Degree spectra of homeomorphism types of Polish spaces
aa r X i v : . [ m a t h . L O ] A p r Degree Spectra of Homeomorphism Types of Polish Spaces
Mathieu HoyrupUniversit´e de Lorraine, CNRS, Inria, LORIA, France [email protected] andTakayuki Kihara ∗ Graduate School of Informatics, Nagoya University, Japan [email protected] andVictor Selivanov † A.P. Ershov Institute of Informatics Systems SB RAS, Russia [email protected]
Abstract
A Polish space is not always homeomorphic to a computably presented Polish space. In this article,we examine degrees of non-computability of presenting homeomorphic copies of Polish spaces. Weshow that there exists a 0 ′ -computable low Polish space which is not homeomorphic to a computableone, and that, for any natural number n , there exists a Polish space X n such that exactly the high n +3 -degrees are required to present the homeomorphism type of X n . We also show that no compact Polishspace has an easiest presentation with respect to Turing reducibility. Key words : computable topology, computable presentation, computable Polish space, degreespectrum.
How difficult is it to describe an explicit presentation of an abstract mathematical struc-ture? Only the isomorphism type of a structure is given to us, and then our task isto present its representative whose underlying set is (an initial segment of) the naturalnumbers ω . However, an isomorphism type of a structure does not necessarily have acomputable presentation. In such a case, our next task is to determine how incomputableit is to present a representative of the isomorphism type. This has long been one of thefundamental questions in computable structure theory, and researchers in this area haveobtained a huge number of interesting results on Turing degrees of presentations of iso-morphism types of groups, rings, fields, linear orders, lattices, Boolean algebras, and soon, cf. [1, 8, 10, 14].In this article we focus on presentations of Polish spaces. The notion of a presentationplays a central role, not only in computable structure theory, but also in computable ∗ Kihara’s research was partially supported by JSPS KAKENHI Grant 19K03602, 15H03634, and the JSPS Core-to-CoreProgram (A. Advanced Research Networks). † Selivanov’s research was partially supported by RFBR-JSPS Grant 20-51-50001. nalysis [3, 4, 27]. In this area, one of the most crucial problems was how to presentlarge mathematical objects (which possibly have the cardinality of the continuum) suchas metric spaces, topological spaces and so on, and then researchers have obtained anumber of reasonable answers to this question. In particular, the notion of a computablepresentation of a Polish space has been introduced around 1950-60s, cf. [20], and sincethen this notion has been widely studied in several areas including computable analysis[3, 23, 27] and descriptive set theory [21].In recent years, several researchers succeeded to obtain various results on Turing degreesof presentations of isometric isomorphism types of Polish spaces, separable Banach spaces,and so on, cf. [7, 18, 19]. However, most of works are devoted to metric structures, andthere seem almost no works on presentations on homeomorphism types of Polish spaces.The investigation of Turing degrees of homeomorphism types of topological spaces (notnecessarily Polish) was initiated in [25] in analogy with the earlier investigation of degreesof isomorphism types of algebraic structures. Some results were obtained for domains butthe case of Polish spaces was apparently not investigated seriously so far.Every Polish space is homeomorphic to the Cauchy completion of a metric on (an initialsegment of) the natural numbers ω , so one may consider any distance function d : ω → Q as a presentation of a Polish space. Then, observe that there are continuum manyhomeomorphism types of Polish spaces. In particular, by cardinality argument, there isa Polish space which is not homeomorphic to any computably presented Polish space.Surprisingly however, it was unanswered until very recently even whether the followingholds: Question 1.
Does there exist a ′ -computably presented Polish space which is not home-omorphic to a computably presented one? Note that countable spaces are useless to solve this problem because of the “hyperarithmetic-is-recursive” phenomenon, cf. [11]; see also Section 2.3. The solution to Question 1 wasvery recently obtained by the authors of this article, and independently by Harrison-Trainor, Melnikov, and Ng [13]. One possible approach to solve this problem is usingStone duality between countable Boolean algebras and zero-dimensional compact metriz-able spaces (where note that compact metrizable spaces are always Polish); see also Section2.4. Combining this idea with classical results on isomorphism types of Boolean algebras[17], one can conclude that every low -presented zero-dimensional compact metrizablespace is homeomorphic to a computable one. This is also noticed by the authors of thisarticle and independently by Harrison-Trainor, Melnikov, and Ng [13].Our next step is to develop new techniques other than Stone duality. More explicitly,the next question is whether there exists a Polish space whose homeomorphism degreespectum is different from that of a zero-dimensional compact Polish space. Here, by thehomeomorphism degree spectrum of a space X we mean the collection of Turing degreeswhich compute a presentation of a homeomorphic copy of X . In particular, it is naturalto ask the following: Question 2.
Does there exist a low -presented Polish space which is not homeomorphicto a computably presented one? One of our main results in this article is that there exists a 0 ′ -computable low infinitedimensional compact metrizable space which is not homeomorphic to a computable one. his solves Question 2. By using similar techniques, we also construct, for any n ∈ ω , aninfinite dimensional compact metrizable space X n whose homeomorphism degree spectrumis the high n +3 -degrees; that is, X n is homeomorphic to a d -computably presented Polishspace if and only if d is high n +3 . This also clarifies substantial differences between zero-dimensional compact metrizable spaces and infinite dimensional ones since the class ofhigh n -degrees is never the degree spectrum of a Boolean algebra [15].We prove the following results: • For every degree d and every n >
0, there exists a space Z d ,n whose compact degreespectrum is { x : d ≤ x (2 n − } (Theorems 3.3, 3.9, 3.11, 3.13). • For every degree d and every n >
0, there exists a space P d ,n whose compact degreespectrum is { x : d ≤ x (2 n ) } and Polish degree spectrum is { x : d ≤ x (2 n +1) } (Theorems 3.16, 3.18, 3.20).Another important question is whether a given Polish space has the least Turing degreein its homeomorphism degree spectrum. In other words, it is natural to ask if the home-omorphism type of a Polish space has an easiest presentation. For instance, it is knownthat the isomorphism types of linear orders, trees, abelian p -groups, etc. have no easiestpresentation whenever they are not computably presentable, cf. [10]. Question 3.
Does there exist a homeomorphism type of a Polish space which is not com-putably presentable, but have an easiest presentation with respect to Turing reducibility?
We partially answers Question 3 in negative. More precisely, we show the cone-avoidancetheorem for compact
Polish spaces, which states that, for any non-c.e. set A ⊆ ω , everycompact Polish space has a presentation that does not enumerate A . Basic terminologies and results on computability theory and computable structure theoryare summarized in [1]. For basics on computable analysis, we refer the reader to [2, 4, 3,27]. For some basic definitions and facts on general topology and dimension theory, seealso [26]. A Polish presentation (or simply a presentation ) of a Polish space X is a distance function d on ω whose Cauchy completion is homeomorphic to X . Some researchers use a slightlydifferent definition, but these definitions are equivalent: Let a n ∈ X be the image of n ∈ ω under such a homeomorphism. Then, ( a n ) n is a dense sequence in X , and d X ( a i , a j ) = d ( i, j ) is the restriction of the metric on X to the dense set { a n } n . We often use the samesymbol d to denote the metric d X on X . For discussion on presentations of Polish spaces,see also [12].A finite union of rational open balls (i.e., balls of the form B ( a i ; r ) for some rational r )is called a rational open set. A code of a finite rational open cover of X is a finite set ⊆ ω × Q such that for any x ∈ X there is ( i, r ) ∈ E such that d ( x, a i ) < r . If X is compact, then a compact presentation of X is a presentation of X equipped with anenumeration of a collection C of codes of all finite rational open covers of X . In particular,a compact presentation contains an information of total boundedness; that is, a function ℓ such that, given s , the 2 − s -net { B ( a n ; 2 − s ) : n < ℓ ( s ) } formally covers the whole space X .For a Turing degree d , a d -computable Polish space is a Polish space which has a d -computable presentation, and such a space is d -computably compact if it has a d -computablecompact presentation. For a Polish space X , the Polish degree spectrum is the set of allTuring degrees d such that X has a d -computable Polish presentation. If such a space X is compact, the compact degree spectrum is the set of all Turing degrees d such that X has a d -computable compact presentation. Lemma 2.1.
Let X be a compact metrizable space. If X has a d -computable Polishpresentation, then X has a d ′ -computable compact presentation.Proof. By compactness of X , one can observe that E is a code of a finite rational opencover of X if and only if there exists ε > x ∈ X we have d ( x, a i ) < r − ε for some ( i, r ) ∈ E . The latter is equivalent to the existence of s ∈ ω such that forall k ∈ ω we have d ( a k , a i ) < r − − s for some ( i, r ) ∈ E . As E is finite, this is a Σ condition relative to a Polish presentation of X . Hence, if X has a d -computable Polishpresentation, then the set of codes of all finite rational open covers is d ′ -c.e. In otherwords, X has a d ′ -computable compact presentation.We will see in Section 5 a more precise relation between Polish and compact presentations.There are another equivalent definitions of Polish and compact presentations. We mayassume that X is a subspace of Hilbert cube Q := [0 , ω w.r.t. the standard metric d on Q . The set Q ◦ := (0 , ω ⊂ Q is called the pseudo-interior of Q . One can alwaysassume that a compact metrizable space X is embedded into the pseudo-interior of Q .In particular, for each open ball B in X , there is an open ball B ∗ in Q with the samecenter and radius with B . Then, B is approximated by B s := B ∗ ∩ X s . One can decideif B s ∩ B t = ∅ , B s ⊆ B t , etc. By A we mean the topological closure cl Q ( A ) of A in Q . Hyperspaces.
We will also use another characterization of Polish and compact presenta-tions of compact Polish spaces, by considering hyperspaces of compact subsets of Q .Let V ( Q ) be the space of compact sets endowed with the lower Vietoris topology. Asubbasis is given by { K ⊆ Q : K ∩ B = ∅} , where B is a rational ball in Q .Let K ( Q ) be the space of compact sets endowed with the Vietoris topology. A subbasis isgiven by a subbasis for the lower Vietoris topology, together with { K ⊆ Q : K ⊆ U = ∅} ,where U is a rational open set in Q . This space can be equivalently obtained by endowingit with the Hausdorff metric.In any such space, a compact set K is d -computable if d computes an enumeration ofthe basic neighborhoods of K . We say that K is d -computably compact if K is a d -computable element of K ( Q ), and that K is d -computably overt if K is a d -computable lement of V ( Q ). In particular, K ⊆ Q is d -computably compact if and only if K is d -computably overt and K ∈ Π ( d ).We now come to the announced characterization. Proposition 2.2.
A compact Polish space X has a d -computable Polish presentation ifand only if it has a d -computable copy in V ( Q ) .A compact Polish space X has a d -computable compact presentation if and only if it hasa d -computable copy in K ( Q ) .Proof. It is the effective version of Theorem 4.14 in [16]. The only modification is thefollowing: if U is an effective open subset of Q and F its complement, then the func-tion d ( x, F ) is not computable in general. However, there exists a computable func-tion f ( x ) which is null exactly on F . Thus the function f should be used in placeof d ( x, F ). Let U be a cover of a topological space X . A refinement V of a cover U of X is a cover suchthat every V ∈ V is included in some U ∈ U . If we moreover require V ⊆ U , then we callit a strict refinement. Given a compact presentation of X , one can obtain a refinementsequence ( U s ) s of finite open coverings of X such that each U ∈ U s is a rational open ball,and U ∈ U s implies X ∩ U = ∅ . Let X s be the closure of S U s . This gives a decreasingsequence ( X s ) of rational open sets in Q such that X = T s X s . One can assume that ( U s ) s is a strict refinement sequence; that is, for any U ∈ U s +1 there is V ∈ U s such that U ⊆ V and that the closure of the union of U s +1 is included in the union of U s (by enlarging every U ∈ U s +1 by 1 + 2 − s times). One can moreover assume that for any U, V ∈ U s , U ∩ V = ∅ if and only if U ∩ V = ∅ . This assumption also ensures that X s +1 ⊆ X s . In summary: Observation 2.3.
Given a compact presentation of X ⊆ Q ◦ , one can effectively obtaina sequence ( X s ) s ∈ ω of rational open subsets of Q such that X s +1 ⊆ X s and X = T s X s . The order of U is the least number n such that any x ∈ X is contained in at most n manysets in U , if such a number exists. The covering dimension of X (written dim( X )) is theleast number n such that every open cover U has an open refinement V of order at most n + 1. If X is normal, it is known that such a V can be a strict refinement. Moreover, if X is a compact metric space, one can moreover assume that V consists of rational opensets. To see this, let V = ( V i ) i ∈ I be given, and δ be a Lebesgue number of V . Then,consider any finite open cover D = ( D j ) j ∈ J of X consisting of rational open balls whosediameters are at most δ , and define V ∗ i = S { D j : D j ⊆ V i } . By the property of δ , every D j is contained in some V i , so V ∗ = ( V ∗ i ) i ∈ I is a cover of X . Moreover, the order of V ∗ is less than or equal to V ; that is, V ∗ is the desired one. Similarly, one can also assumethat U is a rational open cover. In summary, dim( X ) < n if and only if every rationalopen cover of X has a strict refinement of order at most n + 1 consisting of rational opensets. We say that a finite collection V is a finite modification of U s if every member of V is a finite union of elements of U s , and moreover S V = S U s . The above argument showsthat, if a compact presentation of X is fixed, then dim( X ) < n if and only if for any s there is t such that a finite modification of U t refines U s and the order of U t is at most n . emma 2.4. Given a compact presentation of a zero-dimensional compact metrizablespace X , one can effectively find a computable pruned tree T ⊆ <ω such that X ishomeomorphic to [ T ] .Proof. By using a compact presentation of X , we construct a sequence of finite clopencoverings C n of X such that C n is pairwise disjoint and the mesh of C n is at most 2 − n .Given C n , one can effectively find a finite 2 − n − -net U n +1 of X . As X is zero-dimensional,by the above argument, one can find a refinement C n +1 of U n +1 such that the order of C n +1 is at most 1; that is, the collecton C n +1 is pairwise disjoint. Since C n +1 is finite,this means that C n +1 is a clopen cover of X . Clearly the sequence ( C n ) n yields a finitebranching pruned tree T . That is, each member C n is assigned to a node in T of length n ,and then X is homeomorphic to the compact space [ T ] consisting of infinite paths through T . Moreover, T is computably bounded relative to a presentation of X ; that is, one caneffectively compute the number of immediate successors of T . Hence, one can effectivelyfind S ⊆ <ω such that [ S ] is homeomorphic to [ T ]. Let P be a topological space. The Cantor-Bendixson derivative of P is the set P ′ of allnon-isolated points in P . Lemma 2.5.
Let X be a compact Polish space.If X ⊆ Q is d -computably overt then X ′ is Π ( d ′ ) and d ′′ -computably overt.Therefore, if X has a d -computable Polish presentation, then its Cantor-Bendixon deriva-tive X ′ has a d ′′ -computable compact presentation.Proof. Assume that X has a d -computable Polish presentation, or equivalently that X ⊆ Q is d -computably overt.Let B = B ( a i ; r ) be an open ball in X . We claim that B ∩ X ′ is nonempty if andonly if there is s such that B s = B ( a i ; r − − s ) contains infinitely many points. For theforward direction, choose x ∈ B ∩ X ′ . Then, d ( a i , x ) < r , so for any sufficiently large s , d ( a i , x ) < r − − s , i.e., x ∈ B s . Since B s is open and x is of rank 1, it is clear that B s contains infinitely many points. For the backward direction, if B s contains infinitelymany points, but B ∩ X ′ is empty, then B consists of infinitely many isolated points, andso is the closure B s of B s as B s ⊆ B s ⊆ B ; however this is impossible by compactness.By this claim, the property B ∩ X ′ = ∅ is Σ ( d ), or equivalently Σ ( d ′′ ). This means that X ′ is d ′′ -computably overt.Next, let A be the set of all ( i, s ) such that d ( a i , a j ) ≥ − s for any j = i . Then, A is d ′ -computable. One can easily see that x is isolated in X if and only if x ∈ B ( a i ; 2 − s )for some ( i, s ) ∈ A . Thus, the set of isolated points is a Σ ( d ′ ) subset of X ; hence X ′ isΠ ( d ′ ) in X .By Proposition 2.2, we conclude that X ′ has a d ′′ -computable compact presentation.We next note that countable topological spaces are completely useless for constructingnontrivial degree spectra inside the hyperarithmetical hierarchy. Let ω x be the least rdinal which is not computable in x . Observation 2.6.
For any countable ordinal α , there is a compact metrizable space O α whose compact and Polish degree spectrum are both { x : α < ω x } .Proof. Let O α be the compact metrizable space ω α +1 endowed with the order topology. Itis obvious that, if α is x -computable, then O α has an x -computable compact presentation.Conversely, if ω α + 1 has an x -computable Polish presentation, then by Lemma 2.1, it hasan x ′ -computable compact presentation. In particular, there is a countable Π ( x ′ ) class P which is homeomorphic to ω α + 1. Since the Cantor-Bendixson rank of ω α + 1 is α , andthe Cantor-Bendixson rank is a topological invariant, the rank of P is also α . However, asnoted by Kreisel, the Spector boundedness principle implies that the rank of a countableΠ ( x ′ ) class must be x ′ -computable; see also [6, Section 4]. As an x -hyperarithmeticordinal is always x -computable, this implies that α < ω x .This completely characterizes the compact and Polish degree spectra of countable compactmetrizable spaces since every countable compact metrizable space is homeomorphic to theordinal space ω α · n + 1 for some α < ω and n ∈ ω by Mazurkiewicz-Sierpi´nski’s theorem.For more details, see also [11]. Here we show that spectra of compact zero-dimensional spaces are closely related tospectra of Boolean algebras. This follows from an effectivization of Stone duality in [22].Let B be the category formed by the Boolean algebras as objects and the {∨ , ∧ , ¯ , , } -homomorphisms as morphisms. Recall that a Stone space is a compact topological space X such that for any distinct x, y ∈ X there is a clopen set U with x ∈ U y (i.e.,zero-dimensional and T ). Let S be the category formed by the Stone spaces as objectsand the continuous mappings as morphisms.The Stone duality states the dual equivalence between the categories B and S . Moreexplicitly, the Stone space s ( B ) corresponding to a given Boolean algebra B is formed bythe set of prime filters of B with the base of open (in fact, clopen) sets consisting of thesets { F ∈ s ( B ) | a ∈ F } , a ∈ B . (Note that one could equivalently take ideals in placeof filters.) Conversely, the Boolean algebra b ( X ) corresponding to a given Stone space X is formed by the set of clopen sets (with the usual set-theoretic operations). By Stoneduality, any Boolean algebra B is canonically isomorphic to the Boolean algebra b ( s ( B ))(the isomorphism f : B → b ( s ( B )) is defined by f ( a ) = { F ∈ s ( B ) | a ∈ F } ), and anyStone space X is canonically homeomorphic to the space s ( b ( X )).Restricting the Stone duality to the countable Boolean algebras, we obtain their dualitywith the compact zero-dimensional countably based spaces, and in fact with the compactsubspaces of the Cantor space 2 ω . As the nonempty closed subsets of 2 ω coincide withthe sets [ T ] of infinite paths through a pruned tree T ⊆ ω , we obtain a close relationbetween such subspaces and countable Boolean algebras. Fact 2.7. (1)
A Boolean algebra has a d -c.e. (resp. d -co-c.e., d -computable) copy ifand only if it is isomorphic to the Boolean algebra of clopen subsets of [ T ] for some d -co-c.e. (resp. d -c.e., d -computable) pruned tree T . Every d -co-c.e. Boolean algebra is isomorphic to a d -computable Boolean algebra. (3) There is a d -c.e. Boolean algebra which is not isomorphic to a d -computable Booleanalgebra.Proof. The first item follows from [22, Lemma 3]; see also [24]. The second item followsfrom [22, Theorem]. The third item follows from [9].As already noticed by Harrison-Trainer, Melnikov, and Ng [13], one can use Stone dualityto show several results on degree spectra of zero-dimensional compacta. For instance,Stone duality can be used to show the following:
Fact 2.8 (see [13]) . (1) There exists a zero-dimensional compact metrizable space whichhas a ′ -computable Polish presentation, but not homeomorphic to a computable Pol-ish space. (2) If a zero-dimensional compact metrizable space has a low Polish presentation, thenit is homeomorphic to a computable Polish space.
Here we prove another consequence on Stone duality. Let BA and CP be the classes ofcountable Boolean algebras and of the compact zero-dimensional Polish spaces, respec-tively. Let Sp ( BA ), Sp ( CP ), Sp c ( CP ) denote respectively the classes of spectra ofBoolean algebras, CP -spaces w.r.t. Polish presentation, and of CP -spaces w.r.t. com-pact presentation. Theorem 2.9. Sp ( BA ) = Sp c ( CP ) .Proof. We show that for any countable Boolean algebra B , Sp ( B ) = Sp ( s ( B )), andfor any X ∈ CP , Sp ( X ) = Sp ( b ( X )). It suffices to check the first equality becausethe second one follows by Stone duality. If B is a d -computable Boolean algebra, let T ⊆ <ω be a d -computable pruned tree such that b ([ T ]) is isomorphic to B , hence s ( B )is homeomorphic to [ T ]. If { τ , τ , . . . } is a d -computable enumeration of T , let x i ⊒ τ i be the leftmost branch of T . Then { x , x , . . . } is a d -computable dense sequence in [ T ],hence Sp ( B ) ⊆ Sp ( s ( B )). It is clear that [ T ] is d -computably compact.For the converse inclusion, assume that s ( B ) has a d -computable compact presentation.By Lemma 2.4, s ( B ) is homeomorphic to the subspace [ T ] of the Cantor space for some d -computable pruned tree T ⊆ <ω . By Fact 2.7 (1), b ([ T ]) (hence also B ) is d -computablypresentable.The Stone dual of the Cantor-Bendixon derivative is known as the Frech´et derivative B ′ ofa Boolean algebra B which is the quotient of B by the ideal generated by atoms (minimalnon-zero elements). Since the isolated points x of the space s ( B ) (realized as [ F ] above)are precisely the atoms [ τ ] ∩ [ F ] for suitable prefix τ ⊑ x , we obtain the following. Proposition 2.10.
For any countable Boolean algebra B , s ( B ′ ) is homeomorphic to s ( B ) ′ . Precise complexity estimations for the Frechet derivative were obtained in [22]: a count-able Boolean algebra C is d ′′ -computably presentable iff C is isomorphic to B ′ for some -computable Boolean algebra B , and there is a d -computable Boolean algebra B suchthat B ′ is not d ′ -computably presentable. The iterated version is also known for any n >
0: a countable Boolean algebra C is d (2 n ) -computably presentable iff C is isomorphicto the n th derivative B ( n ) for some d -computable Boolean algebra B , and there is a d -computable Boolean algebra B such that the n th derivative B ( n ) is not d (2 n − -computablypresentable.Theorem 2.9 and Proposition 2.10 imply that Lemma 2.5 is almost optimal: Theorem 2.11.
For any n > , a space Y ∈ CP has a d (2 n ) -computable compactpresentation if and only if Y is homeomorphic to the n th derivative X ( n ) for some d -computable compact X ∈ CP , and there is a d -computable compact X ∈ CP such thatthe n th derivative X ( n ) does not have a d (2 n − -computable compact presentation. Given a compact presentation of X , we will construct a finitely branching tree T X ofcomponents and a linear order on components of X . A similar notion has been studiedin Brattka et al. [5].Recall from Section 2.2 that a compact presentation of X yields a strong refinementsequence ( U s ) s of open covers whose members are rational open in the Hilbert cube Q ,and then X s = S U s provides a rational open approximation of X in Q . As disjointnessof rational open sets in Q is decidable, one can effectively decompose X s into finitelymany connected components { C s , . . . , C sℓ ( s ) } . Then, we get the tree of components , T X ,which consists of sequences ( C u (0) , C u (1) , . . . , C ku ( k ) ) of connected components such that C i +1 u ( i +1) ⊆ C iu ( i ) for each i < k . Note that a node of T X can also be considered as arefinement sequence ( V , . . . , V k ) such that V s ⊆ U s and S V s is a component of X s .Then, each infinite path through T X corresponds to a connected component of X . Notethat the construction of T X from a compact presentation of X is effective. If C is acomponent of X s , then there is a unique node σ C ∈ T X of length s + 1 such that thelast entry of σ C is C . We label each component C by the least index i such that a i ∈ C ,where recall that ( a i ) i is a dense sequence of X . Then, we order the components of X s asfollows: For components C and D of X s , define C < D if either the last branching heightof σ C in T X is smaller than that of σ D or the branching heights are the same and the labelof C is smaller than that of D . Roughly speaking, C < D iff either C stabilizes earlierthan D , or else C and D stabilize at the same stage, and C contains a smaller indexedpoint than D . One can also order all infinite paths through T X by a similar argument. .1.2 Learning the dimension of a sphere Consider the following situation: We are informed that, for some d , the d -dimensionalsphere S d is embedded into the Hilbert cube Q , and moreover, a compact presentation ofthe embedded image is given to us, but we do not know the dimension d . How can weguess the correct dimension d ?First, given a finite open covering U s of a compact set X ⊆ Q , let us consider the formalnerve N s of U s , which is an abstract simplicial complex defined as follows: A finite set J ⊆ ℓ ( s ) belongs to N s if and only if the formal intersection of { B ( a n ; 2 − s ) : n ∈ J } is nonempty. As the formal nerve of a given covering of X is a finite abstract simpli-cial complex, one can compute the homology groups of the nerve. However, a compactpresentation of X may give an extremely wild embedding of X into the Hilbert cube.For instance, the embedded image of X in the Hilbert cube may look like the Alexan-der horned sphere (which is homeomorphic to S ). In general, the homology groups ofa compact space can be completely different from the homology groups of its finitaryapproximations. For instance, consider the Warsaw circle, which is approximated by asequence of 1-spheres (i.e., circles). Thus, it is not so obvious whether one can even-tually recognize the correct dimension from information of homology groups of finitaryapproximations of X , so we take a slightly different approach.Let us first describe a geometric idea behind our algorithm Ψ guessing the dimension ofa sphere. If X ≃ S d , for any e = d , any occurrence of an e -dimensional hole (i.e., an e -cycle which is not an e -boundary) will be eventually broken, and only tiny or thin e -dimensional holes may be detected at later stages. On the other hand, we will eventuallydetect a (non-thin portion of) d -dimensional hole which survives forever. Hence, ourlearning algorithm Ψ just returns the dimension of a longest-surviving portion of a holeat each stage.For a given d , we now focus on d -dimensional holes (that look like d -spheres) in U s and N s (or their finite modifications). Formally speaking, a d -dimensional hole or simply a d -hole in the nerve N s is a d -dimensional cycle which is not a d -dimensional boundaryin N s . A face α in the nerve N s is associated with a finite sequence U αs from U s . Thecollection U αs or its intersection is essentially a geometric realization of α , so we call ita formal geometric realization of α . In a similar manner, one can also consider a formalgeometric realization U γs of a chain γ in the nerve N s . More formally, U γs is the union offormal geometric realizations of all faces contained in γ . Note that if γ is a d -hole, evenif we delete duplicated entries in U γs , the resulting set still determines the same formalgeometric d -hole in X s . Thus, one can assume that a d -hole contains no repetition offaces. In particular, one can compute a canonical index of the finite set of all d -holes inthe nerve N s .Now, we say that γ is a d -hole detected at s if it is a d -hole in the induced nerve N s ofa finite modification of U s . For t > s , we say that such a d -hole γ survives at t if, forany face α in γ , the formal geometric realization T U αs is nonempty and connected in X t .Note that one can effectively check whether a given d -hole survives at t or not. Lemma 3.1.
Assume that a compact presentation of a topological sphere X is given (i.e., X ≃ S d for some d ∈ ω ). Then X ≃ S d if and only if there is a d -hole which is detectedat some stage, and survives forever. roof. We first show the forward direction, so assume X ≃ S d . For a sufficiently largestage s , N s must contain a true d -hole γ s : We first consider the d -sphere S d as a subspaceof the Hilbert cube Q . Let h : Q → Q be a continuous extension of a homeomorphism X ≃ S d . Note that such an h exists since X is a closed subset of Q and the Hilbert cube Q is an absolute extensor (by the coordinate-wise application of the Tietze extensiontheorem). Without loss of generality, one can assume that x X implies h ( x ) S d : Thisis because, let us consider { } × S d = { (0 , x , x , . . . ) ∈ Q : ( x , x , . . . ) ∈ S d } insteadof S d . Then, as above, we have a continuous function h : Q → Q such that h ↾ X is ahomeomorphism between X and { } × S d . Since Q is perfectly normal and X is closed,there is a continuous function g : Q → [0 ,
1] such that g ( x ) = 0 if and only if x ∈ X . Now,replace the first coordinate of h ( x ) with g ( x ). This fulfills the desired condition.Let B be a collection of basic open balls in the Hilbert cube Q such that { B ∩ S d : B ∈ B} is a cover of S d of order d + 1, whose nerve gives a rational triangulation of S d . Thenconsider h ∗ B = { h − [ B ] : B ∈ B} , which is a finite open cover of X , and note that h − [ B ∩ S d ] = h − [ B ] ∩ h − [ S d ] = h − [ B ] ∩ X , which is homeomorphic to the connectedset B ∩ S d . Then, let δ be a Lebesgue number of h ∗ B , that is, if an open cover U of X has mesh less than δ then any U ∈ U is a subset of h − [ B ] for some B ∈ B . For anysufficiently large s , the mesh of U s is less than δ , and thus, U s is a refinement of h ∗ B .For each B ∈ B , put U B = S { U ∈ U s : U ⊆ h − [ B ] } , and define U ∗ s = { U B : B ∈ B} .Then, U ∗ s is a finite modification of U s . Now one can obtain a minimal d -hole by listingcollections of members from U ∗ s whose intersection is nonempty and connected in X (notjust in X s ), and we call it a true d -hole. Although this procedure may be not effective,we do not need effectivity. Then, clearly, such a true d -hole survives at any stage t > s .We next show the converse direction; that is, if X S d , then every detected d -holes doesnot survive at some later stage. Let γ be a d -hole in N s , which yields a geometric d -hole U γs of U s . As the shape of any subspace of X is different from S d , if d is bigger than thedimension of X , then such a hole will be eventually broken at some stage by compactness.If d is smaller than the dimension of X , then a d -cycle γ ′ can be described on the surfaceof X , but such a γ ′ must be a boundary, so in particular, γ ′ is not a d -hole. As X S d ,this observation implies that the geometric d -hole U γs must add a new route between twodisjoint locations (which may look like a wormhole) on the surface of the d -sphere X . Butthis route does not exist in X , and moreover, any two disjoint locations on the surface ofa sphere must be separated; hence, by compactness, all such routes are broken at somestage.Formally speaking, in this case, there is a face α in γ whose geometric realization T U αs (which corresponds to a part of a new route in the above sense) is either empty or dis-connected in X , and then it is detected at some stage t ≥ s by compactness. This meansthat any d -hole detected at s does not survive at any stage after t . Lemma 3.2.
There exists a limit computable function which, given a compact presenta-tion of X , returns d ∈ ω such that X ≃ S d whenever such a d exists.Proof. At each stage s , our algorithm Ψ lists all holes detected at some stage t ≤ s , andcheck if a given such a hole still survives at s . Then Ψ returns the least dimension n of alongest-surviving hole at s ; that is, first compute the least t ≤ s such that there is a hole γ detected at t and survives at s , and then the least dimension n of such a hole. This eaning algorithm Ψ converges to the correct dimension d : By (the forward direction in)Lemma 3.1, there is a d -hole detected at some t which survives forever. Similarly, by (thebackward direction in) Lemma 3.1, if e = d , then any e -hole detected at some u ≤ t doesnot survive at some later stage. By finiteness of nerves, there are only finitely many holesdetected at some u ≤ t . Thus, there is s such that any e -hole detected at some u ≤ t doesnot survive at s . This means that a d -hole eventually becomes a longest-surviving hole;hence Ψ returns d at any stages after s . For any degree d , there is a compact metrizable space Z d whose compactdegree spectrum is { x : d ≤ x ′ } .Proof. For D ∈ d , define Z d as the one-point compactification of the following locallycompact space: a n ∈ D S n +1 ∐ a n D S n +2 , where S d is the d -dimensional sphere. Assume that an x -computable presentation of Z d is given. Then, one can effectively construct the tree T Z of components of Z d as in Section3.1.1.We use the learning algorithm Ψ in Lemma 3.2 to find a component of Z d whose dimensionis n . Note that there is unique such a component. At stage s , look at each component C of the height s in the tree T Z , and then apply the learning algorithm Ψ in Lemma3.2 to the component C . We label each node C of the height s in the tree T Z by thecurrent guess n of Ψ; that is, Ψ currently considers that C looks like the n -sphere S n . Atstage s , for any n ≤ s , we check if there is a connected component of height s labeled byeither 2 n + 1 or 2 n + 2. If there are such components, we follow the label of the < -leastcomponent C of height s . If C is labeled by 2 n + 1 then we guess n ∈ D , and if C islabeled by 2 n + 2 then we guess n D , at stage s . If there is no such component, thenwe guess n ∈ D at stage s .We claim that this algorithm eventually produces the correct guess. Now, note that Z d hasexactly one copy of S n +1 or S n +2 . Let C be a connected component in Z d correspondingto such a copy. First note that C has no point which is the limit of other components, andtherefore, by compactness, C corresponds to an isolated path p C in T Z (i.e., for almostall s , the cover U s distinguishes C from other components). Hence, there are only finitelymany infinite paths through T Z which is < -smaller than p C . By Lemma 3.2, the label of p C ↾ s converges to the correct dimension 2 n + 1 or 2 n + 2. As C is the unique componentof dimension n , for any path p < p C (which is always isolated in T Z ), the label of p ↾ s converges to some value different from 2 n + 1 and 2 n + 2. Hence, for any sufficiently large s , p C ↾ s must be the < -least one whose label is either 2 n + 1 or 2 n + 2. This means thatour algorithm eventually follows this correct component, and returns the correct guess. Inother words, this algorithm is an x -computable approximation procedure which decidesthe value of D in the limit. Consequently, we obtain d ≤ x ′ .Conversely, assume d ≤ T x ′ . Then, fix an x -computable approximation procedure ϕ onverging to D ∈ d ; that is, D ( n ) = lim s ϕ ( n, s ). We will construct a presentation of aspace Z . First enumerate a point z in Z , and then prepare for infinitely many pairwiseseparated regions ( R i ) i ∈ ω , where R i is 2 − i -close to z . Inside the region R d , we will constructeither S d +1 or S d +2 . First note that, for any ε , every (2 d + 1)-sphere S d +1 is ε -close to a(2 d + 2)-sphere S d +2 (w.r.t. the Hausdorff distance) and vice versa. At stage s , if we seethat ϕ ( n, s ) = 1, then, inside R d , we start to construct S d +1 which is 2 − s -close to the oneconstructed at the previous stage. Similarly, if we see that ϕ ( n, s ) = 1, then, inside R d , westart to construct S d +2 which is 2 − s -close to the one constructed at the previous stage.As ϕ ( n, s ) stabilizes to the correct value D ( n ) after some stage s , we eventually construct S d +1 if n ∈ D , and S d +2 if n D . This construction clearly produces a space Z which ishomeomorphic to Z d . Moreover, our construction is x -computable, and indeed, it is nothard to make this construction x -effectively compact. Hence, Z d has an x -computablecompact presentation. Theorem 3.4.
For any degree d , there is a compact metrizable space X d whose Polishdegree spectrum is { x : d ≤ x ′′ } , but compact degree spectrum is { x : d ≤ x ′ } . It suffices to show the following:
Lemma 3.5.
For any D ⊆ ω , there is a compact metrizable space X D such that for any Z ⊆ ω , D is Σ relative to Z ⇐⇒ X D has a Z -computable compact presentation. D is Σ relative to Z ⇐⇒ X D has a Z -computable Polish presentation.Proof. We denote by ˜ S n the topological sum of countably many copies of the n -sphere S n .Then, let X D be the one-point compactification of the following locally compact space: a n ∈ D ˜ S n +1 ∐ A, where A is a countable set of isolated points. The same argument as in the proof ofTheorem 3.3 shows that if X D has a Z -computable compact presentation then D is Σ relative to Z : n ∈ D if and only if there is an isolated path through the Z -computabletree of components induced from X D such that some ( n + 1)-hole survives forever alongthis path. This is clearly a Σ ( Z ) condition. Now, assume that X D has a Z -computablePolish presentation. Then, by Lemma 2.1, X D has a Z ′ -computable compact presentation.Therefore, D is Σ relative to Z ′ ; that is, D is Σ relative to Z .Conversely, assume that D is Σ relative to Z . Therefore, there is a Z -computable set S ⊆ ω such that n ∈ D if and only if there is a ∈ ω such that ( n, a, b ) ∈ S for infinitelymany b ∈ ω . We will construct a presentation of a space X . First enumerate a point z in X , and then prepare for infinitely many pairwise separated regions ( R i ) i ∈ ω , where R i is 2 − i -close to z . For any n, a, k ∈ ω , inside the region R h n,a,k i , we will construct either acopy of S n +1 or a finite set of isolated points. Fix a computable homeomorphic copy S n,a,k of S n +1 inside R h n,a,k i , but note that we do not enumerate S n,a,k into X . Let ( q n,a,kb ) b ∈ ω be a computable dense sequence in S n,a,k . Then, we put A n,a,k = { q n,a,kb : ( ∃ s > b ) ( n, a, k, s ) ∈ S } . learly, A n,a,k is c.e. relative to Z . Then we take a Z -computable enumeration of S n,a,k A n,a,k as a dense sequence of our space X .It is not hard to verify that, if n ∈ D , then, by our choice of S , there is a such that, forany k ∈ ω , the completion of A n,a,k is homeomorphic to S n,a,k ≃ S n +1 . In particular, X contains countably many copies of S n +1 . If n D , then for any a and k , the set A n,a,k is a finite set of isolated points. In particular, X contains no copy of S n +1 . If necessary,one can always add a countable set of isolated points. Consequently, X is homeomorphicto X D . Moreover, our construction is Z -computable, and hence, X D has a Z -computablePolish presentation. Proof of Theorem 3.4.
Given D ⊆ ω of Turing degree d , consider X d = X D ∐ X D c . Then,by Lemma 3.5, X d satisfies the desired condition. Remark . One can modify the proof of Theorem 3.4 to ensure that X D is perfect, byusing the one-point compactification of the following: a n ∈ D ˜ S n +1 ∐ a n ∈ ω ˜ I n , where ˜ I n is the topological sum of countably many copies of the n -dimensional cube [0 , n .As [0 , n has no hole, the argument in Theorem 3.3 still works. For the other direction,replace a finite set of isolated points in the proof of Theorem 3.4 with a finite set ofpairwise disjoint small segments (each of which is homeomorphic to [0 , n ) of the surfaceof S n , and put countably many copies of [0 , n at somewhere else.We now solve Question 2 affirmatively. Corollary 3.7.
There exist a low Turing degree d ≤ ′ and a perfect Polish space X such that X is d -computably presentable, but not computably presentable.Proof. Let d ≤ ′ be a low Turing degree which is not low , i.e., d ′′ ′′ . The Polishdegree spectrum of the space X d ′′ in Theorem 3.4 is { x : d ′′ ≤ x ′′ } , which contains d , butdoes not contain since d is not low . In other words, the space X ′′′ is d -computablypresentable, but not computably presentable.We also note that X = X ′′′ is a Polish space such that d is high ⇐⇒ X has a d -computable presentation. Corollary 3.8.
There is a compact perfect Polish space X which has a computable pre-sentation, but has no presentation which makes X computably compact.Proof. The Polish degree spectrum of the space X ′′ in Theorem 3.4 is { x : ′′ ≤ x ′′ } ,which contains all degrees, while the compact degree spectrum of X ′′ is { x : ′′ ≤ x ′ } ,which does not contain . In other words, the space X ′′ has a computable presentation,but has no presentation which makes X ′′ effectively compact. .3 The iterated jumps in compact degree spectra In this section, we combine higher-dimensional spheres (Sections 3.1.2 and 3.2) and count-able topological spaces (Section 2.3) to control the iterated jumps.
Theorem 3.9.
For any degree d , there is a compact metrizable space Z d , whose compactdegree spectrum is { x : d ≤ x ′′′ } . It suffices to show the following:
Lemma 3.10.
For any D ⊆ ω , there is a compact metrizable space Z D, such that forany Z ⊆ ω , D is Σ relative to Z ⇐⇒ Z D, has a Z -computable compact presentation.Proof. Let z be a unique point in the space ω α + 1 whose Cantor-Bendixson rank is α , andlet p be any point in the n -sphere S n . Then, consider the wedge sum of pointed spaces( S n , p ) and ( ω α + 1 , z ). In other words, we combine two spaces S n and ω α + 1 by gluing p and z . More explicitly, let us consider the equivalence relation ∼ with p ∼ z , and takethe quotient ( S n ∪ ( ω α + 1)) / ∼ . We denote the resulting quotient space by S n ∨ ( ω α + 1).Then, we define S n +1 D = ( S n +1 ∨ ( ω + 1) if n ∈ D, S n +1 ∨ ( ω + 1) if n D, Then, let Z d , be the one-point compactification of the following locally compact space: a n ∈ ω S n +1 D ∐ ( ω + 1) . Here we note that ω + 1 contains countably many isolated points, and countably manycopies of ω + 1.Given a Z -computable compact presentation of Z D, , let us first consider the Z -computabletree T Z of components of Z D, (see Section 3.1.1). Recall that a component of Z D, is aninfinite path through T Z . There is a unique component in Z D, which is homeomorphicto S n +1 , and let α n +1 be the corresponding infinite path through T Z . At each stage s ,compute a node η s of T Z which has a longest-surviving ( n + 1)-hole among nodes of T Z of length s . Then, η s is an initial segment of α n +1 for almost all s .The key observation is that if n D , then S n +1 D ≃ S n +1 ∨ ( ω + 1), so any neighborhoodof the ( n + 1)-sphere S n +1 has a copy of ω + 1 which is separated from the sphere. Thismeans that the Cantor-Bendixson rank ρ ( α n +1 ) of α n +1 in the space [ T Z ] is 2. On theother hand, in the case n ∈ D , we have S n +1 D ≃ S n +1 ∨ ( ω + 1), so if a neighborhood ofthe ( n + 1)-sphere S n +1 is sufficiently small, then it only contains isolated points exceptfor the sphere itself. This means that the Cantor-Bendixson rank ρ ( α n +1 ) of α n +1 in thespace [ T Z ] is 1. By this observation, n D if and only if ρ ( α n +1 ) ≥
2. In other words, forany s there are t > s and σ ∈ T Z such that σ (cid:23) η t , σ η t +1 , and σ has some extension ofrank 1 in [ T Z ]; that is, for any ℓ there is a branching node τ ∈ T Z of length ℓ extending σ . This is clearly a Π ( Z ) property. or the other direction, assume that D is Σ relative to Z . Then, there is a computableset S ⊆ ω such that n D if and only if ∃ ∞ a ∃ ∞ b ( n, a, b ) ∈ S ; in other words, infinitelymany sections S a = { b : ( n, a, b ) ∈ S } of S have infinitely many elements. We willconstruct a presentation of a space Z . First enumerate a point z and a copy of ω + 1 into Z , and then we prepare for infinitely many pairwise separated regions ( R i ) i ∈ ω , where R i is 2 − i -close to z . Inside R n , we start by describing S n +1 ∨ ( ω + 1). Let { p a , p ∞ } a ∈ ω be thecopy of ω + 1 in this space, and note that p ∞ also belongs to S n +1 . For each a ∈ ω , weconsider an (imaginary) sequence { p ab } b ∈ ω converging to p a , and for any f , { p a,f ( a ) } a ∈ ω converges to p ∞ . For instance, consider a triangle area, one of whose vertices correspondsto p ∞ ; arrange { p a } a ∈ ω on a side of the triangle, and put { p ab } a,b ∈ ω inside the trianglearea. Then, we only enumerate { p ab : ( n, a, b ) ∈ S } into the region R n .If ( n, a, b ) ∈ S for infinitely many b , then p a is a rank 1 point; otherwise, p a is isolated.Hence, if ∃ ∞ a ∃ ∞ b ( n, a, b ) ∈ S is true (i.e., n D ), there are infinitely many a such that p a is of rank 1, so p ∞ is the limit of rank 1 points. In this case, this space restricted to theregion R n is homeomorphic to S n +1 ∨ ( ω +1). If ∃ ∞ a ∃ ∞ b ( n, a, b ) ∈ S is false (i.e., n ∈ D ), p a is isolated for all but finitely many a . Hence, p ∞ is the limit of rank 0 points, butnot the limit of rank 1 points. There may be other rank 1 points, but they are separatedfrom the sphere S n +1 . Hence, our space restricted to the region R n is homeomorphic tothe separated union of S n +1 ∨ ( ω + 1) and at most finitely many copies of ω + 1. Up tohomeomorphism, the latter part is absorbed into a copy of ω + 1 in some other region.Consequently, Z is homeomorphic to Z D, . Moreover, our construction is Z -computable,and hence Z D, has a Z -computable compact presentation. Theorem 3.11.
For any degree d , there is a compact metrizable space Z d , whose compactdegree spectrum is { x : d ≤ x (5) } . It suffices to show the following:
Lemma 3.12.
For any D ⊆ ω , there is a compact metrizable space Z D, such that forany Z ⊆ ω , D is Σ relative to Z ⇐⇒ Z D, has a Z -computable compact presentation.Proof. Given D , let us consider: S n +1 D,k = ( S n +1 ∨ ( ω k − + 1) if n ∈ D, S n +1 ∨ ( ω k + 1) if n D, Then, let Z D, be the one-point compactification of the following locally compact space: a n ∈ ω S n +1 D, ∐ ( ω + 1) . It is not hard to see that the Cantor-Bendixson derivative of Z D, is homeomorphic to Z D, in Theorem 3.9. Assume that Z D, has a Z -computable presentation. Then, byLemma 2.5, Z D, has an Z ′′ -computable presentation. By Theorem 3.9, we have D is Σ relative to Z ′′ ; hence, Σ relative to Z . or the other direction, assume that D is Σ relative to Z . Then, there is a computableset S ⊆ ω such that n D if and only if ∃ ∞ a ∃ ∞ b ∃ ∞ c ( n, a, b, c ) ∈ S . We will construct apresentation of a space Z . First enumerate a point z and a copy of ω +1 into Z , and thenwe prepare for infinitely many pairwise separated regions ( R i ) i ∈ ω , where R i is 2 − i -closeto z . Inside R n , let { p a , p ∞ } a ∈ ω be the copy of ω + 1 in this space such that p ∞ is a pointin the surface of a copy S of the ( n + 1)-sphere S n +1 . For each a ∈ ω , we consider an(imaginary) sequence P a = { p ab } b ∈ ω converging to p a , and the diameter of P a is at most2 − a . Similarly, consider P ab = { p abc } c ∈ ω converging to p ab , and the diameter of P ab is atmost 2 − a − b . Then, we enumerate S , { p a , p ∞ } a ∈ ω , { p ab } a,b ∈ ω , and { p abc : ( n, a, b, c ) ∈ S } into the region R n .If ( n, a, b, c ) ∈ S for infinitely many c , then p ab is a rank 1 point; otherwise, p ab is isolated.Hence, if ∃ ∞ b ∃ ∞ c ( n, a, b, c ) ∈ S is true, there are infinitely many b such that p ab is ofrank 1, so p a is of rank 2. Otherwise, for almost all b , p ab is isolated, so p a is of rank 1.If ∃ ∞ a ∃ ∞ b ∃ ∞ c ( n, a, b, c ) ∈ S is true (i.e., n D ), there are infinitely many a such that p a is of rank 2, so p ∞ is the limit of rank 2 points. In this case, this space restricted tothe region R n is homeomorphic to S n +1 ∨ ( ω + 1). If ∃ ∞ a ∃ ∞ b ∃ ∞ c ( n, a, b, c ) ∈ S is false(i.e., n ∈ D ), p a is of rank 1 for almost all a . Hence, p ∞ is the limit of rank 1 points, butnot the limit of rank 2 points. There may be other rank 2 points, but they are separatedfrom the sphere S n +1 . Hence, our space restricted to the region R n is homeomorphic tothe separated union of S n +1 ∨ ( ω + 1) and at most finitely many copies of ω + 1. Up tohomeomorphism, the latter part is absorbed into a copy of ω + 1 in some other region.Consequently, Z is homeomorphic to Z D, . Moreover, our construction is Z -computable,and hence Z D, has a Z -computable compact presentation. Theorem 3.13.
For any degree d and n ∈ ω , there is a compact metrizable space Z d ,n whose compact degree spectrum is { x : d ≤ x (2 n − } . It suffices to show the following:
Lemma 3.14.
For any D ⊆ ω and n ∈ ω , there is a compact metrizable space Z D,n suchthat for any Z ⊆ ω , D is Σ n relative to Z ⇐⇒ Z D,n has a Z -computable compact presentation.Proof. Given D ⊆ ω , let Z D,n be the one-point compactification of the following locallycompact space: a d ∈ ω S d +1 D,n ∐ ( ω n + 1) , where S d +1 D,n is the space defined in the proof of Theorem 3.11. Then, the Cantor-Bendixsonderivative of Z D,n is homeomorphic to Z D,n − . The remaining argument is the same asin the proof of Theorem 3.11.Recall that we concluded from Theorem 2.11 that Lemma 2.5 is almost optimal. Asa corollary of Theorem 3.13, one can conclude a similar, but slightly different result.Recall from Theorem 2.9 that the compact degree spectra Sp c ( CP ) of zero-dimensionalcompacta is equal to the degree spectra Sp ( BA ) of Boolean algebras. or a special property of Sp ( BA ), Jockusch-Soare [15] showed that if the isomorphismtype of a Boolean algebra can have n th jump degree for some n ∈ ω then it already hasa computable presentation. Here, (the isomorphism type of) a structure X has n th jumpdegree if there is a least Turing degree d such that the n th jump of a presentation of X bounds d . If such a d is nonzero, we say that X has nontrivial n th jump degree.By the above observation, no homeomorphism type of a zero-dimensional compactum hasnontrivial n th jump degree for any n ∈ ω . On the other hand, Theorem 3.3 shows theexistence of an infinite dimensional compactum which has nontrivial first jump degree.We now combine these observations with Theorem 3.13. Corollary 3.15.
There is a compact metrizable space X which has a computably com-pact presentation, but its n th Cantor-Bendixson derivative X n has no (2 n − -computablycompact presentation, but the homeomorphism type of X n has rd jump degree.Proof. Let us consider the space X = Z d ,n +2 in Theorem 3.13. Then, the compact spec-trum of X is { x : d ≤ x (2 n +3) } . On the other hand, the compact spectrum of its n thderivative X n ≃ Z d , is { x : d ≤ x ′′′ } . If we take d = (2 n +3) then X satisfies the desiredcondition.Recall that a Turing degree x is high n if ( n +1) ≤ x ( n ) . Theorem 3.13 (with d = (2 n ) )shows that the class of high n − -degrees is a compact degree spectrum of a compactmetrizable space. By using the idea in the previous section, we show an analogue of Theorem 3.13 for Polishdegree spectra.Let ` ω X be the separated union of countably many copies of X , and α ω X be its one-pointcompactification, that is, α ω X denotes the one-point compactification of the separatedunion of countably many copies of X . When considering the wedge sum, we often thinkof α ω X as a pointed space ( α ω X ; ∞ ) whose basepoint is ∞ , the point at infinity. Define α ξω as the ξ th iteration of the compactification α ω . For instance, α ω ≃ ω + 1 with thebasepoint ω , and α ξ +1 ω ≃ α ω ( ω ξ + 1) ≃ ω ξ +1 + 1 with the basepoint ω ξ +1 (a unique rank ξ + 1 point).For a connected space S , we define the S -rank of a component C of X as follows: The S -rank of C is > N of C there is an other component in N which is homeomorphic to S . The S -rank of x is > α if for any neighborhood N of C there is an other component in N of rank ≥ α . The S -rank of X is the supremum of the S -ranks of components of X .We inductively define the following compact spaces: S Σ = S Σ +1 = 1 , S Σ n +3 = α ω S Σ +2 n +1 , S Σ +2 n +3 = S Σ n +3 ∐ S Π n +1 , S Π = S , S Π n +3 = α ω ( S Π n +1 ∐ S Σ +2 n +1 ) . or instance, S Σ ≃ ω + 1, S Σ +3 ≃ S ∐ ( ω + 1), and S Π ≃ α ω ( S ∐ S -ranks of S Σ , S Π and S Σ are 0, and the S -ranks of S Π and S Σ are 1. In general, the S -ranksof S Π n +1 and S Σ n +3 are n . Theorem 3.16.
For any degree d , there is a compact metrizable space P d , whose compactdegree spectrum is { x : d ≤ x ′′ } and Polish degree spectrum is { x : d ≤ x ′′′ } . It suffices to show the following:
Lemma 3.17.
For any D ⊆ ω , there is a compact metrizable space P D, such that forany Z ⊆ ω , D is Σ relative to Z ⇐⇒ P D, has a Z -computable compact presentation. D is Σ relative to Z ⇐⇒ P D, has a Z -computable Polish presentation.Proof. Given D ⊆ ω , we define T n +1 D = ( S n +1 ∨ S Σ if n ∈ D, S n +1 ∨ S Π if n D, Then, let P d , be the one-point compactification of the following locally compact space: a n ∈ ω T nD ∐ a ω S Π . As in the proof of Lemma 3.10, given a Z -computable compact presentation of P D, , letus consider the Z -computable tree T Z of components of P D, . Recall that a componentof P D, is an infinite path through T Z . There is a unique component in P D, which ishomeomorphic to S n +1 , and let α n +1 be the corresponding infinite path through T Z . Ateach stage s , compute a node η s of T Z which has a longest-surviving ( n + 1)-hole amongnodes of T Z of length s . Then, η s is an initial segment of α n +1 for almost all s .The key observation is that if n D , then T n +1 D ≃ S n +1 ∨ S Π , so S n +1 is an S -rank 1component of P D, , that is, any neighborhood of the ( n + 1)-sphere S n +1 has a copy of S which is separated from the ( n + 1)-sphere. On the other hand, in the case n ∈ D ,we have T n +1 D ≃ S n +1 ∨ S Σ , so S n +2 is an S -rank 0 component of P D, . Indeed, if aneighborhood of the ( n + 1)-sphere S n +1 is sufficiently small, then it only contains isolatedpoints except for the sphere itself. By this observation, n D if and only if for any s there are t > s and σ ∈ T Z such that σ (cid:23) η t , σ η t +1 , and σ satisfies the Σ ( Z ) propertystating that σ has a 1-hole which survives forever. This is clearly a Π ( Z ) property.If P D, has a Z -computable Polish presentation, then by Lemma 2.1, P D, has a Z ′ -computable compact presentation. Therefore, D is Σ relative to Z ′ ; hence Σ relative to Z .For the other direction, assume that D is Σ relative to Z . We show that P D, has a Z -computable compact presentation. By our assumption, there is a computable set A ⊆ ω such that n D if and only if ∃ ∞ a ∀ b ( n, a, b ) ∈ A . Without loss of generality, we mayassume that ( n, a, A for any a ∈ ω . We will construct a presentation of a space P .First enumerate ω -many copies of S and ω + 1, and compactify them by adding a point z nto P . Then we prepare for infinitely many pairwise separated regions ( R i ) i ∈ ω , where R i is 2 − i -close to z . Inside R n , we start by describing S n +1 . Then, prepare for a sequence ofcountably many separated copies { S n,a } a ∈ ω of the circle S which converge to p ∈ S n +1 ,and let ( S n,a [ s ]) s ∈ ω be an increasing sequence of nonempty sets of finite isolated pointswhich 2 − s -approximates S n,a , that is, d H ( S n,a [ s ] , S n,a ) < − s w.r.t. the Hausdorff distance d H , where we do not enumerate these copies into P at present. For each a , we enumeratethe following set into the region R n :˜ S n,a = [ { S n,a [ s ] : ( ∀ b < s ) ( n, a, b ) ∈ A } . In other words, if ( n, a, b ) ∈ A is true for any b < s , then we describe a 2 − s -approximationof S n,a by enumerating finitely many isolated points into the region R n ; otherwise, we justkeep finitely many isolated points which are already put into R n by the previous stages.If ( n, a, b ) ∈ A for all b , then ˜ S n,a is a dense subset of S n,a , so its completion is S n,a ≃ S ≃ S Π ; otherwise, ˜ S n,a consists of finitely many isolated points (finitely many copiesof S Σ +1 ), where the sentence ( ∀ b <
0) ( n, a, b ) ∈ A is vacuously true, so there is at leastone isolated point. Therefore, if ∃ ∞ a ∀ b ( n, a, b ) ∈ A is true (i.e., n D ), then there are asequence of circles converging to p (that is, α ω ( S Π )), and also a sequence of isolated pointsconverging to p (that is, α ω ( S Σ +1 )) by our assumption that ( n, a, A for any a . Inthis case, this space restricted to the region R n is homeomorphic to α ω ( S Π ) ∨ α ω ( S Σ +1 ) ≃ α ω ( S Π ∐ S Σ +1 ) ≃ S Π . If ∃ ∞ a ∀ b ( n, a, b ) ∈ A is false (i.e., n ∈ D ), then there are atmost finitely many circles (finitely many copies of S Π ), and a sequence of isolated pointsconverging to p (that is, S Σ ≃ α ω ( S Σ +1 )). In this case, this space restricted to the region R n is homeomorphic to the separated union of S n +1 ∨ S Σ and at most finitely many circles S Π . Up to homeomorphism, these finitely many circles are absorbed into ω -many copiesof S Π in some other region. Consequently, P is homeomorphic to P D, . Moreover, ourconstruction is Z -computable, and hence P D, has a Z -computable compact presentation.Next, assume that D is Σ relative to Z . We show that P D, has a Z -computable Polishpresentation. By our assumption, there is a computable set A ⊆ ω such that n D ifand only if ∃ ∞ a ∃ ∞ b ( n, a, b ) ∈ A . We will construct a presentation of a space P . Firstenumerate ω -many copies of S and ω + 1, and compactify them by adding a point z into P . Then we prepare for infinitely many pairwise separated regions ( R i ) i ∈ ω , where R i is2 − i -close to z . Inside R n , we start by describing S n +1 , and enumerating a sequence ofisolated points which converges to a point p ∈ S n +1 . Let S n,a [ s ] be a 2 − s -approximationof a copy S n,a of S as before. We enumerate the following set into the region R n :˜ S n,a [0] ∪ [ { S n,a [ b ] : ( n, a, b ) ∈ S } . If ( n, a, b ) ∈ A for infinitely many b , then ˜ S n,a is a dense subset of S n,a , so its completionis S n,a ≃ S ; otherwise, ˜ S n,a consists of finitely many isolated points. Thus, by thesame argument as above, P is homeomorphic to P D, . Moreover, our construction is Z -computable, and hence P D, has an Z -computable Polish presentation. Theorem 3.18.
For any degree d , there is a compact metrizable space P d , whose compactdegree spectrum is { x : d ≤ x (4) } and Polish degree spectrum is { x : d ≤ x (5) } . t suffices to show the following: Lemma 3.19.
For any D ⊆ ω , there is a compact metrizable space P D, such that forany Z ⊆ ω , D is Σ relative to Z ⇐⇒ P D, has a Z -computable compact presentation. D is Σ relative to Z ⇐⇒ P D, has a Z -computable Polish presentation.Proof. Given D ⊆ ω , we define T n +1 D = ( S n +1 ∨ S Σ if n ∈ D, S n +1 ∨ S Π if n D. Then, let P D, be the one-point compactification of the following locally compact space: a n ∈ ω T n +1 D ∐ a ω S Π . As in Lemmas 3.10 and 3.17, given a compact presentation of P D, , we again make the Z -computable tree T Z of components of P D, . Recall that a component of P D, is aninfinite path through T Z . There is a unique component in P D, which is homeomorphicto S n +1 , and let α n +1 be the corresponding infinite path through T Z . At each stage s ,compute a node η s of T Z which has a longest-surviving ( n + 1)-hole among nodes of T Z of length s . Then, η s is an initial segment of α n +1 for almost all s .We then combine the idea of the proofs of Lemmas 3.10 and 3.17: The key observation isthat if n D , then T n +1 D ≃ S n +1 ∨ S Π , where recall that the S -rank of S Π is 2, so anyneighborhood of S n +1 contains a component of S -rank at least 1. On the other hand,in the case n ∈ D , we have S n +1 D ≃ S n +1 ∨ S Σ , where recall that the S -rank of S Σ is 1, so if a neighborhood of S n +1 is sufficiently small, then it only contains componentsof S -rank 0 except for the sphere itself. By this observation, n D if and only if forany s there are t > s and σ ∈ T Z such that σ (cid:23) η t , σ η t +1 , and σ satisfies the Π ( Z )property stating that σ has some extension in [ T Z ] of S -rank 1; that is, for any ℓ thereare two extensions of τ of length ℓ extending satisfy the Σ ( Z ) property stating that thecorresponding component has a 1-hole which survives forever. In summary, the predicate n D is Π ( Z ).For the other direction, assume that D is Σ relative to Z . We show that P D, has a Z -computable compact presentation. By our assumption, there is a computable set A ⊆ ω such that n D if and only if ∃ ∞ a ∃ ∞ b ∀ c ( n, a, b, c ) ∈ A . Without loss of generality, wemay assume that ( n, a, , c ) ∈ A for all a, c , and ( n, a, b + 1 , , ( n, a, b + 1 , A forall a, b . We will construct a presentation of a space P . First enumerate ω -many copiesof S and ω + 1, and compactify them by adding a point z into P . Then we prepare forinfinitely many pairwise separated regions ( R i ) i ∈ ω , where R i is 2 − i -close to z .Inside R n , we start by describing S n +1 , and enumerating a sequence of isolated points( p a ) a ∈ ω which converges to a point p ∞ ∈ S n +1 . That is, the following is enumerated intothe region R n : S n +1 ∪ { p a : a ∈ ω } . hen, prepare for a sequence of countably many separated copies { S n,a,b } b ∈ ω of thecircle S which also converge to p a ∈ S n +2 , and let ( S n,a,b [ s ]) s ∈ ω be an increasing se-quence of nonempty sets of finite isolated points which 2 − s -approximates S n,a,b , that is, d H ( S n,a,b [ s ] , S n,a,b ) < − s , where we do not enumerate these copies into P at present. Foreach a , we enumerate S n,a, and the following set into the region R n :˜ S n,a,b = [ { S n,a,b [ s ] : ( ∀ c < s ) ( n, a, b, c ) ∈ A } . As before, if ∃ ∞ b ∀ c ( n, a, b, c ) ∈ A is true, then there are a sequence of circles convergingto p a (that is, α ω S Π ), and also a sequence of isolated points converging to p a (that is, α ω S Σ +1 ) by our assumption that ( n, a, b + 1 , A for all b . This part is homeomorphicto S Π as before. If ∃ ∞ b ∀ c ( n, a, b, c ) ∈ A is false, then the a -strategy enumerates atmost finitely many circles (finitely many copies of S Π ), and a sequence of isolated pointsconverging to p a (that is, S Σ ≃ α ω ( S Σ +1 ), so it is homeomorphic to S Σ ∐ ( S Π ) k forsome k , where k is positive by our assumption that ( n, a, , c ) ∈ A for all c .If ∃ ∞ a ∃ ∞ b ∀ c ( n, a, b, c ) ∈ A is true (i.e., n D ), then there are a sequence of copiesof S Π converging to p ∞ , and also a sequence of copies of S Σ ∐ ( S Π ) k converging to p ∞ by our assumption that ( n, a, b + 1 , A for all a, b . It is not hard to check thatthe union of the latter sequence and p ∞ is homeomorphic to α ω S Σ +3 , where recall that S Σ +3 ≃ S Σ ∐ S Π . Therefore, in this case, this space restricted to the region R n ishomeomorphic to the wedge sum of S n +2 and α ω S Π ∨ α ω S Σ +3 ≃ α ω ( S Π ∐ S Σ +3 ) ≃ S Π .If ∃ ∞ a ∃ ∞ b ∀ c ( n, a, b, c ) ∈ A is false (i.e., n ∈ D ), then there are at most finitely manycopies of S Π and a sequence of copies of S Σ ∐ ( S Π ) k converging to p ∞ . The latter partis homeomorphic to α ω S Σ +3 ≃ S Σ as above. Therefore, in this case, this space restrictedto the region R n is homeomorphic to the separated union of S n +2 ∨ S Σ and at mostfinitely many copies of S Π . Up to homeomorphism, these finitely many copies of S Π are absorbed into ω -many copies of S Π in some other region.Consequently, P is homeomorphic to P D, . Moreover, our construction is x -computable,and hence P D, has a Z -computable compact presentation. For a Polish presentation, thesimilar argument as above works. Theorem 3.20.
For any degree d and n > , there is a compact metrizable space P d ,n whose compact degree spectrum is { x : d ≤ x (2 n ) } and Polish degree spectrum is { x : d ≤ x (2 n +1) } . It suffices to show the following:
Lemma 3.21.
For any D ⊆ ω and n > , there is a compact metrizable space P D,n suchthat for any Z ⊆ ω , D is Σ n +1 relative to Z ⇐⇒ P D,n has a Z -computable compact presentation. D is Σ n +2 relative to Z ⇐⇒ P D,n has a Z -computable Polish presentation.Proof. Given D ⊆ ω , we define T d +1 D = ( S d +1 ∨ S Σ n +1 if d ∈ D, S d +1 ∨ S Π n +1 if d D, hen, let P D,n be the one-point compactification of the following locally compact space: a d ∈ ω T d +1 D ∐ a ω S Π n − . Then, proceed the similar argument as above.As a corollary, for any natural number n , the class of high n +1 -degrees is a compact degreespectrum of a compact metrizable space, and the class of high n +3 -degrees is a Polishdegree spectrum of a compact metrizable space. In this section, we solve Question 3 by showing the following:
Theorem 4.1.
Let A ⊆ ω be a non-c.e. set. Every compact Polish space has a Polishpresentation that does not enumerate A . In particular,
Corollary 4.2.
The degree spectrum of a compact Polish space cannot be the uppercone { x : d ≤ x } for any non-computable degree d . Actually the proof also shows that if for each i we choose a non-c.e. set A i , then everycompact Polish space has a presentation that does not enumerate any A i . It implies thatthe degree spectrum of a compact Polish space cannot be a countable union of non-trivialupper cones S i ∈ ω { x : d i ≤ x } .In order to prove the result, we need ideas from computability theory and ideas fromtopology. Overtness argument.
Overtness captures a familiar argument in computably theory,which we describe now.We will apply the technique to the space X = V ( Q ), however it is easier to state theresult for an abstract space X .Let X be a countably-based space with a fixed indexed basis ( B i ) i ∈ ω that is closed underfinite intersections. We say that A is a reducible to x if A is enumeration reducibleto N x = { i ∈ ω : x ∈ B i } . We write M ( x ) = A if M enumerates A from any enumerationof N x . We say that a set S ⊆ X is computably overt if the set { i ∈ ω : S ∩ B i = ∅} is c.e.Given a Turing machine M and a set A , we say that M fails to enumerate A from x if M outputs some n / ∈ A on some enumeration of N x . We denote by F M,A the set of x ’s onwhich M fails to enumerate A . That set is open, because when M outputs some n / ∈ A ,it has only read a finite part of N x , which can be extended to an enumeration of N y forany y in some neighborhood of x , so each such y also belongs to F M,A . Lemma 4.3.
Let X be a countably-based space. Let A ⊆ ω be a non-c.e. set. If x ∈ X enumerates A using machine M , then F M,A intersects every computably overt set contain-ing x . roof. Assume that M ( x ) = A and let V be a computably overt set containing x . If F M,A does not intersect V , then we describe an effective procedure that enumerates A , contra-dicting the assumption that A is not computable. The procedure is as follows: enumerateall the prefixes of names of elements of V (which is possible because V is computablyovert), simulate M on them, and collect all the outputs. As F M,A does not intersect V ,all the outputs are correct, i.e. belong to A , and every element of A appears because M enumerates A on each name of x ∈ V . As a result, this procedure enumerates A , whichis then c.e. The contradiction implies that F M,A intersects V . Perturbations.
We now come to the topological ingredient of the proof, based on thenotion of ǫ -perturbation. The Hilbert cube Q is endowed with the complete metric d ( x, y ) = X i − i | x i − y i | . The proof is a Baire category argument: one can perturb any compact subset of Q sothat its perturbed copy does not enumerate A . Definition 4.4. An ǫ -perturbation is a one-to-one continuous function f : Q → Q suchthat d ( f ( x ) , x ) < ǫ for all x ∈ Q . Lemma 4.5.
Let S = { s , . . . , s n } and T = { t , . . . , t n } be subsets of Q such that d ( s i , t i ) <ǫ for i = 0 , . . . , n . There exists an ǫ -perturbation f such that f ( s i ) = t i for i = 0 , . . . , n .Proof. It is a corollary of the homeomorphism extension theorem (Theorem 5.3.7 in [26]),stating that if
S, T are Z -sets and f : S → T is a homemorphism satisfying d ( f ( x ) , x ) < ǫ for all x ∈ S , then f can be extended to a homeomorphism f : Q → Q satisfying thesame condition for all x ∈ Q . We only need to know that finite sets are Z -sets: in thesame reference, singletons are Z -sets by Remark 5.1.4, and finite unions of Z -sets areagain Z -sets by Lemma 5.1.2.We remind the reader that V ( Q ) is a topological space endowed with the lower Vietoristopology. In the next statement, the notions of computable overtness and closure aremeant in that topology. Lemma 4.6.
Let D ∈ V ( Q ) and ǫ > . There exists a computably overt set A ⊆ V ( Q ) containing D and contained in the closure of the set of ǫ -deformations of D .Proof. Let
F ⊆ V ( Q ) be the family of finite sets of rational points, which can be indexedin an obvious way. We are going to define A in such a way that:(1) F ∩ A is dense in A ,(2) F ∩ A is computably enumerable,(3) Every element of
F ∩ A is contained in some ǫ -deformation of D .The first two conditions imply that A is computably overt, because it is the closure of acomputable dense sequence. he first and third conditions imply that A is contained in the closure of the set of ǫ -deformations of D : A is contained in the closure of F ∩ A , and each element of
F ∩ A is a subset of an ǫ -deformation C of D , so belongs to the closure of { C } (the lowerVietoris closed open sets are upwards closed, equivalently the lower Vietoris closed setsare downwards closed).We now define A satisfying these conditions. If D was perfect then we could just takesome small ball around D in the Hausdorff metric. However we need a bit more work inthe general case.We first show that there exist open sets U , . . . , U n ⊆ Q that cover D , such that forevery x ∈ U , D ∩ B ( x, ǫ ) is infinite and D ∩ U i is a singleton for each i ≥
1. Let D N bethe set of non-isolated points of D and let U = D ǫN . The set D \ U is finite, becauseit is compact and all its points are isolated. Therefore, there exist basic balls U , . . . , U n isolating the points of D \ U . We can make sure that the open sets U i are pairwisedisjoint. By compactness of D , we can assume that U is a finite union of basic balls. Wecan now define our computably overt subset of V ( Q ): let A = ( E ∈ V ( Q ) : E ⊆ [ ≤ i ≤ n U i and | E ∩ U i | = 1 for all i ≥ ) . Conditions 1. and 2. are easily checked.We now prove condition 3.
Claim . For every finite set T ∈ A , there exists and ǫ -deformation of D containing T .Let T = { t , . . . , t n } belong to A . We build a finite set S = { s , . . . , s n } ⊆ D with d ( s i , t i ) <ǫ . For each i , if t i ∈ U then the intersection of B ( t i , ǫ ) with D is infinite, so we can choosea point s i in the intersection, so that s i = s j for i = j . If t i ∈ U k with k ≥
1, then wedefine s i as the unique point of D in U k . The points s i are pairwise distinct, becauseif i = j then s i and s j cannot both belong to a common U k , k ≥
1, as D ∈ A .One has d ( s i , t i ) < ǫ for each i , so we can apply Lemma 4.5 to obtain an ǫ -perturbation f mapping each s i to t i . One has T ⊆ f ( D ) so the claim is proved, and the Lemma aswell.We now have all the ingredients needed to prove the result. Proof of Theorem 4.1.
Let X be a compact Polish space. We prove that some copy of X in V ( Q ) does not enumerate A . It implies the result, because any name of a copy of X in V ( Q ) computes a presentation of X .The space F = { φ : Q → Q continuous one-to-one } , with the topology induced by themetric d ( φ, ψ ) = sup x d ( φ ( x ) , ψ ( x )) is Polish.For any compact set C ⊆ Q and any non-c.e. set A ⊆ ω , we prove that the set { φ ∈ F : A ≤ e φ ( C ) } is meager in F , which implies the existence of a copy D of C such that A (cid:2) e D .It is done by showing that for each Turing machine M , the set { φ ∈ F : M ( φ ( C )) = A } is nowhere dense in F . et φ ∈ F be such that M ( φ ( C )) = A . For ǫ >
0, we prove that there exists ψ ∈ F suchthat d ( φ, ψ ) < ǫ and such that ψ ( C ) ∈ F M,A . It implies the result, because for every ψ ′ sufficiently close to ψ , one also has ψ ′ ∈ F M,A as F M,A is open.Lemma 4.6 provides a computably overt set A containing φ ( C ), in which the set of ǫ -deformations of φ ( C ) is dense. By Lemma 4.3, F M,A intersects A . As F M,A is open, thereexists an ǫ -deformation of φ ( C ) in F M,A . Let f be the corresponding ǫ -perturbation,and ψ = f ◦ φ . One has d ( ψ, φ ) < ǫ and ψ ( C ) ∈ F M,A . Let X be a compact Polish space. In the proofs we frequently use the fact that the jumpof any Polish presentation of X computes a compact presentation of X , which is statedprecisely in Lemma 2.1. Here we investigate whether it can compute more. Of course,it always computes ′ , and we show that if X is perfect, then it does not compute morein general: every compact presentation of X , paired with ′ , computes the jump of aPolish presentation of X . However, it is no more true for non-perfect spaces and we givea counter-example. Theorem 5.1.
Let X be a compact perfect Polish space. The jumps of the Polish presen-tations of X are the compact presentations of X paired with ′ . In other words, one has { d ′ : X has a d -computable Polish presentation } = { ( d , ′ ) : X has a d -computable compact presentation } . Reformulation.
Again, we use some overtness argument to reformulate the problem.We can reformulate the jumps of the Polish presentations of X , i.e. the degrees d ′ suchthat X has a d -computable Polish presentation.As we have already seen the degrees of Polish presentations of X coincide with the degreesof copies of X as elements of V ( Q ). In the same way, the jumps of these degrees are exactlythe jumps of the copies of X in V ( Q ).Again, we abstract away from V ( Q ) to make the results easier to read. Let S be aneffective countably-based space with a fixed index basis ( B i ) i ∈ ω that is closed under fi-nite intersections. Let ( U Si ) i ∈ ω be an effective enumeration of the effective open subsetsof S : U Si = S j ∈ W i B j . Definition 5.2.
The jump of x ∈ S is the set J ( x ) = { i ∈ ω : x ∈ U Si } . Lemma 5.3.
In an effective countably-based space S , computing J ( x ) is equivalent tocomputing d ′ for some d that computes x .Proof. Let δ S : ⊆ ω ω → S be the standard representation of S , mapping p to x if { i : ∃ n, p ( n ) = i + 1 } = { i : x ∈ B i } , in which case we say that p is a name of x . Thefunction δ S is computable and effectively open: the image of an effective open is aneffective open set, uniformly. Observe that d computes x if and only if d computes somename of x . ecause δ S is computable, the preimages of effective open sets are effectively open, so if p is a name of x then p ′ computes J ( x ).Conversely, given J ( x ), we show how to compute p ′ for some name p of x . Let ( U n ) n ∈ ω bethe canonical enumeration of the effective open subsets of ω ω . Let V n be x -effective opensets such that δ − S ( x ) = T n V n .At stage s , we have produced a finite prefix p s of p . Given J ( x ), we can decide whether [ p s ] ∩ δ − X ( x ) intersects U s , because it is equivalent to x ∈ δ S ([ p s ] ∩ U s ) which is an effective openset for which we have an index. If it does, then we extend p s to p s +1 so that [ p s +1 ] ⊆ U s ∩ V s .If it does not, then we simply make sure that [ p s +1 ] ⊆ V s .In the limit, we obtain some p ∈ T s V s so p is a name of x . For each s , we have decidedalong the construction whether p ∈ U s , so we have computed p ′ . Proof of Theorem 5.1.
We are given C ∈ K ( Q ), together with ′ . We progressively com-pute a copy D of C , together with its jump as a point of V ( Q ). Let ( U n ) n ∈ ω be an effectiveenumeration of the effective open subsets of V ( Q ). For each n , we need to decide as longas we build D , whether D ∈ U n .We start from some ǫ > ǫ/ B containing C . It is a computablyovert set in V ( Q ), so we can decide using ′ whether it intersects U . There are two cases.In the first case, B intersects U . As in Lemma 4.6 there exists an ǫ -perturbation f mapping C to C ∈ U (as C is perfect, the computably overt set given by Lemma 4.6can be replaced by the ǫ/ B ). Claim . We can compute such an f .The space P ǫ of ǫ -perturbations is a computable Polish space, the function Φ : P ǫ → V ( Q )mapping f to f ( C ) is C -computable, so Φ − ( U ) is a C -effective open set, in which wecan computably find some f , so the claim is proved.We pick a ball around C , whose closure is contained in U and in which we are going tostay forever, so that in the limit, the copy D of C belongs to U . We declare that D ∈ U .The second case is if B does not intersect U . In that case, we do nothing and proceed.In the sequel, we stay forever in B (and even in some closed ball contained in B ) so thatin the limit, D ∈ B hence D does not belong to U . We declare that D / ∈ U .In both cases, we have decided whether the set D belongs to U . We now iterate thisprocess with U , U , etc., taking ǫ smaller and smaller so that the composition of the ǫ -perturbations converges to a homeomorphism (the Inductive Convergence Criterion [26]tells us that we can always choose the next ǫ sufficiently small to ensure that the limit isa homeomophism, and moreover ǫ can be chosen in a computable way), and taking theclosure of B n +1 contained in B n . In the limit, we have built a copy D of C and computedits jump.Observe that the argument is uniform, assuming that the space is perfect. We now observethat there cannot exist a uniform argument including non-perfect Polish spaces. Indeed,whether X is not perfect is Σ for a Polish presentation, so it is Σ in its jump. If there wasa uniform argument then being non-perfect would be Σ ( ′ ) for compact presentations,in particular it would be open. However, the set of non-perfect compact sets is not open n the Hausdorff metric (witnessed for instance by a sequence of segments shrinking to asingleton).We actually show that Theorem 5.1 simply does not extend to non-perfect Polish spaces.Observe that a corollary of Theorem 5.1 is that if X is a perfect Polish space with a ′ -computable compact presentation, then it has a low Polish presentation. We show thatit fails for some non-perfect Polish space. Proposition 5.4.
There exists a (non-perfect) compact Polish space with a ′ -computablecompact presentation, but no low Polish presentation. Therefore, it has a compact pre-sentation d such that ( d , ′ ) does not compute the jump of any Polish presentation ofit.Proof. Let d = (6) . The space X = Z d , from Theorem 3.11 has a ′ -computable compactpresentation and its Cantor-Bendixon derivative X ′ ∼ = Z d , has no ′′ -computable compactpresentation.Lemma 2.5 implies in particular that if X had a low Polish presentation then X ′ wouldhave a ′′ -computable compact presentation, which is not the case. One of the main questions of this topic (which is probably very hard to answer completely,as also for algebraic structures) is to characterise the spectra of Polish spaces, say byproving its coincidence with the spectra of a natural class of algebraic structures. However,we still do not know the answer to the following question.
Question 4.
Is any Polish (compact) degree spectrum of a Polish space a degree spectrumof an algebraic structure?
Recall also that one of the key ideas of our constructions is using dimension (more explic-itly, high-dimensional holes , i.e., a cycle which is not a boundary) to code a given Turingdegree. As a result, all of our examples in the above results are infinite dimensional . Wedo not know if there are finite dimensional examples satisfying our main results. We alsonote that all of our examples are disconnected, and it is not known if there are connectedexamples.
Question 5.
Does there exist a finite dimensional (connected) low -presented Polish spacewhich is not homeomorphic to a computably presented one? There are many other open questions. For instance, the following is also open.
Question 6.
Does there exist a low -presented Polish space which is not homeomorphicto a computably presented one? The full solution to Question 3 is also yet to be known.
Acknowledgement.
This work was started in September 2019 during visits of T. Kiharaand V. Selivanov to INRIA Nancy. We are grateful to this institute and to M. Hoyrupfor support and excellent research environment. eferences [1] C. J. Ash and J. Knight. Computable structures and the hyperarithmetical hierarchy ,volume 144 of
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