dual of Stone representation theorem

dual of Stone representation theorem

The

Stone representation theorem

characterizes a

Boolean algebra

as a

field of sets

in a

topological space

. There is also a dual to this famous

theorem

that characterizes a

Boolean space

as a topological space constructed from a Boolean algebra.

Theorem 1

.

Let

X

be a Boolean space. Then there is a Boolean algebra

B

such that

X

is

homeomorphic

to

B

*

, the

dual space

(

http://planetmath.org/DualSpaceOfABooleanAlgebra

) of

B

.

Proof.

The choice for

B

is clear: it is the set of

clopen sets

in

X

which, via the set theoretic

operations

of

intersection

, union, and

complement

, is a Boolean algebra.

Next, define a

function

f

:

X

→

B

*

by

f

⁢

(

x

)

:=

{

U

∈

B

∣

x

∉

U

}

.

Our ultimate goal is to prove that

f

is the desired homeomorphism. We break down the proof of this into several stages:

Lemma 1

.

f

is well-defined.

Proof.

The key is to show that

f

⁢

(

x

)

is a

prime ideal

in

B

*

for any

x

∈

X

. To see this, first note that if

U

,

V

∈

f

⁢

(

x

)

, then so is

U

∪

V

∈

f

⁢

(

x

)

, and if

W

is any clopen set of

X

, then

U

∩

W

∈

f

⁢

(

x

)

too. Finally, suppose that

U

∩

V

∈

f

⁢

(

x

)

. Then

x

∈

X

-

(

U

∩

V

)

=

(

X

-

U

)

∪

(

X

-

V

)

, which means that

x

∉

U

or

x

∉

V

, which is the same as saying that

U

∈

f

⁢

(

x

)

or

V

∈

f

⁢

(

x

)

. Hence

f

⁢

(

x

)

is a prime ideal, or a

maximal ideal

, since

B

is

Boolean

. ∎

Lemma 2

.

f

is

injective

.

Proof.

Suppose

x

≠

y

, we want to show that

f

⁢

(

x

)

≠

f

⁢

(

y

)

. Since

X

is

Hausdorff

, there are

disjoint

open sets

U

,

V

such that

x

∈

U

and

y

∈

V

. Since

X

is also

totally disconnected

,

U

and

V

are unions of clopen sets. Hence we may as well assume that

U

,

V

clopen. This then implies that

U

∈

f

⁢

(

y

)

and

V

∈

f

⁢

(

x

)

. Since

U

≠

V

,

f

⁢

(

x

)

≠

f

⁢

(

y

)

. ∎

Lemma 3

.

f

is

surjective

.

Proof.

Pick any maximal ideal

I

of

B

*

. We want to find an

x

∈

X

such that

f

⁢

(

x

)

=

I

. If no such

x

exists, then for every

x

∈

X

, there is some clopen set

U

∈

I

such that

x

∈

U

. This implies that

⋃

I

=

X

. Since

X

is

compact

,

X

=

⋃

J

for some

finite set

J

⊆

I

. Since

I

is an ideal, and

X

is a finite join of elements of

I

, we see that

X

∈

I

. But this would mean that

I

=

B

*

, contradicting the fact that

I

is a maximal, hence a

proper ideal

of

B

*

. ∎

Lemma 4

.

f

and

f

-

1

are

continuous

.

Proof.

We use a fact about continuous functions between two Boolean spaces:

a

bijection

is a homeomorphism iff it maps clopen sets to clopen sets (proof here (

http://planetmath.org/HomeomorphismBetweenBooleanSpaces

)).

So suppose that

U

is clopen in

X

, we want to prove that

f

⁢

(

U

)

is clopen in

B

*

. In other words, there is an

element

V

∈

B

(so that

V

is clopen in

X

) such that

f

⁢

(

U

)

=

M

⁢

(

V

)

=

{

M

∈

B

*

∣

V

∉

M

}

.

This is because every clopen set in

B

*

has the form

M

⁢

(

V

)

for some

V

∈

B

*

(see the lemma in this entry (

http://planetmath.org/StoneRepresentationTheorem

)). Now,

f

⁢

(

U

)

=

{

f

⁢

(

x

)

∣

x

∈

U

}

=

{

f

⁢

(

x

)

∣

U

∉

f

⁢

(

x

)

}

=

{

M

∣

U

∉

M

}

, the last

equality

is based on the fact that

f

is a bijection. Thus by setting

V

=

U

completes

the proof of the lemma.
∎

Therefore,

f

is a homemorphism, and the proof of theorem is complete.
∎

Title

dual of Stone representation theorem

Canonical name

DualOfStoneRepresentationTheorem

Date of creation

2013-03-22 19:08:38

Last modified on

2013-03-22 19:08:38

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

12

Author

CWoo (3771)

Entry type

Theorem

Classification

msc 54D99

Classification

msc 06E99

Classification

msc 03G05

Related topic

BooleanSpace

Related topic

HomeomorphismBetweenBooleanSpaces

Defines

dual space

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