A Fubini theorem

Let I₀ be the σ-ideal of subsets of a Polish group generated by Borel sets which have perfectly many pairwise disjoint translates. We prove that a Fubini-type theorem holds between I₀ and the σ-ideals of Haar measure zero sets and of meager sets. We use this result to give a simple proof of a generalization of a theorem of Balcerzak-Ros?anowski-Shelah stating that I₀ on N² strongly violates the countable chain condition.     

Blockers in hyperspaces

Given a metric continuum X, let X² and C(X) denote the hyperspaces of all nonempty closed subsets and subcontinua, respectively. For A,B∈X² we say that B does not block A if A∩B=∅ and the union of all subcontinua of X intersecting A and contained in X−B is dense in X. In this paper we study some sets of blockers for several kinds of continua. In particular, we determine their Borel classes and, for a large class of locally connected continua X, we recognize them as cap-sets.     

Preservation of the Borel class under countable-compact-covering mappings

The aim of this note is to prove the following result: “Assume that X is a metric Borel space of class ξ, that is continuous, that every fiber f⁻¹(y) is complete and that every countable compact subset of Y is the image by f of some compact subset of X. Then Y is Borel and moreover of class ξ”. We give also an extension to the case where the fibers are only assumed to be Polish.     

Ideal convergence of continuous functions

For a given ideal I⊆P(ω), IC(I) denotes the class of separable metric spaces X such that whenever is a sequence of continuous functions convergent to zero with respect to the ideal I then there exists a set of integers {m₀<m₁<?} from the dual filter F(I) such that limi→∞fmi(x)=0 for all x∈X. We prove that for the most interesting ideals I, IC(I) contains only singular spaces. For example, if I=Id is the asymptotic density zero ideal, all IC(Id) spaces are perfectly meager while if I=Ib is the bounded ideal then IC(Ib) spaces are σ-sets.     

Tychonoff expansions with prescribed resolvability properties

The recent literature offers examples, specific and hand-crafted, of Tychonoff spaces (in ZFC) which respond negatively to these questions, due respectively to Ceder and Pearson (1967) [3] and to Comfort and García-Ferreira (2001) [5]: (1) Is every ω-resolvable space maximally resolvable? (2) Is every maximally resolvable space extraresolvable? Now using the method of KID expansion, the authors show that every suitably restricted Tychonoff topological space (X,T) admits a larger Tychonoff topology (that is, an “expansion”) witnessing such failure. Specifically the authors show in ZFC that if (X,T) is a maximally resolvable Tychonoff space with S(X,T)?Δ(X,T)=κ, then (X,T) has Tychonoff expansions U=Ui (1?i?5), with Δ(X,Ui)=Δ(X,T) and S(X,Ui)?Δ(X,Ui), such that (X,Ui) is: (i=1) ω-resolvable but not maximally resolvable; (i=2) [if κ^′ is regular, with S(X,T)?κ^′?κ] τ-resolvable for all τ<κ^′, but not κ^′-resolvable; (i=3) maximally resolvable, but not extraresolvable; (i=4) extraresolvable, but not maximally resolvable; (i=5) maximally resolvable and extraresolvable, but not strongly extraresolvable.     

Thin-type dense sets and related properties

We build on Gruenhage, Natkaniec, and Piotrowski?s study of thin, very thin, and slim dense sets in products, and the related notions of (NC) and (GC) which they introduced. We find examples of separable spaces X such that X² has a thin or slim dense set but no countable one. We characterize ordered spaces that satisfy (GC) and (NC), and we give an example of a separable space which satisfies (GC) but not witnessed by a collection of finite sets. We show that the question of when the topological sum of two countable strongly irresolvable spaces satisfies (NC) is related to the Rudin-Keisler order on βω. We also introduce and study the concepts of <κ-thin and superslim dense sets.     

On completely nonmeasurable unions

Assume that there is no quasi-measurable cardinal not greater than 2^ω . We show that for a c. c. c. σ -ideal 𝕀 with a Borel base of subsets of an uncountable Polish space, if 𝒜 is a point-finite family of subsets from 𝕀, then there is a subfamily of 𝒜 whose union is completely nonmeasurable, i.e. its intersection with every non-small Borel set does not belong to the σ -field generated by Borel sets and the ideal 𝕀. This result is a generalization of the Four Poles Theorem (see [1]) and a result from [3]. (© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the Laczkovich-Komjáth property of sigma-ideals

Komjáth in 1984 proved that, for each sequence (An) of analytic subsets of a Polish space X, if lim supn∈HAn is uncountable for every H∈ω[N] then _?n∈GAn is uncountable for some G∈ω[N]. This fact, by our definition, means that the σ-ideal [X]^?ω has property (LK). We prove that every σ-ideal generated by X/E has property (LK), for an equivalence relation E⊂X² of type Fσ with uncountably many equivalence classes. We also show the parametric version of this result. Finally, the invariance of property (LK) with respect to various operations is studied.     

σ-collectionwise Hausdorffness at singular strong limit cardinals

Assuming the Singular Cardinals Hypothesis, we prove the following property:

σ-CWH:

For every singular strong limit cardinal κ and ↗-normal space X such that for some χ<κ, every x∈X has a neighborhood base of size ?χ, if every closed discrete subspace of size <κ is σ-separated, then so is every closed discrete subspace of size κ.

So for getting a model of the negation of σ-CWH, we require a large cardinal.     

Tame parts of free summands in coproducts of Priestley spaces

It is well known that a sum (coproduct) of a family of Priestley spaces is a compactification of their disjoint union, and that this compactification in turn can be organized into a union of pairwise disjoint order independent closed subspaces Xu, indexed by the ultrafilters u on the index set I. The nature of those subspaces Xu indexed by the free ultrafilters u is not yet fully understood.In this article we study a certain dense subset satisfying exactly those sentences in the first-order theory of partial orders which are satisfied by almost all of the Xi's. As an application we present a complete analysis of the coproduct of an increasing family of finite chains, in a sense the first non-trivial case which is not a ?ech-Stone compactification of the disjoint union I_?Xi. In this case, all the Xu's with u free turn out to be isomorphic under the Continuum Hypothesis.     

Pseudocompactness of hyperspaces

We consider the following question of Ginsburg: Is there any relationship between the pseudocompactness ofXωand that of the hyperspaceX²? We do that first in the context of Mrówka-Isbell spaces Ψ(A) associated with a maximal almost disjoint (MAD) family A on ω answering a question of J. Cao and T. Nogura. The space Ψω(A) is pseudocompact for every MAD family A. We show that

(1)

(p=c) ²Ψ(A) is pseudocompact for every MAD family A.

(2)

(h<c) There is a MAD family A such that ²Ψ(A) is not pseudocompact.

We also construct a ZFC example of a space X such that Xω is pseudocompact, yet X² is not.     

Covering by discrete and closed discrete sets

Say that a cardinal number κ is small relative to the space X if κ<Δ(X), where Δ(X) is the least cardinality of a non-empty open set in X. We prove that no Baire metric space can be covered by a small number of discrete sets, and give some generalizations. We show a ZFC example of a regular Baire σ-space and a consistent example of a normal Baire Moore space which can be covered by a small number of discrete sets. We finish with some remarks on linearly ordered spaces.     

Interpretation of De Finetti coherence criterion in ?ukasiewicz Logic

De Finetti gave a natural definition of “coherent probability assessment” β:E→[0,1] of a set E={X₁,…,X_m} of “events” occurring in an arbitrary set W⊆[0,1]^E of “possible worlds”. In the particular case of yes-no events, (where W⊆{0,1}^E), Kolmogorov axioms can be derived from his criterion. While De Finetti’s approach to probability was logic-free, we construct a theory Θ in infinite-valued ?ukasiewicz propositional logic, and show: (i) a possible world of W is a valuation satisfying Θ, (ii) β is coherent iff it is a convex combination of valuations satisfying Θ, (iii) iff β agrees on E with a state of the Lindenbaum MV-algebra of Θ, (iv) iff for some Borel probability measure μ on W. Thus ?ukasiewicz semantics, MV-algebraic (finitely additive) states, and (countably additive) Borel probability measures provide a universal representation of coherent assessments of events occurring in any conceivable set of possible worlds.     

Remote points in products under CH

Assume CH. Let I be any index set, and let Xi, for i∈I, be a completely regular ccc topological space of weight ω₂. If X=_∏i∈IXi is ccc and non-pseudocompact, then X has remote points.     

Surfaces meeting porous sets in positive measure

Let X be a Banach space and 2 < n < dimX. We show there exists a directionally porous set P in X for which the set of C ¹ surfaces of dimension n meeting P in positive measure is not meager. If X is separable, this leads to a decomposition of X into the union of a σ-directionally porous set and a set which is null on residually many C ¹ surfaces of dimension n. This is of interest in the study of Γn-null and Γ-null sets and their applications to differentiability of Lipschitz functions.     

On sesquilinear forms over fields with involution

Let k be a field of characteristic ≠2 with an involution σ. A matrix A is split if there is a change of variables Q such that (Q^σ)^TAQ consists of two complementary diagonal blocks. We classify all matrices that do not split. As a consequence we obtain a new proof for the following result. Given a square matrix A there is a matrix S such that (S^σ)^TAS=A^T and S^σS=I.     

The cardinal characteristic for relative γ-sets

For X a separable metric space define p(X) to be the smallest cardinality of a subset Z of X which is not a relative γ-set in X, i.e., there exists an ω-cover of X with no γ-subcover of Z. We give a characterization of p(ω²) and p(ωω) in terms of definable free filters on ω which is related to the pseudo-intersection number p. We show that for every uncountable standard analytic space X that either p(X)=p(ω²) or p(X)=p(ωω). We show that the following statements are each relatively consistent with ZFC: (a) p=p(ωω)<p(ω²) and (b) p<p(ωω)=p(ω²)     

An example on movable approximations of a minimal set in a continuous flow

In the present paper the study of flows on n-manifolds in particular in dimension three, e.g., R³, is motivated by the following question. Let A be a compact invariant set in a flow on X. Does every neighbourhood of A contain a movable invariant set M containing A? It is known that a stable solenoid in a flow on a 3-manifold has approximating periodic orbits in each of its neighbourhoods. The solenoid with the approximating orbits form a movable set, although the solenoid is not movable. Not many such examples are known. The main part of the paper consists of constructing an example of a set in R³ that is not stable, is not a solenoid, and is approximated by Denjoy-like invariant sets instead of periodic orbits. As in the case of a solenoid, the constructed set is an inverse limit of its approximating sets. This gives a partial answer to the above question.     

Ergodic Banach spaces

We show that any Banach space contains a continuum of non-isomorphic subspaces or a minimal subspace. We define an ergodic Banach space X as a space such that E₀ Borel reduces to isomorphism on the set of subspaces of X, and show that every Banach space is either ergodic or contains a subspace with an unconditional basis which is complementably universal for the family of its block-subspaces. We also use our methods to get uniformity results. We show that an unconditional basis of a Banach space, of which every block-subspace is complemented, must be asymptotically c₀ or ?_p, and we deduce some new characterisations of the classical spaces c₀ and ?_p.     

Pasting topological spaces at one point

Let {X _α } _α∈Λ be a family of topological spaces and x _α ∈ X _α , for every α ∈ Λ. Suppose X is the quotient space of the disjoint union of X _α ’s by identifying x _α ’s as one point σ. We try to characterize ideals of C(X) according to the same ideals of C(X _α )’s. In addition we generalize the concept of rank of a point, see [9], and then answer the following two algebraic questions. Let m be an infinite cardinal. (1) Is there any ring R and I an ideal in R such that I is an irreducible intersection of m prime ideals? (2) Is there any set of prime ideals of cardinality m in a ring R such that the intersection of these prime ideals can not be obtained as an intersection of fewer than m prime ideals in R? Finally, we answer an open question in [11].