proof of spaces homeomorphic to Baire space

proof of spaces homeomorphic to Baire space

We show that a

topological space

X

is

homeomorphic

to

Baire space

,

𝒩

, if and only if the following are satisfied.

1.

It is a nonempty

Polish space

.

2.

It is

zero dimensional

.

3.

No nonempty and

open subsets

are

compact

.

As Baire space is easily shown to

satisfy

these properties, we just need to show that if they are satisfied then there exists a homeomorphism

f

:

𝒩

→

X

. By property

1

there is a complete metric

d

on

X

.

We choose subsets

C

⁢

(

n

1

,

…

,

n

k

)

of

X

for integers

k

≥

0

and

n

1

,

…

,

n

k

satisfying the following.

(i)

C

⁢

(

n

1

,

…

,

n

k

)

is a nonempty

clopen set

with

diameter

no more than

2

-

k

.

(ii)

C

⁢

(

)

=

X

.

(iii)

For any

n

1

,

…

,

n

k

then

C

⁢

(

n

1

,

…

,

n

k

,

m

)

are

pairwise disjoint

as

m

ranges over the

natural numbers

and,

⋃

m

=

1

∞

C

⁢

(

n

1

,

…

,

n

k

,

m

)

=

C

⁢

(

n

1

,

…

,

n

k

)

.

(1)

This can be done inductively. Suppose that

S

=

C

⁢

(

n

1

,

…

,

n

k

)

has already been chosen. As it is open, condition

3

says that it is not compact. Therefore, there is a

δ

>

0

such that

S

has no finite

open cover

consisting of sets of diameter no more than

δ

(see here (

http://planetmath.org/ProofThatAMetricSpaceIsCompactIfAndOnlyIfItIsCompleteAndTotallyBounded

)).
However, as Polish spaces are

separable

, there is a

countable

sequence

S

1

,

S

2

,

…

of open sets with diameter less than

δ

and

covering

S

.
As the space is zero dimensional, these can be taken to be clopen. By replacing

S

j

by

S

j

∩

S

we can assume that

S

j

⊆

S

. Then, replacing by

S

j

∖

⋃

i

<

j

S

i

, the sets

S

j

can be taken to be pairwise disjoint.

By eliminating

empty sets

we suppose that

S

j

≠

∅

for each

j

, and since

S

has no finite open cover consisting of sets of diameter less than

δ

, the sequence

S

j

will still be

infinite

. Defining

C

⁢

(

n

1

,

…

,

n

k

,

n

k

+

1

)

=

S

n

k

+

1

satisfies the required properties.

We now define a

function

f

:

𝒩

→

X

such that

f

⁢

(

n

)

∈

C

⁢

(

n

1

,

…

,

n

k

)

for each

n

∈

𝒩

and

k

≥

0

.
Choose any

n

∈

𝒩

there is a sequence

x

k

∈

C

⁢

(

n

1

,

…

,

n

k

)

. This set has diameter

bounded

by

2

-

k

and, so,

d

⁢

(

x

k

,

x

j

)

≤

2

-

k

for

j

≥

k

. This sequence is Cauchy (

http://planetmath.org/CauchySequence

) and, by completeness of the metric, must

converge

to a limit

x

. As

C

⁢

(

n

1

,

…

,

n

k

)

is closed, it contains

x

for each

k

and therefore

⋂

k

C

⁢

(

n

1

,

…

,

n

k

)

≠

∅

.

In fact, as it has zero diameter, this set must contain a single

element

, which we define to be

f

⁢

(

n

)

.

So, we have defined a function

f

:

𝒩

→

X

. If

m

,

n

∈

𝒩

satisfy

m

j

=

n

j

for

j

≤

k

then

f

⁢

(

m

)

,

f

⁢

(

n

)

are both

contained

in

C

⁢

(

m

1

,

…

,

m

k

)

and

d

⁢

(

f

⁢

(

m

)

,

f

⁢

(

n

)

)

≤

2

-

k

. Therefore,

f

is

continuous

.

It only remains to show that

f

has continuous

inverse

. Given any

x

∈

X

then

x

∈

C

⁢

(

)

and equation (

1

) allows us to choose a sequence

n

k

∈

ℕ

such that

x

∈

C

⁢

(

n

1

,

…

,

n

k

)

for each

k

. Then,

f

⁢

(

n

)

=

x

showing that

f

is onto.

If

m

≠

n

∈

𝒩

then, letting

k

be the first integer for which

m

k

≠

n

k

, the sets

C

⁢

(

m

1

,

…

,

m

k

)

and

C

⁢

(

n

1

,

…

,

n

k

)

are

disjoint

and, therefore,

f

⁢

(

m

)

≠

f

⁢

(

n

)

and

f

is one to one.

Finally, we show that

f

is an open map, so that its inverse is continuous. Sets of the form

𝒩

⁢

(

n

1

,

…

,

n

k

)

=

{

m

∈

𝒩

:

m

j

=

n

j

⁢

for

⁢

j

≤

k

}

form a basis for the

topology

on

𝒩

. Then,

f

⁢

(

𝒩

⁢

(

n

1

,

…

,

n

k

)

)

=

C

⁢

(

n

1

,

…

,

n

k

)

is open and, therefore,

f

is an open map.

Title

proof of spaces homeomorphic to Baire space

Canonical name

ProofOfSpacesHomeomorphicToBaireSpace

Date of creation

2013-03-22 18:46:51

Last modified on

2013-03-22 18:46:51

Owner

gel (22282)

Last modified by

gel (22282)

Numerical id

7

Author

gel (22282)

Entry type

Proof

Classification

msc 54E50

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