aarjaneiro/planetmath huggingface dataset:
stereographic projection

stereographic projection

The

n

-dimensional

Euclidean

http://planetmath.org/node/186

unit sphere

S

n

is
defined as a subset
of

ℝ

n

+

1

:

S

n

=

{

(

x

1

,

…

,

x

n

+

1

)

∈

ℝ

n

+

1

∣

∑

k

=

1

n

+

1

x

k

2

=

1

}

.

The

stereographic projection

maps all points of

S

n

to
the

n

-dimensional

Euclidean space

ℝ

n

except one. Let

N

:=

(

0

,

…

,

0

,

1

)

∈

S

n

be this point (it is usually called the

north pole

). Then the stereographic projection is defined by

σ

:

S

n

∖

N

→

ℝ

n

,

(

x

1

,

…

,

x

n

+

1

)

↦

c

-

1

x

n

+

1

-

1

⁢

(

x

1

,

…

,

x

n

)

.

Here,

c

is an arbitrary

real number

. If

c

=

1

, the projection
degenerates; in all other cases, however,

σ

is a smooth
bijective mapping.

The image

P

′

of a point

P

under

σ

can be geometrically
constructed as follows. Embed

ℝ

n

into

ℝ

n

+

1

as a

hyperplane

at

x

n

+

1

=

c

. Unless

c

=

1

, the

straight line

defined by

N

and

P

intersects with

ℝ

n

in
precisely one point,

P

′

. The most common values for

c

are

c

=

-

1

and

c

=

0

, see figures

1

and

2

.

Figure 1:

Stereographic projection of the one dimensional unit
sphere for

c

=

-

1

Figure 2:

Stereographic projection of the one dimensional unit
sphere for

c

=

0

Let

-

id

:

ℝ

n

+

1

→

ℝ

n

+

1

be the map

x

↦

-

x

, then

σ

~

:=

σ

∘

(

-

id

)

(a suitably restricted

composition

) maps all points of

S

n

except the

south pole

S

:=

(

0

,

…

,

0

,

-

1

)

smoothly and
bijectively to

ℝ

n

. Together,

σ

and

σ

~

form an atlas of

S

n

, so

S

n

is an

n

-dimensional

smooth
manifold

.

Title

stereographic projection

Canonical

name

StereographicProjection

Date of creation

2013-03-22 15:18:35

Last modified on

2013-03-22 15:18:35

Owner

GrafZahl (9234)

Last modified by

GrafZahl (9234)

Numerical id

5

Author

GrafZahl (9234)

Entry type

Definition

Classification

msc 54E40

Classification

msc 54C25

Classification

msc 54C05

Classification

msc 51M15

Related topic

CoordinateSystems

Related topic

ClosedComplexPlane

Related topic

RiemannSphere

Defines

north pole

Defines

south pole

Generated on Sat Feb 10 11:28:22 2018 by

LaTeXML
Cantor’s Intersection Theorem

Cantor’s Intersection Theorem

Theorem 1

.

Let

K

1

⊃

K

2

⊃

K

3

⊃

…

⊃

K

n

⊃

…

be a

sequence

of non-empty,

compact subsets

of a metric space

X

. Then the intersection

⋂

i

K

i

is not empty.

Proof.

Choose a point

x

i

∈

K

i

for every

i

=

1

,

2

,

…

Since

x

i

∈

K

i

⊂

K

1

is a sequence in a compact set, by

Bolzano-Weierstrass Theorem

, there exists a

subsequence

x

i

j

which

converges

to a point

x

∈

K

1

. Notice, however, that for a fixed index

n

, the sequence

x

i

j

lies in

K

n

for all

j

sufficiently large (namely for all

j

such that

i

j

>

n

). So one has

x

∈

K

n

.
Since this is true for every

n

, the result follows.
∎

Title

Cantor’s Intersection Theorem

Canonical

name

CantorsIntersectionTheorem

Date of creation

2013-03-22 15:12:35

Last modified on

2013-03-22 15:12:35

Owner

paolini (1187)

Last modified by

paolini (1187)

Numerical id

4

Author

paolini (1187)

Entry type

Theorem

Classification

msc 54E45

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

LaTeXML
Lebesgue number lemma

Lebesgue number lemma

Lebesgue number lemma

:

For every

open cover

𝒰

of a

compact

metric space

X

, there exists a

real number

δ

>

0

such that every open ball in

X

of radius

δ

is contained in some element of

𝒰

.

Any number

δ

satisfying the property above is called a

Lebesgue number

for the covering

𝒰

in

X

.

Title

Lebesgue number lemma

Canonical name

LebesgueNumberLemma

Date of creation

2013-03-22 13:01:05

Last modified on

2013-03-22 13:01:05

Owner

djao (24)

Last modified by

djao (24)

Numerical id

5

Author

djao (24)

Entry type

Theorem

Classification

msc 54E45

Defines

Lebesgue number

Generated on Sat Feb 10 11:28:25 2018 by

LaTeXML
Peano space

Peano space

A

Peano space

is a

compact

,

connected

,

locally connected

metric space

.

Every Peano space is

arcwise connected

.

A

Hausdorff space

is the

continuous

image of the unit interval if and only if
it is a Peano space.

Title

Peano space

Canonical name

PeanoSpace

Date of creation

2013-03-22 15:59:03

Last modified on

2013-03-22 15:59:03

Owner

Mathprof (13753)

Last modified by

Mathprof (13753)

Numerical id

6

Author

Mathprof (13753)

Entry type

Definition

Classification

msc 54E45

Generated on Sat Feb 10 11:28:27 2018 by

LaTeXML
proof of Lebesgue number lemma

proof of Lebesgue number lemma

By way of

contradiction

, suppose that no

Lebesgue number

existed. Then there exists an

open cover

𝒰

of

X

such that for all

δ

>

0

there exists an

x

∈

X

such that no

U

∈

𝒰

contains

B

δ

⁢

(

x

)

(the

open ball

of radius

δ

around

x

). Specifically, for each

n

∈

ℕ

, since

1

/

n

>

0

we can choose an

x

n

∈

X

such that no

U

∈

𝒰

contains

B

1

/

n

⁢

(

x

n

)

. Now,

X

is

compact

so there exists a subsequence

(

x

n

k

)

of the

sequence

of points

(

x

n

)

that

converges

to
some

y

∈

X

. Also,

𝒰

being an open cover of

X

implies that there exists

λ

>

0

and

U

∈

𝒰

such that

B

λ

⁢

(

y

)

⊆

U

. Since the sequence

(

x

n

k

)

converges to

y

, for

k

large enough it is true that

d

⁢

(

x

n

k

,

y

)

<

λ

/

2

(

d

is the metric on

X

) and

1

/

n

k

<

λ

/

2

. Thus after an application of the

triangle inequality

, it follows that

B

1

/

n

k

⁢

(

x

n

k

)

⊆

B

λ

⁢

(

y

)

⊆

U

,

contradicting the

assumption

that no

U

∈

𝒰

contains

B

1

/

n

⁢

(

x

n

)

. Hence a Lebesgue number for

𝒰

does exist.

Title

proof of Lebesgue number lemma

Canonical name

ProofOfLebesgueNumberLemma

Date of creation

2013-03-22 13:09:20

Last modified on

2013-03-22 13:09:20

Owner

scanez (1021)

Last modified by

scanez (1021)

Numerical id

7

Author

scanez (1021)

Entry type

Proof

Classification

msc 54E45

Generated on Sat Feb 10 11:28:29 2018 by

LaTeXML
a closed subset of a complete metric space is complete

a closed subset of a complete metric space is complete

Let

X

be a complete metric space, and let

Y

⊆

X

be a

closed subset

of

X

. Then

Y

is

complete

.

Proof

Let

{

y

n

}

⊆

Y

be a

Cauchy sequence

in

Y

. Then by the completeness of

X

,

y

n

→

x

for some

x

∈

X

. Then every

neighborhood

of

x

contains points in

Y

, so

x

∈

Y

¯

=

Y

.

Title

a closed subset of a complete metric space is complete

Canonical name

AClosedSubsetOfACompleteMetricSpaceIsComplete

Date of creation

2013-03-22 16:31:26

Last modified on

2013-03-22 16:31:26

Owner

ehremo (15714)

Last modified by

ehremo (15714)

Numerical id

4

Author

ehremo (15714)

Entry type

Result

Classification

msc 54E50

Generated on Sat Feb 10 11:28:30 2018 by

LaTeXML
a complete subspace of a metric space is closed

a complete subspace of a metric space is closed

Let

X

be a

metric space

, and let

Y

be a

complete

subspace

of

X

. Then

Y

is closed.

Proof

Let

x

∈

Y

¯

be a point in the

closure

of

Y

. Then by the

definition

of closure, from each ball

B

⁢

(

x

,

1

n

)

centered in

x

, we can select a point

y

n

∈

Y

. This is clearly a

Cauchy sequence

in

Y

, and its limit is

x

, hence by the completeness of

Y

,

x

∈

Y

and thus

Y

=

Y

¯

.

Title

a complete subspace of a metric space is closed

Canonical

name

ACompleteSubspaceOfAMetricSpaceIsClosed

Date of creation

2013-03-22 16:31:29

Last modified on

2013-03-22 16:31:29

Owner

ehremo (15714)

Last modified by

ehremo (15714)

Numerical id

5

Author

ehremo (15714)

Entry type

Result

Classification

msc 54E50

Generated on Sat Feb 10 11:28:32 2018 by

LaTeXML
Baire space

Baire space

Baire space

is the topological product of countably many copies of

ℕ

with

discrete topology

.

It is a

Polish space

.

References

1

Wikipedia’s entry on

http://en.wikipedia.org/wiki/Baire_space_(set_theory)

Baire space (

set theory

)

Title

Baire space

Canonical name

BaireSpace1

Date of creation

2013-03-22 16:04:09

Last modified on

2013-03-22 16:04:09

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

5

Author

mathcam (2727)

Entry type

Definition

Classification

msc 54E50

Generated on Sat Feb 10 11:28:33 2018 by

LaTeXML
Baire space is universal for Polish spaces

Baire space is universal for Polish spaces

The following result says that

Baire space

,

𝒩

, is

universal

for

Polish spaces

.

Theorem

.

Let

X

be a nonempty Polish space. Then, there is a

continuous

and onto map

f

:

N

→

X

.

Title

Baire space is universal for Polish spaces

Canonical name

BaireSpaceIsUniversalForPolishSpaces

Date of creation

2013-03-22 18:46:54

Last modified on

2013-03-22 18:46:54

Owner

gel (22282)

Last modified by

gel (22282)

Numerical id

5

Author

gel (22282)

Entry type

Theorem

Classification

msc 54E50

Related topic

PolishSpace

Related topic

InjectiveImagesOfBaireSpace

Generated on Sat Feb 10 11:28:35 2018 by

LaTeXML
Cantor space

Cantor space

The

Cantor space

, denoted by

ℂ

is the set of all

infinite

binary

sequences

with the

product topology

. It is a

perfect

Polish space

. It is a

compact

subspace

of the

Baire space

, the set of all infinite sequences of integers (again with the natural product topology).

References

.

•

Moschovakis, Yiannis N.

Descriptive Set Theory

. North-Holland Pub. Co. 1980, Amsterdam; New York.

Title

Cantor space

Canonical

name

CantorSpace

Date of creation

2013-03-22 13:44:42

Last modified on

2013-03-22 13:44:42

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

14

Author

mathcam (2727)

Entry type

Definition

Classification

msc 54E50

Generated on Sat Feb 10 11:28:36 2018 by

LaTeXML
complete

complete

A

metric space

X

is

complete

if every

Cauchy sequence

(

http://planetmath.org/CauchySequence

) in

X

is a

convergent sequence

.

Examples:

\PMlinkescapephrase

Cauchy sequence

•

The space

ℚ

of

rational numbers

is not complete: the

sequence

3

,

3.1

,

3.14

,

3.141

,

3.1415

,

3.14159

,

3.141592

⁢

…

consisting of finite decimals converging to

π

∈

ℝ

is a Cauchy sequence in

ℚ

that does not converge in

ℚ

.

•

The space

ℝ

of

real numbers

is complete, as it is the

completion

of

ℚ

with respect to the

standard metric

(other completions, such as the

p

-adic numbers, are also possible). More generally, the completion of any metric space is a

complete metric space

.

•

Every

Banach space

is complete. For example, the

L

p

–space of p-integrable functions is a complete metric space if

p

≥

1

.

Title

complete

Canonical name

Complete

Date of creation

2013-03-22 11:55:11

Last modified on

2013-03-22 11:55:11

Owner

djao (24)

Last modified by

djao (24)

Numerical id

12

Author

djao (24)

Entry type

Definition

Classification

msc 54E50

Generated on Sat Feb 10 11:28:38 2018 by

LaTeXML
completeness principle

completeness principle

The

completeness principle

is a

property

of the

real numbers

, and is one of the

foundations

of

real analysis

. The most common formulation of this principle is that every non-empty set which is

bounded from above

has a

supremum

.

This statement can be reformulated in several ways. Each of the following statements is to the above

definition

of the completeness principle:

1.

The limit of every

infinite

decimal

sequence

is a real number.

2.

Every

bounded

monotonic sequence

is

convergent

.

3.

A sequence is convergent iff it is a

Cauchy Sequence

.

Title

completeness principle

Canonical

name

CompletenessPrinciple

Date of creation

2013-03-22 12:23:06

Last modified on

2013-03-22 12:23:06

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

12

Author

mathcam (2727)

Entry type

Axiom

Classification

msc 54E50

Synonym

completeness Axiom

Synonym

completeness principle

Synonym

least

upper bound

property

Related topic

ConvergentSequence

Related topic

ExistenceOfSquareRootsOfNonNegativeRealNumbers

Related topic

BoundedComplete

Generated on Sat Feb 10 11:28:40 2018 by

LaTeXML
injective images of Baire space

injective images of Baire space

Every

uncountable

Polish space

is, up to a

countable

subset, an

injective image of Baire space

𝒩

.

Theorem

.

Let

X

be an uncountable Polish space. Then, there is a

one-to-one

and

continuous function

f

:

N

→

X

such that

X

∖

f

⁢

(

N

)

is countable.

Although the

inverse

f

-

1

:

f

⁢

(

𝒩

)

→

𝒩

will not generally be

continuous

, it is at least Borel measurable. It can be shown that this is true for all one-to-one and continuous functions between Polish spaces, although here it follows directly from the construction of

f

(

http://planetmath.org/ProofOfInjectiveImagesOfBaireSpace

).

Title

injective images of Baire space

Canonical name

InjectiveImagesOfBaireSpace

Date of creation

2013-03-22 18:47:12

Last modified on

2013-03-22 18:47:12

Owner

gel (22282)

Last modified by

gel (22282)

Numerical id

6

Author

gel (22282)

Entry type

Theorem

Classification

msc 54E50

Related topic

BaireSpaceIsUniversalForPolishSpaces

Related topic

SpacesHomeomorphicToBaireSpace

Generated on Sat Feb 10 11:28:41 2018 by

LaTeXML
Polish space

Polish space

A

Polish space

is a

topological space

that is

homeomorphic

to some

complete

separable

metric space

.

Equivalently, a topological space is a Polish space if it is separable and

completely metrizable

.

Examples of Polish spaces are the

euclidean space

ℝ

n

(with the

usual topology

), the

Cantor space

, and the

Baire space

(

http://planetmath.org/BaireSpace2

).

Title

Polish space

Canonical name

PolishSpace

Date of creation

2013-03-22 14:59:18

Last modified on

2013-03-22 14:59:18

Owner

Koro (127)

Last modified by

Koro (127)

Numerical id

5

Author

Koro (127)

Entry type

Definition

Classification

msc 54E50

Related topic

TopologicalGSpace

Related topic

PolishGroup

Related topic

CategoryOfPolishGroups

Related topic

SpacesHomeomorphicToBaireSpace

Generated on Sat Feb 10 11:28:43 2018 by

LaTeXML
Polish spaces up to Borel isomorphism

Polish spaces up to Borel isomorphism

Two

topological spaces

X

and

Y

are Borel

isomorphic

(

http://planetmath.org/BorelIsomorphism

) if there is a

Borel measurable function

f

:

X

→

Y

with Borel

inverse

. Such a

function

is said to be a

Borel isomorphism

. The following result classifies all

Polish spaces up to Borel isomorphism

.

Theorem

.

Every

uncountable

Polish space

is Borel isomorphic to

R

with the

standard topology

.

As the Borel

σ

-algebra on any

countable

metric space

is just its

power set

, this shows that every Polish space is Borel isomorphic to one and only one of the following.

1.

{

1

,

2

,

…

,

n

}

for some

n

≥

0

, with the

discrete topology

.

2.

ℕ

=

{

1

,

2

,

…

}

with the discrete topology.

3.

ℝ

with the standard topology.

In particular, two Polish spaces are Borel isomorphic if and only if they have the same

cardinality

, and any uncountable Polish space has the cardinality of the

continuum

.

Title

Polish spaces up to Borel isomorphism

Canonical name

PolishSpacesUpToBorelIsomorphism

Date of creation

2013-03-22 18:47:00

Last modified on

2013-03-22 18:47:00

Owner

gel (22282)

Last modified by

gel (22282)

Numerical id

6

Author

gel (22282)

Entry type

Theorem

Classification

msc 54E50

Related topic

CategoryOfPolishGroups

Generated on Sat Feb 10 11:28:44 2018 by

LaTeXML
proof of Baire space is universal for Polish spaces

proof of Baire space is universal for Polish spaces

Let

X

be a nonempty

Polish space

. We construct a

continuous

onto map

f

:

𝒩

→

X

, where

𝒩

is

Baire space

.

Let

d

be a

complete metric

on

X

.
We choose nonempty

closed subsets

C

⁢

(

n

1

,

…

,

n

k

)

⊆

X

for all integers

k

≥

0

and

n

1

,

…

,

n

k

∈

ℕ

satisfying the following.

1.

C

⁢

(

)

=

X

.

2.

C

⁢

(

n

1

,

…

,

n

k

)

has

diameter

no more than

2

-

k

.

3.

For any

k

≥

0

and

n

1

,

…

⁢

n

k

∈

ℕ

then

C

⁢

(

n

1

,

…

,

n

k

)

=

⋃

m

=

1

∞

C

⁢

(

n

1

,

…

,

n

k

,

m

)

.

(1)

This can be done by

induction

. Suppose that the set

S

=

C

⁢

(

n

1

,

…

,

n

k

)

has already been chosen for some

k

≥

0

. As

X

is

separable

,

S

can be covered by a

sequence

of

closed sets

S

1

,

S

2

,

…

. Replacing

S

j

by

S

j

∩

S

, we suppose that

S

j

⊆

S

. Then, remove any

empty sets

from the sequence. If the resulting sequence is finite, then it can be extended to an

infinite

sequence by repeating the last term.
We can then set

C

⁢

(

n

1

,

…

,

n

k

,

n

k

+

1

)

=

S

n

k

+

1

.

We now define the

function

f

:

𝒩

→

X

. For any

n

∈

ℕ

choose a sequence

x

k

∈

C

⁢

(

n

1

,

…

,

n

k

)

. Since this has diameter no more than

2

-

k

it follows that

d

⁢

(

x

j

,

x

k

)

≤

2

-

k

for

j

≥

k

. So, the sequence is Cauchy (

http://planetmath.org/CauchySequence

) and has a limit

x

. As the sets

C

⁢

(

n

1

,

…

,

n

k

)

are closed, they contain

x

and,

⋂

k

=

1

∞

C

⁢

(

n

1

,

…

,

n

k

)

≠

∅

.

(2)

In fact, this has diameter zero, and must contain a single

element

, which we define to be

f

⁢

(

n

)

.

This defines the function

f

:

𝒩

→

X

. We show that it is continuous. If

m

,

n

∈

𝒩

satisfy

m

j

=

n

j

for

j

≤

k

then

f

⁢

(

m

)

,

f

⁢

(

n

)

are in

C

⁢

(

m

1

,

…

,

m

k

)

which, having diameter no more than

2

-

k

, gives

d

⁢

(

f

⁢

(

m

)

,

f

⁢

(

n

)

)

≤

2

-

k

. So,

f

is indeed continuous.

Finally, choose any

x

∈

X

. Then

x

∈

C

⁢

(

)

and equation (

1

) allows us to choose

n

1

,

n

2

,

…

such that

x

∈

C

⁢

(

n

1

,

…

,

n

k

)

for all

k

≥

0

. If

n

=

(

n

1

,

n

2

,

…

)

then

x

and

f

⁢

(

n

)

are both in the set in equation (

2

) which, since it is a singleton, gives

f

⁢

(

n

)

=

x

. Hence,

f

is onto.

Title

proof of Baire space is universal for Polish spaces

Canonical name

ProofOfBaireSpaceIsUniversalForPolishSpaces

Date of creation

2013-03-22 18:46:57

Last modified on

2013-03-22 18:46:57

Owner

gel (22282)

Last modified by

gel (22282)

Numerical id

4

Author

gel (22282)

Entry type

Proof

Classification

msc 54E50

Generated on Sat Feb 10 11:28:47 2018 by

LaTeXML
proof of Polish spaces up to Borel isomorphism

proof of Polish spaces up to Borel isomorphism

We show that every

uncountable

Polish space

X

is Borel

isomorphic

to the

real numbers

.
First, there exists a

continuous

one-to-one

and injective function

f

from

Baire space

𝒩

to

X

such that

X

∖

f

⁢

(

𝒩

)

is

countable

, and such that the

inverse

from

f

⁢

(

𝒩

)

to

𝒩

is

Borel measurable

(see here (

http://planetmath.org/InjectiveImagesOfBaireSpace

)).
Letting

S

be any

countably infinite

subset of

X

, the same result can be applied to

X

∖

S

, which is also a Polish space.
So, there is a continuous and one-to-one function

f

:

𝒩

→

X

∖

S

such that

S

′

≡

X

∖

f

⁢

(

𝒩

)

is countable and such that the inverse defined on

X

∖

S

′

is Borel.
Then,

S

′

contains

S

and is countably infinite.
Hence, there is a invertible function

g

from

ℕ

=

{

1

,

2

,

…

}

to

S

′

. Under the

discrete topology

on

ℕ

this is necessarily a

continuous function

with Borel measurable inverse. By combining the

functions

f

and

g

, this gives a continuous, one-to-one and onto function from the

disjoint union

(

http://planetmath.org/TopologicalSum

)

u

:

𝒩

⁢

∐

ℕ

→

X

with Borel measurable inverse.
Similarly, the set of real numbers

ℝ

with the

standard topology

is an uncountable Polish space and, therefore, there is a continuous function

v

from

𝒩

⁢

∐

ℕ

to

ℝ

with Borel inverse. So,

v

∘

u

-

1

gives the desired

Borel isomorphism

from

X

to

ℝ

.

Title

proof of Polish spaces up to Borel isomorphism

Canonical name

ProofOfPolishSpacesUpToBorelIsomorphism

Date of creation

2013-03-22 18:47:18

Last modified on

2013-03-22 18:47:18

Owner

gel (22282)

Last modified by

gel (22282)

Numerical id

6

Author

gel (22282)

Entry type

Proof

Classification

msc 54E50

Generated on Sat Feb 10 11:28:52 2018 by

LaTeXML
Schauder lemma

Schauder lemma

The following

theorem

is in the

functional analysis

literature generally referred to as the

Schauder lemma

. It is a version of the

open mapping theorem

in Fréchet spaces and is often used to verify the open-ness of linear,

continuous maps

.

Theorem.

Let

E

,

F

be Fréchet spaces. Denote by

𝒰

0

⁢

(

E

)

,

𝒰

0

⁢

(

F

)

the zero

neighborhood

filter of

E

and

F

respectively. Let

T

:

E

→

F

be a linear and continuous map which is

almost open

, i.e.

∀

U

∈

𝒰

0

⁢

(

E

)

T

⁢

(

U

)

¯

F

∈

𝒰

0

⁢

(

F

)

Then

T

is open.

Title

Schauder lemma

Canonical

name

SchauderLemma

Date of creation

2013-03-22 19:09:44

Last modified on

2013-03-22 19:09:44

Owner

karstenb (16623)

Last modified by

karstenb (16623)

Numerical id

4

Author

karstenb (16623)

Entry type

Theorem

Classification

msc 54E50

Classification

msc 46A30

Related topic

OpenMappingTheorem

Defines

almost

open set

Generated on Sat Feb 10 11:28:57 2018 by

LaTeXML
spaces homeomorphic to Baire space

spaces homeomorphic to Baire space

Baire space

,

𝒩

≡

ℕ

ℕ

, is the set of all

functions

x

:

ℕ

→

ℕ

together with the

product topology

. This is

homeomorphic

to the set of

irrational numbers

in the unit interval, with the homeomorphism

f

:

𝒩

→

(

0

,

1

)

∖

ℚ

given by

continued fraction

expansion

f

⁢

(

x

)

=

1

x

⁢

(

1

)

+

1

x

⁢

(

2

)

+

1

⋱

.

Theorem 1

.

Let

I

be an

open interval

of the

real numbers

and

S

be a

countable

dense subset

of

I

. Then,

I

∖

S

is homeomorphic to Baire space.

More generally, Baire space is uniquely characterized up to homeomorphism by the following properties.

Theorem 2

.

A

topological space

X

is homeomorphic to Baire space if and only if

1.

It is a nonempty

Polish space

.

2.

It is

zero dimensional

(

http://planetmath.org/ZeroDimensional

).

3.

No nonempty

open subsets

are

compact

.

In particular, for an open interval

I

of the real numbers and countable dense subset

S

⊆

I

, then

I

∖

S

is easily seen to

satisfy

these properties and

Theorem

1

follows.

Title

spaces homeomorphic to Baire space

Canonical name

SpacesHomeomorphicToBaireSpace

Date of creation

2013-03-22 18:46:48

Last modified on

2013-03-22 18:46:48

Owner

gel (22282)

Last modified by

gel (22282)

Numerical id

7

Author

gel (22282)

Entry type

Theorem

Classification

msc 54E50

Related topic

PolishSpace

Related topic

InjectiveImagesOfBaireSpace

Generated on Sat Feb 10 11:28:59 2018 by

LaTeXML