Universally meager sets, II

We present new characterizations of universally meager sets, shown in [P. Zakrzewski, Universally meager sets, Proc. Amer. Math. Soc. 129 (6) (2001) 1793-1798] to be a category analog of universally null sets. In particular, we address the question of how this class is related to another class of universally meager sets, recently introduced by Todorcevic [S. Todorcevic, Universally meager sets and principles of generic continuity and selection in Banach spaces, Adv. Math. 208 (2007) 274-298].     

Generalized Haar integral

The aim of the paper is to generalize the notion of the Haar integral. For a compact semigroup S acting continuously on a Hausdorff compact space Ω, the algebra A(S)⊂C(Ω,R) of S-invariant functions and the linear space M(S) of S-invariant (real-valued) finite signed measures are considered. It is shown that if S has a left and right invariant measure, then the dual space of A(S) is isometrically lattice-isomorphic to M(S) and that there exists a unique linear operator (called the Haar integral) such that for each f∈A(S) and for any f∈C(Ω,R) and s∈S, , where .     

On affine groups admitting invariant two-point sets

A two-point set is a subset of the plane which meets every line in exactly two points. We discuss previous work on the topological symmetries of a two-point set, and show that there exist subgroups of S¹ which do not leave any two-point set invariant. Further, we show that two-point sets may be chosen to be topological groups, in which case they are also homogeneous.     

On intrinsic invariance of fractal dimensions

We show that intrinsic isometry does not have to preserve a fractal dimension.     

Sum of sets in several dimensions

LetA, B be finite sets in ^d with|A|=m|B|=n, and assume that there is no hyperplane containing both a translation ofA and a translation ofB. Under this condition it is proved that the number of distinct vectors in the form {a+baA, bB} is at leastn+dm–d(d+1)/2. This generalizes results of Freiman (caseA=B) and Freiman, Heppes, Uhrin (caseA=–B). A more complicated estimate is also given which yields the exact bound for alln>2d.Supported by Hungarian National Foundation for Scientific Research, Grant No. 1901.     

On the connectedness of limit net sets

In this paper we introduce and study net sets and limit net sets. The construction and geometry of net sets can be described with the help of substitutions with net matrices which we also introduce here. Limit net sets are a special type of Moran fractals. We study connectedness properties of net sets and limit net sets.     

Euclidean self-similar sets generated by geometrically independent sets

For every integer n>0, we consider all iterated function systems generated by n+1 Euclidean similarities acting on Rn whose fixed points form the set of vertices of an n-simplex, and characterize the nature of attractors of such iterated function systems in terms of contractivity factors of their generators.     

Separable subspaces of affine function spaces on convex compact sets

Let K be a compact convex subset of a separated locally convex space (over R) and let Ap(K) denote the space of all continuous real-valued affine mappings defined on K, endowed with the topology of pointwise convergence on the extreme points of K. In this paper we shall examine some topological properties of Ap(K). For example, we shall consider when Ap(K) is monolithic and when separable compact subsets of Ap(K) are metrizable.     

On the Steinhaus property in topological groups

Let G be a locally compact Abelian group and μ a Haar measure on G. We prove: (a) If G is connected, then the complement of a union of finitely many translates of subgroups of G with infinite index is μ-thick and everywhere of second category. (b) Under a simple (and fairly general) assumption on G, for every cardinal number m such that ℵ₀?m?|G| there is a subgroup of G of index m that is μ-thick and everywhere of second category. These results extend theorems by Muthuvel and Erd?s-Marcus, respectively. (b) also implies a recent theorem by Comfort-Raczkowski-Trigos stating that every nondiscrete compact Abelian group G admits ²|G|-many μ-nonmeasurable dense subgroups.     

Symmetries of two-point sets

A two-point set is a subset of the plane which meets every planar line in exactly two-points. We discuss the problem “What are the topological symmetries of a two-point set?”. Our main results assert the existence of two-point sets which are rigid and the existence of two-point sets which are invariant under the action of certain autohomeomorphism groups. We pay particular attention to the isometry group of a two-point set, and show that such groups consist only of rotations and that they may be chosen to be any subgroup of S¹ having size less than c. We also construct a subgroup of S¹ having size c that is contained in the isometry group of a two-point set.     

Compact sets of continuity for Borel functions

We investigate connections between complexity of a function f from a Polish space X to a Polish space Y and complexity of the set , where K(X) denotes the space of all compact subsets of X equipped with the Vietoris topology. We prove that if C(f) is analytic, then f is Borel; and assuming -determinacy we show that f is Borel if and only if C(f) is coanalytic. Similar results for projective classes are also presented.     

Norm continuity of weakly continuous mappings into Banach spaces

Let T be the class of Banach spaces E for which every weakly continuous mapping from an α-favorable space to E is norm continuous at the points of a dense subset. We show that:

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T contains all weakly Lindelöf Banach spaces;

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l^∞∉T, which brings clarity to a concern expressed by Haydon ([R. Haydon, Baire trees, bad norms and the Namioka property, Mathematika 42 (1995) 30-42], pp. 30-31) about the need of additional set-theoretical assumptions for this conclusion. Also, (l^∞/c₀)∉T.

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T is stable under weak homeomorphisms;

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E∈T iff every quasi-continuous mapping from a complete metric space to (E,weak) is densely norm continuous;

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E∈T iff every quasi-continuous mapping from a complete metric space to (E,weak) is weakly continuous at some point.

     

The compact-open topology: A new perspective

This paper studies the compact-open topology on the set KC(X) of all real-valued functions defined on a Tychonoff space, which are continuous on compact subsets of X. In addition to metrizability, separability and second countability of this topology on KC(X), various kinds of topological properties of this topology are studied in detail. Actually the motivation for studying the compact-open topology on KC(X) lies in the attempt of having a simpler proof for the characterization of a completeness property of the compact-open topology on C(X), the set of all real-valued continuous functions on X.     

On lattices of convex sets in {\mathbb{R}}^{n}

Properties of several sorts of lattices of convex subsets of are examined. The lattice of convex sets containing the origin turns out, for n > 1, to satisfy a set of identities strictly between those of the lattice of all convex subsets of and the lattice of all convex subsets of The lattices of arbitrary, of open bounded, and of compact convex sets in all satisfy the same identities, but the last of these is join-semidistributive, while for n > 1 the first two are not. The lattice of relatively convex subsets of a fixed set satisfies some, but in general not all of the identities of the lattice of “genuine” convex subsets of To the memory of Ivan RivalReceived April 22, 2003; accepted in final form February 16, 2005.This revised version was published online in August 2005 with a corrected cover date.     

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.     

The structure of d-dimensional sets with small sumset

We describe the structure of d-dimensional sets of lattice points, having a small doubling property. Let K be a finite subset of Zd such that dimK=d?2. If and |K|>3⋅d⁴, then K lies on d parallel lines. Moreover, for every d-dimensional finite set K⊆Zd that lies on d?1 parallel lines, if , then K is contained in d parallel arithmetic progressions with the same common difference, having together no more than terms. These best possible results answer a recent question posed by Freiman and cannot be sharpened by reducing the quantity v or by increasing the upper bounds for |K+K|.     

Ambiguous loci of the metric projection onto compact starshaped sets in a Banach space

LetE be a real Banach space andL(E) the family of all nonempty compact starshaped subsets ofE. Under the Hausdorff distance,L(E) is a complete metric space. The elements of the complement of a first Baire category subset ofL(E) are called typical elements ofL(E). ForXL(E) we denote by the metrical projection ontoX, i.e. the mapping which associates to eachaE the set of all points inX closest toa. In this note we prove that, ifE is strictly convex and separable with dimE2, then for a typicalXL(E) the map is not single valued at a dense set of points. Moreover, we show that a typical element ofL(E) has kernel consisting of one point and set of directions dense in the unit sphere ofE.     

On the definition of a probabilistic normed space

Summary In this paper we give a new definition of a probabilistic normed space. This definition, which is based on a characterization of normed spaces by means of a betweenness relation, includes the earlier definition of A. N. erstnev as a special case and leads naturally to the definition of the principal class of probabilistic normed spaces, the Menger spaces.