ω* has (almost) no continuous images

We prove that the following statement follows from the Open Colouring Axiom (OCA): ifX is locally compactσ-compact but not compact and if its Čech-Stone remainderX* is a continuous image ofω*, thenX is the union ofω and a compact set. It follows that the remainders of familiar spaces like the real line or the sum of countably many Cantor sets need not be continuous images ofω*.     

An extension theorem for separable Banach spaces

We construct a totally disconnected ω^*, norming subsetF of the unit ballB ^* of an arbitrary separable Banach space,X, and an operator fromC(F) toC(B^*) that “amost” commutes with the natural embeddings ofX. This is used to give a new proof of Milutin's theorem and to prove some new results on complemented subspaces ofC[0, 1] with separable dual. In particular we show that a complemented subspace ofC(ω^ω), is either isomorphic toC(ω^ω) or toc _u.     

Separable abelianp-groups having certain prescribed chains

An abelianp-groupG is calledp ^ω+1-projective ifp ^ω+1Ext(G, X)=0 for all groupsX. This class of groups constitutes a natural extension of the well-known class of totally projective groups whose members are precisely those groups classifiable by the Ulm-Kaplansky invariants. Fuchs asked whetherp ^ω+1-projective groupsG can be characterized in terms of filtrations ofG. Our Theorem 1 provides counterexamples.     

On the weight-spectrum of a compact space

The weight-spectrumSp(w, X) of a spaceX is the set of weights of all infinite closed subspaces ofX. We prove that ifκ>ω is regular andX is compactT ₂ withω(X)≥κ then some λ withκ≤λ≤2^<κ is inSp(ω, X). Under CH this implies that the weight spectrum of a compact space can not omitω ₁, and thus solves problem 22 of [M]. Also, it is consistent with 2^ω=c being anything it can be that every countable closed setT of cardinals less thanc withω ∈ T satisfiesSp(w, X)=T for some separable compact LOTSX. This shows the independence from ZFC of a conjecture made in [AT]. Research supported by OTKA grant no. 1908.     

C(K) quotients of separable ℒ∞ spaces

It is shown that ifX is an ℒ⫗ space with separable dual, thenX has a quotient isomorphic toC(ω)^α if, and only if, there is anɛ >0, such that theɛ-Szlenk index ofX is at leastα. It was previously known, that ifX is a separable ℒ⫗ space with a non-separable dual, thenX has a quotient isomorphic toC[0, 1]. Research partially supported by NSF Grant MCS-74-07509-A03.     

A predual ofl 1 which is not isomorphic to aC(K) space

We give an example of a Banach spaceX such that (i)X ^* is isometric tol ₁, (ii)X is isometric to a subspace ofC(ω^θ) and (iii)X is not isomorphic to a complemented subspace of anyC(K) space. This is a part of the first author’s Ph. D. Thesis prepared in the Hebrew University of erusalem under the supervision of the second author.     

On Two Questions About Topological (and Nonstandard) Extensions

We construct, under MA, a non-Hausdorff (T₁-)topological extension ^*ω of ω, such that every function from ω to ω extends uniquely to a continuous function from ^*ω to ^*ω. We also show (in ZFC) that for every nontrivial topological extension ^*X of a countable set X there exists a topology τ_f on ^*X, strictly finer than the Star topology, and such that (^*X, τ_f) is still a topological extension of X with the same function extensions ^*f. This solves two questions raised by M. Di Nasso and M. Forti.     

On Two Questions About Topological (and Nonstandard) Extensions

We construct, under MA, a non-Hausdorff (T₁-)topological extension ^*ω of ω, such that every function from ω to ω extends uniquely to a continuous function from ^*ω to ^*ω. We also show (in ZFC) that for every nontrivial topological extension ^*X of a countable set X there exists a topology τ_f on ^*X, strictly finer than the Star topology, and such that (^*X, τ_f) is still a topological extension of X with the same function extensions ^*f. This solves two questions raised by M. Di Nasso and M. Forti.     

Resolvability and monotone normality

A space X is said to be κ-resolvable (resp., almost κ-resolvable) if it contains κ dense sets that are pairwise disjoint (resp., almost disjoint over the ideal of nowhere dense subsets). X is maximally resolvable if and only if it is Δ(X)-resolvable, where Δ(X) = min{|G| : G ≠ open}. We show that every crowded monotonically normal (in short: MN) space is ω-resolvable and almost μ-resolvable, where μ = min{2 ^ω , ω ₂}. On the other hand, if κ is a measurable cardinal then there is a MN space X with Δ(X) = κ such that no subspace of X is ω ₁-resolvable. Any MN space of cardinality < ℵ _ω is maximally resolvable. But from a supercompact cardinal we obtain the consistency of the existence of a MN space X with |X| = Δ(X) = ℵ _ω such that no subspace of X is ω ₂-resolvable. The preparation of this paper was supported by OTKA grant no. 61600     

On families of Lagrangian submanifolds

We study the stability of a compact Lagrangian submanifold of a symplectic manifold under perturbation of the symplectic structure. If X is a compact manifold and the ω _t are cohomologous symplectic forms on X, then by a well-known theorem of Moser there exists a family Φ _t of diffeomorphisms of X such that ω _t =Φ _t ^*(ω₀). If L⊂X is a Lagrangian submanifold for (X,ω₀), L _t =Φ _t ^-1(L) is thus a Lagrangian submanifold for (X,ω _t ). Here we show that if we simply assume that L is compact and ω _t | _L is exact for every t, a family L _t as above still exists, for sufficiently small t. Similar results are proved concerning the stability of special Lagrangian and Bohr–Sommerfeld special Lagrangian submanifolds, under perturbation of the ambient Calabi–Yau structure. Received: 29 May 2001/ Revised version: 17 October 2001     

More on countably compact,locally countable spaces

Following [5], aT ₃ spaceX is called good (splendid) if it is countably compact, locally countable (andω-fair).G(κ) (resp.S(κ)) denotes the statement that a good (resp. splendid) spaceX with |X|=κ exists. We prove here that (i) Con(ZF)→Con(ZFC+MA+2 ^ω is big+S(κ) holds unlessω=cf(κ)<κ); (ii) a supercompact cardinal implies Con(ZFC+MA+2suω>_ω+₁+┐G(ω_ω+₁); (iii) the “Chang conjecture” (ω_ω+₁),→(ω ₁,ω) implies ┐S(κ) for allκ≧k≧ω_ω; (iv) ifP addsω ₁ dominating reals toV iteratively then, in , we haveG(λ^ω) for allλ. Research supported by Hungarian National Foundation for Scientific Research grant no. 1805.     

On banach spaces which containl 1(τ) and types of measures on compact spaces

Two closely related results are presented, one of them concerned with the connection between topological and measure-theoretic properties of compact spaces, the other being a non-separable analogue of a result of Peŀczyński's about Banach spaces containingL ¹. Let τ be a regular cardinal satisfying the hypothesis that κ^ω<τ whenever κ<τ. The following are proved: 1) A compact spaceT carries a Radon measure which is homogeneous of type τ, if and only if there exists a continuous surjection ofT onto [0, 1]^τ. 2) A Banach spaceX has a subspace isomorphic tol ¹(τ) if and only ifX ^∗ has a subspace isomorphic toL ¹({0, 1}^τ). An example is given to show that a more recent result of Rosenthal's about Banach spaces containingl ¹ does not have an obvious transfinite analogue. A second example (answering a question of Rosenthal's) shows that there is a Banach spaceX which contains no copy ofl ¹ (ω₁), while the unit ball ofX ^∗ is not weakly^∗ sequentially compact.     

Hyperfinite Dimensional Representations of Spaces and Algebras of Measures

Let X be a locally compact topological space and (X, E, X_ω) be any triple consisting of a hyperfinite set X in a sufficiently saturated nonstandard universe, a monadic equivalence relation E on X, and an E-closed galactic set X_ω ⊆ X, such that all internal subsets of X_ω are relatively compact in the induced topology and X is homeomorphic to the quotient X_ω/E. We will show that each regular complex Borel measure on X can be obtained by pushing down the Loeb measure induced by some internal function X ? ^*\Bbb CX \rightarrow {}{^{\ast}{\Bbb C}} . The construction gives rise to an isometric isomorphism of the Banach space M(X) of all regular complex Borel measures on X, normed by total variation, and the quotient M_w(X)/M₀(X){\cal M}_{\omega}(X)/{\cal M}_0(X) , for certain external subspaces M₀(X), M_w(X){\cal M}_0(X), {\cal M}_{\omega}(X) of the hyperfinite dimensional Banach space ^*\Bbb C^X{}{^{\ast}{\Bbb C}}^X , with the norm ‖f‖₁ = ∑_{x ∈ X} |f(x)|. If additionally X = G is a hyperfinite group, X_ω = G_ω is a galactic subgroup of G, E is the equivalence corresponding to a normal monadic subgroup G₀ of G_ω, and G is isomorphic to the locally compact group G_ω/G₀, then the above Banach space isomorphism preserves the convolution, as well, i.e., M(G) and M_w(G)/M₀(G){\cal M}_{\omega}(G)/{\cal M}_0(G) are isometrically isomorphic as Banach algebras.     

Modules of vector fields,differential forms and degenerations of differential systems

We deal with (n−1)-generated modules of smooth (analytic, holomorphic) vector fieldsV=(X ₁,..., X_n−1) (codimension 1 differential systems) defined locally on ℝ ⁿ or ℂ ⁿ , and extend the standard duality(X ₁,..., X_n−1)↦(ω), ω=Ω(X₁,...,X_n−1,.,) (Ω−a volume form) betweenV′s and 1-generated modules of differential 1-forms (Pfaffian equations)—when the generatorsX _i are linearly independent—onto substantially wider classes of codimension 1 differential systems. We prove that two codimension 1 differential systemsV and are equivalent if and only if so are the corresponding Pfaffian equations (ω) and provided that ω has1-division property: ωΛμ=0, μ—any 1-form ⇒ μ=fω for certain function germf. The 1-division property of ω turns out to be equivalent to the following properties ofV: (a)fX∈V, f—not a 0-divisor function germ ⇒X∈V (thedivision property); (b) (V ^⊥)^⊥=V; (c)V ^⊥=(ω); (d) (ω)^⊥=V, where ⊥ denotes the passing from a module (of vector fields or differential 1-forms) to its annihilator. Supported by Polish KBN grant N^o 2 1090 91 01. Partially supported by the fund for the promotion of research at the Technion, 100–942.     

Strongly meager sets can be quite big

AssumeCH. There exists a strongly meager setX⊆2^ω and a continuous functionF: 2^ω → 2^ω such thatF″ (X)=2^ω. The analogous statement for the strong measure zero, the notion dual to strongly meager, is false. The first author was partially supported by NSF grant DMS 9971282 and the Alexander von Humboldt Foundation. The second author was partially supported by grant BW 5100-5-0231-2.     

Maximal subspaces for solutions of the second order abstract cauchy problem

For a continuous, increasing function ω: R → R \{0} of finite exponential type, this paper introduces the set Z(A, ω) of all x in a Banach space X for which the second order abstract differential equation (2) has a mild solution such that [ω(t)]-1u(t,x) is uniformly continues on R , and show that Z(A, ω) is a maximal Banach subspace continuously embedded in X, where A ∈ B(X) is closed. Moreover, A|z(A,ω) generates an O(ω(t))strongly continuous cosine operator function family.     

Invariants forω-categorical,ω-stable theories

In this paper we give a complete solution to the classification problem forω-categorical,ω-stable theories. More explicitly, supposeT isω-categorical,ω-stable with fewer than the maximum number of models in some uncountable power. We associate with each modelM ofT a “simple” invariantI(M), not unlike a vector of dimensions, such thatI(M)=I(N) if and only ifM≅N. The spectrum function,I(−,T), for a first-order theoryT is such that for all infinite cardinals λ,I(λ,T) is the number of nonisomorphic models ofT of cardinality λ. As an application of our “structure theorem” we determine the possible spectrum functions forω-categorical,ω-stable theories.     

Hyperfinite Dimensional Representations of Spaces and Algebras of Measures

Let X be a locally compact topological space and (X, E, X_ω) be any triple consisting of a hyperfinite set X in a sufficiently saturated nonstandard universe, a monadic equivalence relation E on X, and an E-closed galactic set X_ω ⊆ X, such that all internal subsets of X_ω are relatively compact in the induced topology and X is homeomorphic to the quotient X_ω/E. We will show that each regular complex Borel measure on X can be obtained by pushing down the Loeb measure induced by some internal function . The construction gives rise to an isometric isomorphism of the Banach space M(X) of all regular complex Borel measures on X, normed by total variation, and the quotient , for certain external subspaces of the hyperfinite dimensional Banach space , with the norm ‖f‖₁ = ∑_{x ∈ X} |f(x)|. If additionally X = G is a hyperfinite group, X_ω = G_ω is a galactic subgroup of G, E is the equivalence corresponding to a normal monadic subgroup G₀ of G_ω, and G is isomorphic to the locally compact group G_ω/G₀, then the above Banach space isomorphism preserves the convolution, as well, i.e., M(G) and are isometrically isomorphic as Banach algebras. Research of both authors supported by a grant by VEGA – Scientific Grant Agency of Slovak Republic.     

Guarding galleries where every point sees a large area

We prove a conjecture of Kavraki, Latombe, Motwani and Raghavan that ifX is a compact simply connected set in the plane of Lebesgue measure 1, such that any pointx∈X sees a part ofX of measure at least ɛ, then one can choose a setG of at mostconst1/ɛ log 1/ɛ points inX such that any point ofX is seen by some point ofG. More generally, if for anyk points inX there is a point seeing at least 3 of them, then all points ofX can be seen from at mostO(k ³ logk) points. Research supported by grants from the Sloan Foundation, the Israeli Academy of Sciences and Humanities, and by G.I.F. Research supported by Czech Republic Grant GAČR 201/94/2167 and Charles University grants No. 351 and 361. Part of the work was done while the author was visiting The Hebrew University of Jerusalem.     

Rational points of bounded height on Del Pezzo surfaces of degree six

LetK be a number field. Denote byV ₃ a split Del Pezzo surface of degree six overK and by ω its canonical divisor. Denote byW ₃ the open complement of the exceptional lines inV ₃. LetN _{W s}(−ω, X) be the number ofK-rational points onW ₃ whose anticanonical heightH _−ω is bounded byX. Manin has conjectured that asymptoticallyN _{W 3}(−ω, X) tends tocX(logX)³, wherec is a constant depending only on the number field and on the normalization of the height. Our goal is to prove the following theorem: For each number fieldK there exists a constantc _K such thatN _{W 3}(−ω, X)≤c_KX(logX)^3+2r , wherer is the rank of the group of units ofO _K. The constantc _K is far from being optimal. However, ifK is a purely imaginary quadratic field, this proves an upper bound with a correct power of logX. The proof of Manin's conjecture for arbitrary number fields and a precise treatment of the constants would require a more sophisticated setting, like the one used by [Peyre] to prove Manin's conjecture and to compute the correct asymptotic constant (in some normalization) in the caseK=ℚ. Up to now the best result for arbitraryK goes back, as far as we know, to [Manin-Tschinkel], who gives an upper boundN _{W 3}(−ω,X)≤cX^l+ε. The author would like to express his gratitude to Daniel Coray and Per Salberger for their generous and indispensable support.