On Higher Effective Descriptive Set Theory

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10307)

Abstract

In the framework of computable topology, we propose an approach how to develop higher effective descriptive set theory. We introduce a wide class \(\mathbb {K}\) of effective \(T_0\)-spaces admitting Borel point recovering. For this class we propose the notion of an \((\alpha ,m)\)-retractive morphism that gives a great opportunity to extend classical results from EDST to the class \(\mathbb {K}\). We illustrate this by several examples.

Keywords

Higher effective descriptive set theory Effective topological space Effective \(T_0\)–space admitting Borel point recovering \((\alpha , m)\)-retractive morphism Effective Borel and Lusin hierarchies Suslin-Kleene Theorem Uniformisation Theorem 

1 Introduction

Having a long term history, classical descriptive set theory (DST) has been based on the fundamental idea that all Polish spaces have common properties related to definability of functions and sets as well as to the resulting Borel and Lusin hierarchies [5, 13]. The similar idea can be used for the computable Polish spaces under consideration of effective versions of the corresponding hierarchies. One of the approaches to effective descriptive set theory (EDST) has been proposed in [10, 12, 14] and developed in [3, 13, 18] among others, where most of results have been obtained for the computable Polish spaces. A comparison of concepts and results from computable analysis and EDST has been presented in [4].

It is worth noting that in the non-effective case a significant progress has been done in [2, 17], where a big part of DST has been developed first for \(\omega \)-continuous domains and then for the wider class of quasi-Polish spaces. Two of the main problems that naturally arise in EDST are the following. The first one is to discover wide classes of effective topological spaces for which the main results of DST hold. Here the class of quasi-Polish spaces looks like a promising candidate. The second problem is to discover wide classes of effective topological spaces admitting results of EDST related to higher levels of the effective Lusin hierarchy.

The main results of our paper concern this second problem. Informally, in the paper we consider a part of EDST that is nearly \(\varDelta ^1_1\)–level or above in the effective Lusin hierarchy and refer to this part as higher effective descriptive set theory (Higher EDST). In order to give a flavor of Higher EDST it is worth noting that an effective version of the Hausdorff theorem does not belong to Higher EDST while the extended Suslin-Kleene theorem does.

The main contributions of the paper are the following. To attack the second problem mentioned above we propose a wide class of effective topological spaces admitting effective Borel point recovering which contains effective quasi-Polish spaces, weakly-effective \(\omega \)–continuous domains as proper subclasses. For this class we provide a fruitful technique that allows us to effectivise classical results from DST. We illustrate this by the Suslin-Kleene Theorem, Uniformisation Theorem.

2 Notations and Preliminaries

We refer the reader to [14, 15] for basic definitions and fundamental concepts of recursion theory, to [3, 5, 13] for basic definitions and fundamental concepts of DST and EDST. We use bold Greek letters \(\varvec{\alpha },\,\varvec{\beta },\varvec{\gamma },\dots \) to denote numberings and light Greek letters to denote ordinals. We use only computable ordinals i.e. that are less than \(\omega ^{\mathbb {CK}}_1\). We work with the Baire space \({\mathcal N}=(\omega ^\omega ,\varvec{\alpha }_{\mathcal N})\), the Cantor space \({\mathcal C}=(2^\omega ,\varvec{\alpha }_{\mathcal C})\) with the standard topologies and numberings of the bases, and the space \(\mathbb {P} =(\mathcal {P}(\omega ),\varvec{\beta })\), where \(\varvec{\beta }\) is the standard numbering of the topology base generated by all open sets of the type \( W_D=\{I\,\subseteq \, \omega \mid D\,\subseteq \, I\}\), where D is a finite subset of \(\omega \).

3 Effective Topological Spaces and Hierarchies

Let \(\left( X,\tau , \varvec{\alpha }\right) \) be a topological space, where X is a non-empty set, \(B_\tau \, \subseteq \, 2^{X}\) is a base of the topology \(\tau \) and \(\varvec{\alpha }:\omega \rightarrow B_\tau \) is a numbering. In notations we skip \(\tau \) since it can be recovered by \(\varvec{\alpha }\). Further on we will often abbreviate \(\left( X,\varvec{\alpha }\right) \) by X if \(\varvec{\alpha }\) is clear from a context.

Definition 1

A topological space \(\left( X,\varvec{\alpha }\right) \) is effective if the following condition holds.

  • There exists a computable function \(g:\omega \times \omega \times \omega \rightarrow \omega \) such that
    $$\begin{aligned} \varvec{\alpha }( i)\bigcap \varvec{\alpha }( j)=\bigcup _{n\in \omega }\varvec{\alpha }( g(i,j,n)). \end{aligned}$$

Now we recall the notion of an effectively enumerable topological space.

Definition 2

[8] An effective topological space \(\left( X, \varvec{\alpha }\right) \) is effectively enumerable if the following condition holds.

It is worth noting that the similar concept of a weakly computable \(cb_0\)–space has been used in [18].

Definition 3

Let \(\left( X,\varvec{\alpha }\right) \) be an effective topological space. A set \(A\,\subseteq \, X\) is effectively open if there exists a computably enumerable set \(V\,\subseteq \,\omega \) such that
$$\begin{aligned} A=\bigcup _{n\in V} \varvec{\alpha }(n). \end{aligned}$$

Let \(\mathcal{O}_X\) denote the set of all open subsets of X and \(\mathcal{O}^e_X\) denote the set of all effectively open subsets of X. The set \(\mathcal{O}^e_X\) is closed under intersection and union since the class of effectively enumerable sets is a lattice. The following proposition is a natural corollary of the definition.

Proposition 1

[7] For every effective topological space X there exists a principal computable numbering \(\varvec{\alpha }^{e}_X\) of \(\mathcal{O}^e_X\).

In this paper we use the notation \(f: X\rightarrow Y\) for a partial function unless the word total is written.

Definition 4

Let \(X=\left( X,\varvec{\alpha }\right) \) be an effective topological space and \(Y=\left( Y,\varvec{\beta }\right) \) be an effective \(T_0\)–space. A total function \(f:X\rightarrow Y\) is called computable if there exists a computable function \(H:\omega ^2\rightarrow \omega \) such that
$$f^{-1}(\beta (m))=\bigcup _{i\in \omega } \alpha (H(m,i)). $$
We adopt the notion of the effective Borel and Lusin hierarchies for computable Polish spaces [13] to effective \(T_0\)–spaces. Put for finite ordinals
$$\begin{aligned}&\varSigma ^0_1= \text{ all } \text{ effecively } \text{ open } \text{ sets, }\\&\varSigma ^0_{n+1}=\exists ^\omega \text{(the } \text{ set } \text{ of } \text{ finite } \text{ boolean } \text{ combinations } \text{ of } \varSigma ^0_n), \\&\text{ where } \exists ^\omega \text{ denotes } \text{ the } \text{ projection } \text{ along } \omega ,\\&\varPi ^0_n=\lnot \varSigma ^0_n,\\&\varDelta ^0_n=\varSigma ^0_n\bigcap \varPi ^0_n. \end{aligned}$$
Following [13], \(\{(\varSigma ^0_n,\varPi ^0_n)\}_{n< \omega }\) is called the Kleene hierarchy. Further on, for \(R\,\subseteq \, X\) and \(R\,\in \, \varSigma ^0_n\) we write \(R\,\in \, \varSigma ^0_n[X]\).

Remark 1

It is worth noting that the difference between our and Moschovakis’s definitions of the Kleene hierarchy [13] is based on the following observation. For the space \(\mathbb {P} \), \(\varSigma ^0_1[\mathbb {P} ]\,\not \subseteq \, \exists ^\omega \varPi ^0_1[\mathbb {P} ]\). To show that it is sufficient to prove that \(A=\{x\,\subseteq \, \omega \mid 0\,\not \in x\}\,\not \in \, \forall ^\omega \varSigma ^0_1[\mathbb {P} ]\). Assume contrary \(A\,\in \, \forall ^\omega \varSigma ^0_1[\mathbb {P} ]\). Then \(A=\bigcap _{n\,\in \,\omega }E_n\), where \(E_n\) are effectively open sets. Therefore, for all Open image in new window and, by the properties of the topology on \(\mathbb {P} \), \(\omega \,\in \, E_n\). So, \(\omega \,\in \, A\). This contradicts the definition of A. In fact this definition of the Kleene hierarchy has been proposed in [18].

One way to introduce the effective Borel hierarchy\(\{(\varSigma ^0_\xi ,\varPi ^0_\xi )\}_{\xi < \omega ^{\mathbb {CK}}_1}\) considered in [10] is to use effective Borel coding [5, 13], i.e., for any B such that \(B=B_\alpha \) for some computable \(\alpha \,\in \,\mathcal {N}\), \(\xi (B)=\,\inf \,\{\xi (\alpha )\mid \alpha \text{ is } \text{ computable } \text{ and } B=B_\alpha \}\), where \(\xi (\alpha )\) is introduced in [5]. Another more direct way proposed in [14] for \(\mathcal {N}\) and in [3] for computable Polish Spaces can be adopted to effective \(T_0\)–spaces [18]. In different situations one of the approaches has its advantages. For inductive considerations the second one is more preferable. In particular, in the framework of the second approach the inclusions \(\varSigma ^0_\alpha \,\subseteq \, \varSigma ^0_{\alpha +1}\), \(\varPi ^0_\alpha \,\subseteq \, \varSigma ^0_{\alpha +1}\) are just parts of the definition. Therefore, \(\varSigma ^0_{\alpha }\cup \varPi ^0_\alpha \,\subseteq \, \varDelta ^0_{\alpha +1}= \varSigma ^0_{\alpha +1}\bigcap \varPi ^0_{\alpha +1}\).

In [3, 18] the effective Lusin hierarchy \(\{(\varSigma ^1_n[X],\varPi ^1_n[X])\}_{n<\omega }\) is defined by induction as follows:
$$\begin{aligned}&\varSigma ^1_1[X]=\{\mathrm{pr}_{\mathcal {N}}(B)\mid B\,\in \,\varPi ^0_2[\mathcal {N}\times X]\},\\&\varSigma ^1_{n+1}[X]=\{\mathrm{pr}_{\mathcal {N}}(B)\mid B\,\in \,\varPi ^1_n[\mathcal {N}\times X]\},\\&\varPi ^1_n[X]=\lnot \varSigma ^1_n[X],\\&\varDelta ^1_n[X]=\varSigma ^1_n[X]\bigcap \varPi ^1_n[X]. \end{aligned}$$
Some properties of the effective Lusin hierarchy on particular spaces (e.g. perfect computable Polish spaces, computable reflective \(\omega \)-algebraic domains) can be found in [18].

Lemma 1

  For an effective \(T_0\)–space X, \(\{(x,y)\mid x=y\}\,\in \,\varPi ^0_2[X\times X]\).

4 The Class \(\mathbb {K}\)

Definition 5

An effective \(T_0\)–space \(\left( X, \varvec{\alpha }\right) \) is said to admit effective Borel point recovering if the following condition holds.

  • The set \(\{A_x\mid x\,\in \, X\}\) is a \(\varDelta ^1_1\)-subset of \(\mathcal {P}(\omega )\), where \(A_x=\{n\mid x\,\in \,\varvec{\alpha }(n)\}\). Here \(\mathcal {P}(\omega )\) is considered as the Cantor space \(\mathcal {C}\).

We denote this class of effective \(T_0\)–spaces admitting Borel point recovering by \(\mathbb {K}\). Now we show the correspondence between spaces from \(\mathbb {K}\) and \(\varDelta ^1_1\)-subsets of \(\mathbb {P} \).

Proposition 2

\(\varDelta ^1_1[\mathcal {C}]=\varDelta ^1_1[\mathbb {P} ]\).

Proof

The inclusion \(\varDelta ^1_1[\mathcal {C}]\supseteq \varDelta ^1_1[\mathbb {P} ]\) follows from the fact that \(\mathcal {C}\) has more \(\varPi ^0_2\)-subsets than \(\mathbb {P} \). Now us take \(\mathrm{id}:(\mathbb {P} ,\beta )\rightarrow (\mathcal {C},\varvec{\alpha }_\mathcal {C})\). Since \(\mathrm{id}^{-1}(\varvec{\alpha }_\mathcal {C}(n))=\varvec{\beta }(h(n))\setminus \varvec{\beta }(t(n))\), where h and t are computable functions, \(\varPi ^0_2[\mathcal {C}]\,\subseteq \, \varPi ^0_3[\mathbb {P} ]\). Therefore, \(\varDelta ^1_1[\mathcal {C}]\,\subseteq \, \varDelta ^1_1[\mathbb {P} ]\).    \(\square \)

Definition 6

We write \(\left( X_0,\varvec{\beta }_0 \right) <(X,\varvec{\beta })\) if
  1. 1.

    \(X_0\) is a subset of X,

     
  2. 2.

    \(\varvec{\beta }_0(n)=\varvec{\beta }(n)\bigcap X\) for any \(n\,\in \,\omega \).

     

Theorem 1

For any effective (effectively enumerable) \(T_0\)–space \( \left( X, \varvec{\alpha }\right) \) the following assertions are equivalent.

  1. 1.

    \(\left( X, \varvec{\alpha }\right) \,\in \, \mathbb {K}\).

     
  2. 2.
    \(\left( X, \varvec{\alpha }\right) \) is computably homeomorphic to some \(\left( X_0,\varvec{\beta }_0 \right) <(\mathbb {P},\varvec{\beta })\) with
    $$\begin{aligned} X_0\,\in \,\varDelta ^1_1[\mathbb {P} ]. \end{aligned}$$
     

Proof

(\(2\rightarrow 1).\) Assume \(X\,\in \,\varDelta ^1_1[\mathbb {P} ]\). Consider \((X,\varvec{\alpha })\), where \(\varvec{\alpha }\) is induced by \(\varvec{\beta }\). It is easy to see that \(\{P_x\mid x\,\in \, \mathbb {P} \}\,\in \, \varDelta ^0_2[\mathcal {C}]\), where \(P_x=\{n\mid x\,\in \,\varvec{\beta }(n)\}\). In particular, \(\mathbb {P} \,\in \, \mathbb {K}\). Let \(A_y=\{n\mid y\,\in \,\varvec{\alpha }(n)\}\). Then
$$\begin{aligned}&J\,\in \,\{A_y\mid y\,\in \, X\}\,\leftrightarrow \\&J\,\in \,\{P_y\mid y\,\in \, \mathbb {P} \}\wedge (\exists y\,\in \, X) J=P_y \,\leftrightarrow \\&J\,\in \,\{P_y\mid y\,\in \, \mathbb {P} \}\wedge (\exists y\,\in \, X) (\forall n\,\in \,\omega ) \big ( y\,\in \,\varvec{\beta }(n)\,\leftrightarrow \, n\,\in \, J \big ) . \end{aligned}$$
This is a \(\varSigma ^1_1\)-condition on \(\mathcal {C}\).
$$\begin{aligned}&J\,\not \in \{A_y\mid y\,\in \, X\}\,\leftrightarrow \\&J\,\not \in \{P_y\mid y\,\in \, \mathbb {P} \}\vee (\exists y\,\in \, \mathbb {P} )\big ( J=P_y\wedge y\,\not \in X\big ) . \end{aligned}$$
This is also a \(\varSigma ^1_1\)-condition on \(\mathcal {C}\). Therefore, \(\{A_y\mid y\,\in \, X\}\,\in \,\varDelta ^1_1[\mathcal {C}]\).

(\(1\rightarrow 2).\) Assume now that \(\{A_x\mid x\,\in \, X\}\,\in \,\varDelta ^1_1[\mathcal {C}]\), where \(A_x=\{n\mid x\,\in \, \varvec{\alpha }(n)\}\). We take the function \(F:X\xrightarrow {1-1} \mathbb {P} \) defined by the rule \(F(x)=A_x\). It is clear that F is effectively continuous, i.e. computable. Indeed, \(F^{-1}(W_D)=\bigcap _{m\in D}\varvec{\alpha }(m)\). Let \(X_0=F(X)\) and \(\varvec{\beta }_0(m)=W_{D_m}\bigcap X_0\). By definition, \((X_0,\varvec{\beta }_0)\) is a subspace of \(\mathbb {P} \). In order to prove that \((X_0,\varvec{\beta }_0)\) is computably homeomorphic to \((X,\varvec{\alpha })\) we show that \(F^{-1}:(X_0,\varvec{\beta }_0)\rightarrow (X,\varvec{\alpha })\) is computable. For that we check \(F(\mathbf { \varvec{\alpha }}(m))=W_{\{m\}}\bigcap X_0\). The inclusion \(F(\varvec{\alpha }(m))\,\subseteq \, W_{\{m\}}\bigcap X_0\) is trivial. Let us show the inclusion \(F(\varvec{\alpha }(m))\supseteq W_{\{m\}}\bigcap X_0\). Let \(I\,\in \, W_{\{m\}}\bigcap X_0\), i.e., \(m\,\in \, I\) and \(I\,\in \, X_0\). Since \(I\,\in \, X_0\), by definition, there exists \(x\,\in \, X\) that \(F(x)=I\), so \(I=A_x\). Since \(m\,\in \, A_x\), \(x\,\in \, \varvec{\alpha }(m)\), so \(I\,\in \, F(\varvec{\alpha }(m))\).

Now we show that \(X_0\,\in \,\varDelta ^1_1[\mathbb {P} ]\). Since \((X_0,\varvec{\beta }_0)\) is computably homeomorphic to \((X,\varvec{\alpha })\), \(\{P_x\mid x\,\in \, X_0\}\,\in \,\varDelta ^1_1[\mathcal {C}]=\varDelta ^1_1[\mathbb {P} ]\). For \(x\,\in \, \mathbb {P} \), we have
$$\begin{aligned}&x\,\in \, X_0 \,\leftrightarrow \\&(\exists I\,\in \, \{P_y\mid y\,\in \, X_0\})\, I=P_x \,\leftrightarrow \\&(\exists I\,\in \, \{P_y\mid y\,\in \, \mathbb {P} \})\, (\forall n\,\in \,\omega ) \big ( x\,\in \,\varvec{\beta }(n)\,\leftrightarrow \, n\,\in \, I \big ) \end{aligned}$$
and
$$\begin{aligned}&x\,\not \in X_0 \,\leftrightarrow \\&(\exists I\,\in \, \{P_y\mid y\,\in \, \mathbb {P} \})\big ( I=P_x\wedge I\,\not \in \{P_y\mid y\,\in \, X_0\}\big ). \end{aligned}$$
Therefore \(X_0\,\in \,\varDelta ^1_1[\mathcal {C}]=\varDelta ^1_1[\mathbb {P} ]\).    \(\square \)

Taking into account results from [2], where the class of quasi-Polish spaces has been introduced, we define in a natural way the notion of a computable quasi-Polish space.

Definition 7

The space \(\left( X, \varvec{\alpha }\right) \) is called computable quasi-Polish if it is computably homeomorphic to some \(\left( X_0,\varvec{\beta }_0 \right) <(\mathbb {P},\varvec{\beta })\) with
$$\begin{aligned} X_0\,\in \,\varPi ^0_2[\mathbb {P} ]. \end{aligned}$$

Now we show that there exists \(X\,\in \,\mathbb {K}\) that is not computable quasi-Polish.

Proposition 3

Let \(\left( X, \varvec{\alpha }\right) \) be a computable quasi-Polish space. Then
$$ \{A_x\mid x \,\in \, X\}\,\in \,\varPi ^0_3[\mathcal {C}]\bigcap \varPi ^0_4[\mathbb {P} ]. $$

Proof

It is sufficient to consider \(\left( X_0,\varvec{\beta }_0 \right) <(\mathbb {P},\varvec{\beta })\). Let us recall \(A_I=\{n\mid D_n\,\subseteq \, I\}\), where \(I\,\in \, X_0\) and \(D_n\) is finite. Assume that \(X_0\,\in \,\varPi ^0_2[\mathbb {P} ]\). This means that \(X_0=\bigcap _{n\in \omega } E_n\), where \(\{E_n\}_{n\in \omega }\) is an effective sequence of boolean combinations of effectively open sets in \(\mathbb {P} \). Let us observe that if we denote \(\mathcal {A}_n\,\rightleftharpoons \,\{A_x\mid x \,\in \, E_n\}\) then \(\mathcal {A}\,\rightleftharpoons \,\{A_x\mid x \,\in \, X\}= \bigcap _{n\in \omega }\mathcal {A}_n\). Therefore, it is sufficient to understand the level of \(\{A_x\mid x\,\in \,\varvec{\beta }(i)\}\). Since \(\mathcal {B}_i\,\rightleftharpoons \,\{A_x\mid x\,\in \,\varvec{\beta }(i)\}=\{\{n\mid D_n\,\subseteq \, x\}\mid x\,\supseteq \, D_i\}\), by a routine computation it can be shown that \(\{\mathcal {B}_i\}_{i\in \omega }\) is a computable sequence of \(\varPi ^0_1[\mathcal {C}]\)–sets (\(\varPi ^0_2[\mathbb {P} ]\)–sets). Finally, \(\mathcal {A}_n\) is a boolean combination of \(\varSigma ^0_2[\mathcal C]\)–sets (\(\varPi ^0_3[\mathbb {P} ]\)–sets). Therefore \(\mathcal {A}\,\in \, \varPi ^0_3[\mathcal C]\) (\(\mathcal {A}\,\in \,\varPi ^0_4[\mathbb {P} ] \)).    \(\square \)

Theorem 2

  1. 1.

    There exists a space from the class \(\mathbb {K}\) which is not computable quasi-Polish.

     
  2. 2.

    There exists a perfect space from the class \(\mathbb {K}\) which is not computable quasi-Polish.

     

Proof

We use the notations from the proposition above.

1. Let us consider \(\left( X_0,\varvec{\beta }_0 \right) <(\mathbb {P},\varvec{\beta })\) such that \(X_0\,\in \,\varDelta ^1_1[\mathcal {C}]\setminus \varPi ^0_4[\mathcal {C}]\). Such space exist since effective hierarchies on \(\mathcal {C}\) are strict. We take the function \(F:\mathbb {P} \xrightarrow {1-1} \mathbb {P} \) defined by the rule \(F(x)=A_x\). The function F is effectively continuous, i.e. computable. Therefore this function is continuous and computable from \(\mathcal {C}\) to \(\mathbb {P} \). It is clear that if \(I\,\in \,\varPi ^0_4[\mathbb {P} ]\) then \(F^{-1}(I)\,\in \, \varPi ^0_4[\mathcal {C}]\). Now if we assume that \(X_0\) is quasi-Polish then by Proposition 3\(\{A_x\mid x\,\in \, X_0\}\,\in \,\varPi ^0_4[\mathbb {P} ]\). So \(X_0=F^{-1}(\{A_x\mid x\,\in \, X_0\})\,\in \, \varPi ^0_4[\mathcal {C}]\). This leads to a contradiction.

2. In order to construct a perfect space we consider \(\tilde{X_0}=\{2\cdot I\mid I\,\in \, X_0\}\). Take \(X_0^*=\{I\oplus J\mid I\,\in \, X_0,\, J\,\subseteq \,\omega \}\). It is easy to see that \(X_0^*\) is computably homeomorphic to \(\tilde{X_0}\times \mathcal {P}(2\cdot \omega +1)\). So \(X_0^*\,\in \,\mathbb {K}\) is perfect but not computable quasi-Polish.    \(\square \)

5 \((\alpha ,m)\)–Retractive Morphisms

In this section we assume that all our spaces belong to the class \(\mathbb {K}\). We introduce a useful tool that provides a fruitful technique to get effective versions of classical theorems from DST that hold for all spaces from \(\mathbb {K}\).

Definition 8

Let \(f:X\rightarrow Y\) and \(g:Y\rightarrow X\), where \((X,\varvec{\alpha })\,\in \,\mathbb {K}\) and \((Y,\varvec{\gamma })\,\in \,\mathbb {K}\). The pair (fg) is called a \(( \alpha ,m)\)–retractive morphism from X to Y (denoted by \(X \underset{\underset{g}{\hookleftarrow }}{\mathop {\rightharpoonup }\limits ^{f}}Y\)) if the following conditions hold.

  1. 1.

    \(f\circ g=\mathrm{id}_Y\),

     
  2. 2.

    \(\mathrm{dom}(f)\,\in \, \varSigma ^0_\mathbf { \alpha }[X]\),

     
  3. 3.

    \(f^{-1}(\varvec{\gamma }(m))= \bigcup _{i\in I_m}\varvec{\alpha }(i)\bigcap \mathrm{dom}(f)\), where \(I_m\) is a computable sequence of c.e. sets,

     
  4. 4.

    For \(\mathcal { O}_n=\varvec{\alpha }^e_X(n)\), \(\{g^{-1}({\mathcal O}_n)\}_{n\in \omega }\) is a computable sequence of elements of \( \varSigma ^0_{m+1}[X]\), i.e., \(\bigcup _{n\in \omega }\{n\}\times g^{-1}({\mathcal O}_n)\,\in \, \varSigma ^0_{m+1}[\omega \times X]\).

     

Remark 2

The mappings f and g are effective Borel functions, i.e., their preimages of effectively open sets are effective Borel sets [13, 18]. From the first condition it follows that the function f is onto and the function g is a total injection the third condition corresponds to the effective continuity of f.

Remark 3

The notion of a computable sequence of elements of \( \varSigma ^0_m[X]\) can be defined in terms of good parametrisations for \( \varSigma ^0_m[X]\) (see e.g. [13]).

Lemma 2

Let \( X \underset{\underset{g_1}{\hookleftarrow }}{\mathop {\rightharpoonup }\limits ^{f_1}} X_1 \underset{\underset{g_2}{\hookleftarrow }}{\mathop {\rightharpoonup }\limits ^{f_2}} X_2 \), where \((f_1,g_1)\) is an \(( \alpha ,n)\)–retractive morphism from X to \(X_1\) and \((f_2,g_2)\) is a \(( \beta ,m)\)–retractive morphism from \(X_1\) to \(X_2\) then there exist ordinals \(\gamma <\omega ^{\mathbb {CK}_1}\) and \(k<\omega \) such that \((f_2\circ f_1,g_1\circ g_2)\) is a \(( \gamma ,k)\)–retractive morphism from X to \(X_2\). The ordinal k can be chosen as \(n+m\).

Lemma 3

Let \( X \underset{\underset{g_1}{\hookleftarrow }}{\mathop {\rightharpoonup }\limits ^{f_1}} Y_0 \underset{\underset{\varphi }{\hookleftarrow }}{\mathop {\rightharpoonup }\limits ^{\varphi ^{-1}}} Y, \) where \(Y_0\,\subseteq \, \mathbb {P} \) is a computable homeomorphic copy of Y under \(\varphi \), (fg) is an \((\alpha ,n)\)-retractive morphism between X and Y. Then \((\varphi ^{-1}\circ f,g\circ \varphi )\) is an \((\alpha ,n)\)-retractive morphism between X and Y.

Theorem 3

For any space \(Y\,\in \, \mathbb {K}\) there exists an \(( \alpha ,1)\)–retractive morphism from \(\mathcal {N}\) to Y for some ordinal \(\alpha <\omega ^{\mathbb {CK}}_1\).

Proof

By Theorem 1 and Lemma 3, without loss of generality we assume that \(Y\,\in \,\varDelta ^1_1[\mathbb {P} ]\). A required \(( \alpha ,1)\)–retractive morphism from \(\mathcal {N}\) to Y is the composition of the following \(( \alpha ,1)\)–retractive morphisms:
$$ \mathcal {N} \underset{\underset{g_1}{\hookleftarrow }}{\mathop {\rightharpoonup }\limits ^{f_1}} \mathcal {C} \underset{\underset{g_2}{\hookleftarrow }}{\mathop {\rightharpoonup }\limits ^{f_2}} \mathbb {P} \underset{\underset{g_3}{\hookleftarrow }}{\mathop {\rightharpoonup }\limits ^{f_3}} Y, $$
where the functions are defined as follows.
  1. 1.

    The function \(f_1\) and \(g_1\) are standard well known computable mappings: \(f(\chi )(n)=\chi (n)\,\mathrm{mod}\, 2\) and \(g=\mathrm{id}\).

     
  2. 2.

    Put \(f_2=\mathrm{id}\) and \(g_2=\mathrm{id}\).

     
  3. 3.

    Put \(f_3=\left\{ \begin{array}{ll} x, &{} x\,\in \, Y\\ \uparrow , &{} x\,\not \in \, Y \end{array} \right. \) and \(g_3=\mathrm{id}\).

     
The functions satisfy the requirements of Definition 8. Let us show non-trivial parts. It is easy to see that
$$ g^{-1}_2(\alpha _{\mathcal {C}}(n))=\{f\mid (\forall j\,\in \, D_2) f(j)=1\}\setminus \{f\mid (\forall i\,\in \, D_1) f(i)=1\}, $$
where \(\alpha _{\mathcal {C}}(n)=\{f:\omega \rightarrow \{0,1\}\mid (\forall i\,\in \, D_1) f(i)=0\wedge (\forall j\,\in \, D_2) f(j)=1 \}\) for some finite sets \(D_1\) and \(D_2\) such that Open image in new window. Therefore, \(g^{-1}_2(\alpha _{\mathcal {C}}(n))\,\in \, \varDelta ^0_2[\mathbb {P} ]\) uniformly in n. Since \(Y\,\in \,\varDelta ^1_1[\mathbb {P} ]=\varDelta ^1_1[\mathcal {C}]\) and the Suslin-Kleene Theorem holds for all perfect computable Polish spaces (see e.g. [18]), \( Y\,\in \, \varSigma ^0_\alpha [\mathcal {C}]\). So, \( Y\,\in \, \varSigma ^0_{\alpha ^\prime }[\mathcal {\mathbb {P} }]\) for some \(\alpha ^\prime \ge \alpha \).    \(\square \)

Proposition 4

Let \(Y\,\in \,\mathbb {K}\) and \( \mathcal {N} \underset{\underset{g}{\hookleftarrow }}{\mathop {\rightharpoonup }\limits ^{f}} Y \) be an \(( \alpha ,1)\)–retractive morphism from \(\mathcal {N}\) to Y. For any \(A\,\in \, \varPi ^1_1[Y]\), \(g(A)\,\in \, \varPi ^1_1[\mathcal {N}]\).

Proof

By the definition of (fg), \(g(A)=f^{-1}(A)\bigcap g(Y)\). Let us denote \(Y_0=g(Y)\) and assume \(A\,\in \,\varPi ^1_1[Y]\). Since f is an effective Borel function, \(f^{-1}(A)\,\in \varPi ^1_1[\mathcal {N}]\). It is easy to see that since \(\{(x,y)\mid x=y\}\,\in \,\varPi ^0_2[X\times X]\), \(Y_0=\{x\,\in \,\mathcal {N}\mid g(f(x))=x\}\,\in \,\varPi ^0_{\alpha ^\prime }\) for some \(\alpha ^\prime \ge \alpha \), Therefore, \( g(A)\,\in \,\varPi ^1_1[\mathcal {N}]\).    \(\square \)

Proposition 5

Let \(X\,\in \,\mathbb {K}\) and \(B\,\in \,\varDelta ^1_1[X\times X]\). Then \(A=\mathrm{pr}_X(B)\,\in \, \varSigma ^1_1[X] \).

Proof

Let us fix an \(( \alpha ,1)\)–retractive morphism \( \mathcal {N} \underset{\underset{g}{\hookleftarrow }}{\mathop {\rightharpoonup }\limits ^{f}} X .\) Put \(B^\prime =\{(x,y)\,\in \, X\times \mathcal {N}\mid B(x,f(y))\}\). It is clear that \(B\,\in \, \varSigma ^0_\alpha [X\times \mathcal {N}]\) for some \(\alpha <\omega ^{\mathbb {CK}}_1\). Let us show that \(x\,\in \, A \,\leftrightarrow \, (\exists y\,\in \, \mathcal {N}) B^\prime (x,y)\).

(\(\rightarrow ).\) Assume B(xz) for some \(z\,\in \, X\). Put \(y=g(z)\). By definition, \(f(y)=z\). We have B(xf(y)), i.e. \(B^\prime (x,y)\).

(\(\leftarrow ).\) Assume \(B^\prime (x,y)\), i.e. B(xf(y)) for some \(y\,\in \, \mathcal {N}\). Put \(z=f(y)\). Then B(xz), i.e. \(x\,\in \, A\). So, the projections of \(\varSigma ^1_1\)–sets along X are again \(\varSigma ^1_1\)–sets.    \(\square \)

6 Transferring EDST Theorems

In this section we show how \((\alpha ,m)\)–retractive morphisms can be used to make an effectivisation of classical results from DST that hold on the spaces from this class \(\mathbb {K}\). The following proposition is an extension of the classical Suslin-Kleene Theorem.

Theorem 4

For any \(Y\,\in \, \mathbb {K}\),
$$ \varDelta ^1_1[Y]=\bigcup _{ \alpha <\omega ^{\mathbb {CK}}_1}\varSigma ^0_\mathbf { \alpha }[Y].$$

Proof

The inclusion \(\varDelta ^1_1\supseteq \bigcup _{ \alpha <\omega ^{\mathbb {CK}}_1}\varSigma ^0_\mathbf { \alpha }\) follows from the observation that \(\varSigma ^1_1[Y]\) is closed under effective infinite unions, intersections. In order to show the inclusion \(\varDelta ^1_1\,\subseteq \,\bigcup _{ \alpha <\omega ^{\mathbb {CK}}_1}\varSigma ^0_\mathbf { \alpha }\) let us fix an \(( \alpha ,1)\)–retractive morphism \( \mathcal {N} \underset{\underset{g}{\hookleftarrow }}{\mathop {\rightharpoonup }\limits ^{f}} Y .\) Let us denote \(Y_0=g(Y)\) and assume \(A\,\in \,\varDelta ^1_1[Y]\). Since f is an effective Borel function, \(f^{-1}(A)\,\in \,\varDelta ^1_1[\mathcal {N}]\). By the Suslin-Kleene Theorem for \(\mathcal {N}\) there exists a computable ordinal \(\alpha \) such that \(f^{-1}(A)\,\in \, \varDelta ^0_\alpha [\mathcal {N}]\). Since \(f^{-1}(A)\bigcap g(Y)=g(A)\), \(\{(x,y)\mid x=y\}\,\in \,\varPi ^0_2[X\times X]\), \(Y_0=\{x\,\in \,\mathcal {N}\mid g(f(x))=x\}\,\in \,\varPi ^0_{\alpha ^\prime }[\mathcal {N}]\) for some ordinal \(\alpha ^\prime \ge \alpha \), Therefore, \(f^{-1}(A)\bigcap g(Y)\,\in \,\varSigma ^0_\beta [\mathcal {N}]\) for some \(\beta \ge \alpha \). Finally, \(g^{-1}(f^{-1}(A)\bigcap g(Y))=A\). Therefore, \(A\,\in \,\varSigma ^0_\gamma [X]\) for some \(\gamma <\omega ^{\mathbb {CK}}_1\).    \(\square \)

The following proposition is an extension of the classical Novikov-Kondo-Addison Uniformisation Theorem.

Theorem 5

(Uniformisation). Let \(X\,\in \,\mathbb {K}\). If \(Y\,\in \,\varPi ^1_1[X\times X]\) then there exists a function \(F:X\rightarrow X\) such that
  1. 1.

    The graph \(\varGamma _F\) of the function F is a subset of Y.

     
  2. 2.

    \(\delta (F)=\delta ( Y)=\{x\mid (\exists y\,\in \, X)\, (x,y)\,\in \, Y\},\)

     
  3. 3.

    \(\varGamma _F\,\in \, \varPi ^1_1[X\times X].\)

     

Proof

Let us fix an \(( \alpha ,1)\)–retractive morphism \( \mathcal {N} \underset{\underset{g}{\hookleftarrow }}{\mathop {\rightharpoonup }\limits ^{f}} Y .\) Let \(Y\,\in \,\varPi ^1_1[X]\). By Proposition 4, \(g(Y)\,\in \, \varPi ^1_1[\mathcal {N}]\). From the Novikov-Kondo-Addison Theorem for \(\mathcal {N}\) it follows that there exists G that is a \(\varPi ^1_1\)–uniformisation of g(Y). Put \(F=g^{-1}(G)\). Since g is an effective Borel function and a bijection between Y and g(Y), \(\varGamma _F\,\in \,\varPi ^1_1[X]\) and F is a required function.    \(\square \)

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Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.A.P. Ershov Institute of Informatics Systems, SbRASNovosibirskRussia
  2. 2.Sobolev Institute of Mathematics, SbRASNovosibirskRussia

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