Compact Metric Spaces as Minimal Subspaces of Domains of Bottomed Sequences Hideki Tsuiki Kyoto University, Japan

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






ω-algebraic cpo D

I

X






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.≦ ≦




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



Corollary: ind M(D) 　≦　 length(L(D))

M(D)


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


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 １１ …

StateWorktapes

Execusion Rules

IM2-machine

1 0 1 0



0

0

⊥

⊥0 …


1 0 1 0



0

0 1

⊥

⊥0 …

⊥0 1⊥


1 0 1 0



0

1 0 1

⊥

⊥0

⊥0 1⊥

10 1⊥

1



• 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⊥


1

10


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.


•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


0 0.5

bit 0

bit 1

bit 2

bit 3bit 4

0 1 1

0 1 1 11 0

⊥100000…

0100000… 1100000…

RD


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


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


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


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


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


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]

Documents

Compact Metric Spaces as Minimal Subspaces of Domains of Bottomed Sequences Hideki Tsuiki Kyoto University, Japan