Embeddings into the Medvedev and Muchnik lattices of Π<Superscript>0</Superscript><Subscript>1</Subscript> classes

Let _w and _M be the countable distributive lattices of Muchnik and Medvedev degrees of non-empty ₁⁰ subsets of 2, under Muchnik and Medvedev reducibility, respectively. We show that all countable distributive lattices are lattice-embeddable below any non-zero element of _w. We show that many countable distributive lattices are lattice-embeddable below any non-zero element of _M.Simpsons research was partially supported by NSF Grant DMS-0070718. We thank the anonymous referee for a careful reading of this paper and helpful comments.     

Existentially closed CSA-groups

We study existentially closed CSA-groups. We prove that existentially closed CSA-groups without involutions are simple and divisible, and that their maximal abelian subgroups are conjugate. We also prove that every countable CSA-group without involutions embeds into a finitely generated one having, up to conjugacy, the same maximal abelian subgroups, except maybe the infinite cyclic ones. We deduce from this that there exist 2^₀ countable existentially closed CSA-groups without involutions and that their first-order theories have 2^₀ types over .     

On lightly and countably compact spaces in ZF

Abstract

Given a topological space X = (X, T ), we show in the Zermelo-Fraenkel set theory ZF that:

1. Every locally finite family of open sets of X is finite iff every pairwise disjoint, locally finite family of open sets is finite.

2. Every locally finite family of subsets of X is finite iff every pairwise disjoint, locally finite family of subsets of X is finite iff every locally finite family of closed subsets of X is finite.

3. The statement “every locally finite family of closed sets of X is finite” implies the proposition “every locally finite family of open sets of X is finite”. The converse holds true in case X is T₄ and the countable axiom of choice holds true.

   We also show:

4. It is relatively consistent with ZF the existence of a non countably compact T₁ space such that every pairwise disjoint locally finite family of closed subsets is finite but some locally finite family of subsets is infinite.

5. It is relatively consistent with ZF the existence of a countably compact T₄ space including an infinite pairwise disjoint locally finite family of open (resp. closed) sets.

     

Comparison of Some Classical Topologies on Hyperspaces within the Framework of Point Spaces

S. Dolecki, G. Greco and A. Lechicki call a space X consonant if the co-compact topology and the upper Kuratowski topology on the set of closed subsets of X coincide. We call a space X hyperconsonant if Fell's topology and the (Kuratowski) convergence topology coincide. Recently, we proved that a first countable, locally paracompact, T ₃-space is hyperconsonant if and only if the space possesses at most one point without a compact neighbourhood, extending the same result of D. Fremlin obtained for metrizable spaces. In this paper, we pursue the study of hyperconsonance within the framework of point spaces (countable T ₁-spaces with exactly one accumulation point) and we compare consonance and hyperconsonance in such spaces. In particular, we answer a question of T. Nogura and D. Shakhmatov: does there exist a nonconsonant point space? We provide a Fréchet, -point space which is not consonant. Moreover, this example proves that the consonance is not preserved by continuous closed compact-covering maps of separable complete metrizable spaces onto Hausdorff spaces.     

Two theorems about projective sets

In this paper we prove two (rather unrelated) theorems about projective sets. The first one asserts that subsets of ℵ₁ which are in the codes are constructible; thus it extends the familiar theorem of Shoenfield that subsets of ω are constructible. The second is concerned with largest countable sets and establishes their existence under the hypothesis of Projective Determinacy and the assumption that there exist only countably many ordinal definable reals. Y. N. Moschovakis is a Sloan Foundation Fellow. During the preparation of this paper, both authors were partially supported by NSF Grant GP-27964.     

Topological Duality for Boolean Algebras with a Normal <Emphasis Type="Italic">n</Emphasis>-ary Monotonic Operator

In this paper we shall give a topological duality for Boolean algebras endowed with an n-ary monotonic operator (BAMOs). The dual spaces of BAMOs are structures of the form , such that is a Boolean space, and R is a relation between X and a finite sequences of non-empty closed subsets of X. By means of this duality we shall characterize the equivalence relations of the dual space of a BAMO A that correspond biunivocally to subalgebras of A. We shall prove that there exist bijective correspondences between the lattice of congruences, the lattice of closed filters, and the lattice of certain closed subsets of the dual space of a BAMO. These correspondences are used to study the simple and the subdirectly irreducible algebras.      

Sets in a euclidean space which are not a countable union of convex subsets

We prove that if a closed planar setS is not a countable union of convex subsets, then exactly one of the following holds:

More on P-Stable Convex Sets in Banach Spaces

We study the asymptotic behavior and limit distributions for sums S _n =b_n ^-1 _i=1 ⁿ _i,where _i, i 1, are i.i.d. random convex compact (cc) sets in a given separable Banach space B and summation is defined in a sense of Minkowski. The following results are obtained: (i) Series (LePage type) and Poisson integral representations of random stable cc sets in B are established; (ii) The invariance principle for processes S _n(t) =b_n ^-1 _i=1 ^[nt] _i, t[0, 1], and the existence of p-stable cc Levy motion are proved; (iii) In the case, where _i are segments, the limit of S _n is proved to be countable zonotope. Furthermore, if B = R ^d , the singularity of distributions of two countable zonotopes Y_{p 1}, ₁,Y_{p 2}, ₂, corresponding to values of exponents p ₁, p ₂ and spectral measures ₁, ₂, is proved if either p ₁ p ₂ or _{1 2}; (iv) Some new simple estimates of parameters of stable laws in R ^d , based on these results are suggested.     

Uncountable constructions for B.A. e.c. groups and banach spaces

This paper has two aims: to aid a non-logician to construct uncountable examples by reducing the problems to finitary problems, and also to present some construction solving open problems. We assume the diamond forℵ ₁ and solve problems in Boolean algebras, existentially closed groups and Banach spaces. In particular, we show that for a given countable e.c. groupM there is no uncountable group embeddable in everyG -equivalent toM; and that there is a non-separable Banach space with noℵ ₁ elements, no one being the closure of the convex hull of the others. Both had been well-known questions. We also deal generally with inevitable models (§4). The author would like to thank the NSF for partially supporting this research by grants H144-H747 and MCS-76-08479, and the United States-Israel Binational Science Foundation for partially supporting this research.     

Metric and Complete Metric σ-Frames

A -frame is a lattice in which countable joins exist and binary meets distribute over countable joins. In this paper, the category MFrm, of metric -frames, is introduced, and it is shown to be equivalent to the category MLFrm _u, of metric Lindelöf frames.Finally, it is shown that the complete metric -frames are exactly the cozero parts of complete metric Lindelöf frames.     

Some remarks on entropy numbers of sets with non-integer dimension

Abstract  We study entropy numbers of sets and generalize some results shown by B. Carl and I. Stephani in [2]. It is well known that if Ω is the closed unit ball in the Euclidean space , then for every ,

Remarks on absolutely star countable spaces

We prove the following statements: (1) every Tychonoff linked-Lindelöf (centered-Lindelöf, star countable) space can be represented as a closed subspace in a Tychonoff pseudocompact absolutely star countable space; (2) every Hausdorff (regular, Tychonoff) linked-Lindelöf space can be represented as a closed G _δ-subspace in a Hausdorff (regular, Tychonoff) absolutely star countable space; (3) there exists a pseudocompact absolutely star countable Tychonoff space having a regular closed subspace which is not star countable (hence not absolutely star countable); (4) assuming $2^{\aleph _0 } = 2^{\aleph _1 }$ , there exists an absolutely star countable normal space having a regular closed subspace which is not star countable (hence not absolutely star countable).     

Remarks on star countable discrete closed spaces

In this paper, we prove the following statements:

1. There exists a Tychonoff star countable discrete closed, pseudocompact space having a regular-closed subspace which is not star countable.
2. Every separable space can be embedded into an absolutely star countable discrete closed space as a closed subspace.
3. Assuming $2^{\aleph _0 } = 2^{\aleph _1 } $ , there exists a normal absolutely star countable discrete closed space having a regular-closed subspace which is not star countable.

     

The isomorphism problem for classes of computable fields

Theories of classification distinguish classes with some good structure theorem from those for which none is possible. Some classes (dense linear orders, for instance) are non-classifiable in general, but are classifiable when we consider only countable members. This paper explores such a notion for classes of computable structures by working out several examples. One motivation is to see whether some classes whose set of countable members is very complex become classifiable when we consider only computable members. We follow recent work by Goncharov and Knight in using the degree of the isomorphism problem for a class to distinguish classifiable classes from non-classifiable. For real closed fields we show that the isomorphism problem is ¹₁ complete (the maximum possible), and for others we show that it is of relatively low complexity. We show that the isomorphism problem for algebraically closed fields, Archimedean real closed fields, or vector spaces is ⁰₃ complete.     

The completeness of the isomorphism relation for countable Boolean algebras

We show that the isomorphism relation for countable Boolean algebras is Borel complete, i.e., the isomorphism relation for arbitrary countable structures is Borel reducible to that for countable Boolean algebras. This implies that Ketonen's classification of countable Boolean algebras is optimal in the sense that the kind of objects used for the complete invariants cannot be improved in an essential way. We also give a stronger form of the Vaught conjecture for Boolean algebras which states that, for any complete first-order theory of Boolean algebras that has more than one countable model up to isomorphism, the class of countable models for the theory is Borel complete. The results are applied to settle many other classification problems related to countable Boolean algebras and separable Boolean spaces. In particular, we will show that the following equivalence relations are Borel complete: the translation equivalence between closed subsets of the Cantor space, the isomorphism relation between ideals of the countable atomless Boolean algebra, the conjugacy equivalence of the autohomeomorphisms of the Cantor space, etc. Another corollary of our results is the Borel completeness of the commutative AF -algebras, which in turn gives rise to similar results for Bratteli diagrams and dimension groups.

     

A two-parameter spectral theorem

Summary We study the characteristic set of a couple (A, B) of selfadjoint compact operators on a real Hilbert spaceH. We prove thatC is the union of a sequence of characteristic curvesC _n in the (, ) plane. Each curve is the analytic image of an open interval and it is either closed or it goes to infinity at both ends of the interval. Moreover, it may intersect either itself or other characteristic curves in an at most countable set of points, which may accumulate only at infinity. Finally, to each characteristic curve one can associate an analytic function E_n, which gives the eigenprojection onto the eigenspace attached to each point of the characteristic curve, except at the intersection points, where the eigenspace is the direct sum of the projection relevant to each branch passing through the point. The dimension of the eigenprojection is constant along each curve and it is called the multiplicity of the characteristic curve.     

A note on a theorem of Kanovei

We give a short proof of a theorem of Kanovei on separating induction and collection schemes for _n formulas using families of subsets of countable models of arithmetic coded in elementary end extensions.Mathematics Subject Classification (2000): 03C62     

On the sampling theorem for wavelet subspaces

In [13], Walter extended the classical Shannon sampling theorem to some wavelet subspaces. For any closed subspace V₀/L² (R), we present a necessary and sufficient condition under which there is a sampling expansion for everyf V₀-Several examples are given.     

Classical solvability of the Dirichlet problem for the Monge-Ampère equation

It is proved that the problemdet(u _xx)=f(x,u,u _x)>0, is solvable in spaces , provided a natural connection between the curvature of the closed surface and the growth of the functionf(x,u,p) in¦p¦ is valid.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 131, pp. 72–79, 1983.It is my pleasure mentioning that I have discussed the above material many times with O. A. Ladyzhenskaya and that for the clear understanding of all aspects of the problem I am deeply indebted to her for her remarks and advice.