Volume 26, Issue 4

We shall show that under Martin’s axiom, there exist absolutely Baire nonmeasurable additive functions. This provides a Baire category counterpart of an analogous measure-theoretic result of A. B. Kharazishvili.

We show that, in any ordered plane with a symmetric orthogonality relation which allows for a meaningful definition of acute and obtuse angles, in which all points are colored with three colors, such that each color is used at least once, there must exist both an acute triangle whose vertices have all three colors and an obtuse triangle with the same property. We also show that, in both a geometry endowed with an orthogonality relation, in which there is a reflection in every line, in which all right angles are bisectable, which satisfies Bachmann’s Lotschnittaxiom (the perpendiculars raised on the sides of a right angle intersect), and in plane absolute geometry, in which all points are colored with three colors, such that each color is used at least once, there exists a right triangle with all vertices of different colors.

We examine images of Bernstein sets via continuous mappings. Among other results, we prove that there exists a continuous function f:ℝ→ℝ{f\colon\mathbb{R}\to\mathbb{R}} that maps every Bernstein subset of ℝ{\mathbb{R}} onto the whole real line. This gives the positive answer to a question of Osipov.

We observe a connection between the existence of square-integrable representations of a locally compact group G and the existence of nonzero translation invariant operators from its Fourier–Stieltjes algebra B⁢(G){B(G)} into L2⁢(G){L^{2}(G)} or, equivalently, from L2⁢(G){L^{2}(G)} into its enveloping von Neumann algebra C*⁢(G)**{C^{*}(G)^{**}}.

In this article, we demonstrate how the Vitali and Bernstein constructions together with a simple theory on semigroups and ideals of sets can be used for producing different semigroups of sets without the Baire property and/or non-measurable in the Lebesgue sense.

The paper deals with two ideals of subsets of ℕ{\mathbb{N}} defined with the help of density points and its category analogue. We present basic properties of these ideals and show that they are incomparable under inclusion.

Some questions are discussed concerning the existence of nonzero σ-finite invariant Borel measures in infinite-dimensional Polish vector spaces.

We consider the family 𝒞⁢𝒮{\mathcal{CS}} of symmetric Cantor subsets of [0,1]{[0,1]}. Each set in 𝒞⁢𝒮{\mathcal{CS}} is uniquely determined by a sequence a=(an){a=(a_{n})} belonging to the Polish space X:=(0,1)ℕ{X\mathrel{\mathop{:}}=(0,1)^{\mathbb{N}}} equipped with probability product measure μ. This yields a one-to-one correspondence between sets in 𝒞⁢𝒮{\mathcal{CS}} and sequences in X . If 𝒜⊂𝒞⁢𝒮{\mathcal{A}\subset\mathcal{CS}}, the corresponding subset of X is denoted by 𝒜∗{\mathcal{A}^{\ast}}. We study the subfamilies ℋ0{\mathcal{H}_{0}}, 𝒮⁢𝒫{\mathcal{SP}} and ℳ{\mathcal{M}} of 𝒞⁢𝒮{\mathcal{CS}}, consisting (respectively) of sets with Haudsdorff dimension 0, and of strongly porous and microscopic sets. We have ℳ⊂ℋ0⊂𝒮⁢𝒫{\mathcal{M}\subset\mathcal{H}_{0}\subset\mathcal{SP}}, and these inclusions are proper. We prove that the sets ℳ∗{\mathcal{M}^{\ast}}, ℋ0∗{\mathcal{H}_{0}^{\ast}}, 𝒮⁢𝒫∗{\mathcal{SP}^{\ast}} are residual in X , and μ⁢(ℋ0∗)=0{\mu(\mathcal{H}_{0}^{\ast})=0}, μ⁢(𝒮⁢𝒫∗)=1{\mu(\mathcal{SP}^{\ast})=1}.

The well-known Gitik–Shelah theorem (1989) touches the problem of existence isomorphisms between some quotient algebras. In this paper, we study a relation between the existence of such isomorphisms and the existence of so-called Kuratowski partitions of adequate Baire spaces. For this purpose, we give strictly combinatorial methods.

The equidistant set of a collection F of lines in 3-space is the set of those points whose distances to the lines in F are all equal. We present many examples and results related to the lines possibly contained in the equidistant set of F . In particular, we determine the possible numbers of lines in the equidistant set of a collection of n lines for every n>0{n>0}. For example, if n=3{n=3}, then the possible number of such lines is either 4 or 2 or 1 or 0. In a natural way, our results are connected with properties of special types of (ruled) surfaces. For example, we obtain also results on the number of lines in the intersection of quadratic surfaces.

In the first part of the paper we study the sets of boundedness and of convergence and divergence to infinity of sequences of real closed-graph functions. Generalization on ideal convergence of such sequences is discussed. Limits and ideal-limits of sequences of functions with closed graphs are considered in the last part of the article.

In this paper, theorems are stated regarding geometrical realizations of finite families of sets, for a given countable family of sets the existence of families of point sets combinatorially strictly ω-isomorphic to this given family of sets, independent families of triangles in the Euclidean plane, also, the existence and extensions of ρ-at-sets, ρ-rt-sets, ρ-ot-sets and ot-sets. These theorems are a continuation of the corresponding results of [6, 7, 8, 5, 3, 4].

In this paper, we study the question of the existence of Kuratowski partitions in tree structures and Ellentuck topology. On the examples of Marczewski and Laver structures and the structure of Ellentuck topology, we show that non-small sets of these structures do not admit Kuratowski partitions.

A characterization of all cardinal numbers that are cardinals of automorphism groups of mono-unary algebras is given. Some connections are also established between the notion of an automorphism of mono-unary algebras and the continuum hypothesis. Close connections of the obtained results with one problem of Ulam are also considered.

The aim of this paper is the investigation of topologies generated by operators related to a complete extension of the Lebesgue measure over the real line. Some properties of such topologies provide their structure and separation axioms.

Here we give an abstract formulation (in uncountable commutative groups) of a classical theorem of Sierpiński on the nonmeasurability of the algebraic sum of measure zero sets.

It is well known that not every convex set-valued function admits an affine selection. One could ask whether there exists at least one local affine selection. The answer is positive in the finite-dimensional case. The main part of the note comprises two examples of the non-existence of local affine selections of convex set-valued functions defined on certain infinite-dimensional Banach spaces.

The following question is considered: when an uncountable commutative group of homeomorphisms of a second category topological space contains a subgroup, no orbit of which possesses the Baire property?

Let (X,∥⋅∥){(X,\lVert\,\cdot\,\rVert)} be a Banach space. Let C be a nonempty, bounded, closed and convex subset of X and let T:C→C{T:C\rightarrow C} be a G -monotone nonexpansive mapping. In this work, it is shown that the Mann iteration sequence defined by xn+1=tn⁢T⁢(xn)+(1-tn)⁢xn,n=1,2,…,x_{n+1}=t_{n}T(x_{n})+(1-t_{n})x_{n},\quad n=1,2,\dots, proves the existence of a fixed point of G -monotone nonexpansive mappings.

We show that a coded system is mixing if and only if it is totally transitive. If in addition it has a generator the length of whose elements are relatively prime, then it has strong property P . We continue by showing that a mixing half-synchronized system has such a generator. Moreover, we give an example of a mixing coded system which does not have any generator the length of whose elements are relatively prime.

We consider some families of real functions endowed with the metric of uniform convergence. In the main results of our work we present two methods of comparison of families of real functions in porosity terms. The first method is very general and may be applied to any family of real functions. The second one is more convenient but can be used only in the case of path continuous functions. We apply the obtained results to compare in terms of porosity the following families of functions: continuous, absolutely continuous, Baire one, Darboux, also functions of bounded variation and porouscontinuous, ρ-upper continuous, ρ-lower continuous functions.