Point degree spectra of represented spaces Arno Pauly Swansea - PowerPoint PPT Presentation

Point degree spectra of represented spaces Arno Pauly Swansea University Joint work with Takayuki Kihara, Nagoya

The talk in a nutshell The notion of a point degree spectrum of a space links: ◮ Descriptive Set Theory ◮ Computable Analysis ◮ Recursion Theory and allows us to use techniques from one field to solve problems in others.

Small inductive dimension Definition Let the (small inductive) dimension of a Polish space be defined inductively via dim( ∅ ) = − 1 and: dim( X ) = sup sup U ∈O ( X ) , x ∈ U ⊆ B ( x , 2 − n ) dim( δ U ) + 1 inf x ∈ X n ∈ N ◮ If dim( X ) exists, it is a countable ordinal and we call X countably dimensional. ◮ Otherwise X is infinite-dimensional.

A question from DST Definition Call Polish spaces X , Y piecewise homeomorphic , if there are partitions X = � i ∈ N X i and Y = � i ∈ N Y i such that ∀ i ∈ N X i ∼ = Y i . (denote this by X ∼ = ω Y ) Theorem (Hurewicz & Wallmann) = ω { 0 , 1 } N iff X is countably dimensional. We have X ∼ Question (Jayne & Rogers; Motto-Ros, Schlicht & Selivanov) Is there more than one ∼ = ω -equivalence class of infinite-dimensional Polish spaces?

Represented spaces and computability Definition A represented space X is a pair ( X , δ X ) where X is a set and δ X : ⊆ N N → X a surjective partial function. Definition F : ⊆ N N → N N is a realizer of f : X → Y , iff δ Y ( F ( p )) = f ( δ X ( p )) for all p ∈ δ − 1 X (dom( f )) . Abbreviate: F ⊢ f . F N N → N N − − − −    � δ X  � δ Y f − − − − → X Y Definition f : X → Y is called computable (continuous), iff it has a computable (continuous) realizer.

Computable metric spaces We define a computable metric space with its Cauchy representation as follows: Definition 1. An effective metric space is a tuple M = ( M , d , ( a n ) n ∈ N ) such that ( M , d ) is a metric space and ( a n ) n ∈ N is a dense sequence in ( M , d ) . 2. The Cauchy representation δ M : ⊆ N N M associated with the effective metric space M = ( M , d , ( a n ) n ∈ N ) is defined by d ( a p ( i ) , a p ( k ) ) ≤ 2 − i for i < k � δ M ( p ) = x : ⇐ ⇒ and x = lim i →∞ a p ( i )

Computable metric spaces, cont. Theorem (Weihrauch) For maps between effective metric spaces, metric continuity and realizer continuity coincide. Question (Pour-El & Richards) For which effective metric spaces do points have Turing degrees?

Point degree spectra Definition For A , B ⊆ N N , say A ≤ M B iff ∃ F : B → A , F computable. Definition For a represented space X , let Spec ( X ) = { δ − 1 X ( x ) / ≡ M | x ∈ X } ⊆ M be the point degree spectrum of X . ◮ Spec ( N N ) = Spec ( R ) = T (Turing degrees) ◮ Spec ([ 0 , 1 ] N ) =: C (continuous degrees, M ILLER ) ◮ Spec ( O ( N )) = E (enumeration degrees)

Alternatively definition of the spectrum Definition For x ∈ X , y ∈ Y write x ≤ T y iff there is a computable partial function F : ⊆ Y → X with F ( y ) = x . Definition For x ∈ X , let ◮ Spec ( x ) = { p ∈ { 0 , 1 } N | x ≤ T p } . ◮ Spec ( X ) = { Spec ( x ) | x ∈ X } ◮ coSpec ( x ) = { p ∈ { 0 , 1 } N | p ≤ T x } . ◮ coSpec ( X ) = { coSpec ( x ) | x ∈ X } (works equivalently for countably based spaces)

What are the continuous degrees? For A ∈ A ( { 0 , 1 } N ) , let T ( A ) ⊆ { 0 , 1 } N be all codes of trees for A . Theorem The following are equivalent for B ⊆ { 0 , 1 } N : 1. ∃ A ∈ A ( { 0 , 1 } N ) B ≡ M A ≡ M T ( A ) 2. ∃ X effective Polish, x ∈ X Spec ( x ) ≡ M B

What are the continuous degrees?, contd Theorem (M ILLER ) If X is effective Polish and x ∈ X , then coSpec ( x ) is either a principal or a Scott-ideal. Corollary Let A ∈ A ( { 0 , 1 } N ) be such that A ≡ M T ( A ) and A / ∈ T . If T ( B ) ≤ M { p } ≤ M A for B ∈ A ( { 0 , 1 } N ) , p ∈ { 0 , 1 } N , then B ≤ M A. Question Is there a direct proof?

The main theorem Theorem The following are equivalent for a represented space X : 1. Spec ( X ) ⊆ Spec ( Y ) n ∈ N X n where there are Y n ⊆ Y with X n ∼ 2. X = � = c Y n Theorem (Alternate form) The following are equivalent for a represented space X : 1. ∃ t ∈ T t × Spec ( X ) = t × Spec ( Y ) 2. N × X ∼ = ω N × Y Corollary For Polish X the following are equivalent: 1. ∃ p ∈ T p × Spec ( X ) ⊆ T 2. X has countable dimension.

A question and an answer Question (M OTTO -R OS , S CHLICHT , S ELIVANOV ) Is there a Polish space X s.t. for any p ∈ T we find that: T � ( p × Spec ( X )) � p × C Theorem There is an embedding of the inclusion ordering ([ ω 1 ] ≤ ω , ⊆ ) of countable subsets of the smallest uncountable ordinal ω 1 into the piecewise-embeddability ordering of Polish spaces.

Towards the proof Definition Ψ : { 0 , 1 } N → [ 0 , 1 ] N is generated by a An ω -left-CEA operator J ω computable function Ψ : { 0 , 1 } N × [ 0 , 1 ] <ω × N 2 → Q ≥ 0 such that r n = lim sup s →∞ Ψ( x , r 0 , . . . , r n − 1 , n , s ) Observation There is an effective enumeration ( J ω e ) e ∈ N of the ω -left-CEA operators. Definition The ω -left-computably-enumerable-in-and-above space ω CEA is a subspace of N × { 0 , 1 } N × [ 0 , 1 ] N defined by ω CEA = { ( e , x , r ) ∈ N × { 0 , 1 } N × [ 0 , 1 ] N : r = J ω e ( x ) } = “the graph of a universal ω -left-CEA operator.”

The first partial result Theorem 1. ω CEA is a Polish space. 2. { 0 , 1 } N ≇ ω ω CEA ≇ ω [ 0 , 1 ] N Side note: Theorem The following spaces are all piecewise homeomorphic to each other. 1. The ω -left-CEA space ω CEA . 2. Rubin-Schori-Walsh’s totally disconnected strongly infinite dimensional space RSW . 3. Roman Pol’s counterexample RP to Alexandrov’s problem.

The intuition Figure: The upper and lower approximations of { 0 , 1 } N , ω CEA and [ 0 , 1 ] N

Some weird spaces Theorem There exists a nondegenerated continuum A ⊆ [ 0 , 1 ] N in which no point has Turing degree. Theorem (Following H UREWICZ ) The following are equivalent: 1. CH 2. There is an infinite dimensional space X ⊂ [ 0 , 1 ] N , such that any countably dimensional Z ⊂ X is countable. 3. There is an ascending chain Ω in the Turing degrees, such that ∀ t ∈ T ∃ s ∈ Ω t ≤ T s.

The lower reals Definition In R < , real numbers are represented as limits of increasing sequences of rationals. Definition U ∈ O ( N ) is called semi-recursive, if there is a computable function f : N × N → { 0 , 1 } such that if n 0 ∈ U ∨ n 1 ∈ U , then n f ( n 0 , n 1 ) ∈ U . Let S ⊆ E be all degrees of semi-recursive sets. Proposition (G ANCHEV & S OSKOVA ) Spec ( R < ) = S

An application Theorem Let X be a countably-based T 1 space. Then R n + 1 does not < piecewise embed into X × R n < . Let Λ n = ( { 0 , 1 } n , ≤ ) be a partial order on { 0 , 1 } n obtained as the n -th product of the ordering 0 < 1. Lemma For every countable partition ( P i ) i ∈ ω of the n-dimensional hypercube [ 0 , 1 ] n (endowed with the standard product order), there is i ∈ ω such that P i has a subset which is order isomorphic to the product order Λ n . Corollary For any n ∈ N there exists an enumeration degree which is expressible as the product of n + 1 semirecursive degrees, but not of n semirecursive degrees.

Outlook ◮ What other degree structures arise? (Partial answer: Looking at non-countably based spaces gets us beyond enumeration degrees) ◮ Which recursion-theoretic results hold there? (Kihara & Ng: The Shore-Slaman join theorem holds in the degree spectrum of any Polish space) ◮ Are there connections to other topological properties?

The article T. Kihara & A. Pauly. Point degree spectra of represented spaces. arXiv 1405.6866, 2014.