Effective inseparability in a topological setting

ANNAUOF PUREAND APPLIEDLOGIC ELSEYIER

Annals of Pure and Applied Logic 80 (1996) 257-275

Effective inseparability in a topological setting Dieter Spreen Fachbereich

Mathematik,

Theoretische Informatik, Universitiit-GH Siegen, D-57068 Siegen, Germany

Received 1 October 1995 Communicated by A. Nerode

Abstract Effective inseparability of pairs of sets is an important notion in logic and computer science. We study the effective inseparability of sets which appear as index sets of subsets of an effectively given topological To-space and discuss its consequences. It is shown that for two disjoint subsets X and Y of the space one can effectively find a witness that the index set of X cannot be separated from the index set of Y by a recursively enumerable set, if X intersects the topological closure of an effectively enumerable subset of Y. As a consequence of a more general parametric inseparability result a theorem of Rice-Shapiro type is obtained. Moreover, under some additional requirements it follows that nonopen subsets have productive index sets. This implies a generalized Rice theorem: Connected spaces have only trivial completely recursive subsets. As application some decision problems in computable analysis and domain theory are studied. It follows that the complement of the halting problem can be reduced to the problem to decide of a number whether it is a computable irrational. The same is true for the problems to decide whether two numbers are equal, whether one is not greater than the other, and whether a number is equal to a given number. In the case of an effectively given continuous complete partial order the complexity of the last problem depends on whether the given element is the smallest element, in which case the complement of the halting problem is reducible to it, whether it is a base element and maximal, then the decision problem is recursively isomorphic to the halting problem, or whether it is none of these. In this case, both the halting problem and its complement are reducible to the problem. The same is true in nontrivial cases for the problems whether an element belongs to the basis, whether two elements of the partial order are equal, or whether one approximates the other. In general, for any nonempty proper subset of the partial order either the halting problem or its complement can be reduced to the membership problem of the subset.

*E-mail: spreenQinformatik.uni-siegen.de. BRA 3230.

This research

has partially

0168-0072/96/$15.00 @ 1996 Elsevier Science B.V. All rights reserved SSDZ 0168-0072(95)00067-4

been supported

by ESPRIT

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D. Spreenl Annals of Pure and Applied Logic 80 (1996) 257-275

0. Introduction Effective inseparability

of pairs of sets is an important

science. A careful discussion

of various inseparability

by Case [6]. In this paper we consider

the effective inseparability

which appear as index sets of subsets of an effectively the other by a recursively

enumerable

notion in logic and computer

notions has recently

been given

of one of two sets

given topological

space from

set and study some of its consequences.

Topological spaces appear quite naturally in computer science. Scott [36] and Smyth [37] pointed out that data types can be thought of as countably based topological To-spaces with basic open sets the finitely describable properties of the data objects. Most structures considered in programming are equipped with a canonical topology. Prominent

examples

are Scott domains

and metric spaces. As is shown in Stoltenberg-

Hansen and Tucker [46] many algebraic structures e.g. all term algebras over a finite signature can be canonically embedded in complete ultrametric spaces as well as Scott domains. Moreover, since set theory is only applicable in computer science when developed constructively, all sets are codable, i.e. (partially) numberable. On such sets there are natural topologies generated by some or all of their completely enumerable subsets. They have first been considered by Mal’cev [25]. Therefore, we call them Mal’cev topologies. If Scott domains or metric spaces are effectively given, their canonical

topologies

43, 501). Topological

can be characterized

spaces that satisfy

certain

in terms of Mal’cev topologies natural

effectivity

(cf. e.g. [33,

requirements

have been

studied by various authors (cf. [8, 14-24, 28-32, 4245, 47, 48, 511). We consider countable (i.e. (partially) numbered) topological To-spaces (T, z) with a countable basis, on which a relation of strong inclusion is defined such that the property of being a topological

basis holds effectively

with respect to this relation

instead

of normal

set

inclusion. The further effectivity conditions that have to be satisfied are very natural. We require that elements of each basic open set can uniformly be enumerated as well as all basic open sets containing

a given point. This implies that for each point we can

enumerate a base of its neighbourhood filter. A second condition demands that we can also do the converse, i.e. from a (normed) enumeration of a filter base of basic open sets compute a point for which the corresponding filter is a neighbourhood filter. The numbering generation

of the space can be thought of as a programming system for the effective of approximations. The indices are obtained by coding the programs.

For such spaces we then show our inseparability

result: Let X and Y be two disjoint

subsets of T. Then the index set of X is effectively recursively enumerably (r.e.) inseparable from the index set of Y, i.e., there is a total recursive function which witnesses that no recursively enumerable set separates the first from the second, if X intersects the topological closure of some effectively enumerable subset of Y. As we shall see, for singletons X the condition is also sufficient. Moreover, under more complex, but less restricting assumptions it is shown that the function witnessing r.e. inseparability depends uniformly on the enumeration of the enumerable subset and the point in the intersection of a weakening of its closure with X. All this is done in

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259

Section 2. Section 1 contains basic definitions and properties. In the following sections some consequences of the insep~abili~ result are studied. In Section [3] we derive a generalization of the classical Rice-Shapiro theorem (cf. [34]). If T is effectively pointed, which means that with respect to the specialization order each basic open set of topology z has a lower bound in a certain neighbourhood of it, then z can be generated by all completely enumerable subsets of T, i.e., it is equivalent to the finest Mal’cev topology on T. This topology is also called ErSov topology. As is well-known, the characterization implies that effective operators between such spaces are effectively continuous, which generalizes the classical Myhill-Shepherdson theorem. The more general result that r is the recursively finest Mal’cev topology on T compatible with it follows from the more general parametric inseparability result. Here, topology r is recursively finer than topology q, if given a basic open set in n and a point contained in it, we can effectively find a basic open set in r which shows that the open set in yl is also open with respect to r. Moreover, saying that topology n is compatible with topology r roughly means that, if some basic open set of topology z is not included in a given basic open set of topology 11,then we must be able to find effectively a witness for this, i.e. an element of the basic open set of topology r not contained in the basic open set of topology 4. As is shown in Spreen [43], from this result one not only obtains the just mentioned characterization of effectively pointed topologies, but also a characterization of all effectively given semi-regular topologies and separable metric topologies; moreover, Spreen and Young’s general theorem on the effective continui~ of effective operators [45]. As a further consequence of our inseparability result, in Section 4 we obtain a lower bound for the complexity of some decision problems. From the above-mentioned effectivity requirements it follows that every effectively enumerable union of basic open sets is completely enumerable. If, on the other hand, a subset of T is not open and its complement contains a dense enumerable subset, then its index set is productive, which by Myhill’s theorem means that the complement of the halting set can be reduced to it. It follows that the index set of any singleton (y} which is not closed is coproductive. If T is recursively separable, i.e., if it contains a proper dense enumerable subset D, and {v} is not open, then its index set is productive, if either y 4 D, or y E D, D\ {y} is dense in T\{ y} and it is decidable for an element of D whether it is equal to y. Moreover, the index set of T \D is productive. As a further consequence we obtain a generalization of Rice’s theorem: If T is connected, then 0 and T are its only completely recursive subsets. As is shown in ErSov [l 11, Rice’s theorem holds for any precomplete numbered set. We present a necessary and sufficient condition for the numbering of T being precomplete. If z is effectively pointed, the index set of any nonopen subset is productive. Thus, for any subset of T the index set of at least the set or its complement is productive. Moreover, the index set of any singleton which is neither open nor closed is productive and coproductive. This improves Lemma 3.3 in [13]. Effectively pointed spaces are recursively separable. A singleton is neither open nor closed, if its element does either

260

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257-275

not belong to the dense base or, if it does, is neither minimal nor maximal with respect to the specialization order. If the dense base is a proper subset of T or T is not discrete, it is not a Tt-space. Hence, both equality and specialization order are not open in the product topology nor are their complements. This implies that their index sets are also productive and coproductive. In the remaining sections the results obtained so far are applied to some important, standard examples of topological spaces. As is shown in [42, 431 the above effectivity requirements can be verified for effectively given separable metric spaces (cf. 17, 26, 491). In Section 5 we consider the special case of the computable real numbers. If they are indexed in such a way that from a normed recursive sequence of rationals one can effectively pass to its limit, it follows that the problems to decide whether a number is a computable irrational or whether it equals some given number is at least as hard as to decide of a computation whether it does not terminate. The same is true for the problems whether two numbers are equal or whether one is not greater than the other. Note that since the numbering of the computable reals is partial, the complement of the index set of a set is not the index set of the complement of this set. We therefore work with a refinement of the usual many-one reduction, which maps the complement of the halting set to the index set of a problem and the halting set to the index set of its complement. Topological spaces consisting of the computable elements of an effectively given continuous complete partial order are an example of effectively pointed spaces, They are considered in Section 6. Since the space is not discrete in this case and the partial order coincides with the specialization order, it follows that both the halting problem as well as its complement can be reduced to the problems to decide the equality of two elements and whether one element approximates the other. The same is true for the problem to decide whether an element equals a given element y, if y is neither the smallest element nor both maximal and a base element. If y is the smallest element, {y} is closed. Hence, the complement of the halting problem is reducible to the decision problem for y in this case. This improves Theorem 3.4 in [35]. If y is both maximal and a base element, (y} is open, Because one can choose the numbering of the space as totai for this kind of spaces, it follows from the generalized Rice-Shapiro theorem that the decision problem for y is recursively isomorphic to the halting problem. If a programming language semantics is based on continuous complete partial orders, this decision problem is the problem to decide whether a program computes a specified function, i.e. whether it is correct. The results in this paper partly generalize results in [5, 411.

1. Basic definitions and properties

In what follows, let ( , ) : co24 w be a recursive pairing function with corresponding projections 7~1and 7~2such that ~j((u~,~2)) = ai, let P@) (R(“)) denote the set of all n-at-y partial (total) recursive functions, and let Wi be the domain of the ith partial recur-

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261

sive function vi with respect to some Giidel numbering cp. We let cpi(a)J mean that the computation of vi(a) stops, cpi(a)l EC that it stops with value in C, and qoi(a)ln that it stops within IEsteps. In the opposite cases we write cpi(a)T and qi(a)tn, respectively. A pair (Ai ,A2) of sets of natural numbers is m-reducible to another such pair such that for i = 1,2 and all a E w, if a E Aj (Bi,&), if there is a function PER then f(a) E Bj. In the case that AZ and &, respectively, are the complements Ai and 81 of A, and BI, this definition reduces to the well-known many-one reducibiIi~. We denote both kinds of reducibilities as usual by d ,,,. Moreover, 6 1 denotes one-to-one reducibility, =i one-to-one equivalence, and Ai $ A2 is the join of Al and AZ. A C w is recursively en~er~bly (r.e. ) ~n~epar~b~efrom B C w if there is no r.e. set C with A C C and 3 C C. A is qfictively r.e. inseparable from B if there is some function PER such that for all iE w, either h(i)EA\W, or h(i)EBn lVi. The function h witnesses the r.e. inseparability of A from B. Obviously, A is effectively r.e. inseparable from 2, iff A is completely productive, hence, iff A is productive, and thus, by Myhill’s theorem, iff K < IA, where K is the halting set. By using the function witnessing r.e. inseparability instead of that witnessing productivi~ in the proof of Myhill’s theorem (cf. [40]), an analogous characterization of effective r.e. inseparability can be obtained also in the general case. Lemma 1.1 Let A and B be disjoint. Then A is effectively r.e. inseparable from B $f (6OG(A,B). Note that the above proof shows only that (J?,K)<, (A, B). One-to-one reducibility follows by an ardent used in ErSov [9, Theorem 3’1. Let v and v’, respectively, map K and B to 1 and g and i? to 0. Then v and v’ are total numberings of (0, I}. Now, let g E P(l). Then g-‘(K) is r.e. and hence g-‘(K)GmK. Let this be witnessed by k f I?(‘). Then for i E dam(g) we have that Vs(j) = vk(i), which means that v is precomplete. Let the function f E R(i) witness that (&,K)<,(A,B). Then it also witnesses that there is a morphism from the numbered set ((0, l}, v) into the mxmbered set ((0,l},v’). S’mce v is precomplete, the morphism can be turned into a morphism with a one-to-one witness unction h f R(‘1 (cf. [ll, pp. 33113321). To this end a function t E R@) is constructed such that Am.f( t(n, m)) is one-to-one and vfcn,m) = v,, for all 12, m E w. Then the function h is defined by h(0) = f(0) and h(n + 1) = f(t(~~ + l,pm : f(t(n f 1,m)) > h(n))). Because f(g) CA, it also witnesses that (R,K)< ,(A$). Now, let $r’ = (Z”,z) be a countable topological To-space with a countable basis g. The closure of a subset X of T is denoted by cl(X). In the special cases we have in mind, a relation between the basic open sets can be defined which is stronger than usual set inclusion, and one has to use this relation in order to derive the results we talked about in the introduction. We call a relation 4 on a strong inclusion if for all X, YE &?‘,from X + Y it follows that X C Y. Furthermore, we say that %?is a strung basisifforallzETandX,YE~withz~XnYthereisaVE~suchthatzEV, V -:X and V -: Y.

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If one considers basic open sets as vague descriptions, then strong inclusion relations can be considered as “definite refinement” relations. Strong inclusion relations that satisfy much stronger requirements have also been used in Smyth’s work on topological foundations of programming language semantics (cf. [38, 391). Compared with these conditions, the above requirements seem to be rather weak, but as we go along, we shall meet further requirements, which in applications prevent us from choosing + to be ordinary set inclusion. For what follows we assume that -: is a strong inclusion on g and &? is a strong basis. Let x:wT (onto) and B : o - @ (onto), respectively, be (partial) indexings of T and # with domains dam(x) and dam(B). The value of x at i E dam(x) is denoted, interchangeably, by xi or x(i). The same holds for the indexing B. For a subset X of T and a relation R C_T x T, respectively, S&Y) = {if dam(x) [xi EX} and B(R) = are the sets of indices under x. If X is a singleton ((i,j> /i,jEdom(x) A (xi,xj)ER} {y}, we write O(y) instead of sZ({y}). X is completely enumerable if there is an r.e. set W,, such that xi EX iff i E W,,, for all i E dam(x). Set A4, = X, for any such n and X, and let M,, be undefined, otherwise. Then M is an indexing of the class CE of all completely enumerable subsets of T. If W, is creative, X is called completely creative, and if Wn is recursive, it is said to be completely recursive. X is enumerable if there is an r.e. set A C dam(x) such that X = {xi]iEA}. The function (P,,, enumerates X in a steps if range(q,,, ]a) & dam(x) and X = {xi 1i E range(cp, la)}. Here qrn ]a denotes the restriction of rp, to arguments not greater than a. We say that x is computable if there is some r.e. set L such that for all if domfx) and n E dam(B), (i,n) f L ifI xi E B,. Furthermore, the space S is called efictive if B is a total indexing and the property of being a strong basis holds effectively, which means that there exists a function p EP(~) such that for iEdom(x) and n, mEw with xi E & n & p(i, m, n)l, xi E Bp(i,m,n)+Bp(i,m,n)+ B,, and Bp(i,m,n)+ &. As it is shown in [42], 9 is effective if x is computable, B total, and { (m,n) I& 4 B,) r.e. Note that very often the totality of B can easily be achieved if the space is ~ec~rs~ue~y separable, which means that T has a dense enumerable subset, called its dense base. As it is well-known, each point of a To-space is uniquely determined by its neighbourhood filter and/or a base of it. If x is computable, a base of basic open sets can effectively be enumerated for each such filter. An enumeration (B_(a))aeo with f : o --+ u such that range(f) 2 dam(B) is said to be normed, if it is decreasing with respect to 4. If f is recursive, it is also called recursive and any Gijdel number off is said to be an index of it. In case (Bf(*)) enumerates a base of the neighbourhood filter of some point, we say it converges to that point. The following result is proved in [42]: Lemma 1.2 Let .F be effective and x be ~omputabIe, Then there are functions qEP(‘) and pi PC21such that for all i E dam(x) and all n E w with xi EB,,, q(i) and p(i,n) are indices of normed recursive enumerations of basic open sets which converge to xi. Moreover, BVp(,n)(~)--xB,.

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D. Spreen I Annals of Pure and Applied Logic 80 (1996) 257-275

x is said to allow efictive limit passing if there is a function

The numbering

such that, if m is an index of a normed recursive converges effective

to some point limit passing

and is computable,

always assume that the space y By definition

over an effectively This means

one is only interested

enumerable

of basic open sets which

and xptcm) = y. If x allows

it is called acceptable. In the sequel we

is effective and the numbering

each open set is the union

of effective topology

enumeration

y E T, then pt(m)L E dam(x)

pt EP(‘)

x is acceptable.

of certain basic open sets. In the context in such open sets where the union is taken

class of basic open sets. They are called Lacombe sets.

U E r is a Lacombe

set if there is some r.e. set AC dam(B)

such that

U = U{B, IaEA}.

2. On inseparability In this section we present sufficient conditions

for the effective r.e. inseparability

of

the index set of one of two disjoint subsets X and Y of T from the other. Obviously, X is inseparable from Y by an open set if X intersects the closure of Y, i.e., in this case for no open set U one can have X & U and Y & 0. Thus, X is also inseparable from Y by a Lacombe set. If x is computable, every basic open set and hence each Lacombe set is completely enumerable. It follows that if X intersects the closure of Y, then Q(X) is inseparable from Q(Y) by index sets of Lacombe sets. As is pointed out in [27, p. 451, this does not imply that Q(X) is r.e. inseparable from s2( Y). In order to obtain a sufficient condition for r.e. inseparability, the assumption has thus to be more constructive. The modifications we present already guarantee effective r.e. inseparability. The first result is a parametric inseparability theorem. Let qER(‘) be as in Lemma 1.2 and for n E w define cl,(Y) to be the set of all xj for which B(q,(j)(a)) all cl,(Y).

intersects

Y, for all abn.

Then cl(Y) is the intersection

over

Theorem 2.1 There is a function t E Rc2) with the following property for j, m E CO: For all disjoint subsets X and Y of T with j E Q(X) and such that, if qt(j,,,)(i)E Wi, then m is an index of an enumeration in g(i, j, m) (= uc : qi(qt(j,,)(i))lc) steps of a subset Y’ of Y with xj E ~l~(~,~,,,)(Y’), qt(j,m) witnesses the efSective r.e. inseparability of Q(X) from Q(Y). Proof. The function

t is constructed

such that the following

properties

hold:

1. If g(i, j, mYT, then cpt(j,,)(i)lEdom(x) andX('Pt(j,m)(i)) = Xj. 2. If g(i, j, m)l and m is an index of an enumeration in g(i, j, m) steps of a subset Y’ of T which intersects B(cpq(j)MC_iTm)))n

B(q,(j)(g(i, j, m))), then qt(j,,)(i)J ~dom(x)

and x(qt(j,,)(i))

E

Y'.

From this it follows that for all X and Y as in the theorem, qt(j,,) witnesses the effective r.e. inseparability of O(X) from Q(Y). To obtain a function t with these properties we construct a self-referential normed

D. SpreenlAnnais of Pure and Applied Logic 80 (1996) 257-275

264

enumeration

of basic open sets which follows the enumeration

to Xi, as long as an index of the point it converges

(~(~~~j~(~)))

converging

to, if it does that at all, has not

been found in I+$. If such an index has been found, say after a steps, then it follows a normed

enumeration

converging

to some point in the intersection

of B(cpgcjj(a))

and Y’. LetLCoandptcP(‘) Moreover,

witness that x is acceptable,

and be PEP(~) as in Lemma 1.2.

define s(j, n, m) to be the first element enumerated

in {b [ (b, q+&n))

EL}n

range( (PmIn). Then s EP(~), and for j E dam(x) and m, n E w such that m is an index of an enumeration in n steps of a subset Y’ of T which intersects B(~~~j)(~)~, we have that s(j, n, m)l E dam(r) and x~(~,~,~Q•~~~~~j)(~))n and let k E ipc4) be such that

cPk(j,m,i,a)(n) =

Y’. NOW, set i(i,a)

cPqdn) ‘pp(s(j,g(Ca),rn),rp,,,)(~(i,,)))(n - $(i, a>>

= PC : &pt(a))J.,

if n < &i, a),

otherwise.

By the recursion theorem there is then a function ,~ER(~) with qf(j,m,i) = (Pk(j,,,i,f(j,,,i)), Let j E dam(x). Then f(j, m, i) is an index of a normed recursive enumeration of basic open sets. If &i,f(j, m, i))?, it converges to Xj. On the other hand, if @(i,f(j, m, i))J and m is an index of an enumeration in i(i, f(j, m, i)) steps of a subset Y’ of T which intersects ~(~~~j~(~(~, f(j, m, i)))), it converges to Xsij,~(i,~(j,Rl,it),m), an element of this intersection. Thus, in both cases, pt(f(j, m, i))J.. Let t E Rc2) be such that m, i)). Then gfij, m) = i(i,f(j, m, i)), and (1) and (2) hold. q 41)ttj,,)(i) = P(f(j, The above result has a complex

assumption.

Next we derive a consequence

which

uses a simpler but very natural condition. Theorem 2.2 Let X and Y be disjoint subsets of T. If Y has an enumerable subset Y’ the closure of which intersects X, then Q(X) is effectively r.e. inseparable from

Q(Y). Proof. Let L 2 LI)witness that x is compu~ble range(s) ). Moreover,

and s E

Rfif such that Y’ = {x, / a E

let z&C n cl(Y’) and j be an index of z. For n ECOdefine f(n)

to be the first element enumerated in {bl (6, pq(j)(n)} EL} n range(s). Then f E P(l) with range(f) 5 dam(n). Since z E cl(Y’), each basic open set containing z intersects Y’. Thus, f is even total. For n ECOlet Y,’ be the subset of Y’ enumerated by f in n steps. Then z E cl,(Y,‘). By the preceding theorem it now follows that pt(j,,) where a III is a Godel number of f witnesses the r.e. inseparability of Q(X) from Q(Y). As is well-known, on To-spaces there is a canonical partial order: the specialization order, which we denote by d 7. For u, z E T, it is defined by u d r z iff u E cl({z}). If X and Y, respectively, contain elements u and z with ~1<%z, the assumptions of the theorem can easily be verified. Set Y’ = {z}. Then Y’ is an enumerable subset of Y and by the definition of the specialization order u E cl( Y’). This shows:

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265

Corollary 2.3 Let X and Y be disjoint subsets of T. If there are u E X and z E Y with u < ‘Tz, then Q(X) is e~ctively r.e. inseparub~e horn sZ(Y). An obvious question is whether one of the conditions in the above theorems characterizes effective r.e. inseparability. This is not known in general, but is true if X is a singleton. Theorem 2.4 Let Y be a subset of T and z E P,Y. Gael Y has an enumerable subset Y’ the closure of which contains z, t! B(z) is effectively r.e, inseparable from G?(Y). Proof. Because of Theorem 2.2 it remains to show the “only if’ part. Let to this end L C o witness that x is computable, be j an index of Z, and k = qq(j). Then k E R(l). Let g E R(lf with ~~(~) = {i I(~,~(~)} EL}. By the ass~ption there exists a action h E Rf’) such that for n E w either &g(n)) E ~(z)\~~~~~ or h(g(n)) E sZ(Y) n I&,). In the first case we have x~(~(~J)= z, but XQ(~)) $ Bko,), which contradicts the definition of k and Lemma 1.2. Hence the second case holds. Set Y’ = {xa 1u E range(h o g)}. Then Y’ is an enumerable subset of Y. Furthermore, for each n E w we have that Q(~(,,))EBQ). Since the set of all &fn) is a base of the neighbourhood filter of z, this means that ZECl(Y’). cl In the following sections we study some consequences of the above inseparability results.

3. Generalized RicMhapiro

theorems

As is well-known, the classical Rice-Shapiro theorem has been lifted to effectively given Scott domains, saying that on the computable domain elements the Scott topology can be generated by the completely enumerable subsets of the domain. In [43] a theorem of this type is proved for effective topological spaces. From this result one obtains the just mentioned characterization of the Scott topology as a special case. Moreover, it shows that also other spaces can be characterized with the help of completely enumerable subsets. If the topology is semi-regular, it can be generated by the regular elements in the lattice of all completely en~erable subsets of the space, Any effectively given separable metric space is an example of such a space. A further consequence of this result is the theorem on the effective continuity of effective operators in [45]. In this section we show that this general result can be derived from the parametric inseparability theorem. Moreover, we shall see that for domain-like spaces the generalized ice-Shapiro theorem follows already from Theorem 2.2. Any topology t? on T with a basis of completely enumerable subsets of 7’ is said to be a Mal’cev topology and the basis a Mal’cev basis. The topology CRgenerated by CE is called Eriov topology. With the help of the numbering A4 of CE every Mal’cev basis can be indexed in such a way that n is computable with respect to

D. Spreen I Annals of Pure and Applied Logic 80 (1996) 257-275

266

this numbering. reducible

Moreover,

each numbering

of a Mal’cev

to M (cf. [42]). Thus, each Mal’cev topology

basis with this property

is recursively

is

coarser than the

ErSov topology. A topology

g on T with countable

basis Gf:and an indexing

C :o -

%T(onto) of w,

is said to be recursively coarser than r and r recursively finer than 4, if there is some function

such that g(i, m)l

g EP

ldom(B)

and xi E Bg(i,,) & C,,,, for all iEdom(x)

and m E dam(C) with x, E C,. If q is both recursively finer than r and recursively coarser than r, then v] and r are called recursively equivalent. In order to characterize

the given

we have to study which Mal’cev

topology

topologies

r by a class of Mal’cev 4 are recursively

coarser

topologies, than r. The

verification of this proceeds indirectly, i.e., for some xi, B, and C,,, with xj E C,,, we assume that xj E B,, but B, $ C,,, and derive a contradiction. To this end we need to compute a witness for B,, not being included in C,. The applications in [43, 441 show that one has to work with a witness that a certain neighbourhood of B,, is not included in a completely enumerable subset of C,. Note that by work of Beeson [l-4] a direct proof seems impossible. For X G T, let hi(X)=

n{OE~~130’XCO’

+ 0).

Then topology q is compatible with r, if there are functions SE Pc2) and rEPc3) such that for all i E dam(x), 12E dam(B) and m E dam(C) the following hold: 1. If xi EC,, then s(i, m)J. E dam(M) and xi E M&m) C C,. 2. Moreover if B,, $ C,, then also r(i, n, m)J Edom(x)

and ~,.(i,~,~) E hl(Bn)\Ms(i,m).

Theorem 3.1 Let the topology z be compatible with itself: Then z is the recursively finest Mal’cev topology on T that is compatible with z. Proof. Since x is computable, r is a Mal’cev topology. compatible

each basic open set in r is completely

Let q be a Mal’cev topology

with r. We have to show that 4 is recursively Lemma

1.2 and Theorem

2.1. Moreover,

i.e.,

coarser than z. Let to this end

s E Pc2), r E Pc3), q ER(‘), g E Pc3) and t ERc2), respectively, compatibility,

enumerable,

with Mal’cev basis %?which is be as in the definition

of

let k E Rc2) with qk(j,,,,)(a) =

r(j, p4(j)(a + l),m) and set @(j, m) = g(s(j, m),j,k(j, m)). We want to show for jEdom(x) and m E dam(C) with xj E C, that &j, m)l and B(pqcj,(i(i, m)+ 1)) G C,,,. Assume to the contrary that if @(j, m)l then B(q,(j,(i(j, m) + 1)) $! C,. Since the sequence (B(q,(j)(n))) is decreasing with respect to strong inclusion, it follows from the compatibility

of q with r that for all a
m)

Thus, k(j, m) is an index of an enumeration in &, m) steps of a subset Y’ of Ms(j,m) with xj E c$(~,~)(Y’). With Theorem 2.1 we obtain that B(MJ(j,,)) is effectively r.e.

D. Spreeni Annals of Pure and Applied Logic 80 (1996)

inseparable

from sZ(Ms(j,,,),

which contradicts

the complete

It follows that 1j, m.qe(j)(g( j, m) + 1) witnesses

In the case of domain-like included

in a completely

enumerability

that n is recursively

of MKi,m). coarser than r. 0

spaces a witness which shows that a basic open set is not

enumerable

in this case the generalized

267

257-275

set can easily be given. As we shall show now,

Rice-Shapiro

theorem

can be obtained

as a consequence

of Theorem 2.2. An essential property of Scott domains just as of Er$.ov’s f-spaces [lo, 121 is that their canonical topology has a basis with each basic open set being an upper set generated by a single element.

We say that a subset X of T is pointed if there is some

y E hi(X) such that y <,z, for all z E X. F is said to be efictiuely pointed if there is some function pd E I’(‘) such that for all nEdom(B) with B, # 0, pd(n)J, ldom(x), xp+) E hl(B,) and xpd(,,) <,z, for all ZEB,. In [43] it is shown that completely enumerable

subsets of T are upwards closed un-

der the specialization order. Thus, if X is completely enumerable and B, $ X, this is witnessed by ~pd(~). Now, assume that the Ersov topology 8 is not recursively coarser than r. Then there are j E dam(x) and m E dam(M) with Xj E Mm such that for all n E w B(q,(j)(n)) $ Mm. It follows for Y’ = {x, la~range(pd o q,(j))} that Y’ is disjoint from IV,. Because the sequence (B(q,(j)(n))) is decreasing with respect to strong inclusion, we have for all a, n co with a > n that x(pd(qq(j)(a)))E B(q,(j)(n)). This shows that Xj E Mm n cl(Y’). By Theorem 2.2 it follows that sZ(M,) is effectively r.e. inseparable from C&V,), which contradicts the complete enumerability of Mm. Hence, there is some nEo with B(rp,(i)(n)) CM,. It follows that x(pd(q,&n + ~)))EM,. Since x is computable with respect to M, a number n with this property can effectively be found, uniformly in j and m. This shows that d is recursively have already seen if x is computable, z is a Mal’cev topology

coarser than z. As we and hence recursively

coarser than 8. This shows:

Theorem 3.2 Let F

be effectively pointed. Then z is recursively equivalent to the

Ersov topology on T. 4. Some decision problems We have already seen that the index set of a Lacombe set is r.e. if x is computable. As a consequence of Theorem 2.2 we now obtain a classification of the index sets of nonopen sets.

Theorem 4.1 Let X be a nonopen subset of T such that its complement contains an enumerable dense subset. Then (Z?,K) d 1(G?(X), l&J?)), in particular Q(X) is productive.

Proof. Let Z be a dense enumerable subset of 2 and assume that X is not open. Then there is some point y E X such that each open set containing y intersects the

268

D. Spreen I Annals of Pure and Applied Logic 80 (1996) 257-275

complement

of X. Since Z is dense in X, each such intersection

of Z. Thus, X intersects therefore

the closure

of Z. With Theorem

obtain that (R, K) d 1(Q(X), L’(x)).

Obviously,

the above requirement

contains

some element

2.2 and Lemma

1.1 we

cl

on X is satisfied if this set is enumerable.

Corollary 4.2 For any nonopen subset X of T with enumerable complement, (Z?,K) < l(Q(X), Q(X)). In particular, C?(X) is productive. Zf”X is completely enumerable, then (Q(X), O(X)) ~1 (K, K). Proof. By the above theorem, Q(X) is productive and (K,K) < l(Q(X), Q(X)). If X is completely enumerable, there is some r.e. set A with O(X) = A n dam(x). Since A is r.e., we have that Ad IK, which (G?(X), Q(X)) ~1 (K, K).

implies

that (G(X), G(X))<

l(K,K).

Thus,

q

In the case that F is also connected, it follows for each Lacombe set X that (a(X),Q(X)) ~1 (K,Z?), which in particular means that X is completely creative. Moreover, if F is recursively separable and D is its proper enumerable dense base, then (x;, K) < 1(a( T \D), Q(D)). As a further consequence we obtain a lower bound for the complexity of the problem to decide for a given element y E T whether xj = y. If one thinks of the index j as coding a program for the effective generation of approximations, then this is the problem to decide of a program whether it computes the approximation of a specified datum, i.e., whether it is correct. Corollary 4.3 (1) Let YET. Then (K,K)QI(B(~),S~(T\{~})), if(y) is not closed. (2) Zf Y is recursively separable and for some g E R(l), D = {x~ (a orange} is its dense base, then (K, K) < 1(0(y), Q(T\{ y})), zf {y} is not open and either y E T\D or LED, D\(y) is dense in T\(y) and {aIx,(,) = y} is recursive. Proof. If {y} is not closed, T\(y) Corollary

4.2 is applicable.

is not open. Moreover, {y} is enumerable. Thus, In the case that D is dense in T and enumerable, ye T\D

and {y} is not open, the conditions

in Theorem

4.1 are satisfied, for then D is dense

in T\{ y} as well. If y E D and {a Ixsca) = y} is recursive, Theorem 4.1 applies also in the remaining case. q If F

is a Ti-space,

the requirement

D\{ y} is enumerable.

that D\{ y} is dense in T\(y)

Thus,

is redundant.

Lemma 4.4 Let r be a separable Tl -space with dense base D. Moreover, let y ED such that {y} is not open. Then D\{ y} is dense in T\(y). Proof. Assume that D\(y) is not dense in T\(y). Then there is an open set U which contains y, but no element from D. Since {y} is not open, U cannot contain only y. Let z E U\{ y}. Since F satisfies the Tl -axiom, there is some open subset V of U with z E V but y 4 V. Because of the density of D, V must contain an element of D. This

269

D. Spreenl Annals of Pure and Applied Logic 80 (I 996) 257-275

contradicts

the fact that U contains

dense in T\(y).

from D other than y. Thus, D\(y)

is

0

Let io E B(y) and define f(i) (Q(=), a(#)).

no element

If y

= (io, i) . Then f wi t nesses that (8(y),sZ(Z’\{y}))
is recursively

separable

with dense base D and {y} is open, then

y E D. Hence, if D # T, there is some y E T \ D such that {y} is not open. With r--.1,,.-. n 1/?\ __.^ ,.L&.:-. ~“l”llaly ‘t.,(L) WE;&L.., L11llS““Lillll.

Corollary 4.5 Let F be recursively separable such that the dense base is properly included in T. Then (I?, K) Q 1(Q( =), sZ(#)). An additional outcome of Theorem 4.1 is a generalization of Rice’s theorem: connected effective spaces there are no nontrivial decidable properties.

On

Theorem 4.6 Let F be connected and X be a subset of T. Then X is completely recursive, ifsX is empty or the whole space. Proof. The “if” part is obvious. For the “only if’ part assume that X is neither empty nor the whole space, and completely recursive. Then there is some recursive set A such that for i E dam(x), xi EX iff i EA. Thus, both X and its complement are completely enumerable. If X were not open, Corollary 4.2 would imply that Kd IA. This means A would be productive,

a contradiction. Hence X is open. In the same way it follows that X is open, which is impossible by the connectedness of T. 0 As is shown in [l 11, Rice’s theorem is true for arbitrary precomplete numbered sets. (Ergov considers only total numberings, but the proof remains valid also for partial numberings.)

Here, for a numbered

set (Y, v) the numbering

v: co -

Y(onto)

is

called precomplete if for any function g E P(l) there is a function f E R(l) such that vf(i) = vs(i), for i E dam(g) with g(i) E dam(v). It is not known, whether acceptable numberings of effective topological spaces are also precomplete. The next lemma presents

a necessary

and sufficient

condition.

Lemma 4.7 The numbering x is precomplete, ifs there is a total function pt E R(l) which witnesses that x allows effective limit passing. Proof. For the “if’ part let g E P(l). Moreover, let q E k E R(l) with qk(i) = qa(q(ijj. .~“, // Then, for i E dam(g) such index of a normed recursive enumeration of basic open Thus, pt(k(i)) E dam(x) and xpt(k(i)) = xq(i). Set f =

P(l) be as in Lemma 1.2 and that g(i) E dam(x), k(i) is an sets which converges to xs(i). pt o k. Since pt E R(l), also

f ER(‘). For the “only if” part assume that x is precomplete, and let pt E P(l) witness that x allows effective limit passing. Then there is a function PER with XS;TC~, = xPtcm), for M E dom(pt) with pt(m) E dam(x). Then also 3 witnesses that x allows effective limit passing. 0

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D. Spreen I Annals of Pure and Applied Logic 80 (1996) 257-275

If Y is effectively

Theorem 4.8 Let 5

pointed,

the above results can be improved.

be effectively pointed

Then for any nonopen subset X of T,

(ZC~)
Proof. If X is not upwards closed with respect to the specialization reducibility assume

statement

follows

that X is upwards

from Corollary

closed

2.3 and Lemma

with respect

order, the above

1.1. Let us therefore

to the specialization

order,

and let

q EP(‘) and pd ER(‘), respectively, be as in Lemma 1.2 and the definition of being effectively pointed. If X is not open, there is some Xj E X such that for no n E w, B(q,(j)(n)) is contained in X. Hence for all n, x(pd(cp,(j)(n))) ~2. Since the set of all B(q,(j,(n)) is a base of the neighbourhood filter of Xi, it follows that xj is an accumulation point of the set of all x(pd(cp,(j,(n))). The above reducibility statement is now a consequence of Theorem 2.2 and Lemma 1.l. ci Obviously, an effectively pointed space is recursively separable. The set D of all points ~pd(~) is a dense base. If D is not already the whole space, it is not open. With the remark in the beginning of this section we thus have that (Z?@X,K@Z?) < 1(!2( T\D), a(D)).

If Y is also connected,

it follows that for any nonempty

proper subset X of T

at least one of the index sets Q(X) and s2(J?) is productive. For the decision problem whether xj is equal to a given element y E T we obtain the following improvement.

Corollary 4.9 Let F be effectively pointed and y E T. Then the following hold 1. Zf ~ET\D, then (K$K,K~~)~~(~(~),S~(T\{Y})). 2. Zf y E D and not maximal with respect to the specialization order, then (~,K)~~(~(Y),S~(T\{Y})).

If in addition y is not minimal, then also (K,Z?) < ,(g( y),

Q(T\{yI)). 3. Zf y E D with y = x&n), B, is not empty, and y is maximal and not minimal with respect to the specialization order, then (g(y), Q(T\ { y})) ~1 (K,Z?).

Proof. Let y E T be such that y is not maximal with respect to the specialization

order, if y ED, and assume that {y} is open. Then there is some basic open set B, which

equals { y}. It follows that y is maximal. By Lemma 1.2 there is a further basic open set y and is strongly included in B,. Hence, y = X@(m). This means that

B, which contains

y ED, a contradiction. With Theorem 4.8 we obtain that (Z?, K) < 1(g(y), G?(T\ { y})). Since y is contained in some basic open set B,,, we have that y = X@(n) and hence that LED, if y is minimal with respect to the specialization order. If y is not minimal, it follows from Corollary 2.3 and Lemma 1.1 that (Z?,K) < l(Q(T\{y}), g(y)). If y = X@(n), B, is not empty and y is maximal with respect to the specialization order, then {y} is a basic open set. Moreover, {y} is not open. Since y is not minimal with respect to the specialization order, there is some z E T with z & y. Then z E {y}. If {y} were open, we would have that y E {y}, a contradiction. With Corollary 4.2 it thus follows that (g(y),Q(T\{y})) -_I (K,Z?). 0

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Logic 80 (1996) 257-275

271

Assume that F is a Tt-space, then the specialization order coincides with equality. Thus, for each basic open set B, we have that B, = (~pd(~)}. It follows that T contains only the points of the dense base D and the topology is discrete. Thus, if either D is a proper subset of T or the topology is not discrete, r does not satisfy the Tt-axiom. This implies that there are two different points y and z in T with yGTz. It is now easy to see that {(u,~)EZ’ x Tlu= u} and {(u,v)~T x TIuGzv} as well as their complements are not open in the product topology. With Theorem 4.8 we therefore obtain: Corollary 4.10 Let 5

be efS^ectiuelypointed such that either I) is a proper subset

of T or the topology is not discrete.

Then (I? @ K,K @ k?),<1(52(=),S1(#))

and

(K~~,:,K&,R)~,(a(~*),SZ(p,)). In the next two sections we appfy the results of this and the preceding section to two important special cases: the computable real numbers and constructive domains.

5. The computable real numbers Let Iw denote the real nmber set, and v be a canonical indexing of the set Q of rational numbers. Then {(a, b) 1v&&,) is recursive. A real number z is said to be computable if there is a function f E R(‘) such that for all m, n E w with m
IVi -

Vjl

+

2-” < 2-“.

Using the triangular inequation it is readily verified that -+ is a strong incl~io~ and {B, I a E w) is a strong basis. In [43] it is shown that 88, is effective if x allows effective limit passing. Let (yajaEo be a sequence of rational nabers. Then (ya) is said to be normed if Iy,,, - y,,l < 2-“, for all m, n E w with m
limit passing, isf there is a function Ii E P(l) such that if m is an index of a normed recursive sequence

(y,)

of ratio~l

n~bers,

then li(m).l E dam(x) and x+)

= lim ye

Proof. The “if” part has been shown in [43]. For the “only if” part let (v(~~(a~))~~~ be a nonned recursive sequence of rational numbers, and let k f R(I) with (ok = is a normed recursive enumeration of basic open sets (9&r+ 11,a>. Then ~3~~~(~)(a)))

272

D. SpreenIAnnals

which converges

to lim v(cp,(a)).

effective limit passing,

of Pure and Applied Logic 80 (1996) 257-275

Let pt EP (l) be the function witnessing

and set li = pt ok. Then x/i(m) = lim v(cp,(a)).

It follows that the numbering

y allows effective limit passing.

there is some a E w with Iv(i) - v(qj(a))j remainder

+2P

be strictly increasing,

0

Since yj E Bci,,) iff

< 2-m, y is also computable.

of this section, x is assumed to be an acceptable

Let feR(‘)

that x allows

numbering

For the

of [w,.

and k ER(‘) with qk(i)(m) = (i,f(m)).

Then, for

any i E o, (B(qk(i)(m))) is a normed recursive enumeration of basic open sets that converges to vi. Hence vi = +(k(i)). It follows that Iw, is recursively separable. We now apply the results of the last section to obtain lower bounds for the complexity of some decision problems. Since [w, is connected, we first obtain from Theorem 4.6 that there are no nontrivial decidable properties of computable real numbers, i.e. Rice’s theorem for [w,. Moreover,

we have:

Theorem 5.2 (1) (K,K)6,(S2([W,\Q),S2(~)). (2) (K,K)~1(B(y),SZ(T\{y})),for al (3) (K,K)&(S2(=)$(#)),

YEK

K~Klw~uwn

Proof. (1) follows from Corollary 4.2, and (2) from Corollary 4.3(2) and Lemma 4.4, since no singleton is open in the topology Corollary 4.5, the second is a consequence

on [w,. The first part of (3) follows from of Theorem 4.1, because the set {(y,z) E

[w, x [w,1y
and {(y, z) E (EJx Q 1y > z} is dense

6. Constructive domains Let % = (Q, L) be a partial order with smallest

element

1. A nonempty

subset S

of Q is directed if for all yi, y2 E S there is some u E S with yi, y2 5 U. 9 is a directed-complete partial order (cpo) if every directed subset S of Q has a least upper bound sup S in Q. Let < denote the way-below relation on Q, i.e., let yi < y2 iff for directed subsets S of Q the relation

y2 L sups

always implies the existence

of a

UES with yt E u. A subset Z of Q is a basis of % if for any YEQ the set Z, = {ZEZ 1 z < y} is directed and y = supZ,. A cpo that has a basis is called continuous. Every basis contains all elements y E Q with y < y. They are called compact. As is well-known, on each cpo there is a canonical topology: the Scott topology. A subset X of Q is open if X is upwards closed with respect to E and with each u E X there is some y EX with y < u. In case that 9 is continuous, this topology is generated by the sets 0, = { y E Q )z < y} with z E Z. Note that the specialization order on 9 coincides with the partial order E. Let 9 be continuous, then it is called efictive if there exists an indexing e : o -+ Z (onto) of Z such that {(i,j)j ei < ei} is r.e. An element y E Q is said to be computable,

D. Spreen f Annals of Pure and Applied Logic 80 (19%)

2f7-275

273

if {i / ej < y} is r.e. Let Qc denote the set of all computable elements of Q, then (Qc, L,Z, e) is called constructive domain. Let G be the induced Scott topology on Qc. Then & = (Qc,c) is a countable connected To-space with basic open sets B, = O+) fl Qc. Moreover, Z is dense in 2,. An indexing x : o + QC (onto) of the computable cpo elements is called admissible if { (i,j)lei << xi} is r.e. and there is a function d E I@) with xd(i) = supe(Wi), for all indices i E COsuch that e( Wi) is directed. As it is shown in [50] such total numberings exist. In what follows, we always assume that constructive domains are admissibly indexed. Define B, -iB,uee,

<
Then + is a strong inclusion and {B,, [no CO}is a strong basis. In [43] it is shown that 2$ is effective and effectively pointed, and any admissible numbering x is acceptable. From the results of the preceding sections we thus obtain that there are no nontrivial decidable properties on Qc and that the induced Scott topology on Qc is recursively equivalent to the ErSov topology, i.e. the Rice and the Rice-Shapiro theorem for constructive domains. Moreover, we have: Theorem 6.1 Let (Qc,L,Z, e) be a ~o~st~~tive domain that contains at least two elements. Then the following hold: 1. The index set of any subset of QC which is not open in the induced Scott topology is productive, and for each nonempty proper subset S of QC at least one of the index sets a(S) and G?(s) is productive. 2. If Z is properly contained in QC, then K @ I? < 1G?(Z). 3. KEB~<,~(=) andK$I?<~~(C). 4. If ycQC\Z, then K @~G~B(y). 5. Ify~Z\{l} and not maximal with respect to L, then K@I?<,e(y). 6. Zg.re(i). 7. rf’ y E Z is compact and maximal with respect to C, then B(y) is re~rsively isomorphic to K. Proof. The first part of (1) follows from Theorem 4.8. The second as well as (2) are a consequence of a remark preceding Corollary 4.9. (3) is a special case of Corollary 4.10, and (4)-(7) is a consequence of Corollary 4.9. If y is compact, y E 0,. Hence 0, n Qc is not empty. 0 References [ 1] M. Beeson, The nonde~vabili~ in intuitionistic formal systems of theorems on the continuity of effective operations, J. Symbolic Logic 40 (1975) 321-346. [Z] M. Beeson, The unpmvabili~ in in~itionistic formal systems of the continui~ of effective operations on the reals, 1. Symbolic Logic 41 (1976) 18-24.

274

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257-275

[3] M. Beeson, Continuity and comprehension in ~n~itionistic forma1 systems, Pacific J. Math. 68 (1977)

29-40. [4] M. Beeson and A. SEedrov, Church’s thesis, continuity, and set theory, J. Symbolic Logic 49 (1984) 630-643. [S] J. Case, M. Fulk and K. Ebcioglu, Re. inseparable general and subrecursive index sets, J. Symbolic Logic 48 (1983) 1235. [6] J. Case, Effectivizing inseparability, Z. Math. Logik Grundl. Math. 37 (1991) 97-l 11. [7] G.S. Ceitin, Algorithmic operators in constructive metric spaces, Trudy Mat. Inst. Steklov 67 (1962) 295-361 (in Russian); English transl., Amer. Math. Sot. Transl. (2) 64 (1967) I-80. [S] E.Z. Dyment, Recursive met&ability of numbered topological spaces and bases of effective linear topological spaces, Izv. Vyssh. Uchebn. Zaved, Mat. (Kazan) 8 (1984) 59-61 (in Russian); English transl., Sov. Math. (1~. VUZ) 28 (1984) 74-78. 191Ju.L. ErSov, On a hierarchy of sets I, Algebra Logika 7 (1968) 47-74 (in Russian); English trawl., Algebra and Logic 7 (1968) 25-43. [IO] Ju.L. ErSov, Computable Rmctionals of finite type, Algebra Logika 11 (1972) 367-437 (in Russian); English transl., Algebra and Logic 11 (1972) 203-242. [l l] Ju.L. ErSov, Theorie der Numerierungen I, Z. Math. Logik Grundl. Math. 19 (1973) 289-388. [12] Ju.L. Emov, Theorie der Numerierungen II, Z. Math. Logik Grundl. Math. 21 (1975) 473-584. [13] P. Giannini and G. Longo, Effectively given domains and lambda-calculus models, Inform. and Control 62 (1984) 36-63. [14] J. Hauck, Konstmktive Darstellungen in topologischen R&tmen mit rekursiver Basis, Z. Math. Logik Grundl. Math. 26 (1980) 565-576. [15] J. Hauck, Berechenbarkeit in topologischen Raumen mit rekursiver Basis, Z. Math. Logik Grundl. Math. 27 (1981) 473-480. [16] Ph. Hingston, Non-complements open sets in effective topology, J. Austral. Math. Sot. Ser. A 44 (1988) 129-137. 1171I. Kalantari, Major subsets in effective topology, in: G. Metakides, ed., Patras Logic Symp. (NorthHolland, Amsterdam, 1982) 77-94. [IS] I. Kalantari and A. Leggett, Simplicity in effective topology, J. Symbolic Logic 47 (1982) 169-183. [19] I. Kalantari and A. Leggett, Maximality in effective topology, J. Symbolic Logic 48 (1983) 100-l 12. [20] I. Kalantati and J.B. Remmel, Degrees of recursively enumerable topological spaces, J. Symbolic Logic 48 (1983) 610-622. [21] I. Kalantari and A. Retzlaff, Recursive constructions in topological spaces, J. Symbolic Logic 44 (1979) 609-625. (221 I. Kalantari and G. Weitkamp, Effective topological spaces I: a definability theory, Amt. Pure Appl. Logic 29 (1985) l-27. [23] I. Kalantari and G. Weitkamp, Effective topological spaces II: a hierarchy, Ann. Pure Appl. Logic 29 (1985) 207-224. 1241I. Kalantari and G. We&, Effective topological spaces III: forcing and definability, Ann. Pure Appl. Logic 36 (1987) 17-27. 1251A.I. Mal’cev, The Me~mathematics of Algebraic Systems, in: B.F. Wells III, ed., Collected papers: 1936-1967 (Nosh-Holland Amsterdam, 1971). 1261Y.N. Moschovakis, Recursive metric spaces, Fundam. Math. 55 (1964) 215-238. 1271J. Myhill and J.C.E. Dekker, Some theorems on classes of recursively enumerable sets, Trans. Amer. Math. Sot. 89 (1958) 29-59. [28] E.Ju. Nogina, On effectively topological spaces, Dokl. Akad. Nauk SSSR 169 (1966) 28-31 (in Russian); English transl., Soviet Math. Dokl. 7 (1966) 865-868. [29] E.Ju. Nogina, Relations between certain classes of effectively topological spaces, Mat. Zametki 5 (1969) 483-495 (in Russian); English transl., Math. Notes 5 (1969) 288-294. [30] E.Ju. Nogina, Enumerable topological spaces, Z. Math. Logik Grundl. Math. 24 (1978) 141-176 (in Russian). [3 I] E.Ju. Nogina, On completely enumerable subsets of direct products of numbered sets, in: Mathematical Linguistics and Theory of Algorithms, 130-132 {in Russian), Interuniv. thematic Coilect., Kalinin Univ., 1978. 1321E.Ju. Nogina, The relation between sepambiIi~ and traceability of sets, in Majestical Logic and Mathematical Linguistics, 135-144 (in Russian), Kalinin Univ., 1981.

D. Spreen I Annals of Pure and Applied Logic 80 (1996) 257-275

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[33] G. Plotkin, Domains, Course Notes, Dept. of Computer Science, Univ. of Edinburgh, 1983. [34] H. Rogers, Jr., Theory of Recursive Functions and Effective Computability (McGraw-Hill, New York. 1967). [35] D. Scott, Data types as lattices, SIAM J. Comput. 5 (1976) 522-587. [36] D. Scott, Lectures on a mathematical theory of computation, Techn. Monograph PRG-19, Oxford Univ. Comp. Lab., 1981. [37] M.B. Smyth, Power domains and predicate transformers, in: J. Diaz, ed., Automata, Languages and Programming, Lecture Notes in Computer Science, Vol. 154 (Springer, Berlin, 1983) 662-675. [38] M.B. Smyth, Completeness of quasi-uniform spaces in terms of filters, manuscript, 1987. [39] M.B. Smyth, Quasi-uniformities: reconciling domains with metric spaces, in: M. Main et al., eds., Mathematical Foundations of Programming Language Semantics, 3rd Workshop, Lecture Notes in Computer Science, Vol. 298 (Springer, Berlin, 1988) 236-253. [40] R.I. Soare, Recursively Enumerable Sets and Degrees (Springer, Berlin, 1987). [41] D. Spreen, On r.e. inseparability of cpo index sets, in: E. Borger et al., eds., Logic and Machines: Decision Problems and Complexity, Lecture Notes in Computer Science, Vol. 171 (Springer, Berlin, 1984) 103-l 17. [42] D. Spreen, A characterization of effective topological spaces, in: K. Ambos-Spies et al., eds., Recursion Theory Week, Proc. Oberwolfach, 1989, Lecture Notes in Math., Vol. 1432 (Springer, Berlin, 1990) 363-388. [43] D. Spreen, A characterization of effective topological spaces II, in: G.M. Reed et al., eds., Topology and Category Theory in Computer Science (Oxford Univ. Press, Oxford, 1991) 231-255. [44] D. Spreen, Effective operators and continuity revisited, in: A. Nerode et al., eds., Logical Foundations of Computer Science - Tver ‘92, Lecture Notes in Computer Science, Vol. 620 (Springer, Berlin, 1992) 459-469. [45] D. Spreen and P. Young, Effective operators in a topological setting, in: M.M. Richter et al., eds., Computation and Proof Theory, Proc. Logic Colloquium ‘83, Lecture Notes in Math., Vol. 1104 (Springer, Berlin, 1984) 437 -45 1. [46] V. Stoltenberg-Hansen and J.V. Tucker, Algebraic and fixed point equations over inverse limits of algebras, Theoret. Comput. Sci. 87 (1991) l-24. [47] Ju.R. Vainberg and E.Ju. Nogina, Categories of effectively topological spaces, in: Studies in Formalized Languages and Nonclassical Logics (Izdat. ‘Nauka’, Moscow, 1974) 253-273 (in Russian). [48] Ju.R. Vainberg and E.Ju. Nogina, Two types of continuity of computable mappings of numerated topological spaces, in: Studies in the Theory of Algorithms and Mathematical Logic, Vol. 2 (Vycisl. Centr Akad. Nauk SSSR, Moscow, 1976) 84-99, 159 (in Russian). [49] K. Weibrauch, Computability on computable metric spaces, manuscript, 1991. [50] K. Weihrauch and Th. Deil, Berechenbarkeit auf cpo-s, SchriRen Angew. Math. Informatik Nr. 63, Rheinisch-Westfilische Technische Hochschule Aachen, 1980. [51] Li Xiang, Everywhere nomecursive r.e. sets in recursively presented topological spaces, J. Au&al. Math. Sot. Ser. A 44 (1988) 105-128.