J

}

for some index set

J

. Hence

B

*

=

⋃

{

M

⁢

(

a

k

)

∣

k

∈

I

∪

J

}

. Since

B

*

is compact, there is a finite subset

K

of

I

∪

J

such that

B

*

=

⋃

{

M

⁢

(

a

k

)

∣

k

∈

K

}

. Let

V

=

⋃

{

M

⁢

(

a

i

)

∣

i

∈

K

∩

I

}

. Then

V

⊆

U

. But

B

*

-

V

⊆

B

*

-

U

also. So

U

=

V

. Let

y

=

⋁

{

a

i

∣

i

∈

K

∩

I

}

, which exists because

K

∩

I

is finite. As a result,

U

=

V

=

⋃

{

M

⁢

(

a

i

)

∣

i

∈

K

∩

I

}

=

M

⁢

(

⋁

{

a

i

∣

i

∈

K

∩

I

}

)

=

M

⁢

(

y

)

∈

ℬ

.

∎

Finally, based on the result of this entry (

http://planetmath.org/RepresentingABooleanLatticeByFieldOfSets

),

B

is isomorphic to the

field of sets

F

:=

{

F

⁢

(

a

)

∣

a

∈

B

}

,

where

F

⁢

(

a

)

=

{

P

∣

P

⁢

prime in

⁢

B

,

and

⁢

a

∉

P

}

. Realizing that

prime ideals

and maximal ideals coincide in any Boolean algebra, the set

F

is precisely

ℬ

.
∎

Remark

. There is also a dual version of the

Stone representation theorem

, which says that every Boolean space is

homeomorphic

to the dual space of some Boolean algebra.

Title

M. H. Stone’s representation theorem

Canonical name

MHStonesRepresentationTheorem

Date of creation

2013-03-22 13:25:34

Last modified on

2013-03-22 13:25:34

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

19

Author

rspuzio (6075)

Entry type

Theorem

Classification

msc 54D99

Classification

msc 06E99

Classification

msc 03G05

Synonym

Stone representation theorem

Synonym

Stone’s representation theorem

Related topic

RepresentingABooleanLatticeByFieldOfSets

Related topic

DualSpaceOfABooleanAlgebra

Generated on Sat Feb 10 11:24:02 2018 by

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proof of Lindelöf theorem

proof of Lindelöf theorem

Let

X

be a

second countable

topological space

,

A

⊆

X

any subset and

𝒰

an

open cover

of

A

. Let

ℬ

be a

countable basis

for

X

; then

ℬ

′

=

{

B

∩

A

:

B

∈

ℬ

}

is a countable basis of the

subspace topology

on A. Then for each

a

∈

A

there is some

U

a

∈

𝒰

with

a

∈

U

a

, and so there is

B

a

∈

ℬ

′

such that

a

∈

B

a

⊆

U

a

.

Then

{

B

a

∈

ℬ

′

:

a

∈

A

}

⊆

ℬ

is
a countable open cover of

A

. For each

B

a

, choose

U

B

a

∈

𝒰

such that

B

a

⊆

U

B

a

. Then

{

U

B

a

:

a

∈

A

}

is a countable subcover of

A

from

𝒰

.

□

Title

proof of Lindelöf theorem

Canonical name

ProofOfLindelofTheorem

Date of creation

2013-03-22 12:56:31

Last modified on

2013-03-22 12:56:31

Owner

Evandar (27)

Last modified by

Evandar (27)

Numerical id

5

Author

Evandar (27)

Entry type

Proof

Classification

msc 54D99

Generated on Sat Feb 10 11:24:04 2018 by

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