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

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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

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(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

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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)

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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)

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K(D) L(D)

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K(D) L(D) X

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K(D) L(D) X

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K(D) L(D) X

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K(D) L(D) X

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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)

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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).

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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).

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Minimal subspace I I is representing r(lim I) lim 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).

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When minimal subspace exists? D ∽ 、 Pω 、Ｔ ω do not have. X Definition 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 K0K0 K1K1 K2K2 finite Theorem When D is a fb-domain, L(D) has the minimal subspace.

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

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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.

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ind: Small Inductive Dimension. B X (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.

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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.

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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)) 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.

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Top. space X bababb fb-domain admissible proper representation b a d a y lim I y abc Type 2 machineComputation

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1010 cell: peace of information filling a cell: increase the information and go to an adjacent element. 0 101 Domains of bottomed sequences ⊥ ⊥0⊥0 ⊥0⊥1⊥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.

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Computation by IM2-machines.[Tsuiki] ⊥ ⊥0⊥0 ⊥0⊥1⊥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.

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Top. space X 110111 fb-domain admissible proper representation 1 1⊥11⊥1 y lim I y Type 2 machine Computation IM2 machine 101 101 ⊥ 1

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

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Binary expansion of [0,1] 00.51.0 bit 0 bit 1 bit 2 bit 3 bit 4

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Gray-code Expansion 00.51.0 bit 0 bit 1 bit 2 bit 3 bit 4

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Binary expansion of [0,1] 00.51.0 bit 0 bit 1 bit 2 bit 3 bit 4 0 1 1 1 1 1 0 0 0 0

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Gray-code Expansion 00.51.0 bit 0 bit 1 bit 2 bit 3 bit 4 0 0 1 0 0 1 0 1 0 0

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Gray-code embedding from [0,1] to M(RD) IM （ G ）＝ Σ ω － Σ ＊ ０ ω ＋ Σ ＊ ⊥１０ ω 00.51.0 bit 0 bit 1 bit 2 bit 3 bit 4 ⊥ 0 1 0 0

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RD realized as bottomed sequences 01 … 1 00010111111010 ⊥ 100000… 0100000…1100000… … Σ ＊ ＋ Σ ＊ ⊥１０ ＊ 100000…00000… 010101… M(RD) is homeo. to [0,1] through Gray-code Signed digit representation[Gianantonio] Gray code [Tsuiki] Σ ω ＋ Σ ＊ ⊥１０ ω

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Synchronous product of fb-domains. XYX ×Y D1D2 D1× s D2 I ×I can be embedded in RD× s RD as the minimal subdomain. I n can be embedded in RD (n) as the minimal subdomain. L(D1) ×L(D2)

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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

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Nobeling’s universal space N m n : subspace of I m in which at most m dyadic coordinates exist. a dyadic number … s/2 m t G m : I m = M(RD (m) ) G m : N m n 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) ) NmnNmn RD (m) n

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Fact. n-dimensional separable metric space can be embedded in N 2n+1 n Fact. ∽ -dimensional separable metric space can be embedded in Π ∽ I When X is compact

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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

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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.

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Top. space X 110111 fb-domain admissible proper representation 1 1⊥11⊥1 lim I y Type 2 machine Computation IM2 machine 101 101 ⊥ 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.

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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

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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 )

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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. 0101000… 0 １１ … State Worktapes Execusion Rules IM2-machine

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1010 cell: peace of information filling a cell: increase the information and go to an adjacent element. 0 0 ⊥ ⊥0⊥0 … Domains of bottomed sequences

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1010 cell: peace of information filling a cell: increase the information and go to an adjacent element. 0 01 ⊥ ⊥0⊥0 … ⊥0⊥1⊥0⊥1 Domains of bottomed sequences

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1010 cell: peace of information filling a cell: increase the information and go to an adjacent element. 0 101 ⊥ ⊥0⊥0 ⊥0⊥1⊥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.

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10100 ⊥ ⊥0⊥0 ⊥0⊥1⊥0⊥1 10 ⊥ 1 the order the cells are filled is arbitrary. 1 10 Domains of bottomed sequences 10 ⊥ 10 ⊥ 0… cf. Σ ω : 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.

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Domains of bottomed sequences Σ = {0,1} Σ ⊥ ω : Infinite sequences of Σ in which undefined cells are allowed to exist. 10100 K(Σ ⊥ ω ):Finite cells filled. L(Σ ⊥ ω ):Infinite cells filled.

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

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

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Gray-code Embedding 00.5 bit 0 bit 1 bit 2 bit 3 bit 4 011 011110 ⊥ 100000… 0100000…1100000… RD

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1.0 Gray-code Embedding 00.5 bit 0 bit 1 bit 2 bit 3 bit 4 011 011110 ⊥ 100000… 0100000…1100000… RD

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Gray-code Expansion 00.51.0 bit 0 bit 1 bit 2 bit 3 bit 4 011 011110 ⊥ 100000… 0100000…1100000… RD

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1.0 Gray-code Embedding 00.5 bit 0 bit 1 bit 2 bit 3 bit 4 011 011110 ⊥ 100000… 0100000…1100000… RD

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1.0 Gray-code Embedding 00.5 bit 0 bit 1 bit 2 bit 3 bit 4 011 011110 ⊥ 100000… 0100000…1100000… RD

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1.0 Gray-code Embedding 00.5 bit 0 bit 1 bit 2 bit 3 bit 4 011 011110 ⊥ 100000… 0100000…1100000… RD

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1.0 Gray-code Embedding 00.5 bit 0 bit 1 bit 2 bit 3 bit 4 011 011110 ⊥ 100000… 0100000…1100000…

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1r.0 Gray-code Embedding 00.5 bit 0 bit 1 bit 2 bit 3 bit 4 011 011110 ⊥ 100000… 0100000…1100000… RD I = [0.1] is homeo to M(RD) IM2-machine which I/O bottomed sequences [Tsuiki]

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Future Works Properties of the representations. (Proper) Relation with uniformity. (Uniformity-like condition on domains.) Topology in Matsue (June

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fb-domain RD

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

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