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Compact Metric Spaces as Minimal Subspaces of Domains of Bottomed Sequences

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.

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Compact Metric Spaces as Minimal Subspaces of Domains of Bottomed Sequences

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

  2. 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

  3. An ideal of K(D) as a filter of L(D) (Increasing sequence of K(D)) ⇔Ideal I of K(D) ⇔ filterbase F(I) = {↑d∩L(D) | d∈I} of L(D) which converges to 　　　　　　　　　　　　　　↓(lim I) ∩L(D) lim I L(D) I K(D)

  4. K(D) as a base of each subspace of L(D) K(D) ・・・Base of X X 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 ???? I 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).

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

  6. L(D) K(D)

  7. X L(D) K(D)

  8. X L(D) K(D)

  9. X L(D) K(D)

  10. X L(D) K(D)

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

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

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

  14. Minimal subspace • Theorem. When X is a dense minimal Hausdorff subspace of L(D), • 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). lim I I I is representing r(lim I)

  15. When minimal subspace exists? Definition P is a finitely-branching poset if each element of P has finite number of adjacent elements. • D∽ 、Pω、Ｔωdo not have. X finite level 3 Definitionω-algebraic cpo D is a fb-domain if K(D) is a finite branching ω-type coherent poset. level 2 K2 level 1 K1 Theorem When D is a fb-domain, L(D) has the minimal subspace. level 0 K0

  16. Representations via labelled fb-domains. （Adjacent elements of d∈K(D) labelled by Γ） lim I y 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) y a a b c d a b bada… represents y

  17. Dimension and Length of domains Theorem 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.

  18. 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.

  19. Dimension and Length of domains Theorem 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.

  20. Dimension and Length of domains Theorem 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. M(D) Corollary: ind M(D) 　≦　 length(L(D))

  21. fb-domain admissible proper representation Top. space X lim I y b a b a b b y a a b c d a b Type 2 machine Computation

  22. Domains of bottomed sequences 10⊥10⊥0… 1 0 1 0 0 • cell: peace of information • filling a cell: increase the information and go to an adjacent element. 10⊥1 ⊥0⊥1 10 ⊥0 1 1 0 1 ⊥ • 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.

  23. Computation by IM2-machines.[Tsuiki] 10⊥10⊥0… 10⊥1 ⊥0⊥1 10 ⊥0 1 ⊥ • We can consider a machine (IM2-machine) which input/output bottomed sequences. • Computation over M(D) defined through IM2-machines.

  24. fb-domain admissible proper representation Top. space X lim I y 1 1 0 1 1 1 y 101⊥1 101 1⊥1 1 Type 2 machine IM2 machine Computation

  25. 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 We show that every compact metric space has such an embedding. X First consider the case X =[0,1].

  26. Binary expansion of [0,1] bit 4 bit 3 bit 2 bit 1 bit 0 0 0.5 1.0

  27. Gray-codeExpansion bit 4 bit 3 bit 2 bit 1 bit 0 0 0.5 1.0

  28. Binary expansion of [0,1] 1 0 bit 4 1 bit 3 0 1 0 bit 2 1 0 bit 1 1 0 bit 0 0 0.5 1.0

  29. Gray-codeExpansion 0 0 bit 4 bit 3 0 0 0 0 bit 2 1 1 bit 1 1 0 bit 0 0 0.5 1.0

  30. Gray-codeembedding from [0,1] to M(RD) 0 bit 4 bit 3 0 0 bit 2 1 bit 1 ⊥ bit 0 0 0.5 1.0 • IM（G）＝ Σω－Σ＊０ω＋Σ＊⊥１０ω

  31. 0100000… 1100000… … … 00000… 100000… ⊥100000… 010101… 0 0 0 1 0 1 1 0 1 1 1 1 1 0 0 1 1 Σ＊ ＋Σ＊⊥１０ ＊ RD realized as bottomed sequences Σω＋Σ＊⊥１０ω M(RD) is homeo. to [0,1] through Gray-code Signed digit representation[Gianantonio] Gray code [Tsuiki]

  32. Synchronous product of fb-domains. D1×s D2 D1 D2 L(D1) ×L(D2) X Y X ×Y • I ×I can be embedded in RD×sRD as the minimal subdomain. • In can be embedded in RD(n)as the minimal subdomain.

  33. Infinite synchronous product of fb-domains. … … … … … • Infinite dimensional. • The number of branches increase as the level goes up Π∽I （Hilbert Cube) = M(Π∽sRD).

  34. Nobeling’s universal space Nmn : 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 RD (m) n:Restrict the structure of RD(m) so that the limit space is upper-n(RD (m) ) Nmn

  35. Fact. n-dimensional separable metric space can be embedded in N2n+1n Fact. ∽-dimensional separable metric space can be embedded in Π∽I When X is compact

  36. D X 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.

  37. D as domain of Bottomed sequences • RD as bottomed sequences • When 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.

  38. admissible proper representation fb-domain Top. space X lim I 1 1 0 1 1 1 y 101⊥1 101 1⊥1 1 Type 2 machine IM2 machine Computation • Important thing is to find a D which induces good notion of computation for each X. • When X = [0,1], such a D exists.

  39. 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

  40. 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)

  41. Computation by IM2-machines. 0 1 0 1 0 0 0 … IM2-machine State Worktapes Execusion Rules 0 １ １ … • 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.

  42. Domains of bottomed sequences 1 0 1 0 0 • cell: peace of information • filling a cell: increase the information and go to an adjacent element. 0 … ⊥0 ⊥

  43. Domains of bottomed sequences 1 0 1 0 0 • cell: peace of information • filling a cell: increase the information and go to an adjacent element. 0 1 ⊥0⊥1 … ⊥0 ⊥

  44. Domains of bottomed sequences 1 0 1 0 0 • cell: peace of information • filling a cell: increase the information and go to an adjacent element. 1 0 1 10⊥1 • the order the cells are filled is arbitrary. • At each time, the next cell to fill is selected from a finite number of candidates. ⊥0⊥1 ⊥0 1 ⊥

  45. Domains of bottomed sequences 1 0 1 0 0 • the order the cells are filled is arbitrary. cf. Σω: cells are filled from left to right induce tree structure and Cantor space. 10⊥10⊥0… 10⊥1 ⊥0⊥1 10 • Σ⊥ω forms an ω-algebraic domain. • It is not finite-branching, no minimal subspaces. ⊥0 1 ⊥

  46. 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.

  47. fb-domains of bottomed sequences At each time, the next information (the next cell) is selected from a finite number of candidates.

  48. fb-domains of bottomed sequences ⇒Restrict the number of cells skipped. BD1 Σ⊥n＊: finite sequences of Σ in which at most n ⊥ are allowed. Σ⊥nω: infinite sequences of Σ in which at most n ⊥ are allowed. BDn: the domain Σ⊥n＊+ Σ⊥nω fb-domain, M(BDn) not Hausdorff Σ⊥1ω 0101000… 0⊥010… 01⊥1000… 01⊥10 01⊥1 01 ０ １ ⊥1 ⊥0 Σ⊥1＊ ⊥

  49. Gray-codeEmbedding bit 4 bit 3 bit 2 RD bit 1 0100000… 1100000… ⊥100000… 0 1 1 0 1 1 bit 0 0 0.5 0 1 1

  50. Gray-codeEmbedding bit 4 bit 3 bit 2 RD bit 1 0100000… 1100000… ⊥100000… 0 1 1 0 1 1 bit 0 0 0.5 1.0 0 1 1

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