First Order Theories of Some Lattices of Open Sets

TL;DR

This paper investigates the logical complexity of the lattice of open sets in various topological spaces, showing equivalences to second order arithmetic and undecidability results for computable spaces.

Contribution

It establishes the first order theories of these lattices as equivalent to second order arithmetic or undecidable, depending on the space.

Findings

Lattice of open sets in some spaces is m-equivalent to second order arithmetic.

For many computable spaces, the theory of effectively open sets is undecidable.

In spaces like ^n, the theory is m-equivalent to first order arithmetic.

Abstract

We show that the first order theory of the lattice of open sets in some natural topological spaces is $m$-equivalent to second order arithmetic. We also show that for many natural computable metric spaces and computable domains the first order theory of the lattice of effectively open sets is undecidable. Moreover, for several important spaces (e.g., $\mathbb{R}^n$, $n\geq1$, and the domain $P\omega$) this theory is $m$-equivalent to first order arithmetic.

Equations

$$\{(i,j,k,l)\mid {\varkappa }_k<d(\nu (i),\nu (j))<{\varkappa }_l\}$$

$$R=\{(x_1,\dots ,x_k)\in A^k\mid \mathbb{A}\models \varphi (x_1,\dots ,x_k,p_1,\dots ,p_l)\}.$$

$$R=\{(x_1,\dots ,x_k)\in A^k\mid \mathbb{A}\models \varphi (x_1,\dots ,x_k)\}.$$

$${\mathbb{A}}^{mso}:=(A\cup P(A);A,P(A),\in ,\sigma )$$

$$\{(m,n,p)\mid {\kappa }_n\subseteq {\beta }_{p(0)}\cup \cdots \cup {\beta }_{p(m)}\})$$

Full text

1. Introduction

From the very beginning of Model Theory, the study of (un)decidability of first order theories became a central and popular topic. As a result, for virtually all structures $\mathbb{A}=(A;\sigma)$ of a given signature $\sigma$ (in the Russian literature structures are also known as algebraic systems) occurring naturally in mathematics their first order theories $FO(\mathbb{A})$ has been shown to be decidable or undecidable (among vast literature on the subject we mention [tmr53, eltt, er80], as examples).

More recently, several researchers in Computability Theory have been working on the problem of characterizing the algorithmic complexity of undecidable first order theories (the complexity is usually measured by the $m$-degree [ro67, so87] of the theory that in this context coincides with the 1-1-degree, i.e. with the type of computable isomorphism of the theory). For many natural structures $\mathbb{A}$ with undecidable theory the theory $FO(\mathbb{A})$ turns out to be $m$-equivalent either to first order arithmetic $FO(\mathbb{N})$ or to second order arithmetic $FO(\mathbb{N}_{2})$ where $\mathbb{N}:=(\omega;+,\times)$ and $\mathbb{N}_{2}:=(\omega\cup P(\omega);\omega,P(\omega),\in,+,\times)$, see e.g. [ns80, nss96, ni98]. As is well known (see e.g. [ro67]), $FO(\mathbb{N})$ is $m$-equivalent to the $\omega$’th iteration $\emptyset^{(\omega)}$ of the Turing jump starting from the empty set.

Decidability issues for topological spaces seem to have been studied less systematically than for structures arising in algebra, logic and discrete mathematics, probably because first order language is not well suited for topology. Nevertheless, there was some important work for structures related to a topological space $X$, the most natural of which is the lattice $\mathbf{\Sigma}^{0}_{1}(X)$ of open sets in $X$. To our knowledge, A. Grzegorczyk [gr51] was the first to consider decidability issues in topology. One of the results in [gr51] (Corollary 2) interprets first order arithmetic in $\mathbf{\Sigma}^{0}_{1}(\mathbb{R}^{n})$ for each $n\geq 2$ which implies that $FO(\mathbb{N})$ is $m$-reducible to $FO(\mathbf{\Sigma}^{0}_{1}(\mathbb{R}^{n}))$ and hence the latter theory is undecidable. The question of whether $FO(\mathbf{\Sigma}^{0}_{1}(\mathbb{R}))$ is decidable was left open. M. Rabin [ra69] answered the question affirmatively (as well as the analogous question for the Cantor and Baire spaces) as a corollary of his result on decidability of the monadic second order theory of the binary tree.

A systematic model-theoretic study of structures arising in a topological setting was undertaken in [hjrt77] where it is shown, in particular, that $FO(\mathbb{N}_{2})\leq_{m}FO(\mathbf{\Sigma}^{0}_{1}(X))$ for many Hausdorff spaces $X$, and the above mentioned Grzegorczyk’s estimate was improved to $FO(\mathbb{N}_{2})\equiv_{m}FO(\mathbf{\Sigma}^{0}_{1}(\mathbb{R}^{n}))$ for each $n\geq 2$. Note that in fact the papers [gr51, hjrt77] work with the lattices of closed sets rather than with the lattices of opens but for our purposes this is clearly equivalent.

Another facet of the relationship between Topology and Computability Theory is the study of effectivity in a topological setting as developed in Computable Analysis [wei00] and Effective Descriptive Set Theory [mo09, s06]. An important object of study here is the lattices $\Sigma^{0}_{1}(X)$ of the so called effectively open sets in topological spaces $X$ satisfying some effectivity conditions (see the next section for more details). The lattice $\Sigma^{0}_{1}(X)$ is certainly the most important sublattice of the lattice $\mathbf{\Sigma}^{0}_{1}(X)$ of open sets, hence it is natural and instructive to study also definability and (un)decidability issues for the lattices of effectively open sets.

This study is interesting and non-trivial even for the discrete space $\omega$ of natural numbers since in this case the lattice $\Sigma^{0}_{1}(\omega)$ coincides with the lattice $\mathcal{E}$ of computably enumerable (c.e.) subsets of $\omega$ which is an important and popular object of study in Computability Theory [ro67, so87]. A principal fact about this lattice is the undecidability of $FO(\mathcal{E})$ [he83, he84]. Moreover, $FO(\mathcal{E})$ is known [hn98] to be $m$-equivalent to first-order arithmetic $FO(\mathbb{N})$. Note that the first order theory of $\mathbf{\Sigma}^{0}_{1}(\omega)=P(\omega)$ is decidable (because $P(\omega)$ is a Boolean algebra).

It seems that not much is known about (un)decidability of first order theories of the lattices of effectively open sets except for what is known about the lattice $\Sigma^{0}_{1}(\omega)$ and its relativizations. To our knowledge, only the cases of the Cantor space $\mathcal{C}$ and the Baire space $\mathcal{N}$ have been studied to some extent, in the context of the theory of $\Pi^{0}_{1}$-classes. In [ni00] (see the discussion of Main Theorem in the Introduction and Section 3) it is shown that $FO(\Pi^{0}_{1}(\mathcal{C}))$ is $m$-equivalent to $FO(\mathbb{N})$. Since the lattice $\Pi^{0}_{1}(\mathcal{C})$ formed by the complements of effectively open sets is anti-isomorphic to $\Sigma^{0}_{1}(\mathcal{C})$, this settles the question for $\mathcal{C}$. To our knowledge, similar questions for $\mathcal{N}$ (even the decidability of $FO(\Sigma^{0}_{1}(\mathcal{N}))$) are open.

In this paper, we make further steps in the study of (un)decidability issues for the theories of $\Sigma^{0}_{1}(X)$ and $\mathbf{\Sigma}^{0}_{1}(X)$. After recalling some necessary preliminaries we reprove in Section 3 the estimate from [hjrt77] $FO(\mathbb{N}_{2})\equiv_{m}FO(\mathbf{\Sigma}^{0}_{1}(\mathbb{R}^{n}))$, $n\geq 2$, using the original approach of A. Grzegorczyk which is different from the approach in [hjrt77]. We also establish the same estimate for some natural domains. In Section LABEL:efopen we first show that for many natural effective spaces $X$ (including computable metric spaces without isolated points and many natural computable domains) the theory $FO(\Sigma^{0}_{1}(X))$ is undecidable. Then we show that $FO(\Sigma^{0}_{1}(\mathbb{R}^{n}))$, $n\geq 1$, is $m$-equivalent to first order arithmetic. The same estimate also holds for some natural domains. We conclude in Section LABEL:con with a discussion of remaining open questions.

The methods of this paper apply mainly to second countable locally compact spaces. A precise estimate of the complexity of $FO(\mathbf{\Sigma}^{0}_{1}(X))$ and $FO(\Sigma^{0}_{1}(X))$ turns out to be subtle and depends strongly on the topology of $X$. For many natural spaces $X$ we still have a big gap between the known lower and upper bounds for $FO(\mathbf{\Sigma}^{0}_{1}(X))$ and $FO(\Sigma^{0}_{1}(X))$. In particular, for the Baire space we currently only know the estimate $\emptyset^{\prime}\leq_{m}FO(\Sigma^{0}_{1}(X))\leq_{m}O^{(\omega)}$ where $O$ is the Kleene ordinal notation system which is a $\Pi^{1}_{1}$-complete set.

Our upper bounds for the $m$-degree of $FO(\mathbf{\Sigma}^{0}_{1}(X))$ and $FO(\Sigma^{0}_{1}(X))$ are obtained by a straightforward application of the Tarski-Kuratowski algorithm, while the lower bounds use a suitable interpretation of one of the structures $\mathcal{E},\mathbb{N},\mathbb{N}_{2}$ in the lattice under consideration.

This paper is an extended version of the conference paper [ks16] that contains, in particular, new results on the lattices of all open sets and an essentially modified proof of Theorem LABEL:euclide for $n>1$.

2. Preliminaries

Here we briefly recall some notions and notation relevant to this paper. We freely use the standard set-theoretic notation like $|X|$ for the cardinality of $X$, $X\times Y$ for the cartesian product of sets and topological spaces, $P(X)$ for the set of all subsets of $X$, $\overline{A}$ for the complement $X\setminus A$ of a subset $A$ of a space $X$.

We assume the reader to be familiar with basic notions of topology (see e.g. [en89]). We often abbreviate “topological space” by “space”. By $Cl(S)$ (resp. $Int(S)$) we denote the closure (resp. the interior) of a set $S\subseteq X$ in a space $X$. A space $X$ is Polish if it is separable and metrizable with a metric $d$ such that $(X,d)$ is a complete metric space. We denote the set of open subsets of a space $X$ by $\mathbf{\Sigma}^{0}_{1}(X)$. This is the first among the finite levels $\{\mathbf{\Sigma}^{0}_{n}(X)\}$ of the Borel hierarchy [ke95, br13] which is formed by applying the operations of complementation and countable union to the open sets.

Let $\omega$ be the space of non-negative integers with the discrete topology. The space $\omega\times\omega=\omega^{2}$ is homeomorphic to $\omega$, the homeomorphism being realized by the Cantor pairing function $\langle x,y\rangle$.

Let $\omega^{\omega}$ be the set of all infinite sequences of natural numbers (i.e., of all functions $\xi:\omega\rightarrow\omega$). Let $\omega^{<\omega}$ be the set of finite sequences of elements of $\omega$, including the empty sequence. For $\sigma\in\omega^{<\omega}$ and $\xi\in{\mathcal{N}}$, we write $\sigma\sqsubseteq\xi$ to denote that $\sigma$ is an initial segment of the sequence $\xi$. By $\sigma\xi=\sigma\cdot\xi$ we denote the concatenation of $\sigma$ and $\xi$, and by $\sigma\cdot\mathcal{N}$ the set of all extensions of $\sigma$ in $\mathcal{N}$. For $x\in\omega^{\omega}$, we can write $x=x(0)x(1)\cdots$ where $x(i)\in\omega$ for each $i<\omega$. For $x\in\mathcal{N}$ and $n<\omega$, let $x[n]=x(0)\cdots x(n-1)$ denote the initial segment of $x$ of length $n$. Notations in the style of regular expressions like $0^{\omega}$, $0^{<\omega}1$ or $0^{m}1^{n}$ have the obvious standard meaning. Define the topology on $\omega^{\omega}$ by taking arbitrary unions of sets of the form $\sigma\cdot\omega^{\omega}$, where $\sigma\in\omega^{<\omega}$, as the open sets. The space $\mathcal{N}=\omega^{\omega}$ with this topology known as the Baire space, is of primary importance for Descriptive Set Theory and Computable Analysis.

For any finite alphabet $A$ (usually we assume without loss of generality that $A=k=\{0,\ldots,k-1\}$ where $0<k<\omega$), let $A^{\omega}$ be the set of $\omega$-words over $A$. This set may be topologized similarly to the Baire space. The resulting spaces, which for $k\geq 2$ are all (computably) homeomorphic among themselves, are known as Cantor spaces (usually the term Cantor space is applied to the space $\mathcal{C}=2^{\omega}$ of infinite binary sequences). Note that the Cantor space is compact while the Baire space is not.

Next we recall some definitions related to domain theory (for more details see e.g. [aj, er93, gh03]).

Let $X$ be a $T_{0}$-space. For $x,y\in X$, let $x\leq y$ denote that $x\in U$ implies $y\in U$, for all open sets $U$. The relation $\leq$ is a partial order known as the specialization order. Let $F(X)$ be the set of finitary elements of $X$ (known also as compact elements), i.e. elements $p\in X$ such that the upper cone $\mathord{\uparrow}{p}=\{x\mid p\leq x\}$ is open. Such open cones are called $f$-sets. The space $X$ is called a $\varphi$-space if every open set is a union of $f$-sets. Note that every non-discrete $\varphi$-space is not Hausdorff. A $\varphi$-space is complete if any non-empty directed set has a supremum w.r.t. the specialization order.

A $\varphi$-space $X$ is an $f$-space if any compatible elements $c,d\in F(X)$ have a least upper bound w.r.t. $\leq$ (compatibility means that $c,d$ have an upper bound in $F(X)$). An $f$-space $X$ is an $f_{0}$-space if $F(X)$ has a least element.

Let $\omega^{\leq\omega}=\omega^{<\omega}\cup\omega^{\omega}$ be the set of all finite and infinite strings of natural numbers with the topology generated by the sets $\{x\mid u\sqsubseteq x\}$ where $u\in\omega^{<\omega}$. For every $1\leq k<\omega$, the space $k^{\leq\omega}$ of all finite and infinite words over the alphabet $\{0,\ldots,k-1\}$ is defined in the same way. Let $P\omega$ be the powerset of $\omega$ with the topology generated by the sets $\mathord{\uparrow}{F}:=\{A\mid F\subseteq A\subseteq\omega\}$ where $F$ is a finite subset of $\omega$.

Let $\omega_{\bot}=\omega\cup\{\bot\}$ be the the space with the topology generated by $\{n\}$, $n\in\omega$. Let $\omega_{\bot}^{\omega}$ be the space of partial functions on $\omega$ with the topology generated by the sets $\{g\mid f\subseteq g\}$ where $f$ is a function with finite graph and $\subseteq$ is the subgraph relation (as usual, we identify a partial function $g$ on $\omega$ with the total function $\tilde{g}:\omega\rightarrow\omega_{\bot}=\omega\cup\{\bot\}$ where $g(x)$ is undefined iff $\tilde{g}(x)=\bot$). For each $k$, $2\leq k<\omega$, let $k^{\omega}_{\bot}$ be the space of partial functions $g:\omega\rightharpoonup\{0,\ldots,k-1\}$ defined similarly to $\omega_{\bot}^{\omega}$.

As is well known, $\omega^{\leq\omega},k^{\leq\omega},P\omega,\omega_{\bot},\omega_{\bot}^{\omega},k^{\omega}_{\bot}$ are complete $f_{0}$-spaces where the sets of $f$-elements are respectively $\omega^{<\omega},k^{<\omega}$,the finite subsets of $\omega$, $\omega_{\bot}$, the finite partial functions on $\omega$, the finite partial functions from $\omega$ to $\{0,\ldots,k-1\}$.

As is well known (see e.g. [er72]), for any (complete) $f_{0}$-spaces $X,Y$ the space $Y^{X}$ of continuous functions from $X$ to $Y$ with the topology of pointwise convergence is again a (complete) $f_{0}$-space. Therefore, any space of continuous partial functionals over $\omega$ of a finite type is a complete $f_{0}$-space. In particular, this applies to the spaces $\mathbb{F}_{n}$ defined by induction as $\mathbb{F}_{0}:=\omega_{\bot}$, $\mathbb{F}_{n+1}:=\omega_{\bot}^{\mathbb{F}_{n}}$.

Next we explain what we mean by effectively open sets. For any countably based topological space $X$ and any numbering $\beta$ of a base of $X$, define a function $\pi:\omega\to P(X)$ by $\pi(n)=\bigcup\beta[W_{n}]$ where $\{W_{n}\}$ is the standard numbering of the c.e. sets [ro67, so87]) and $\beta[W_{n}]=\{\beta(a)\mid a\in W_{n}\}$. The sets in $\pi[\omega]$ are called effectively open sets in $X$. Thus, the set of effectively open sets $\Sigma^{0}_{1}(X)$ is always equipped with the induced numbering $\pi$, hence it makes sense to speak about computable sequences of effectively open sets.

For many reasonable spaces $X$ with effectivity conditions, one can define in a natural way the (finite levels of the) effective Borel hierarchy $\{\Sigma^{0}_{n}(X)\}$ and the effective Luzin hierarchy $\{\Sigma^{1}_{n}(X)\}$ (see e.g. [mo09, s06, s15] for details) which are reasonable effective versions of the classical Borel and Luzin hierarchies. (Note that the definition of effective hierarchies in a given effective space depends on the chosen numbering of a base of the space.) In particular, $\{\Sigma^{0}_{n}(\omega)\}$ and $\{\Sigma^{1}_{n}(\omega)\}$ (taken with a natural numbering of a base in $\omega$) coincide with the arithmetical and analytical hierarchies of subsets of $\omega$ which are central objects of study in Computability Theory [ro67].

We define some particular classes of effective spaces relevant to this paper. A computable metric space [wei00] is a triple $(X,d,\nu)$, where $(X,d)$ is a metric space and $\nu:\omega\rightarrow X$ is a numbering of a dense subset $rng(\nu)$ of $X$ such that the set

[TABLE]

is c.e. Here $\varkappa$ is the conventional numbering of the set $\mathbb{Q}$ of rationals. Any computable metric space $(X,d,\nu)$ gives rise to a numbering $\beta$ of the standard base $\beta_{\langle m,n\rangle}=B(\nu_{m},\varkappa_{n})$ where $\langle m,n\rangle$ is the Cantor pairing and $B(\nu_{m},\varkappa_{n})$ is the basic open ball with center $\nu_{m}$ and radius $\varkappa_{n}$ (if $\varkappa_{n}\leq 0$ the “ball” is empty).

By a strongly computable metric space (SCMS) we mean a computable metric space such that there exists an infinite computable sequence $\{B_{n}\}$ of pairwise disjoint basic open balls. The metric spaces $\omega$, $\mathbb{Q}$, $\mathcal{C}$, $\mathcal{N}$, $\mathbb{R}^{n}$ (where $\mathbb{R}$ is the space of real numbers) equipped with the standard metrics and with natural numberings of dense subsets are SCMS. Any computable metric space without isolated points is an SCMS. Working with the Euclidean spaces $\mathbb{R}^{n}$, we denote by $d$ the Euclidean metric, by 0 the zero-vector $(0,\ldots,0)\in\mathbb{R}^{n}$, non-empty open (resp. closed) rational balls by $B(a,r)$ (resp. $C(a,r)$) where $a\in\mathbb{Q}^{n},r\in\mathbb{Q}^{+}$. Sometimes it is convenient to use also the empty ball $B(a,0)$.

By a computable $\varphi$-space we mean a pair $(X,\delta)$ consisting of a $\varphi$-space $X$ and a numbering $\delta:\omega\rightarrow F(X)$ of all the finitary elements such that the specialization order is c.e. on the finitary elements (i.e., the relation $\delta_{x}\leq\delta_{y}$ is c.e.). Setting $\beta(n):=\mathord{\uparrow}{\delta_{n}}$ we obtain a numbering of a topological base of $X$. Thus, we have a notion of an effective open set in every computable $\varphi$-space.

By a strongly computable $\varphi$-space (SC$\Phi$S) we mean a computable $\varphi$-space $X$ such that the specialization order is computable on the finitary elements, and there is a computable sequence $\{c_{n}\}$ of pairwise incomparable finitary elements. An SC$\Phi$S $X$ is a strongly computable $f_{0}$-space (SCF0S) if it is an $f_{0}$-space, the relation of compatibility is computable on $F(X)$, and the supremum of compatible finitary elements is computable. Although the restrictions imposed on SC$\Phi$Ss and SCF0Ss are rather strong, many popular domains are SCF0Ss. In particular, this applies to all concrete examples of $\varphi$-spaces mentioned above in this section. (For the space $Y^{X}$ of continuous functions, a close inspection of the corresponding proofs [er72] shows that if $X,Y$ are (complete) SCF0Ss then so is $Y^{X}$. Therefore, any space of continuous partial functionals over $\omega$ of a finite type is a complete SCF0S.) Note that the “strong” variations above are rather ad hoc and do not pretend to be fundamental notions in the field.

We conclude this section by briefly recalling of some notions from logic. We consider only structures of finite relational signatures (when a functional symbol is used, as e.g. in the structure $\mathbb{N}$, we identify the corresponding function with its graph). For a $\sigma$-structure $\mathbb{A}=(A;\sigma)$, a relation $R\subseteq A^{k}$ is definable in $\mathbb{A}$ if there is a first order $\sigma$-formula $\phi(x_{1},\ldots,x_{k},p_{1},\ldots,p_{l})$ and (possibly) some values $p_{1},\ldots,p_{l}\in A$ of parameters such that

[TABLE]

If the list of parameters is empty then we speak about definability without parameters. Thus, $R\subseteq A^{k}$ is definable in $\mathbb{A}$ without parameters if there is a first order $\sigma$-formula $\phi(x_{1},\ldots,x_{k})$ with

[TABLE]

A function on $A$ is definable (with or without parameters) if its graph is definable. An element of $A$ is definable if the corresponding singleton set $\{a\}$ is definable. A structure is definable if its universe and all signature predicates are definable.

E.g., if $\mathbb{A}=(A;\cup,\cap,0,1)$ is a bounded distributive lattice then any of $\cup,\cap,0,1$ is definable without parameters in $(A;\leq)$ where $\leq$ is the induced partial order on $A$, and $\leq$ is definable in $(A,\cup)$. Moreover, in this case we can even speak about arbitrary Boolean terms of elements of $A$ (meaning their values in a Boolean algebra extending $\mathbb{A}$). Thus, dealing with our lattices $\mathbf{\Sigma}^{0}_{1}(X)$ and $\Sigma^{0}_{1}(X)$ we can mean any of the signatures $\{\subseteq\}$, $\{\cup,\cap,\emptyset,X\}$, or even $\{\cup,\cap,\bar{},\emptyset,X\}$. For simplicity, we omit the signature symbols in the notation of these structures. We use in our formulas some standard abbreviations, in particular the bounded quantifiers $(\forall x)_{\phi(x)}\psi:=\forall x(\phi(x)\to\psi)$ and $(\exists x)_{\phi(x)}\psi:=\exists x(\phi(x)\wedge\psi)$ or the “quantifier” $\exists!x\psi$ meaning “there exists a unique $x$ satisfying $\psi$”.

The first order theory $FO(\mathbb{A})$ of the structure $\mathbb{A}$ is the set of $\sigma$-sentences true in $\mathbb{A}$. Along with first order logic, in logical theories some other logics are considered, in particular the (monadic) second order logic where one can use, along with the usual variables, also variables ranging over the (unary) relations on $A$. Accordingly, one can consider the monadic second order theory $\mathit{MSO}(\mathbb{A})$, or the full second order theory $SO(\mathbb{A})$, of $\mathbb{A}$ (in the latter case one needs variables for relations of any arity). Since the Cantor coding is definable in $\mathbb{N}$ without parameters, $SO(\mathbb{N})\equiv_{m}\mathit{MSO}(\mathbb{N})$. The theory $\mathit{MSO}(\mathbb{A})$ may be considered as the first order theory $FO(\mathbb{A}^{mso})$ of the extended structure

[TABLE]

obtained from $\mathbb{A}$ by adjoining the powerset of $A$ to the universe, the unary predicates for $A$ and $P(A)$, and the membership relation to the list of relations. In particular, we have $\mathbb{N}_{2}=\mathbb{N}^{mso}$ and $\mathit{SO}(\mathbb{N})\equiv_{m}FO(\mathbb{N}_{2})$.

An important tool to compare algorithmic complexity of theories is the notion of interpretability of one theory or structure in another. In fact, there are many versions of this notion (see e.g. [tmr53, eltt, er80] of which we briefly recall a couple of those used in the sequel.

A $\tau$-structure $\mathbb{B}$ is interpretable in a $\sigma$-structure $\mathbb{A}$ without parameters if some isomorphic copy of $\mathbb{B}$ is definable in $\mathbb{A}$ without parameters. A weaker version of this is the notion of $c$-interpretability (where $c$ comes from “congruence”). We say that $\mathbb{B}$ is $c$-interpretable in $\mathbb{A}$ without parameters if there exist a $\tau$-structure $\mathbb{C}$ and a congruence $\sim$ on $\mathbb{C}$ such that both $\mathbb{C}$ and $\sim$ are definable in $\mathbb{A}$ without parameters and the quotient-structure $\mathbb{C}/\mathord{\sim}$ (whose elements are the equivalence classes $c/\mathord{\sim},c\in C$) is isomorphic to $\mathbb{B}$. Interpretability with parameters is introduced in the same manner.

As is well known (see e.g. [tmr53, eltt]), if $\mathbb{B}$ is $c$-interpretable in $\mathbb{A}$ without parameters then $FO(\mathbb{B})\leq_{m}FO(\mathbb{A})$. The same is true for definability with parameters provided that the set of “defining” parameters may be chosen definable. The latter notion means that there is a non-empty definable (without parameters) set $P\subseteq A^{l}$ of parameters such that for any value of parameters in $P$ the corresponding structure is isomorphic to $\mathbb{B}$. For future reference, we formulate some of the mentioned facts as a lemma.

Lemma 1**.**

Let $\mathbb{B}$ be $c$-interpretable in $\mathbb{A}$ without parameters (or with a non-empty set of parameters which is itself definable without parameters). Then $FO(\mathbb{B})\leq_{m}FO(\mathbb{A})$.

When no non-empty set of eligible parameters is definable, the relation $FO(\mathbb{B})\leq_{m}FO(\mathbb{A})$ is not true in general but there is some version of undecidability which is preserved also by such interpretations. A theory (not necessarily complete) of signature $\sigma$ is hereditarily undecidable if any of its subtheories of signature $\sigma$ is undecidable. It is well known (see e.g. [er80]) that if $FO(\mathbb{B})$ is hereditarily undecidable and $\mathbb{B}$ is $c$-interpretable in $\mathbb{A}$ with parameters then $FO(\mathbb{A})$ is hereditarily undecidable.

Additional information about interpretations may be found on page 215 of [ho93].

3. The lattices of open sets

Here we give precise estimates of the algorithmic complexity of $FO(\mathbf{\Sigma}^{0}_{1}(X))$ for some spaces $X$.

3.1. An upper bound

First we establish a natural upper bound that applies to many countably based locally compact spaces. We need a technical notion related to local compactness. By analytically locally compact space (AnLCS) we mean a triple $(X,\beta,\kappa)$ consisting of a topological space $X$, a numbering $\beta$ of a base in $X$ containing the empty set (the presence of the empty set is not principal and maybe removed by using slight modification of the notion of AnLCS), and a numbering $\kappa$ of some compact sets in $X$ such that any set $\beta_{n}$ is a union of some sets in $\{\kappa_{i}\mid i<\omega\}$, and the relation $\kappa_{i}\subseteq\bigcup\beta[D_{n}]$ (where $\{D_{n}\}$ is the canonical numbering of finite subsets of $\omega$ [ro67]) is analytical, i.e. it is in $\bigcup_{n}\Sigma^{1}_{n}(\omega)$.

Note that although AnLCSs are not automatically locally compact, many locally compact spaces may be considered as AnLCSs. In particular, the computable $\varphi$-spaces, the finite dimensional Euclidean spaces, and the Cantor space are AnLCSs (for instance, for a computable $\varphi$-space $(X,\delta)$ we can set $\kappa_{n}:=\beta_{n}:=\mathord{\uparrow}{\delta_{n}}$ which is compact; the relation $\kappa_{i}\subseteq\bigcup\beta[D_{n}]$ in this case is c.e.).

Proposition 2**.**

If $(X,\beta,\kappa)$ is an AnLCS then $FO(\mathbf{\Sigma}^{0}_{1}(X))\leq_{m}FO(\mathbb{N}_{2})$.

Proof 3.1**.**

Define a surjection $\tau:\mathcal{N}\to\mathbf{\Sigma}^{0}_{1}(X)$ by $\tau(p)=\bigcup_{n}\beta_{p(n)}$. The surjection $\tau$ has several nice properties, in particular it is an admissible representation of the hyperspace of open sets in $X$ (see e.g. [s13] for additional details).

It suffices to show that the relation $\tau(p)\subseteq\tau(q)$ is an analytical subset of $\mathcal{N}\times\mathcal{N}$ because then the elementary diagram of the represented structure $(\mathbf{\Sigma}^{0}_{1}(X);\subseteq,\tau)$, and hence also $FO(\mathbf{\Sigma}^{0}_{1}(X))$, are $m$-reducible to $FO(\mathbb{N}_{2})$.

Obviously, $\tau(p)\subseteq\tau(q)$ is equivalent to $\forall n(\kappa_{n}\subseteq\tau(p)\rightarrow\kappa_{n}\subseteq\tau(q))$, hence it suffices to show that the relation $\kappa_{n}\subseteq\tau(p)$ is analytical. We have $\kappa_{n}\subseteq\tau(p)$ iff $\kappa_{n}\subseteq\bigcup_{i}\beta_{p(i)}$ iff $\exists m(\kappa_{n}\subseteq\beta_{p(0)}\cup\cdots\cup\beta_{p(m)})$, by compactness of $\kappa_{n}$. The set

[TABLE]

is, by the definition of AnLCS, an analytical subset of $\omega\times\omega\times\mathcal{N}$, hence the relation $\kappa_{n}\subseteq\tau(p)$ is analytical. ∎

3.2. Lattices of opens in Euclidean spaces

The next result improves the estimate from [gr51] mentioned in the Introduction, providing a different proof of this result compared with [hjrt77].

Theorem 3**.**

For any $n\geq 2$, $FO(\mathbf{\Sigma}^{0}_{1}(\mathbb{R}^{n}))\equiv_{m}FO(\mathbb{N}_{2})$.