Compact Metric Spaces as Minimal Subspaces of Domains of

Bottomed Sequences

Hideki Tsuiki

Kyoto University, Japan

ω-algebraic cpo --- topological space with a base

Limit elements L(D) ・・・ Topological space 　Finite elements K(D) ・・・ Base of L(D)

d

identifying d with ↑d 　∩　L(D)D

(Increasing sequence of K(D))

⇔ 　 Ideal I of K(D)

⇔ filter 　 base 　　　　　　　　　　　　 F(I) = {↑d∩L(D) | d∈I} of L(D)

which converges to 　　　　　　　　　　　　　　↓ (lim I) ∩L(D)

An ideal of K(D) as a filter of L(D)

L(D)

K(D)

I

lim I

I

X

K(D) ・・・ Base of X

We consider conditions so that each infinite ideal I of K(D) (infinite incr. seq. of K(D)) is representing a unique point of X as the limit of F(I).

identifying d with ↑d 　∩　 X

　 Ideal I of K(D) (⇔ Incr. seq. of K(D))

⇔ F(I) = {↑d∩X | d∈I } of X which converges to ????

K(D) as a base of each subspace of L(D)

ω-algebraic cpo D

I

X

each inf. ideal I of K(D) is representing a unique point of X as the limit of F(I).

•F(I) = {↑d∩X | d∈I } is a filter base

X is dense in D•F(I) converges to at most one point

X is Hausdorff•F(I) always converges, the limit is a limit in L(D).

X is a minimal subspace of L(D)

K(D)

L(D)

K(D)

L(D)X

K(D)

L(D)X

K(D)

L(D)X

K(D)

L(D)X

ω-algebraic cpo D

I

X

each inf. ideal I of K(D) is representing a unique point of X as the limit of F(I).

•F(I) = {↑d∩X | d∈I } is a filter base

X is dense in D•F(I) converges to at most one point

X is Hausdorff•F(I) always converges, the limit is a limit in L(D).

X is a minimal subspace of L(D)

ω-algebraic cpo D

I

X

•F(I) = {↑d∩X | d∈I } is a filter base

X is dense in D•F(I) converges to at most one point

X is Hausdorff•F(I) always converges, the limit is a limit in L(D).

X is a minimal subspace of L(D)

each inf. ideal I of K(D) is representing a unique point of X as the limit of F(I).

Minimal subspace

I

Theorem. When X is a dense minimal Hausdorff subspace of L(D),

(1) X is a retract of L(D) with the retract map r.

(2) Each filter base F(I) converges to r(lim I).

(3) ∩F(I) = {lim I} if lim I ∈ 　 X

(4) ∩F(I) = φ 　　　 if not lim I ∈ 　X

(5) ∩{cl(s) | s ∈F(I)} = {r(lim I)}

i.e., r(lim I) is the unique cluster point of F(I).

Minimal subspace

I

I is representing r(lim I)

lim ITheorem. When X is a dense minimal H

ausdorff subspace of L(D),

(1) X is a retract of L(D) with the retract map r.

(2) Each filter base F(I) converges to r(lim I).

(3) ∩F(I) = {lim I} if lim I ∈ 　 X

(4) ∩F(I) = φ 　　　 if not lim I ∈ 　X

(5) ∩{cl(s) | s ∈F(I)} = {r(lim I)}

i.e., r(lim I) is the unique cluster point of F(I).

When minimal subspace exists?• D ∽ 、 Pω 、Ｔ ω 　 do not have.

XDefinition P is a finitely-branching poset if each element of P has finite number of adjacent elements.

Definition ω-algebraic cpo D is a fb-domain if K(D) is a finite branching ω-type coherent poset.

level 0

level 1

level 2

level 3

K0

K1

K2

finite

Theorem When D is a fb-domain, L(D) has the minimal subspace.

Representations via labelled fb-domains.

b

　 representations of X by Γω

each point y of X

⇔ 　 infinite ideals with limit in r-1(y)

⇔ 　 infinte increasing 　 　 sequences of K(D)

⇔ 　 infinite strings of Γ

(Γ ： alphabet of labels)

a

da

bada… represents y

y

lim I

y

（ Adjacent elements of d K(D) l∈abelled by Γ ）

a b c

Dimension and Length of domainsTheorem When D is (1) an ω-algebraic consistently complete domain 　　 or (2) a mub-fb-domain,

ind(L(D)) = length(L(D))

length(P): the maximal length of a chain in P.

mub-domain: a finite set of minimal upper bounds exists for each finite set.

ind: Small Inductive Dimension.BX(A) : the boundary of A in X.

ind(X) : the small inductive dimension of the space X.

– ind(X) = -1 if X is empty.

–ind(X) n if for all p U X. p ≦ ∈ ⊂ ∈ ∃V X ⊂ 　 s.t. ind B(V) n-1.≦

–ind(X) = n if ind(X) n and not ind B(V) n-1.≦ ≦

Dimension and Length of domainsTheorem When D is (1) an ω-algebraic consistently complete domain 　　 or (2) a mub-fb-domain,

ind(L(D)) = length(L(D))

length(P): the maximal length of a chain in P.

mub-domain: each finite set has a finite set of minimal upper bounds.

Dimension and Length of domainsTheorem When D is (1) an ω-algebraic consistently complete domain 　　 or (2) a mub-fb-domain,

ind(L(D)) = length(L(D))

Corollary: ind M(D) 　≦　 length(L(D))

M(D)

length(P): the maximal length of a chain in P.

mub-domain: each finite set has a finite set of minimal upper bounds.

Top. space X

b a b a b b

fb-domainadmissible proper representation

ba

da

y

lim I

y

ab c

Type 2 machine Computation

1 0 1 0

• cell: peace of information• filling a cell: increase the information

and go to an adjacent element.

0

1 0 1

Domains of bottomed sequences

⊥

⊥0

⊥0 1⊥

10 1⊥

1

10

10 10 0…⊥ ⊥

• the order the cells are filled is arbitrary.

• finite-branching: At each time, the next cell to fill is selected from a finite number of candidates.

Computation by IM2-machines.[Tsuiki]

⊥

⊥0

⊥0 1⊥

10 1⊥

1

10

10 10 0…⊥ ⊥

•We can consider a machine (IM2-machine) which input/output bottomed sequences.•Computation over M(D) defined through IM2-machines.

Top. space X

1 1 0 1 1 1

fb-domainadmissible proper representation

1

1⊥1

y

lim I

y

Type 2 machine Computation

IM2 machine

101

101⊥1

Goal: For each topological space X , find a fb-domain D such that

(1) X = M(D)

(2) X dense in D

(3) ind X = length(L(D))

(4) D is composed of bottomed sequences

XWe show that every compact metric space has such an embedding.

First consider the case X =[0,1].

Binary expansion of [0,1] 　

0 0.5 1.0

bit 0

bit 1

bit 2

bit 3bit 4

Gray-code Expansion 　

0 0.5 1.0

bit 0

bit 1

bit 2

bit 3bit 4

Binary expansion of [0,1] 　

0 0.5 1.0

bit 0

bit 1

bit 2

bit 3bit 4

0

1

1

11

1

0

0

00

Gray-code Expansion 　

0 0.5 1.0

bit 0

bit 1

bit 2

bit 3bit 4

0

0

1

00

1

0

1

00

Gray-code embedding from [0,1] to M(RD)

•IM （ G ）＝ Σω － Σ ＊０ ω ＋ Σ ＊⊥１０ ω

0 0.5 1.0

bit 0

bit 1

bit 2

bit 3bit 4

⊥

0

1

00

RD realized as bottomed sequences

0 1

…

1

0 0 0 1 0 1 1 1 1 1 1 01 0

⊥100000…

0100000… 1100000…

…

Σ ＊ ＋ Σ ＊⊥１０ ＊

100000…00000… 010101…

M(RD) is homeo. to [0,1] through Gray-codeSigned digit representation[Gianantonio] Gray code [Tsuiki]

Σω ＋ Σ ＊⊥１０ ω

Synchronous product of fb-domains.

X Y X ×Y

D1 D2D1×s D2

• I ×I can be embedded in RD×s RD as the minimal subdomain.

• In can be embedded in RD(n) as the minimal subdomain.

L(D1) ×L(D2)

Infinite synchronous product of fb-domains.

Π∽I （ Hilbert Cube) = M(Π∽s RD).

…… … … …

•Infinite dimensional.

•The number of branches increase as the level goes up

Nobeling’s universal space Nm

n : subspace of Im in which at most m dyadic coordinates exist. a dyadic number … s/2mt

Gm : Im = M(RD (m) )

Gm : Nmn M(RD (m) ) ∩upper-n(RD (m) )

RD (m) n: Restrict the structure of

RD(m) so that the limit space is upper-n(RD (m) ) Nm

n

RD (m) n

Fact. n-dimensional separable metric space can be embedded in N2n+1

n

Fact. -dimensional separable metric space ∽can be embedded in Π∽I

When X is compact

Theorem. 1) When X is a compact metric space, there is a fb-domain D such that X = M(D).

2) D is composed of bottomed sequences and the number of ⊥ 　 which appears in each element of D is the dimension of X.

X

D

D as domain of Bottomed sequences

•RD as bottomed sequencesWhen X is a compact metric space, there is a fb-domain D of bottomed sequences such that X = M(D).The number of bottomes we need is equal to the dimension of X.

Top. space X

1 1 0 1 1 1

fb-domain

admissible proper representation

11⊥1

lim I

y

Type 2 machine

ComputationIM2 machine

101101⊥1

•Important thing is to find a D which induces good notion of computation for each X.

•When X = [0,1], such a D exists.

Further Works

• Properties of the representations.

(Proper)

• Relation with uniform spaces.

(When D has some uniformity-like condition, then M(D) is always metrizable.)

CCA 2002

Uniformity-like conditions

f(n) = The least level of the maximal lower bounds of elements of level n .

f(n) ∽ 　 as n ∽

n

f(n)

Computation by IM2-machines.

•Extension of a Type-2 machine so that each input/output tape has n heads.•Input/output -sequences with n+1 heads.•Indeterministic behavior depending on the way input tapes are filled.

0 1 0 1 0 0 0 …

0 １ １ …

StateWorktapes

Execusion Rules

IM2-machine

1 0 1 0

• cell: peace of information• filling a cell: increase the information

and go to an adjacent element.

0

0

⊥

⊥0 …

Domains of bottomed sequences

1 0 1 0

• cell: peace of information• filling a cell: increase the information

and go to an adjacent element.

0

0 1

⊥

⊥0 …

⊥0 1⊥

Domains of bottomed sequences

1 0 1 0

• cell: peace of information• filling a cell: increase the information

and go to an adjacent element.

0

1 0 1

⊥

⊥0

⊥0 1⊥

10 1⊥

1

Domains of bottomed sequences

• the order the cells are filled is arbitrary.

• At each time, the next cell to fill is selected from a finite number of candidates.

1 0 1 0 0

⊥

⊥0

⊥0 1⊥

10 1⊥

• the order the cells are filled is arbitrary.

1

10

Domains of bottomed sequences

10 10 0…⊥ ⊥

cf. Σω: cells are filled from left to right induce tree structure and Cantor space.

•Σ⊥ω forms an ω-algebraic domain.

•It is not finite-branching, no minimal subspaces.

Domains of bottomed sequences

•Σ = {0,1}•Σ⊥

ω: Infinite sequences of Σ in which undefined cells are allowed to exist.

1 0 1 0 0

•K(Σ⊥ω):Finite cells filled.

•L(Σ⊥ω):Infinite cells

filled.

fb-domains of bottomed sequences

At each time, the next information (the next cell) is selected from a finite number of candidates.

fb-domains of bottomed sequences

⇒ 　 Restrict the number of cells skipped.

Σ n⊥＊ : finite sequences of Σ in whi

ch at most n are allowed.⊥

Σ n⊥ω : infinite sequences of Σ in whi

ch at most n are allowed.⊥

BDn: the domain Σ n⊥＊ + Σ n⊥

ω fb-domain, M(BDn) not Hausdorff

⊥

１０⊥1 ⊥0

01

01 1 ⊥01 10⊥

01 1000…⊥

Σ 1⊥＊

Σ 1⊥ω

BD1

0101000…

0 010…⊥

Gray-code Embedding 　

0 0.5

bit 0

bit 1

bit 2

bit 3bit 4

0 1 1

0 1 1 11 0

⊥100000…

0100000… 1100000…

RD

1.0

Gray-code Embedding 　

0 0.5

bit 0

bit 1

bit 2

bit 3bit 4

0 1 1

0 1 1 11 0

⊥100000…

0100000… 1100000…

RD

Gray-code Expansion 　

0 0.5 1.0

bit 0

bit 1

bit 2

bit 3bit 4

0 1 1

0 1 1 11 0

⊥100000…

0100000… 1100000…

RD

1.0

Gray-code Embedding 　

0 0.5

bit 0

bit 1

bit 2

bit 3bit 4

0 1 1

0 1 1 11 0

⊥100000…

0100000… 1100000…

RD

1.0

Gray-code Embedding 　

0 0.5

bit 0

bit 1

bit 2

bit 3bit 4

0 1 1

0 1 1 11 0

⊥100000…

0100000… 1100000…

RD

1.0

Gray-code Embedding 　

0 0.5

bit 0

bit 1

bit 2

bit 3bit 4

0 1 1

0 1 1 11 0

⊥100000…

0100000… 1100000…

RD

1.0

Gray-code Embedding 　

0 0.5

bit 0

bit 1

bit 2

bit 3bit 4

0 1 1

0 1 1 11 0

⊥100000…

0100000… 1100000…

1r.0

Gray-code Embedding 　

0 0.5

bit 0

bit 1

bit 2

bit 3bit 4

0 1 1

0 1 1 11 0

⊥100000…

0100000… 1100000…

RD

I = [0.1] is homeo to M(RD)IM2-machine which I/O bottomed sequences [Tsuiki]

Future Works

• Properties of the representations.

(Proper)

• Relation with uniformity.

(Uniformity-like condition on domains.)

•Topology in Matsue (June

fb-domain RD

fb-domain RD

fb-domain RD

fb-domain RD

fb-domain RD

fb-domain RD

M(RD) is homeomorphic to I=[0,1] Signed digit representation[Gianantonio] Gray code [Tsuiki]