proof of injective images of Baire space

proof of injective images of Baire space

We show that, for an

uncountable

Polish space

X

, there exists a

continuous

and one-to-one function

f

:

𝒩

→

X

such that

X

∖

f

⁢

(

𝒩

)

is

countable

. Furthermore, the

inverse

of

f

defined on

f

⁢

(

𝒩

)

is Borel measurable.

The construction of the

function

relies on the following result. We let

d

be a complete metric on

X

with respect to which the

diameter

of any subset is defined.

Lemma

.

Let

S

be an uncountable subset of

X

which can be written as the

difference

of two

closed sets

, and choose any

ϵ

>

0

.

Then, there exists a

sequence

S

1

,

S

2

,

…

of

pairwise disjoint

and uncountable subsets of

S

with diameter no more than

ϵ

and such that,

1.

for each

n

>

0

,

⋃

k

≤

n

S

k

is closed.

2.

S

∖

⋃

n

S

n

is countable.

Proof.

As

S

=

A

∖

B

is the difference of closed sets, it is a Polish space under the

subspace topology

.
In fact, if

B

is nonempty, the

topology

is generated by the complete metric

d

S

⁢

(

x

,

y

)

=

d

⁢

(

x

,

y

)

+

sup

⁡

{

|

d

⁢

(

x

,

z

)

-

1

-

d

⁢

(

y

,

z

)

-

1

|

:

z

∈

B

}

,

otherwise we may take

d

S

=

d

. In either case,

d

S

≥

d

, so it is enough to choose the sets

S

n

to have diameter no more than

2

-

n

with respect to

d

S

. Note also that any

bounded

and closed set with respect to this metric is also closed as a subset of

X

.

Let

S

c

be the

condensation points

of

S

, which are the points whose

neighborhoods

all contain uncountably many points of

S

. Then,

S

∖

S

c

is a union of countably many countable and

open subsets

of

S

, so is countable and open. Hence,

S

c

is uncountable and closed in

S

, and every open subset is uncountable.
Choosing any

p

∈

S

c

then

S

c

∖

p

will not be a closed subset of

X

. So, replacing

S

by

S

∖

{

p

}

if necessary, we may suppose that

S

c

is not closed as a subset of

X

and, therefore is not

compact

.

So, for some

δ

>

0

,

S

c

cannot be covered by finitely many sets with

d

S

-diameter no more than

δ

(see here (

http://planetmath.org/ProofThatAMetricSpaceIsCompactIfAndOnlyIfItIsCompleteAndTotallyBounded

)). By separability, there is a sequence

T

1

,

T

2

,

…

of

open balls

in

S

c

with diameter less than

min

⁡

(

ϵ

,

δ

)

, and covering

S

c

. Writing

T

¯

n

for the

d

S

-

closure

of

T

n

and eliminating any terms such that

T

n

⊆

⋃

k

<

n

T

¯

k

, then

S

n

≡

T

¯

n

∖

⋃

k

<

n

T

¯

k

have nonempty

interior

and hence are uncountable, and

S

∖

⋃

n

S

n

=

S

∖

S

c

is countable, as required.
∎

Note that

S

n

=

⋃

k

≤

n

S

k

∖

⋃

k

≤

n

-

1

S

k

is also a difference of closed sets. So, the lemma allows us to inductively choose sets

C

⁢

(

n

1

,

…

,

n

k

)

⊆

X

for

integers

k

≥

0

and

n

1

,

…

,

n

k

≥

1

such that

C

⁢

(

)

=

X

and the following are satisfied.

1.

C

⁢

(

n

1

,

…

,

n

k

,

m

)

are uncountable,

contained

in

C

⁢

(

n

1

,

…

,

n

k

)

, and pairwise disjoint as

m

runs through the

positive integers

.

2.

⋃

j

≤

m

C

⁢

(

n

1

,

…

,

n

k

,

j

)

is closed for all

m

≥

1

.

3.

C

⁢

(

n

1

,

…

,

n

k

)

∖

⋃

m

C

⁢

(

n

1

,

…

,

n

k

,

m

)

is countable and, for

k

≥

1

, has diameter no more than

2

-

k

.

For any

n

∈

𝒩

we may choose a sequence

x

k

∈

C

⁢

(

n

1

,

…

,

n

k

)

. Since, for

k

≥

1

, this set has diameter no more than

2

-

k

, then

d

⁢

(

x

j

,

x

k

)

≤

2

-

k

whenever

j

≥

k

≥

1

. So, the sequence is Cauchy (

http://planetmath.org/CauchySequence

) and hence has a limit

x

.
Furthermore, as

x

j

is contained in the closed set

⋃

m

≤

n

k

+

1

C

⁢

(

n

1

,

…

,

n

k

,

m

)

for

j

>

k

, then

x

must also be contained in it and hence is in

C

⁢

(

n

1

,

…

,

n

k

)

. So

C

⁢

(

n

)

≡

⋂

k

=

0

∞

C

⁢

(

n

1

,

…

,

n

k

)

contains

x

and is nonempty. Furthermore, as it has zero diameter, it is a

singleton

. So

f

:

𝒩

→

X

is uniquely defined by

f

⁢

(

n

)

∈

C

⁢

(

n

)

.

Given any

m

,

n

∈

𝒩

such that

m

j

=

n

j

for

j

≤

k

, then

f

⁢

(

m

)

and

f

⁢

(

n

)

are both in the set

C

⁢

(

m

1

,

…

,

m

k

)

, which has diameter no more than

2

-

k

. So,

d

⁢

(

f

⁢

(

m

)

,

f

⁢

(

n

)

)

≤

2

-

k

and

f

is continuous.

If

m

and

n

are distinct elements of

𝒩

and

k

is the smallest integer such that

m

k

≠

n

k

, then

f

⁢

(

m

)

and

f

⁢

(

n

)

are in the

disjoint sets

C

⁢

(

m

1

,

…

,

m

k

)

and

C

⁢

(

n

1

,

…

,

n

k

)

respectively. So

f

⁢

(

m

)

≠

f

⁢

(

n

)

, and

f

is

one-to-one

.

Now let

A

be the

countable set

A

=

⋃

k

⋃

n

1

,

…

,

n

k

(

C

⁢

(

n

1

,

…

,

n

k

)

∖

⋃

m

C

⁢

(

n

1

,

…

,

n

k

,

m

)

)

⊆

X

.

Also define

𝒩

⁢

(

n

1

,

…

,

n

k

)

≡

{

m

∈

𝒩

:

m

j

=

n

j

⁢

for

⁢

j

≤

k

}

.

These sets form a basis for the topology on

𝒩

. Clearly,

f

⁢

(

𝒩

⁢

(

n

1

,

…

,

n

k

)

)

⊆

C

⁢

(

n

1

,

…

,

n

k

)

∖

A

.

Choosing any

x

∈

C

⁢

(

n

1

,

…

,

n

k

)

∖

A

then we can inductively find

n

j

for

j

>

k

such that

x

∈

C

⁢

(

n

1

,

…

,

n

j

)

. Then, setting

n

=

(

n

1

,

n

2

,

…

)

gives

f

⁢

(

n

)

=

x

. This shows that

f

⁢

(

𝒩

⁢

(

n

1

,

…

,

n

k

)

)

=

C

⁢

(

n

1

,

…

,

n

k

)

∖

A

.

(1)

In particular,

f

⁢

(

𝒩

)

=

X

∖

A

and, therefore

X

∖

f

⁢

(

𝒩

)

is countable.
Finally, as

C

⁢

(

n

1

,

…

,

n

k

)

is a difference of closed sets, it is Borel, and equation (

1

) shows that the inverse of

f

is Borel measurable.

Title

proof of injective images of Baire space

Canonical

name

ProofOfInjectiveImagesOfBaireSpace

Date of creation

2013-03-22 18:47:15

Last modified on

2013-03-22 18:47:15

Owner

gel (22282)

Last modified by

gel (22282)

Numerical id

5

Author

gel (22282)

Entry type

Proof

Classification

msc 54E50

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