Baire Property Research Articles

Baire measurable paradoxical decompositions via matchings

We show that every locally finite bipartite Borel graph satisfying a strengthening of Hall's condition has a Borel perfect matching on some comeager invariant Borel set. We apply this to show that if a group acting by Borel automorphisms on a Polish space has a paradoxical decomposition, then it admits a paradoxical decomposition using pieces having the Baire property. This strengthens a theorem of Dougherty and Foreman who showed that there is a paradoxical decomposition of the unit ball in R3 using Baire measurable pieces. We also obtain a Baire category solution to the dynamical von Neumann–Day problem: if a is a nonamenable action of a group on a Polish space X by Borel automorphisms, then there is a free Baire measurable action of F2 on X which is Lipschitz with respect to a.

Effros, Baire, Steinhaus and non-separability

We give a short proof of an improved version of the Effros Open Mapping Principle via a shift-compactness theorem (also with a short proof), involving ‘sequential analysis’ rather than separability, deducing it from the Baire property in a general Baire-space setting (rather than under topological completeness). It is applicable to absolutely-analytic normed groups (which include complete metrizable topological groups), and via a Steinhaus-type Sum-set Theorem (also a consequence of the shift-compactness theorem) includes the classical Open Mapping Theorem (separable or otherwise).

The Algebra of Semigroups of Sets

We study the algebra of semigroups of sets (i.e. families of sets closed under finite unions) and its applications. For each $n > 1$ we produce two finite nested families of pairwise different semigroups of sets consisting of subsets of $\mathsf{R}^n$ without the Baire property.

On topologies in the family of sets with the Baire property

Abstract Starting with an operator defined on the family of open sets in a topological Baire space 〈 X , τ 〉 ${\langle X, \tau \rangle }$ we can define a lower density operator on the family of sets having the Baire property with respect to the τ-topology. The investigation of the abstract density topologies generated by such operators is the main subject of this paper.

Weak Baire Spaces

In this paper, we study Baire property of a family of spaces which contains properly the family of all topological spaces and generalize the existing results. Also, we study the images and inverse images of such spaces.

Density topology involving microscopic sets and category

Abstract Analogously to the method used for c-density point, we introduce the notion of cμ-density point for arbitrary set A having the Baire property, replacing the convergence in measure with the convergence with respect to the σ-ideal of microscopic sets. Then, we consider the density type topology Tcμ generated by cμ-density operator and study basic properties of Tcμ in comparison with the classical density, I-density and c-density topologies

Baire property in product spaces

We show that if a product space $\mathit\Pi$ has countable cellularity, then a dense subspace $X$ of $\mathit\Pi$ is Baire provided that all projections of $X$ to countable subproducts of $\mathit\Pi$ are Baire. It follows that if $X_i$ is a dense Baire subspace of a product of spaces having countable $\pi$-weight, for each $i\in I$, then the product space $\prod_{i\in I} X_i$ is Baire. It is also shown that the product of precompact Baire paratopological groups is again a precompact Baire paratopological group. Finally, we focus attention on the so-called \textit{strongly Baire} spaces and prove that some Baire spaces are in fact strongly Baire.

On Borel Hull Operations

We show that some set-theoretic assumptions (for example Martin’s Axiom) imply that there is no translation invariant Borel hull operation on the family of Lebesgue null sets and on the family of meager sets (in \(\mathbb{R}^{n}\)). We also prove that if the meager ideal admits a monotone Borel hull operation, then there is also a monotone Borel hull operation on the \(\sigma\)-algebra of sets with the property of Baire.

Analyticity in Spaces of Convergent Power Series and Applications

We study the analytic structure of the space of germs of an analytic function at the origin of \ww C^{\times m} , namely the space \germ{\mathbf{z}} where \mathbf{z}=\left(z\_{1},\cdots,z\_{m}\right) , equipped with a convenient locally convex topology. We are particularly interested in studying the properties of analytic sets of \germ{\mathbf{z}} as defined by the vanishing locus of analytic maps. While we notice that \germ{\mathbf{z}} is not Baire we also prove it enjoys the analytic Baire property: the countable union of proper analytic sets of \germ{\mathbf{z}} has empty interior. This property underlies a quite natural notion of a generic property of \germ{\mathbf{z}} , for which we prove some dynamics-related theorems. We also initiate a program to tackle the task of characterizing glocal objects in some situations.

Bases of measurability in Boolean algebras

Abstract In the paper we find characterizations of some notions studied by Kułaga [5] and we generalize his results. In particular, we characterize regularity and completeness of factor subalgebras via stability of the decidability operator and we discuss some possibilities in defining the notions of first category and Baire property in Boolean algebras.

Pointwise density topology

Abstract The paper presents a new type of density topology on the real line generated by the pointwise convergence, similarly to the classical density topology which is generated by the convergence in measure. Among other things, this paper demonstrates that the set of pointwise density points of a Lebesgue measurable set does not need to be measurable and the set of pointwise density points of a set having the Baire property does not need to have the Baire property. However, the set of pointwise density points of any Borel set is Lebesgue measurable.

On partitions of Ellentuck-large sets

It is proved that no non-meager subspace of the space [ω]ω equipped with the Ellentuck topology does admit a Kuratowski partition, that is such a subset cannot be covered by a family F of disjoint relatively meager sets such that ⋃F′ has the Baire property (relatively) for every subfamily F′⊆F. Some remarks concerning continuous restrictions of functions with domain in the Ellentuck space are made.

On the measurability of functions with quasi-continuous and upper semi-continuous vertical sections

Abstract Let f: ℝ2 → ℝ be a function with upper semicontinuous and quasi-continuous vertical sections f x(t) = f(x, t), t, x ∈ ℝ. It is proved that if the horizontal sections f y(t) = f(t, y), y, t ∈ ℝ, are of Baire class α (resp. Lebesgue measurable) [resp. with the Baire property] then f is of Baire class α + 2 (resp. Lebesgue measurable and sup-measurable) [resp. has Baire property].

A convexity property and a new characterization of Euler’s gamma function

Our main results are: (I) Let α ≠ 0 be a real number. The function (Γ ◦ exp)α is convex on \({\mathbf{R}}\) if and only if $$\alpha \geq \max_{0<{t}<{x_0}}\Big(-\frac{1}{t\psi(t)} - \frac{\psi'(t)}{\psi(t)^2}\Big) = 0.0258... .$$ Here, x0 = 1.4616... denotes the only positive zero of \({\psi = \Gamma'/\Gamma}\). (II) Assume that a function f: (0, ∞) → (0, ∞) is bounded from above on a set of positive Lebesgue measure (or on a set of the second category with the Baire property) and satisfies $$f(x+1) = x f(x) \quad{\rm for}\quad{x > 0}\quad{\rm and}\quad{f(1) = 1}.$$ If there are a number b and a sequence of positive real numbers (an) \({(n \in \mathbf{N})}\) with \({{\rm lim}_{n\to\infty} a_n =0}\) such that for every n the function \({(f \circ {\rm exp})^{a_n}}\) is Jensen convex on (b, ∞), then f is the gamma function.

THE BAIRE PROPERTY IN THE REMAINDERS OF SEMITOPOLOGICAL GROUPS

AbstractIt is proved that every remainder of a nonlocally compact semitopological group $G$ is a Baire space if and only if $G$ is not Čech-complete, which improves a dichotomy theorem of topological groups by Arhangel’skiǐ [‘The Baire property in remainders of topological groups and other results’, Comment. Math. Univ. Carolin. 50(2) (2009), 273–279], and also gives a positive answer to a question of Lin and Lin [‘About remainders in compactifications of paratopological groups’, ArXiv: 1106.3836v1 [Math. GN] 20 June 2011]. We also show that for a nonlocally compact rectifiable space $G$ every remainder of $G$ is either Baire, or meagre and Lindelöf.

On the regularity of topologies in the family of sets having the Baire property

The paper concerns the topologies introduced in the family of sets having the Baire property in a topological space (X, ?) and in the family generated by the sets having the Baire property and given a proper ?-ideal containing ? -meager sets. The regularity property of such topologies is investigated.

On countable families of sets without the Baire property

We suggest a method of constructing decompositions of a topological space $X$ having an open subset homeomorphic to the space ($\mathbb R^n, \tau )$, where $n$ is an integer $\geq 1$ and $\tau $ is any admissible extension of the Euclidean topology of $\m

On topologies generated by some operators

Abstract The paper concerns topologies introduced in a topological space (X, τ) by operators which are much weaker than the lower density operators. Some properties of the family of sets having the Baire property and the family of meager sets with respect to such topologies are investigated.

Feebly compact paratopological groups and real-valued functions

We present several examples of feebly compact Hausdorff paratopological groups (i.e., groups with continuous multiplication) which provide answers to a number of questions posed in the literature. It turns out that a 2-pseudocompact, feebly compact Hausdorff paratopological group $$G$$ can fail to be a topological group. Our group $$G$$ has the Baire property, is Fréchet–Urysohn, but it is not precompact. It is well known that every infinite pseudocompact topological group contains a countable non-closed subset. We construct an infinite feebly compact Hausdorff paratopological group $$G$$ all countable subsets of which are closed. Another peculiarity of the group $$G$$ is that it contains a nonempty open subsemigroup $$C$$ such that $$C^{-1}$$ is closed and discrete, i.e., the inversion in $$G$$ is extremely discontinuous. We also prove that for every continuous real-valued function $$g$$ on a feebly compact paratopological group $$G$$ , one can find a continuous homomorphism $$\varphi $$ of $$G$$ onto a second countable Hausdorff topological group $$H$$ and a continuous real-valued function $$h$$ on $$H$$ such that $$g=h\circ \varphi $$ . In particular, every feebly compact paratopological group is $$\mathbb{R }_3$$ -factorizable. This generalizes a theorem of Comfort and Ross established in 1966 for real-valued functions on pseudocompact topological groups.

Pontryagin duality in the class of precompact Abelian groups and the Baire property

We present a wide class of reflexive, precompact, non-compact, Abelian topological groups G determined by three requirements. They must have the Baire property, satisfy the open refinement condition, and contain no infinite compact subsets. This combination of properties guarantees that all compact subsets of the dual group G∧ are finite. We also show that many (non-reflexive) precompact Abelian groups are quotients of reflexive precompact Abelian groups. This includes all precompact almost metrizable groups with the Baire property and their products. Finally, given a compact Abelian group G of weight ≥2ω, we find proper dense subgroups H1 and H2 of G such that H1 is reflexive and pseudocompact, while H2 is non-reflexive and almost metrizable.