Countable dense homogeneous spaces under Martin’s axiom

We show that Martin’s axiom for countable partial orders implies the existence of a countable dense homogeneous Bernstein subset of the reals. Using Martin’s axiom we derive a characterization of the countable dense homogeneous spaces among the separable metric spaces of cardinality less thanc. Also, we show that Martin’s axiom implies the existence of a subset of the Cantor set which isλ-dense homogeneous for everyλ <c.     

On universal locally finite groups

We deal with the question of existence of a universal object in the category of universal locally finite groups; the answer is negative for many uncountable cardinalities; for example, for 2^ℵ ₀, and assuming G.C.H. for every cardinal whose confinality is >ℵ₀. However, if λ>κ when κ is strongly compact and of λ=ℵ₀, then there exists a universal locally finite group of cardinality λ. The idea is to use the failure of the amalgamation property in a strong sense. We shall also prove the failure of the amalgamation property for universal locally finite groups by transferring the kind of failure of the amalgamation property from LF into ULF. We would like to thank Simon Thomas for reading carefully a preliminary version of this paper, proving Lemma 20 and making valuable remarks. Also we thank the United States—Israel Binational Science Foundation for partially supporting this work.     

Covering properties which,under weak diamond principles,constrain the extents of separable spaces

We show that separable, locally compact spaces with property (a) necessarily have countable extent — i.e., have no uncountable closed, discrete subspaces — if the effective weak diamond principle ⋄(ω,ω,<) holds. If the stronger, non-effective, diamond principle Φ(ω,ω,<) holds then separable, countably paracompact spaces also have countable extent. We also give a short proof that the latter principle implies there are no small dominating families in ^ω ₁ ω.     

Violating the singular cardinals hypothesis without large cardinals

We extend a transitive model V of ZFC+GCH cardinal preservingly to a model N of ZF + “GCH holds below ℵ _ω ” + “there is a surjection from the power set of ℵ _ω onto λ”, where λ is an arbitrarily high fixed cardinal in V. The construction can be described as follows: add ℵ _n +1 many Cohen subsets of ℵ _n+1 for every n < ω, and adjoin λ many subsets of ℵ _ω which are unions of ω-sequences of those Cohen subsets; then let N be a choiceless submodel generated by equivalence classes of the λ subsets of ℵ _ω modulo an appropriate equivalence relation.     

Large continuum,oracles

Our main theorem is about iterated forcing for making the continuum larger than ℵ₂. We present a generalization of [2] which deal with oracles for random, (also for other cases and generalities), by replacing ℵ₁,ℵ₂ by λ, λ ⁺ (starting with λ = λ ^<λ > ℵ₁). Well, we demand absolute c.c.c. So we get, e.g. the continuum is λ ⁺ but we can get cov(meagre) = λ and we give some applications. As in non-Cohen oracles [2], it is a “partial” countable support iteration but it is c.c.c.     

The problem of envelopes for Banach spaces

LetX be a Banach space. A Banach spaceY is an envelope ofX if (1)Y is finitely representable inX; (2) any Banach spaceZ finitely representable inX and of density character not exceeding that ofY is isometric to a subspace ofY. Lindenstrauss and Pelczynski have asked whether any separable Banach space has a separable envelope. We give a negative answer to this question by showing the existence of a Banach space isomorphic tol ₂, which has no separable envelope. A weaker positive result holds: any separable Banach space has an envelope of density character ≦ℵ₁ (assuming the continuum hypothesis).     

Counting models in universal Horn classes

Definen _K (λ) to be either ω, or the number of non-isomorphic models inK having cardinality α, whichever cardinal is larger. This paper contains a proof that for a congruence modular variety ⋎ of algebras of countable similarity type, there are only six possible functionsn _⋎. It is also proved that ifn _K (λ)≠2^λ for some λ, andK is a universal Horn class of models for a countable language, thenK must satisfy two conditions, one of which is quite restrictive and requires that the members ofK are all in a certain sense Abelian. Presented by B. Jonsson.     

Some remarks on universal graphs

LetΓ be a class of countable graphs, and let ℱ(Γ) denote the class of all countable graphs that do not contain any subgraph isomorphic to a member ofΓ. Furthermore, letTΓ andHΓ denote the class of all subdivisions of graphs inΓ and the class of all graphs contracting to a member ofΓ, respectively. As the main result of this paper it is decided which of the classes ℱ(TK ⁿ ) and ℱ(HK ⁿ ),n≦ℵ₀, contain a universal element. In fact, for ℱ(TK ⁴)=ℱ(HK ⁴) a strongly universal graph is constructed, whereas for 5≦n≦ℵ₀ the classes ℱ(TK ⁿ ) and ℱ(HK ⁿ ) have no universal elements. Dedicated to Klaus Wagner on his 75^th birthday     

Gently killing S-spaces

We produce a model of ZFC in which there are no locally compact first countable S-spaces, and in which 2^ℵ ₀<2^ℵ ₁. A consequence of this is that in this model there are no locally compact, separable, hereditarily normal spaces of size ℵ₁, answering a question of the second author [9]. The first author would like to thank The Hebrew University of Jerusalem for their support while the research in this paper was being carried out. The research of the second author was partially supported by NSF Grant DMS-9322613. The research of the third author was partially supported by NSF grant DMS-9704477 and the Israel Science Foundation founded by the Israel Academy of Sciences and Humanities. This is publication number 690 in the list of the third author.     

The number of non-isomorphic models of an unstable first-order theory

It is proved that ifT is an unstable (first-order) theory,λ>|T|+ℵ₀, thenT has exactly 2 ^λ non-isomorphic models of cardinalityλ. In fact we have stronger results: this is true for pseudo-elementary classes, and for almost everyλ≧|T|+ℵ₁. The preparation of this paper was sponsored in part by NSF Grant GP-22937. This work was supported in part by NSF Grant GP-22794.     

A combinatorial principle and endomorphism rings I: Onp-groups

Two lines of research are involved here. One is a combinatorial principle, proved in ZFC for many cardinals (e.g., any λ = λ^ℵ ₀) enabling us to prove things which have been proven using the diamond or for strong limit cardinals of uncountable cofinality. The other direction is building abelian groups with few endomorphisms and/or prescribed rings of endomorphisms. We prove that for a ringR, whose additive group is thep-adic completion of a freep-adic module,R is isomorphic to the endomorphism ring of some separable abelianp-groupG divided by the ideal of small endomorphisms, withG of power λ for any λ = λ^ℵ ₀≧|R|. Dedicated to the memory of Abraham Robinson on the tenth anniversary of his death The author would like to thank the United States-Israel Binational Science Foundation for partially supporting this research.     

On the Countable Tightness of Product Spaces

In this paper, we discuss the countable tightness of products of spaces which are quotient simages of locally separable metric spaces, or k-spaces with a star-countable k-network. The main result is that the following conditions are equivalent： （1） b = ω1; （2） t（Sω×Sω1） 〉 ω; （3） For any pair （X, Y）, which are k-spaces with a point-countable k-network consisting of cosmic subspaces, t（X×Y）≤ω if and only if one of X, Y is first countable or both X, Y are locally cosmic spaces. Many results on the k-space property of products of spaces with certain k-networks could be deduced from the above theorem.     

Notes on combinatorial set theory

We shall prove some unconnected theorems: (1) (G.C.H.) \omega _{\alpha + 1} \to \left( {\omega _\alpha + \xi } \right)_2^2 when ℵ_α is regular, │ξ│⁺<ωα. (2) There is a Jonsson algebra in ℵ_α+n, and \aleph _{a + n} \not \to \left[ {\aleph _{a + n} } \right]_{\aleph _{a + n} }^{n + 1} if 2^{\aleph _{ - - } } = \aleph _{a + n} \cdot (3) If λ>ℵ₀ is a strong limit cardinal, then among the graphs with ≦λ vertices each of valence <λ there is a universal one. (4)(G.C.H.) If f is a set mapping on \omega _{a + 1} (ℵ_α regular) │f(x)∩f(y│<ℵ_α, then there is a free subset of order-type ζ for every ζ<ω_α+1.     

The extent of strength in the club filters

In this paper we consider whether the minimal normal filter onP _κλ, the club filter, can have strong properties like saturation, pre-saturation, or cardinal preserving. We prove in a number of cases that the answer is no. In the case of saturation, Foreman and Magidor prove the answer is always no (except in the caseκ =λ = ℵ₁—and in this case saturation is known to be consistent). The first author was partially supported by NSF grant DMS-9626212.     

Some ℵ0-bounded subsets of stone-čech compactifications

We characterize the maximalm-bounded extension of an arbitrary completely regular Hausdorff spaceX. The other principal results are:Theorem. LetX be a locally compact, σ-compact non-compact space with no more than 2^ℵ0 zero-sets. Then assuming the continuum hypothesis,βX − X can be written as the union of 2^2ℵ0 pairwise disjoint, dense ℵ₀-bounded subspaces.Theorem. LetX be a locally compact, σ-compact metric space without isolated points. Then both the set of remote points ofβX and the complement of this set inβX −X are ℵ₀-bounded.     

On universality of the Lerch zeta-function

It is known that the Lerch zeta-function L(λ, α, s) with transcendental parameter α is universal in the Voronin sense; i.e., every analytic function can be approximated by shifts L(λ, α, s + iτ) uniformly on compact subsets of some region. In this paper, the universality for some classes of composite functions F(L(λ, α, s)) is obtained. In particular, general theorems imply the universality of the functions sin(L(λ, α, s)) and sinh(L(λ, α, s)).     

For what filters is every reduced product saturated?

In this paper we shall characterize the filtersD such that for everyM _i,M _i,Π_i∈I M_i/D is λ-saturated (where λ>ℵ₀). The characterization is:D is λ-good,D is ℵ₀ incomplete andS(I)/D is λ-saturated.     

Iterated forcing in quadratic form theory

In [Sp1] and [B/Sp] it has been shown that the existence of quadratic spaces of uncountable dimension over finite or countable fields sharing the property that every infinite dimensional subspace has its orthogonal complement of at most countable dimension is independent of the axioms of ZFC set theory. Such a space will be called astrong Gross space in the sequel. Cardinal invariants of the continuum decide whether strong Gross spaces exist or not. Namely, when b=ω₁ a strong Gross space of dimension ℵ₁ exists. When p>ω₁ such spaces do not exist. Here we answer the question what happens with strong Gross spaces in case b>ω₁ or p=ω₁. This work forms part of the author’s Habilitationsschrift at the ETH Zürich. The author is supported by the Basic Research Foundation of the Israel Academy of Sciences.     

Two cardinal versions of diamond

Roughly speaking, ◇_K,λ asserts the existence of a sequence of size <κ sets that captures every subset ofλ on a stationary set. The paper is devoted to the study of ◇_K,λ and related principles, which are for instance obtained by considering sequences of larger sets, or by requesting the simultaneous capture of many subsets ofλ. Our main result is that ◇_K,λ holds in caseλ>2^<K .     

On Countable-Dimensional Spaces with the Menger Property, Rational Dimension and a Question of S. D. Iliadis

 This paper deals with the class of spaces which are countable unions of zero-dimensional sets and with the larger class of Haver’s C-spaces. All spaces are assumed to be separable and metrizable. We are concerned with various aspects of universality of these classes when they are combined with the covering analogue for σ-compactness defined by Menger and the rational dimension introduced by Menger and N?beling. A solution of a problem of S. D. Iliadis [16] concerning universal spaces for rational dimension will result. Received 11 September 1998; in revised form 12 January 1999