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]